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Comparison of Einstein-Maxwell theory to an alternative theory. Jim Shifflett shifflet@hbar.wustl.edu WUGRAV group Advisor: Clifford Will. Overview: Einstein-Maxwell theory Lambda-renormalized Einstein-Schr ö dinger theory (LRES theory) Motivation for LRES theory - PowerPoint PPT Presentation
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Jim Shifflettshifflet@hbar.wustl.eduWUGRAV groupAdvisor: Clifford Will
Graduate student seminar, 16 Feb. 2007
Overview:• Einstein-Maxwell theory• Lambda-renormalized Einstein-Schrödinger theory (LRES theory)• Motivation for LRES theory• Comparison of LRES theory to Einstein-Maxwell theory
• Greek indices μ, ע,α,β etc. always go from 0…3
• Geometrized units: c=G=1
interval) time-(space ),(),,,( 3210 xddtdxdxdxdxdx
3
0
3
0
dxgdxdxgdxds
Some conventions:
• Einstein summation convention: paired indices imply summation
, potential vectornetic electromag
AAAAA
A
3
2
1
0
The fundamental fields of Einstein-Maxwell theory
• The electromagnetic vector potential Aμ is the fundamental field
x
A
x
A
BBEBBE
BBEEEE
F
xyz
xzy
yzx
zyx
00
00
• Electric and magnetic fields (E and B) are defined in terms of Aμ
The fundamental fields of Einstein-Maxwell theory
• Metric determines distance between points in space-time
scoordinate zy,x,t,and space flat for
1-1-
1-1
gggggggggggggggg
tensormetric
33231303
23221202
13121101
03020100
000000000000
μνg
dxgdxds dx1
dx2
• Connection determines how vectors change when moved
3
2
1
0
connection
dxvvv
θ
r
vvv
θ
dxαr
2D radial coordinates(x1,x2)=(r,θ)
generalized Pythagorean theorem (ds)2=(dx1)2+(dx2)2
•
•
Almost all field theories can be derived from a Lagrangian
μνν x/A , A potentialvector
neticelectromag examplefor s,derivative& fields offunction
density Lagrangian
L
)(
α
να
νν x/AxAδA
δ LLL0
310 dxdxdx S L
• The field equations are derived from the Euler-Lagrange equations
which minimizes the “action”
• Lagrangian is also necessary for quantization via path integral methods.
• Guarantees field equations are coordinate independent and self consistent
Einstein-Maxwell theory = General Relativity + Electromagnetism
)( , ),A,,,(u
16π
1
2Λ( 16π
1
νν
M
νμ1μ-
μννμμν gggψFFg
Rgg
det
)
L
L
xxR
g μν
)( , connection
, tensormetric
0 ,0
0
eqs. Lagrange-Euler
LL
L
g
A
0B andlaw sFaraday'
, law sGauss' andlaw sAmpere'
equations sMaxwell'
Lorentz-force equationEinstein equations
,potential vectorA ν ν
μ
μ
νμναβ
μβ-να-νμ
x
A
x
A, FFggF
11
wasteful.seems This
μν. all for Γ Γ ,gg is that symmetric; are Γ and g ανμ
αμννμμν
αμνμν
simple.very really not is This
.
for short is term netismelectromag The
3
0
3
0
3
0
3
0
11)det(
x
A
x
Ag
x
A
x
Agg
FFg νμμν
Some Nitpicking of Einstein-Maxwell theory
• Measurement gives Λ~10-56 cm-2. QED indicates Λz~-1066 cm-2 from zero-point fluctuations. This suggests “bare” Λb cancels Λz so that “physical” Λ=Λb+Λz.
• Gravitation and electromagnetism are the two long range forces. They are not unified in Einstein-Maxwell theory.
