Luca Bai Otti

8182019 Luca Bai Otti

httpslidepdfcomreaderfullluca-bai-otti 1100

Luca Baiotti

Osaka University Institute for Laser Engineering

Introduction to numerical relativity



bull Overview of general relativity

bull Gravitational waves

bull Gravitational-wave detection

bull Numerical relativity bull 3+1 decomposition

bull Einstein equations

bull gauge conditions

bull gravitational-wave extraction

bull non-vacuum spacetimesbull general-relativistic magnetohydrodynamics

bull Some applications binary neutron-star merger



bull General Relativity is Einsteinrsquos theory of gravity (1915)

bull At the beginning of the XX century there were no observational inconsistencies with

Newtonrsquos theory of gravitation

bull But Newtonian gravity is inconsistent with the principle of causality of special

relativity nothing can move faster than the speed of light

bull Indeed Newtonrsquos inverse square law implies action at a distance If one object moves

the other one knows about it instantaneously due to the change in the gravitational force

no matter how big their separation is

bull The work of Einstein to make gravity consistent with special relativity brought to a

major revision of how we think of space and time It started with a simple principle now

known as the equivalence principle



bull The equivalence principle in few common words ldquoAll things fall in the same wayrdquo

bull Or slightly more wordy ldquoall objects have the same acceleration in a gravitational

fieldrdquo (eg a feather and bowling ball fall with the same acceleration in the absence of

air friction)

bull The fact that ldquoAll things fall in the same wayrdquo is true because the ldquoinertialrdquo mass

that enters Newtonrsquos law of motion F=ma is the same as the ldquogravitationalrdquo mass

that enters the gravitational-force law The principle of equivalence is really a

statement that inertial and gravitational masses are equal to each other for anyobject



bull Einstein reasoned that a uniform gravitational field in some

direction is indistinguishable from a uniform acceleration in

the opposite direction

bull And so postulated his version of the equivalence principlewhich puts gravity and acceleration on an equal footing

bull ldquowe [] assume the complete physical equivalence of a

gravitational field and a corresponding acceleration of

the reference system



bull Einstein started to think of the path of an object as a property of spacetime itself rather than

being related with the specific properties of the object

bull The idea is that gravity is a manifestation of the fact that objects in free fall follow

geodesics in curved spacetimes

bull What are geodesics

bull We know in our ordinary experience (flat spacetime) that in the absence of any forces

objects follow straight lines and we also know that straight lines are the shortest possible

paths that connect two points in such conditions

bull The generalization of the notion of a ldquostraight linerdquo valid also in curved spacetimes is called

geodesic



bull A famous story to simply illustrate the idea of general relativity is the Parable of the Apple

by Misner Thorne and Wheeler [Gravitation (1973)]

bull The parable tries to explain the nature of gravitation in terms of the curvature of spacetime

The spacetime of the parable is the two-dimensional curved surface of an apple



bull The tale goes like this One day a student reflecting on the difference between Einsteins and

Newtons views about gravity noticed ants running on the surface of an apple

bull By advancing alternately and of the same amount the left and right legs the ants seemed to take

the most economical path ldquowow they are going along geodesics on this surfacerdquo

bull The student followed the path of an ant tracing it and then cutting with a knife a small stripe

around the trace ldquoIndeed when put on a plane the path is a straight linerdquo

bull Each geodesic may be regarded as a path (world line) of a free particle on this surface (taken as

a two-dimensional spacetime)



bull Then the student looked at two ants going from the same spot onto initially divergent paths but

then when approaching the top (near the dimple) of the apple the paths crossed and continued

into different directions

bull The reason of the curved trajectories is

bull According to Newton gravitation is acting at a distance from a center of attraction (the

dimple)

bull According to Einstein the local geometry of the surface around the dimple is curved



bull Comments

bull Einstein interpretation dispenses with any action-at-a-distance

bull Although the surface of the apple is curved if you look at any local spot closely (with amagnifying glass) its geometry looks like that of a flat surface (the Minkowski spacetime of

the special relativity)

bull The interaction of spacetime and matter is summarized in John Archibald Wheelers favorite

words spacetime tells matter how to move and matter tells spacetime how to curve

bull This reciprocal influence (matter spacetime) makes Einsteinrsquos field equation non-linearand so very hard to solve



bull Summary of the parable

bull 1) objects follow geodesics and locally geodesics appear straight

bull 2) over more extended regions of space and time geodesics originally receding from eachother begin to approach at a rate governed by the curvature of spacetime and this effect of

geometry on matter is what was called ldquogravitationrdquo

bull 3) matter in turn warps geometry



bull Important quantities in general relativity

bull the metric (the metric tensor g) which may be regarded as a machinery for measuring

distances

dS 2 = gmicroν dxmicrodxν

Rmicroν = Rα

microαν bull the Ricci tensor and the curvature (Ricci) scalar R = gmicroν Rmicroν

