James A. ShifflettDissertation Presentation For Degree of

Doctor of Philosophy in PhysicsWashington University in St. Louis

April 22, 2008Chairperson: Professor Clifford M. Will

Extensions of the Einstein-SchrodingerNon-Symmetric Theory of Gravity

• Einstein-Maxwell theory• -renormalized Einstein-Schrodinger (LRES) theory - Lagrangian - Field equations• Exact solutions - Electric monopole - Electromagnetic plane-wave• Equations of motion - Lorentz force equation - Einstein-Infeld-Hoffman method• Observational consequences - Pericenter advance - Deflection of light - Time delay of light - Shift in Hydrogen atom energy levels• Application of Newman-Penrose methods - Asymptotically flat 1/r expansion of the field equations• LRES theory for non-Abelian fields• Conclusions

Overview

• Greek indices , , , etc. always go from 0…3

• Geometrized units: c=G=1

interval) time-(space ),(),,,( 3210 xddtdxdxdxdxdx

Some conventions

• Einstein summation convention: paired indices imply summation

• comma=derivative, [ ]=antisymmetrization, ( )=symmetrization,

3

0

3

0

dxgdxdxgdxds

],[,, 2

AAA

x

A

x

AF

Einstein-Maxwell theory

, potential vector

netic electromag

AAAAA

A

3

2

1

0

The fundamental fields of Einstein-Maxwell theory

• The electromagnetic vector potential A 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 A

The fundamental fields of Einstein-Maxwell theory

• Metric determines distance between points in space-time

g metric

tensor

g00 g01 g02 g03

g01 g11 g12 g13

g02 g12 g22 g23

g03 g13 g23 g33

1 0 0 0

0 -1 0 0

0 0 -1 0

0 0 0 -1

for flat space and

t,x,y, z coordinates

dxgdxds dx1

dx2

• Connection determines how vectors change when moved

3

2

1

0

connection

dxvvv

v vr

v

dxr

2D radial coordinates(x1,x2)=(r,)

generalized Pythagorean theorem (ds)2=(dx1)2+(dx2)2

•

•

Almost all field theories can be derived from a Lagrangian

L Lagrangian density

function of fields & derivatives,for example electromagneticvector potential A , A/x

0 LA

LA

x

L(A /x )

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

)det(),,,,,(

2)(1

ggAguFFg

Rgg

M

b

L161

-

161

-L

g metric tensor ,

connection , R ()

x

x

Einstein-Maxwell theory = General Relativity + Electromagnetism

00

0

LL

L

,

eqs. Lagrange-Euler

g

A 0B and

law sFaraday' ,

law sGauss' andlaw sAmpere'

equations sMaxwell'

Lorentz-force equationEinstein equations

A vector potential , F g-1g-1F, F A

x A

x

Early attempts to unify General Relativity and Electromagnetism

particles. chargedbetween force Lorentz nopredict to shownr theory Schrodinge-Einstein and theory Straus-Einstein - 1953

.))(det(R- Lagrangian simple very a from derived beCan

: ath theory wiStrausEinsteinr theorySchrodingeEinstein - 1947

or , relativity general vacuumicnonsymmetr : theoryStraus-Einstein - 1946

, relativity general vacuum5D :ryKlein theo-Kaluza -1920s

forces. range long twounify thereally t Doesn' : theoryMaxwellEinstein - 1916

b

][][)(

)5(5

)5(

L

NFNFNg

gAgg

-renormalized Einstein-Schrodinger (LRES) theory

LRES theory vs. Einstein-Maxwell theory

theory MaxwellEinstein

L -1

16 g g 1 R () 2b

-1

16 gFgg F LM (g ,A ,u ,,), gdet(g )

where

F A, A,

LRES theory allows nonsymmetric N and

, excludes Fgg F term,

and includes an additional cosmological constant z,

L -1

16 N N 1 R () 2b

-1

16 g2z LM (g , A ,u ,,), N det(N )

where the "bare" b -z so that b z matches measurement, and

A

[ ] / 18b , gg N N 1( )

• Einstein-Schrödinger theory is non-symmetric generalization of vacuum GR

• LRES theory basically includes a z term in the ES theory Lagrangian - gives the same Lorentz force equation as in Einstein-Maxwell theory

• z term might be expected to occur as a 0th order quantization effect - zero-point fluctuations are essential to Standard Model and QED - demonstrated by Casimir force and other effects • = b+z resembles mass/charge/field-strength renormalization in QED - “physical” mass of an electron is sum of “bare” mass and “self energy” - a “physical” is needed to represent dark energy!