Λ-renormalized Einstein-Schrödinger theory (LRES theory)
)( ),(
)(
M μννz
bνμμν
NN,Agψug
ΛΓRNN
det,,216
1
2ˆ16
1 1
L
L
• Einstein-Maxwell Lagrangian again
• LRES theory excludes FμעFעμ part, uses non-symmetric and , and uses “bare” Λb≈ -Λz so that “physical” Λ =Λb+Λz matches measurement
N
where gμע and Aμ are defined by
rizationantisymmet Ationsymmetriza
] [
) (
,6//2ˆ,
][
)(1
b
NNgg
)( , )(
M μνννμμν g g,Ag,ψuFFg
ΓRgg
det,16
1
2)(16
1 1
L
L
theory
MaxwellEinsteintheoryLRESlim
zΛ
LRES theory can be used to derive Einstein-Maxwell theory
10~1
~~Λ 2662
24z
cmL
LP
Pc
• But because of quantum gravity effects we probably have instead
• We will assume ωc~1/LP and Λz~1/LP2 for the rest of this presentation
where Λz = (cosmological constant from zero-point fluctuations)
where ωc=(cutoff frequency)~1/Lp
Lp=(Planck length)
using“geometrized”
unitsc=G=1
• Einstein-Schrödinger theory is non-symmetric generalization of GR - developed by two Nobel prize winning physicists
• LRES theory matches measurement as well as Einstein-Maxwell theory
• LRES theory reduces to ordinary GR without E&M for symmetric fields
• LRES theory is a vacuum energy renormalization of ES theory • Λz term should be expected to occur as a quantization effect
• Zero-point fluctuations are essential to QED - demonstrated by Casimir effect, etc.
• Λ = Λb+Λz is like mass/charge/field-strength renormalization in QED - “physical” mass of an electron is sum of “bare” mass and “self energy” - it’s fine tuning but this is not new; its already in Standard Model + GR
• Λz modification of ES theory has never been considered before
LRES theory is well motivated
g
g
Electron Self Energy→ mass renormalizationm = mb- mb·ln(ћωc/mc2)3α/2π
Photon Self Energy (vacuum polarization)→ charge renormalizatione = eb - eb·ln(M/m)α/3π
Zero-Point Energy (vacuum energy density)→ cosmological constant renormalizationΛ = Λb - LP
2ωc4(fermions-bosons)/2π
ωc= (cutoff frequency) LP = (Planck length) M= (Pauli-Villars cutoff mass) α = (fine structure constant)
γe-
γ
e+
e-γ
γ e-
e-
Λ = Λb+ Λz is similar to mass/charge renormalization in QED
2
2
0
22
2
r
q
r
2M1a
where
,
θsinr0000r00001/a0000a
r
qA
g
Exact charged black hole solution of Einstein-Maxwell theory
• Called the Reissner-Nordström solution
• Becomes Schwarzschild solution for q=0
• -2M/r term is what causes gravitational force
1000010000100001
g
is scoordinate txyz inspace Flat s.coordinate
spherical in space flatgives 1a Setting
Exact charged black hole solution of LRES theory
• The charged solution is very close to the Reissner-Nordström solution,
70-763
64-7642
17
10 10 | /10 10 | /
,,10 |
rMrq
MMeqcmrMMqr
b
b
esun
4b
22
b4b
2
2
2
266b
2b4
b
2
3b
0
22
2
rΛ
2q1b , )O(Λ
r10Λ
q1
r
q
r
2M1a
where
10~ , )O(Λr5Λ
4q
rΛ
M1
θsinr0000r00001/ab0000a
b
cmr
qA
g
• Extra terms are tiny for worst-case radii accessible to measurement:
Charged solution of Einstein-Maxwell theory vs. LRES theory
LRESEinstein-Maxwell Event horizon conceals interior(disappears for Q>M as is the case for elementary particles)
r+
r-r+
r-
cmLq Pb334/1 10~~)/2(
~
e
μνμν
μνμνμ
11
r
at is it 0r of instead0;area) (surface whereis origin
finite all are R g- ,N g-
,g g- , g- N- ,N ,F , A
y,singularit r1/ has g
,iessingularit have also
fields relevant other ally,singularit has1/r g 2
00
• Additional fields couple to gμע and Aע just as in Einstein-Maxwell theory
• may contain all of the Standard Model (excluding FμעFμעterm)
LRES theory allows other fields to be included in the Lagrangian
Aiq
x
�
D , ψψmψ)γDψψDγψ(
2
1g :QED σ
σσ
ML
ML
- gives fractional changes of <10-50 in Hydrogen atom energy levels