Rαβmicroν = Γ

αβν micro minus Γ

αβmicroν + Γ

ασmicroΓ

σβν minus Γ

ασν Γ

σβmicro

Γαβmicro =

12

gασ(gβσmicro + gσmicroβ minus gβmicroσ)

bull curvature expressed by the Riemann curvature tensor

(where are the Christoffel symbols)

bull covariant derivative a derivative that takes into account the curvature of the spacetime



bull From these quantities the path of any particle can be calculated This is how geometry tells

matter how to move

bull The other direction (matter tells spacetime how to curve) requires to know the distribution of

matter (massenergymomentum) described through the stress-energy tensor T

bull After many years of thinking Einstein reached a satisfactory form for the equations relating

geometry and matter

Einstein_tensor = constant x T

bull The Einstein tensor (usually called G) is a tensor in 4D spacetime that has the wanted properties

of

bull being a symmetric tensor (it must because the stress-energy tensor is symmetric)

bull having vanishing (covariant) divergence (it must because the stress-energy tensor has

vanishing divergence)

bull the weak-field limit of the Einstein equations gives the Newtonian Poisson equation (from the

comparison to which the value of the above constant is found)



ie six second-order-in-time second-order-in-space coupled highly-

nonlinear quasi-hyperbolic partial differential equations (PDEs)

four second-order-in-space coupled highly-nonlinear elliptic PDEs

The Einstein equations

Matter and other

fieldsCurvature scalar

Metric(measure of spacetime distances)

Einstein

tensor

(spacetime)

Rest-mass density Internal energy density Pressure

4-velocity

Ricci tensor



bull Gravitational redshift

bull change in the frequency of light moving in regions of different curvature

bull measured in experiments and taken into account by the GPS

bull Periastron shift

bull first measured in the perihelion advance of Mercury (the GR prediction coincides with

the ldquoanomalousrdquo advance if computed in Newtonian theory)bull binary pulsar (strong fields so larger shifts)

bull Bending of light

bull first measured in the bending of photons traveling near the Sun

bull gravitational lensing

bull



bull A black hole is literally a region from which light cannot escape

bull The boundary of a black hole is called the event horizon When something passes through the

event horizon it can no longer communicate in any way with the world outside However it is

not a concrete surface there is no local measurement that can tell an observer that heshe is on

the event horizon

bull General relativity predicts that black holes can form by gravitational collapse Once a star has

burned up its supply of nuclear fuel it can no longer support itself against its own weight and it

collapses It will reach a critical density at which an explosive re-ignition of burning occurs

called a supernova

bull There are several candidate objects that are thought to be black holes (because they are very

compact) but there has been no direct observation of black holes up to now

bull The definitive way to detect black holes is through gravitational waves (more on this later)

because gravitational waves from black holes are distinct from those of other objects



bull In general relativity disturbances in the spacetime curvature (the ldquooldrdquo ldquogravitational

fieldrdquo) propagate at the speed of light as gravitational radiation or gravitational waves

(also known as gravity waves but this term was already in use in fluid dynamics with a

different meaning so I recommend to avoid it)

bull Gravitational waves are a strong point of general relativity which solves the action-at-a-

distance problem of Newtonian gravity

bull Analogously to light which is produced by the motion of electric charges gravitational

waves are produced by the motion of anything (mass-energy) When you wave yourhand you make gravitational waves

bull The point is that the amplitude of gravitational radiation is very small



T R3(GM )

part 3Qij

part t3

MR2

T 3

Mv2

T

LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

LGW = 15Gc5

part 3

Qij

part t3part 3

Qij

part t3 1048575



LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

G4

c5

82times

10

minus74

cgs units

c5 27 times 10minus60 cgs units



bull In fact gravitational waves have not been detected yet even if there are strong

indications that gravitational waves exist as predicted by general relativity

bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each

other at least one of which is a pulsar The curvature is large so it is an interesting place