• Non-Abelian LRES theory requires –z ≈ b ≈ 1063 cm-2 ~ 1/(Planck length)2

- this is what would be expected if z was caused by zero-point fluctuations

• z term could also result from the minimum of the potential of some additional scalar field in the theory, like the Weinberg-Salam field

• z modification is a new idea, particularly the non-Abelian version

LRES theory is well motivated

g

g

g

g

The field equations

• Ampere’s law is identical to Einstein-Maxwell theory

• The electromagnetic field tensor f can be defined by

• Other field equations have tiny extra terms

,2/2/1][1biNNfg

jgfg 4)( ,

114)(

113],[

)''()(4

128

)''()(2

bb

bb

fffgffgffTG

ffAf

rays) gamma Hz10 eV,10 (e.g. tmeasuremen to accessible

f| case- worstfor terms usual of 10 are terms Extra 3420

-13 |||,||, ,,, ff

Possible Proca field ghost with M/ 2b ~1/LP , but probably not.

Exact Solutions

g

a 0 0 00 1/a 0 00 0 r2 00 0 0 r2sin2

A0 q

r,

where

a 1 2M

r q2

r2

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,

• Extra terms are tiny for worst-case radii accessible to measurement:

| r q M Msun r 10 17cm,q e,M Me

q2/br4 | 10 73 10 61

M/br3 | 10 73 10 67

g b

a 0 0 00 1/ab 0 00 0 r2 00 0 0 r2 sin2

A0 q

r1

M

br3 4q2

5br4 O(b 2)

, b~1063cm 2

where

a 12M

r

q2

r2 1q2

10br4 O(b 2)

, b 1

2q2

br4

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-

g11 has 1/ r singularity,

A, F , N , -N , -g , -g g ,

-g N , -g R are all finite

origin is where (surface area)0;instead of r 0 it is at

re q(2 /b )1/ 4 ~ LP ~ 10 33cm

iessingularit have alsofields relevant other all

y,singularit has1/r g 200

)(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

Equations of Motion

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

mgu

x u

u qF u

where u ( ,

v ) (4 - velocity)

• 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 g, and A

• 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

Observable Consequences

M1, Q1

M2, Q2

Pericenter Advance

M1 Q1Q2/M2

Kepler’s third law

3

2

ro

periodfrequencyorbital

This ignoresradiation reaction

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

Comparison to

Einstein-Maxwell theory

extremal charged black hole

Q=M=Msun,r=4M

atomic parameters

Q1=-Q2=e, M=MP, r=a0

fractional difference 10-75 10-85

frequencypericenter

p

Deflection of Light

photon

M, Q

parameter

impact b

4

2

2

2

8

3

4

34

b

Q

b

Q

b

M

b

Einstein-Maxwell theory

LRES theorymodification

Comparison to

Einstein-Maxwell theory

extremal charged black hole

Q=M=Msun,r=4M

atomic parameters

Q=e, M=MP, r=a0

fractional difference 10-76 10-54

Time Delay of Light

radio signal

M, Q t=d/c+t

t=0

satellite

parameter

impact b

–(

)–

3

22

2

3ln4

b

Q

b

Q

b

dMt

b

Einstein-Maxwell theory

LRES theorymodification

d

Comparison to

Einstein-Maxwell theory

extremal charged black hole

Q=M=Msun,r=4M

atomic parameters

Q=e, M=MP, r=a0

fractional difference 10-75 10-55

• may contain all of the Standard Model (excluding FFterm)