velocity)(4vu ugu2
μAu
m
μq :fluid Charged σ
ασα
νν
M ),(,L
),(
2Λ)(
M
b
)det(,216
1
ˆ16
1 1
NN,Ag,ψug
ΓRNN
νz
νμμν
L
L
Lorentz force equation is identical to that of Einstein-Maxwell theory
• Usual Lorentz force equation results from divergence of Einstein equations
BvqEqdt
pd
+q/r2 -q/r2 +q/r2
B
velocity)-(4) v ( where
γγ,u
uqFuux
umg
• Lorentz force equation in 4D form
• Also includes gravitational “force”; it becomes geodesic equation when q=0
• Requires no sources (no in the Lagrangian)
• LRES theory and Einstein-Maxwell theory are both non-linear so two stationary charged solutions summed together is not a solution
• EIH method finds approximate two-particle solutions for gμע, Γαμע and Aμ
• Motion of the particles agrees with the Lorentz force equation
q/r2 q/r2
Lorentz force also results from Einstein-Infeld-Hoffman (EIH) method
ML
LRES theory avoids ghosts
ghost no meaning ,ω then ω witherenormalizfully weIf
ωLω2~2Λω have they would because off cut are Ghosts
1)Gc (with Lω~Λ have we1/L~ωfrequency cutoff a With
fieldsenergy negative effective are Ghosts
ghostc
cP2czghost
2P
4czPc
M1, Q1
M2, Q2
Periastron Advance
rMQ
MQ
r
Q
rr
Q
r
M
rM
bo
p
21
12
3
21
211
22
22
21 6
361
2
3
2
Einstein-Maxwell theory LRES theory modification
2211 /MQQMμ
Kepler’s third law
frequencyperiastron
p
3period
2frequencyorbital
ro
This ignoresradiation reaction
test case → extremal charged black hole
Q1=M1=Msun,
Q2=0,
r=4M1=5.9x105 cm
ωo = 2.6x104 rad/s
Bohr atom
Q2= -Q2=e,
M1=MP, M2=Me,
r=a0=.53x10-8 cmωo = 4.1x1016 rad/s
periastron frequency (ωp) periastron frequency (ωp)
Einstein-Maxwell
theory
1.6x104 rad/s 1.1x1012 rad/s
LRES theory
modification
5.5x10-74 rad/s 2.1x10-76 rad/s
fractional
difference10-78 10-88
Periastron Advance – LRES theory vs. Einstein-Maxwell theory
)(uf
• EM plane wave solution is identical to that of Einstein-Maxwell theory
function) arbitrary
(
)(ˆ
)(ˆ
f
ctxfzB
ctxfyE
)(
10000100001001
222 zyfh
hhhh
g
Exact Electromagnetic Plane Wave Solution of LRES theory
ctxu
who’s profile is this?
Deflection of Light
photon
M, Q
ΔΦ
parameter
impact b
4
2
2
2
8
3
4
34
b
Q
b
Q
b
M
b
Einstein-Maxwell theory
LRES theorymodification
Deflection of Light – LRES theory vs. Einstein-Maxwell theory
test case → extremal charged black hole
Q=M=Msun,
b=4M=5.9x105 cm
atomic charges/masses/radii
Q=e, M=MP,
b=a0=.53x10-8 cm
deflection of light (ΔΦ) deflection of light (ΔΦ)
Einstein-Maxwell
theory
.85 rad 9.4x10-44 rad
LRES theory
modification
2.1x10-79 rad 2.9x10-101 rad
fractional
difference10-79 10-57
Time Delay of Light
radio signal
M, Qt=d/c+Δt
t=0
satellite
parameter
impact b
–(
)–
3
22
2
3ln4
b
Q
b
Q
b
dMt
b
Einstein-Maxwell theory
LRES theorymodification
d
Time Delay of Light – LRES theory vs. Einstein-Maxwell theory
test case → extremal charged black hole
Q=M=Msun,
b=4M=5.9x105 cm
atomic charges/masses/radii
Q=e, M=MP,
b=a0=.53x10-8 cm
time delay of light (Δt) time delay of light (Δt)
Einstein-Maxwell
theory
2.1x10-5 sec 2.7x10-62 sec
LRES theory
modification
4.2x10-83 sec 6.3x10-120 sec
fractional
difference10-78 10-58
Why pursue LRES theory if it’s so close to Einstein-Maxwell theory
• It unifies gravitation and electromagnetism !
• Quantization of LRES theory is untried approach to quantization of gravity - LRES theory gets much different than Einstein-Maxwell theory as k→1/LP
- This could possibly fix some infinities which spoil the quantization of GR
• LRES theory suggests untried approaches to a complete unified field theory - Higher dimensions, but with LRES theory instead of vacuum GR? - Non-abelian fields, but with LRES theory instead of Einstein-Maxwell?