to study general relativistic effects (Incidentally their orbit may precess by 4 degree per

year)bull The spin axis of pulsars is not aligned with

their magnetic axis so at each rotation theyemit pulses of radio waves and we can

detect them from the Earth Pulsars behave

as very accurate clocks



bull Actually such measurements are very precise and

confirm that the orbit is shrinking with time at a rate

in complete agreement with the emission of

gravitational waves predicted by general relativity





radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs

It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed

The same will happen with GW-astronomy

GSFCNASA





Large-scale

Cryogenic

Gravitational-waveTelescope



o Binary neutron stars

o Binary black-holes

o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars

o Cosmological stochastic background

o Mixed binary systems

o Gravitational collapse (supernovae neutron stars)

o Extreme-Mass-Ratio Inspirals (EMRI)

o





TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina

and LCGT will measure LL ~ h lt 10-21 with SN~1

Numericalrelativity

h

Knowledge of thewaveforms cancompensate for the

very small SN(matched-filtering) and

so enhance detection and allow for source-characterization

possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor

The fundamental equations standard numerical relativity aims at solving

are the Einstein equations



General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics

How to solve the Einstein equations numerically

Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions

There is a successful recipe though



We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step

Also assign a normalizationsuch that

Define therefore

So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the

time coordinate t

(ldquothe direction of timerdquo)

(As mostly in numer ical relativity the signature is here -+++)

The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces

α



so that

ii) and the spatial metric

Letrsquos also define

i) the unit normal vector to the hypersurface



The spatial part is obtained by contracting with the spatialprojection operator defined as

By using we can decompose any 4D tensor into a

purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )

while the timelike part is obtained by contracting with the timelikeprojection operator

The two projectors are obviously orthogonal



The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative

All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice

In particular the 3D connection coefficients

the 3D Riemann tensor

and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and



Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic

curvatureThe extrinsic curvature is defined in terms of the unit normal to as

where is the Lie derivative along

This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric

We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic

curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold

and this can be shown to be equivalent also to



Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a

tensor field along the flow of a vector fieldFor a scalar function this is given by

For a vector field this is given by the commutator

For a 1-form this is given by

As a result for a generic tensor of rank this is given by

= X micropart microV

ν

minus V micropart microX

ν

= X micropart microων + ωmicropart microX

ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α



Consider a vector at one position and then parallel- transport it to a new location

The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative

P P + δ P

The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed

parallel

transport

Hence it measures how the 3Dhypersurface is ldquobentrdquo with

respect to the 4D spacetime



Next we need to decompose the Einstein equations in the spatial

and timelike parts

To this purpose it is useful to derive a few identities

Gauss equations decompose the 4D Riemann tensorprojecting all indices

Codazzi equations take 3 spatial projections and a timelike one



Ricci equations take 2 spatial projections and 2 timelike ones

Another important identity which will be used in the following is

and which holds for any spatial vector



Letrsquos define the double timelike projection of the stress-energy tensor as

We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)

Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection

And similarly for the space-space projection

jmicro = minus

γ

α

micron

β

T αβ



Now we can decompose the Einstein equations in the 3+1splitting

We will get two sets of equations

1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces



We first time-project twice the left-hand-side of the Einstein

equations to obtain

Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the

definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation

Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface



Similarly with a mixed time-space projection of the left-hand-

side of the Einstein equations we obtain

Doing the same for the right-hand-side using the contracted

Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations

which are also 3 elliptic equations

The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime



we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface

t+ δ t

tΣ1

2 1

Σ2

The most general of suchvectors that connect twohypersurfaces is

where is any spatial ldquoshiftrdquo vector Indeed we see that

so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface

t tmicro

δ t = tmicronablamicrot = 1

Before proceeding to the derivation of the evolution equations





We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations

As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones

Using the Ricci equations one then obtains

where



In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is

i) three purely spatial coordinates with unit vectors

So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight

the ldquospatialrdquo nature of and

ii) one coordinate unit vector along the vector



As a result

ie the Lie derivative along is a simple partial derivative

ie the space covariant components of a timelike vector arezero only the time component is different from zero

ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero

Putting things together and bearing in mind that



Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and

that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are

Similarly since the covariant components are

Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors

ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν



We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by

It is now clearer that

bull the lapse measures proper time between two adjacent hypersurfaces

bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces

bull the spatial metric measures distances between points onevery hypersurface








normal linecoordinate line



First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces

Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric

Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how

to relate coordinates between two slices the lapse measures the proper

time while the shift measures changes in the spatial coordinates

Fourth step select a coordinate basis and express all equationsin 3+1 form

The 3+1 or ADM (Arnowitt Deser Misner) formulation



NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime

Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo

While there are infinite possible choices not all of them are

equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions



Different recipes for selecting lapse and shift are possible

i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing

obviously not a good idea



Suppose you want to follow the gravitational

collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)

That would lead to a code crash as soon as asingularity forms No chance of measuring gws

One needs to use smarter temporal gauges

In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal

slicing)

Some chance of measuring gravitational waves



Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time

ii) fix the lapse and shift by requiring they satisfy some

condition eg maximal slicing for the lapse

which has the desired ldquosingularity-avoidingrdquo properties






iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations

This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations

A family of slicing conditions that works very well to obtain

both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing

where and is a positive but otherwisearbitrary function



A high value of the metric

components means that thedistance between numerical grid

points is actually large and this

causes problems

In addition large gradients my benumerically a problem

Choosing a ldquobadrdquo shift may even

lead to coordinates singularities



where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant

A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition

Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-

driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes

B

i

equiv part tβ i



In practice the actual

e x c i s i o n r e g i o n i s a

ldquolegosphererdquo (black region) and

is placed well inside theapparent horizon (which is

found at every time step) and

is allowed to move on the grid

apparent horizon

The region of spacetime inside a horizon

( yellow region) is causally disconnected

from the outside (blue region)

So a region inside a horizon may be

excised from the numerical domain

This is successfully done in pure

spacetime evolutions since the work

of Nadeumlzhin Novikov Polnarev (1978)

Baiotti et al [PRD 71 104006

(2005)] and other groups [Duez et al

PRD 69 104016 (2004)] have shown

that it can be done also in non-vacuum

simulations



There is an alternative to explicit excision Proposed independently by Campanelli et

al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very

popular method for moving--black-hole evolution

It consists in using coordinates that allow the punctures (locations of the

singularities) to move through the grid but do not allow any evolution at the puncture

point itself (ie the lapse is forced to go to zero at the puncture though not the

shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The

conditions that have so far proven successful are modifications to the so-called 1+log

slicing and Gamma-driver shift conditions

In practice such gauges take care that the punctures are never located at a grid

point so that actually no infinity is present on the grid This method has proven to be

stable convergent and successful A few such gauge options are available with

parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without

excisionrdquo) It has been shown that in the coordinates implied by the employed gauges

the singularity is actually always outside the numerical domain



The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation

Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-

posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)

In practice numerical instabilities rapidly appear that destroy the solution exponentially

However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions



Instead we would like to have only terms of the type

In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on

Think of the above system as a second-order system for

because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for

Letrsquos inspect the 3+1 evolution equations again

γ ij

part micropart microγ ij



We care to have wave equations like

because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of

this school)

We can make the equations manifestly hyperbolic indifferent ways including

i) using a clever specific gauge

ii) introducing new variables which follow additional equations

(imposed by the original system)

These methods aim at removing the mixed-derivative term



Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic

The generalized harmonic formulation is based on ageneralization of the harmonic coordinates

When such condition is enforced in the Einstein equations the principal part of the equations for each metric element

becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms



where the a are a set of source functions

The and the equations for their evolution must be suitably

chosen

The harmonic condition is known to suffer frompathologies

However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form



conformal factor

conformal 3-metric trace of extrinsic curvature

trace-free conformalextrinsic curvature

ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i

New evolution variables are introduced to obtain from the

ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is



Dtγ ij = minus2α Aij

Dtφ = minus1

6αK

Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α

K Aij minus 2 Ail

Alj

DtK = minusγ ijnablainablajα + α

Aij Aij + 1

3K 2 + 1

2 (ρ + S )