Shift in Hydrogen Atom Energy Levels

ML

Charged fluid : LM q

mu A

2

ug u , u (,v )(4 velocity)

L 116

N N 1 R () 2b

1

16 g2z LM(u,,g,A), N det(N )

levelsenergy atom H in 10 of change fractional gives solution Charged-

unchanged is equationDirac -

:QED

49-

A

iq

xDmDDgM

�

,)(

2

1L

Application of Newman Penrose Methods

• 1/r expansion shows that: a) LRES theory has no continuous wave Proca solutions like τ≈sin(kr-t)/r b) LRES theory = Einstein-Maxwell theory to O(1/r2) for k= propagation

• 1/r expansion may not necessarily rule out wave-packet Proca solutions. Perhaps a Proca field with M/ħ~1/LP could be a built-in Pauli-Villars field?

Asympotically flat 1/r expansion of the field equations

Assume all fields depend on u t-kr/, not on r or t separately Expand the fields and field equations in a Newman- Penrose frame as

0 0th orderequations

1st orderequations

1

r 2nd order

equations

1

r2

One of the field equations is f 2A[, ] (f 3)b-1 (f ")b

-1. Taking the curl of this gives something similar to the Proca equation ( ; ;

apparently negligible terms)/2b where f[, ]/ 4.

This suggests that Proca waves with mass M/ 2b ~1/LP might exist. If they exist, a rough calculation suggests they might have negative energy.

Non-Abelian LRES theory

Non-Abelian LRES theory vs. Einstein-Weinberg-Salam theory

L 1

16 g g 1 R () 2b

1

32 g tr(Fgg F )LM (g ,A ,e ,), g det(g )

where A Ia ibi is composed of 2x2 Hermitian matrix components and

F A, A, i

2LP sinw

[A ,A], [A,B] AB-BA

theory Salam-Weinberg-Einstein

,)(

ˆ

z a includes and , excludes ,componentsmatrix Hermitian

2x2 with and Nic nonsymmetr allowstheory LRES Abelian-Non

FggFtr

L 116

N1/ 4 tr( N 1 R ()) 4b

1

4g1/ 4z LM (g , A ,e ,,), N det(N )

where the "bare" b -z so that b z matches measurement, and

A

[ ] / 18b , g1/ 4g N1/ 4N 1( ), (assume g Ig )

The non-Abelian field equations

• Ampere’s law is identical to Weinberg-Salam theory

• The electro-weak field tensor f is defined by

• Other field equations have tiny extra terms

,2/2/1][14/14/1biNNfg

(g1/ 4 f ), 2b g1/ 4[ f ,A ]4g1/ 4 j ,

-z b

8LP2 sin2w

~ 1063cm 2 consistent with a z causedby zero - point fluctuations

12/13)(

12/12],[

)''()(4

18

)''()(],[22

bb

bbb

fffgffgfftrTG

ffAAAf

rays) gamma Hz10 eV,10 (e.g. tmeasuremen to accessible

f| case- worstfor terms usual of 10 are terms Extra 3420

-13 |||,||, ,,, ff

• LL under SU(2) gauge transformation, with 2x2 matrix U

• LL under U(1) gauge transformation, with scalar

• L*=L when A and f are Hermitian

theory Salam- Wienberglike invariance SU(2)U(1) has Lagrangian

A A 1

2b

, ,

2iI[

, ],

R

R

, N N , g g, f f .

A UAU 1

i

2b

U,U 1,

U

U 1 2[

U, ]U 1,

R

U R

U 1, N UNU 1, g UgU

1, f UfU 1.