• We still don’t have a unified field theory, 50 years after Einstein: - Standard Model: excludes gravity, 25 parameters, not very “beautiful” - String theory: background dependent, spin-2 particle → GR? ~ 10500 versions, none of which have been shown to give Standard Model, problems accounting for Λ>0 and broken symmetry, little predictive ability, read “The Trouble With Physics” by Lee Smolin.
Summary of Λ-renormalized Einstein-Schrödinger theory
224
zc
1~~Λ
1 lim but
theoryMaxwellEinstein
theoryLRES
zΛP
PcP L
LL
•
• Matches measurement as well as Einstein-Maxwell theory.
• Reduces to ordinary GR without electromagnetism for symmetric fields.
• Other Standard Model fields can be added just like Einstein-Maxwell theory.
• Avoids the problems of the original Einstein-Schrödinger theory.
• Well motivated – it’s the ES theory but with a quantization effect.
• Unifies gravitation and electromagnetism in a classical sense.
• Suggests untried approaches to a complete quantized unified field theory.
• For the details see my papers: www.arxiv.org/abs/gr-qc/0310124, www.arxiv.org/abs/gr-qc/0403052, www.arxiv.org/abs/gr-qc/0411016.
Backup charts
LRES theory avoids the problems of Einstein-Schrödinger theory
• Matches measurement as well as Einstein-Maxwell theory
• Definitely predicts a Lorentz force
• Allows other fields in the Lagrangian, which couple to symmetric metric
• Avoids ghosts
LRES theory matches measurement as well as Einstein-Maxwell theory
• Exact solutions: - EM plane-wave solution is identical to that of Einstein-Maxwell theory - Charged solution and Reissner-Nordström sol. have tiny fractional difference: 10-76 for extremal charged black hole; 10-64 for atomic charges/masses/radii.
Standard tests
extremal charged black hole atomic charges/masses/radii
periastron advance 10-78 10-88
deflection of light 10-79 10-57
time delay of light 10-78 10-58
• Other Standard Model fields can be added just like Einstein-Maxwell theory:- Energy levels of Hydrogen atom have fractional difference of <10-50.
• fractional difference from Einstein-Maxwell result
• Reduces to ordinary GR without electromagnetism for symmetric fields
• Lorentz force equation is identical to that of Einstein-Maxwell theory
• Extra terms in Einstein and Maxwell equations are <10-16 of usual terms.
Electromagnetism in 4-dimensional form
x
A
x
A
BBEBBEBBEEEE
F
xyz
xzy
yzx
zyx
00
00
tensorfieldneticelectromag
vector)-4 potential gnetic(electroma),(
vector)-4(velocity )v,( vector)-4rrent density/cu (charge),(
vector)-4tion (time/posi ),( x
AA
ujj
xt
0B and law sGauss' and law sFaraday' law sAmpere'
0 ,
equations sMaxwell'
x
F
x
F
x
Fj
x
Fg
1000010000100001
tensormetricg
Greek indicesμ,ע,α,β = 0…3
implied is
:conventionsumEinstein
3
0,
implies
amBvqEq
ux
umuFqg
:3D
equation force Lorentz
Derivation of Electromagnetism from a Lagrangian
jAFgFgtrL jAgFgF)(
00
00
tensorfieldneticelectromag
x
A
x
A
BBEBBEBBEEEE
F
xyz
xzy
yzx
zyx
)(0
equations LagrangeEuler
αν
ανν x/A
L
xA
L
δA
δL law sGauss' and
law sAmpere'
0B andlaw sFaraday'
Maxwell’s equations
1000010000100001
tensormetric
gg
vector)-4 potential gnetic(electroma),( vector)-4rrent density/cu (charge),(
vector)-4tion (time/posi ),( x
AAjj
xt
Greek indicesμ,ע,α,β = 0…3
Derivation of Vacuum General Relativity from a Lagrangian
)det( , 216
1),(
ggRggg L
xxR tensor
Ricci
0
0
equationsLagrange-Euler
L
Lg
Einstein equations
x
g
x
g
x
gg
gR
2
1
n)(connectio , tensor)(metricg Greek indices
μ,ע,α,β = 0…3
vector)-4tion (time/posi ),( x xt
implied is
:conventionsumEinstein
3
0
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