DtΓi = minus2 Aijpart jα + 2α

ΓijkAkj

minus 2

3γ ijpart jK minus γ ijS j + 6 Aijpart jφ

minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l

Dt equiv part t minus Lβwhere

These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations

And the ADM equations are then rewritten as



Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the

1st-order in time 2nd-order in space formulation

scalar wave equation

conformal tracelessformulation

The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated

to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes



In addition to the (6+6+3+1+1=17) hyperbolic evolution equations

to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations

and 5 additional constraints are introduced by the new variables

NOTE most often these equations are not solved but only monitored to verify that





Computing the waveforms is the ultimate goal of a large

portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are

Both are based of finding some gauge invariant quantities or

the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform

bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)

bullperturbative matching to a Schwarzschild background



In both approaches ldquoobserversrdquo are placed on nested 2-spheres

and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole

Once the waveforms

are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply



The ADM equations are ill posed and not suitable for

numerical integrations

Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations

Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions

Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations

Gravitational waves are present in simulations and can beextracted with great accuracy



A first course in general relativity Bernard Schutz Cambridge University Press (2009)

Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)

Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)

Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)



ie 4 conservation equations for the energy and momentum

ie conservation of baryon number and equation of state (EOS)

(microphysics input)

The additional set of equations to solve is



The evolution of the magnetic field obeys Maxwellrsquos equations

part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B

that give the divergence-free condition

and the equations for the evolution of the magnetic field

nablaν

lowastF microν

= 0

nablamiddot radic γ

B = 0

for a perfect fluid with infinite conductivity (ideal MHD approximation)

lowastF microν

=1

2microνσρ

F σρ



T microν = (ρ + ρε + b2)umicrouν + p + 1

2

b2 gmicroν

minus bmicrobν

where

is the rest-mass density

is the specific internal energy

u is the four-velocity

p is the gas pressure

vi is the Eulerian three-velocity of the fluid (Valencia formulation)

W the Lorentz factor

b the four-vector of the magnetic field

Bi the three-vector of the magnetic field measured by an Eulerian observer



Letrsquos recall the equations we are dealing with

This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo

Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla

lowast

ν F microν = 0 (Maxwell eqs induction zero div)

The complete set of equations is solved with theCactusCarpetWhisky codes

+1 + 3 + 1



Cactus (wwwcactuscodeorg) is a computational

ldquotoolkitrdquo developed at the AEICCT and provides a general

infrastructure for the solution in 3D and on parallel

computers of PDEs (eg Einstein equations)

Whisky (wwwwhiskycodeorg) developed at the AEI

Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes

Carpet (wwwcarpetcodeorg) provides box-in-box

adaptive mesh refinement with vertex-centered grids

CactusCarpetWhisky



is a driver mainly developed by E

Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids

Carpet follows a (simplified) Berger-Oliger

[J Comput Phys 53 484 (1984)] approach to

mesh refinement that is

bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids

bull refined subdomains have boundaries alignedwith the grid lines

bull the refinement ratio between refinement levels

is constant

While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement



Many codes for numerical relativity are publicly available from the recent project of

the which aims at providing computational toolsfor the community

It includes

bull spacetime evolution code

bull GRHydro code (not completely tested yet see also the Whisky webpage)

bull GRMHD code (in the near future)

bull mesh refinement

bull portability

bull simulation factory(an instrument to managesimulations)





Definition in general for a hyperbolic system of equations a

Riemann problem is an initial-value problem with initial condition

given by

where UL and UR are two constant vectors representing the ldquoleftrdquo

and ldquorightrdquo state

For hydrodynamics a (physical) Riemann problem is the evolution of

a fluid initially composed of two states with different and constant

values of velocity pressure and density



I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local

Riemann problem

There are powerful numericalmethods to solve Riemann

problems both exactly andapproximately



bull High-order (up to 8th) finite-difference techniques for the field equations

bull Flux conservative form of HD and MHD equations with constraint transport

or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)

bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)

bull AMR with moving grids

bull Accurate measurements of BH properties through apparent horizons (IH)

bullUse excision (matter andor fields) if needed

bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)

bull Single-fluid description no superfluids nor crusts

bull Only inviscid fluid so far (not necessarily bad approximation)

bull Radiation and neutrino transport totally neglected (work in progress)