.*,,,

,,,ˆˆ,ˆˆ***

****

ggggffAA

NNNNRRTTT

TTT

For the details see

Refereed Publications • “A modification of Einstein-Schrodinger theory that contains both general relativity and electrodynamics”, General Relativity and Gravitation (Online First), Jan. 2008, gr-qc/0801.2307.Additional Archived Papers• “A modification of Einstein-Schrodinger theory which closely approximates Einstein-Weinberg-Salam theory”, Apr. 2008, gr-qc/0804.1962• “Lambda-renormalized Einstein-Schrodinger theory with spin-0 and spin-1/2 sources”, Apr. 2007, gr-qc/0411016.• “Einstein-Schrodinger theory in the presence of zero-point fluctuations”, Apr. 2007, gr-qc/0310124.• “Einstein-Schrodinger theory using Newman-Penrose tetrad formalism”, Jul. 2005, gr-qc/0403052.Other material on http://www.artsci.wustl.edu/~jashiffl/index.html• Check of the electric monopole solution (MAPLE)• Check of the electromagnetic plane-wave solution (MAPLE)• Asymptotically flat Newman-Penrose 1/r expansion (REDUCE)

Why pursue LRES theory?

• It unifies gravitation and electro-weak theory in a classical sense

• It is vacuum GR generalized to non-symmetric fields and Hermitian matrix components, with a well motivated z modification

• It suggests untried approaches to a complete unified field theory - Higher dimensions, but with LRES theory instead of vacuum GR? - Larger matrices: U(1)xSU(5) instead of U(1)xSU(2)?

Conclusion: Non-Abelian LRES theory ≈ Einstein-Weinberg-Salam

• Charged solution and Reissner-Nordström sol. have tiny fractional difference: 10-73 for extremal charged black hole; 10-61 for atomic charges/masses/radii.

Standard tests

extremal charged black hole atomic charges/masses/radiipericenter advance 10-75 10-85

deflection of light 10-76 10-54

time delay of light 10-75 10-55

• Other Standard Model fields included like Einstein-Weinberg-Salam theory: - Energy levels of Hydrogen atom have fractional difference of <10-49.

• fractional difference from Einstein-Maxwell result

• Extra terms in the field equations are <10-13 of usual terms.

• Lorentz force equation is identical to that of Einstein-Maxwell theory

• EM plane-wave solution is identical to that of Einstein-Maxwell theory.

Possible Proca field ghost with M/ 2b ~1/LP , but probably not.

.invariance SU(2)U(1) has Lagrangian

Backup charts

The non-Abelian/non-symmetric Ricci tensor

• We use one of many non-symmetric generalizations of the Ricci tensor

• Because it has special transformation properties

• For Abelian fields the third and fourth terms are the same

.relativity generalordinary in occurs as 0, and 0 for tensor Ricciordinary the to Reduces ],[][

ˆˆ

R

,

(( ), )

1

2

( )

1

2

( )

1

3

[ ]

[ ]

R

(

T) R

T (

) (transposition symmetric)

R

(U

U 1 2[

U, ]U 1) U R

(

)U 1 (almost SU(2) invariant)

R

(

2iI[

, ]) R

(

) (U(1) invariant)

Proca waves as Pauli-Villars ghosts?

• For the Standard Model this difference is about 60

• Non-Abelian LRES theory works for dd matrices as well as 22 matrices

• Maybe 4πsin2w/ or its “bare” value at c works out correctly for some “d”

• SU(5) almost unifies Standard Model, how about U(1)xSU(5)?

• If wave-packet Proca waves exist and if they have negative energy, perhaps the Proca field functions as a built-in Pauli-Villars ghost

c cutofffrequency

Pr oca 2b , -z b

8LP

2 sin2w

z c

4LP2

2 fermionspin states boson

spin states

fermionspin states boson

spin states4 sin2w

412.82

Electron Self Energy mass renormalizationm = mb- mb·ln(ћωc/mc2)3/2

Photon Self Energy (vacuum polarization) charge renormalizatione = eb - eb·ln(M/m)/3

Zero-Point Energy (vacuum energy density) cosmological constant renormalization = b - LP

2c4(fermions-bosons)/2

c= (cutoff frequency) LP = (Planck length) M= (Pauli-Villars cutoff mass) = (fine structure constant)

e-

e+

e-

e-

e-

= b+ z is similar to mass/charge renormalization in QED