2008 May 31 5th Italian-Sino Workshop

Yu-Huei Wu and Chih-Hung WangYu-Huei Wu and Chih-Hung Wang http://arxiv.org/abs/0803.2194 Accept by Accept by Classical and Quantum GravityClassical and Quantum Gravity without correction without correction Provisionally scheduled to publish for June 2008Provisionally scheduled to publish for June 2008

Quasi-local mass in the covariant Newtonian space time

Yu-Huei Wu 吳育慧Institute of Astronomy, NCU

Mass in General Relativity• Equivalence principle forbids us to have a localized gravitational energy-

momentum expression in GR.

• Unlike Newtonian theory, we cannot separate background and dynamics in GR

• General relativity is not a gauge field theory and highly non-linear.

• All other fields can have proper energy density definitions in curved space-time.

Gab = 8 G /c^4 Tab

• Unlike Newtonian theory, there’s no unique definition of quasi-local mass in GR. Mass is formulated in various ways.

Dark energy: back to Newtonian --------Lucy Calder and Ofer Lahavn 2008

Quasi-local mass construction• Quasi-local idea is to define gravitational energy-

momentum associated to a closed 2-surface

• We examine (1) Komar integral, (2) Brown-York expression and (3) Dogan-Mason mass expression in this work

source

sphere

Quasi-local mass energy? What does it mean physically?

• Bondi's prolate and oblate mechanism.

• Gravitational wave can transfer energy locally (GW carry mass loss and radiate radiate to null infinity.)

• Bondi put it: 'one can heat water with them'.

• Tidal heating Io & Jupiter [Both + Creighton 2000]

General Relativity v.s. Newtonian

General Relativity:• 10 field equations and 10

potentials • Non-linear equations• Intrinsically geometrical• Horizons of black holes• Gravitational waves

Newtonian Gravity:• 1 field equation and 1

potential• Linear equation• Absolute space and time• No horizons or no black hole• No gravitational waves• Have localized mass and

energy

What does it mean by the Newtonian limit?

• Post-Newtonian theory depends on the choice of the coordinate and background Minkowski metric is required. Not a covariant fashion !

• Can these quasi-local expressions directly go back to the unique surface integration in Newtonian space time?

• Can spinor field has a Newtonian correspondence?

• Newtonian and post-Newtonian approximations are asymptotic to general relativity by using the Newtonian sequence method [Futamase and Schutz ’83].

• Hence Jeffryes use it to verify the Newtonian limit of Penrose quasi-local mass. Unfortunately, it can return to the Newtonian mass and energy only by using transaction between 2-surface twistors and 3-surface twistors.

• All these are from GR aspect

• Can a purely Newtonian type of star distort the geometry of 3-space , i.e., generate pseudo-Riemannian curvature on it?

• A more consistent transition between relativistic and Newtonian theory of gravity is needed.

Neutron starOuter crust: Newtonian theory or flat background

V.S. Inner core: General Relativity

Equation of State (EOS) of inner core ??

?

1. For the ordinary pulsars, the magnetic field is sufficiently low. a purely Newtonian framework.

2. For the magnetars, the magnetic field is sufficiently strong. relativistic treatment will be indispensable even locally.

Covariant Newtonian Theory How to write the Newtonian theory in terms of space-

time geometry?

Newtonian space time is handicap, therefore cannot give an unique connection unless we introduce the ether frame e.

• The covariant Newtonian theory is developed by Kunzle ’72, and Carter ’04.

• The purpose is to show how to set up and apply a fully covariant formulation of the kinds of non-relativistic multiconstituent fluid dynamical models and relativistic correspondence.

Newtonian (Galilean) space-time is considered as a direct product of a flat Euclidean 3-space and one dimensional Euclidean time line t, i.e., four dimensional fiber bundle. Each fiber t contains a Euclidean (contravariant) flat metric γ.

Each fiber t contains a Euclidean (contravariant) flat metric γ and one-form dt where γis a degenerate metric and cannot upper or lower indices, i.e. no metric-dual definition of tensor fields. It has been found that conditions ∇γ = 0 and ∇dt = 0 with vanishing torsion cannot give an unique connection ∇ in the Newtonian space-time.

Milne gauge in the Newtonian space time

• Equivalence principle in the Newtonian space time [Milne 1934]. • In the Milne structure of Newtonian space-time, ∇ is not gauge invariant and has the

following gauge transformations

• Milne gauge invariant connection D (Newton-Cartan connection)

• Newtonian gravitational equation Gauge invariant

Mass expression in the covariant Newtonian space time

From volume integration to surface integration

Gauge invariant!!

Relativistic correspondence

From Newtonian to GR

Carter et al.'03 considered the following four dimensional non-degenerate metric

where

The generalization of can be written as

We further restrict deformations of as following

It is sufficient to represent a deformation generated by a star.

Einstein field equation gives

(1) Komar integral• In terms of a asymptotic time-like Killing vector

field , Komar integral yields

• It gives exactly the same covariant expression as Newtonian surface integration without choosing a specific closed 2-surface.

(2) Brown-York quasi-local energy

Einstein field equation gives BY mass in Newtonian limit then becomes

If it is spherical symmetry with , then

(3) Dogan-Mason mass:Twistor program

• Twistor is a geometrical object that is more fundamental than spinors. Space-time themselves can be regarded as derived objects [Penrose ‘67].

• We can use the spinors to define mass or angular momentum.

• The Newtonian limit of spinor field is shown here.• Metric:

Spin frame in Newtonian limit

The asymptotic constant spinor

can be expressed as

They satisfy the holomorphic condition

The key NP coefficients are:

The first order spinor solutions are

The second order spinor solutions are

The mass expression can be expanded as

Conclusion

• Since general relativity has a well known Newtonian limit, it is reasonable for us to argue that it should yield the Newtonian expression for all of quasi-local expressions in the covariant Newtonian space-time.

• In this work, we verify that Komar integral can yield the Newtonian quasi-local mass expression without choosing a specific 2-sphere or referring to spherical symmetry of the Newtonian potential, however, the Brown-York expression and Dougan-Mason mass can give Newtonian expression with spherical symmetry

• Whether this problem is due to the expressions themselves or some other technical problems requires a further investigation.

Brown D and York J W Quasilocal energy and conserved charges derived from the gravitational action 1993 Phys. Rev. D 47 1407 (Preprint gr-qc/9209012)

Carter B, Chachoua E and Chamel N Covariant Newtonian and Relativistic dynamics of (magneto)-elastic solid model for neutron star crust 2006 Gen. Rel. Grav. 38 83-119

Carter B and Chamel N Covariant analysis of Newtonian multi-fluid models for neutron stars: I Milne-Cartan structure and variational formulation 2004 Int. J. Mod. Phys. D 13 291-325 (Preprint astro-ph/0305186)

Dougan A J and Mason L J Quasi-local mass constructions with positive energy 1999 Phys. Rev. Lett. 67 2119-22

Künzle H P Galilei and Lorentz structures on space-time: comparison of the corresponding geometry and physics 1972 Annales de l’institut Henri Poíncare

Futamase T and Schutz B 1983 Phys. Rev. D 28 2363

Jeffryes B 1986 Class. Quantum Grav. 3 841-852

Escher: Relativity