bull Very coarse resolution far from regimes where turbulencedynamos develop

BINARY NEUTRON STAR MERGERS



BINARY NEUTRON-STAR MERGERS

bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005

bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168

bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105

THE INITIAL GRID SETUP FORTHE BINARY



V i s u a l i z a t i o n b y G i a c o m a z z o

K a e h l e r R e z z o l l a

THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS



M d i



Matter dynamicshigh-mass binary

soon after the merger the torus isformed and undergoes oscillations

Merger

Collapse toBH



W f l i EOS



Waveforms polytropic EOShigh-mass binary

The full signal from the inspiral to theformation of a BH has been computed

Merger Collapse to BH

I i t f th EOS id l fl id l i E S



Imprint of the EOS ideal-fluid vs polytropic EoS

After the merger a BH is producedover a timescale comparable with the

dynamical one

After the merger a BH is producedover a timescale larger or much

larger than the dynamical one

W f id l fl id E S



Waveforms ideal-fluid EoS

low-mass binary

C ti ti f th d



Conservation properties of the code

conservation of energy

low-mass binary

conservation of angular momentum

Imprint of the EOS frequency domain




The pre-merger dynamics are verysimilar the post-merger phase isvery different

Contributions from themerger






Contributions from thebar-deformed HMNS






Contributions from thecollapse to BH

E t di th k t MHD



We have considered the same models also when an initially

poloidal magnetic field of ~108

or ~1012

G is introduced

The magnetic field is added by hand using the vector potential

where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field

The initial magnetic fields are therefore fully contained inside the

stars ie no magnetospheric effects

Ab P cut = 004timesmax(P )

Aφ = Abr2[max(P minus P cut 0)]

n

Extending the work to MHD



Idealminus

fluid M = 165 M B = 10

12

G

Note that the torus is much less dense and a large

plasma outflow is starting to be launchedValidations are needed to confirm these results

Typical evolution for a magnetised binary



bull Overview of general relativity

bull Gravitational waves

bull Gravitational-wave detection

bull Numerical relativity bull 3+1 decomposition

bull Einstein equations

bull gauge conditions

bull gravitational-wave extraction

bull non-vacuum spacetimesbull general-relativistic magnetohydrodynamics

bull Some applications binary neutron-star merger



















air friction)

























geodesic
























dimple)




bull Comments


















distances


Rmicroν = Rα


Rαβmicroν = Γ


αβmicroν + Γ

ασmicroΓ

σβν minus Γ

ασν Γ

σβmicro

Γαβmicro =

12








matter how to move




geometry and matter



of












Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor












bull







the event horizon




called a supernova


















T R3(GM )

part 3Qij

part t3

MR2

T 3

Mv2

T

LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

LGW = 15Gc5

part 3

Qij

part t3part 3

Qij

part t3 1048575



LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

G4

c5

82times

10

minus74

cgs units























radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G






















air friction)

























geodesic
























dimple)




bull Comments


















distances


Rmicroν = Rα


Rαβmicroν = Γ


αβmicroν + Γ

ασmicroΓ

σβν minus Γ

ασν Γ

σβmicro

Γαβmicro =

12








matter how to move




geometry and matter



of












Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor












bull







the event horizon




called a supernova


















T R3(GM )

part 3Qij

part t3

MR2

T 3

Mv2

T

LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

LGW = 15Gc5

part 3

Qij

part t3part 3

Qij

part t3 1048575



LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

G4

c5

82times

10

minus74

cgs units























radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G









air friction)

























geodesic
























dimple)




bull Comments


















distances


Rmicroν = Rα


Rαβmicroν = Γ


αβmicroν + Γ

ασmicroΓ

σβν minus Γ

ασν Γ

σβmicro

Γαβmicro =

12








matter how to move




geometry and matter



of












Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor












bull







the event horizon




called a supernova


















T R3(GM )

part 3Qij

part t3

MR2

T 3

Mv2

T

LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

LGW = 15Gc5

part 3

Qij

part t3part 3

Qij

part t3 1048575



LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

G4

c5

82times

10

minus74

cgs units























radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G
























geodesic
























dimple)




bull Comments


















distances


Rmicroν = Rα


Rαβmicroν = Γ


αβmicroν + Γ

ασmicroΓ

σβν minus Γ

ασν Γ

σβmicro

Γαβmicro =

12








matter how to move




geometry and matter



of












Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor












bull







the event horizon




called a supernova


















T R3(GM )

part 3Qij

part t3

MR2

T 3

Mv2

T

LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

LGW = 15Gc5

part 3

Qij

part t3part 3

Qij

part t3 1048575



LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

G4

c5

82times

10

minus74

cgs units























radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G















geodesic
























dimple)




