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

QQ

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