bull Comments


















distances


Rmicroν = Rα


Rαβmicroν = Γ


αβmicroν + Γ

ασmicroΓ

σβν minus Γ

ασν Γ

σβmicro

Γαβmicro =

12








matter how to move




geometry and matter



of












Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor












bull







the event horizon




called a supernova


















T R3(GM )

part 3Qij

part t3

MR2

T 3

Mv2

T

LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

LGW = 15Gc5

part 3

Qij

part t3part 3

Qij

part t3 1048575



LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

G4

c5

82times

10

minus74

cgs units























radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G



























dimple)




bull Comments


















distances


Rmicroν = Rα


Rαβmicroν = Γ


αβmicroν + Γ

ασmicroΓ

σβν minus Γ

ασν Γ

σβmicro

Γαβmicro =

12








matter how to move




geometry and matter



of












Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor












bull







the event horizon




called a supernova


















T R3(GM )

part 3Qij

part t3

MR2

T 3

Mv2

T

LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

LGW = 15Gc5

part 3

Qij

part t3part 3

Qij

part t3 1048575



LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

G4

c5

82times

10

minus74

cgs units























radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G





















dimple)




bull Comments


















distances


Rmicroν = Rα


Rαβmicroν = Γ


αβmicroν + Γ

ασmicroΓ

σβν minus Γ

ασν Γ

σβmicro

Γαβmicro =

12








matter how to move




geometry and matter



of












Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor












bull







the event horizon




called a supernova


















T R3(GM )

part 3Qij

part t3

MR2

T 3

Mv2

T

LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

LGW = 15Gc5

part 3

Qij

part t3part 3

Qij

part t3 1048575



LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

G4

c5

82times

10

minus74

cgs units























radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G











dimple)




bull Comments


















distances


Rmicroν = Rα


Rαβmicroν = Γ


αβmicroν + Γ

ασmicroΓ

σβν minus Γ

ασν Γ

σβmicro

Γαβmicro =

12








matter how to move




geometry and matter



of












Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor












bull







the event horizon




called a supernova


















T R3(GM )

part 3Qij

part t3

MR2

T 3

Mv2

T

LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

LGW = 15Gc5

part 3

Qij

part t3part 3

Qij

part t3 1048575



LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

G4

c5

82times

10

minus74

cgs units























radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G






bull Comments


















distances


Rmicroν = Rα


Rαβmicroν = Γ


αβmicroν + Γ

ασmicroΓ

σβν minus Γ

ασν Γ

σβmicro

Γαβmicro =

12








matter how to move




geometry and matter



of












Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor












bull







the event horizon




called a supernova


















T R3(GM )

part 3Qij

part t3

MR2

T 3

Mv2

T

LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

LGW = 15Gc5

part 3

Qij

part t3part 3

Qij

part t3 1048575



LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

G4

c5

82times

10

minus74

cgs units























radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G















distances


Rmicroν = Rα


Rαβmicroν = Γ


αβmicroν + Γ

ασmicroΓ

σβν minus Γ

ασν Γ

σβmicro

Γαβmicro =

12








matter how to move




geometry and matter



of












Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor












bull







the event horizon




called a supernova


















T R3(GM )

part 3Qij

part t3

MR2

T 3

Mv2

T

LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

LGW = 15Gc5

part 3

Qij

part t3part 3

Qij

part t3 1048575



LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

G4

c5

82times

10

minus74

cgs units























radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G








distances


Rmicroν = Rα


Rαβmicroν = Γ


αβmicroν + Γ

ασmicroΓ

σβν minus Γ

ασν Γ

σβmicro

Γαβmicro =

12








matter how to move




geometry and matter



of












Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor












bull







the event horizon




called a supernova


















T R3(GM )

part 3Qij

part t3

MR2

T 3

Mv2

T

LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

LGW = 15Gc5

part 3

Qij

part t3part 3

Qij

part t3 1048575



LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

G4

c5

82times

10

minus74

cgs units























radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G







matter how to move




geometry and matter



of












Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor












bull







the event horizon




called a supernova


















T R3(GM )

part 3Qij

part t3

MR2

T 3

Mv2

T

LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

LGW = 15Gc5

part 3

Qij

part t3part 3

Qij

part t3 1048575



LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

G4

c5

82times

10

minus74

cgs units























radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G










Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor












bull







the event horizon




called a supernova


















T R3(GM )

part 3Qij

part t3

MR2

T 3

Mv2

T

LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

LGW = 15Gc5

part 3

Qij

part t3part 3

Qij

part t3 1048575



LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

G4

c5

82times

10

minus74

cgs units























radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G















bull







the event horizon




called a supernova


















T R3(GM )

part 3Qij

part t3

MR2

T 3

Mv2

T

LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

LGW = 15Gc5

part 3

Qij

part t3part 3

Qij

part t3 1048575



LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

G4

c5

82times

10

minus74

cgs units























radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G










the event horizon




called a supernova


















T R3(GM )

part 3Qij

part t3

MR2

T 3

Mv2

T

LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

LGW = 15Gc5

part 3

Qij

part t3part 3

Qij

part t3 1048575



LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

G4

c5

82times

10

minus74

cgs units























radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G

















T R3(GM )

part 3Qij

part t3

MR2

T 3

Mv2

T

LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

LGW = 15Gc5

part 3

Qij

part t3part 3

Qij

part t3 1048575



LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

G4

c5

82times

10

minus74

cgs units























radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G






T R3(GM )

part 3Qij

part t3

MR2

T 3

Mv2

T

LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

LGW = 15Gc5

part 3

Qij

part t3part 3

Qij

part t3 1048575



LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

G4

c5

82times

10

minus74

cgs units























radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G






LGW 1

5

G

c5

M 1048575v21114111

T

1

5

G4

c5

M

R

2

G4

c5

82times

10

minus74

cgs units























radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G

























radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G














radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G








radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G






radio

far-IR

mid-IR

near-IR

opticalx-ray

gamma-ray

GWs



GSFCNASA





Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G








Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G






Large-scale

Cryogenic











o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G













o







Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G










Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G








Numericalrelativity

h




possible






Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G









Matter and other



Einstein

tensor

(spacetime)


4-velocity

Ricci tensor













Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G














Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G








Define therefore


time coordinate t




α



so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G






so that



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G



































ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G




























ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G





















ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G












ν


ν


ν

= X α

part αT micro

ν minus T

α

ν part αX

micro+ T

micro

αpart ν X

α





P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G








P P + δ P


parallel

transport






and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G







and timelike parts















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G















jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G










jmicro = minus

γ

α

micron

β

T αβ









equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G












equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G







equations to obtain















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G















t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G







t+ δ t

tΣ1

2 1

Σ2




t tmicro










where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G











where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G









where










As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G













As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G






As a result











ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G










ij= γ

ij

γ

α0= 0

gmicroν

= γ microν

minus nmicron

ν











































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G














































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G







































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G































slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G






















slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G
















slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G











slicing)























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G

























causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G
















causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G









causes problems










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G










B

i

equiv part tβ i









apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G












apparent horizon











PRD 69 104016 (2004)] have shown


simulations
































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G



































γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G


















γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G











γ ij






this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G








this school)
















chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G














chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G








chosen





conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G






conformal factor



ldquoGammasrdquo

φ

˜γ ij

K

Aij

˜Γ

i






Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G







Dtφ = minus1

6αK


K Aij minus 2 Ail

Alj


Aij Aij + 1

3K 2 + 1

2 (ρ + S )


ΓijkAkj

minus 2


minuspart jβ l

part lγ ij

minus 2γ m(j

part mβ i)

+

2

3 γ ij

part lβ l
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G
































Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G
























Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G


















Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G
















Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G








Once the waveforms

























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G



























part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G



















part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G













part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G







part

part t

radic γ B

= nablatimes

α v minus β

timesradic

γ B



nablaν

lowastF microν

= 0


B = 0


lowastF microν

=1

2microνσρ

F σρ




2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G







2

b2 gmicroν

minus bmicrobν

where














lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G









lowast



+1 + 3 + 1











CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G














CactusCarpetWhisky











is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G














is constant






It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G








It includes





bull portability








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G










given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G








given by









Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G







Riemann problem
































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G

































M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G


















M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G











M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G






M d i





Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G








Merger

Collapse toBH



W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G






W f l i EOS











dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G














dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G








dynamical one



W f id l fl id E S




low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G







low-mass binary

C ti ti f th d





low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G








low-mass binary




















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G





















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G















E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G









E t di th k t MHD





or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G








or ~1012

G is introduced







n




Idealminus

fluid M = 165 M B = 10

12

G






Idealminus

fluid M = 165 M B = 10

12

G




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Luca Bai Otti