Extensions of the Einstein-Schrodinger
Non-Symmetric Theory of Gravity
James A. Shifflett
Dissertation Presentation For Degree of
Doctor of Philosophy in Physics
Washington University in St. Louis
April 22, 2008
Chairperson: Professor Clifford M. Will
Overview
• 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
Some conventions
• Geometrized units: c=G=1
• Greek indices , , ,  etc. always go from 0…3

dx  (space - time interval)  (dx , dx , dx , dx )  ( dt , dx )

0
1
2
3
• Einstein summation convention: paired indices imply summation
3
3
ds   dx g  dx    dx  g  dx


 0  0
• comma=derivative, [ ]=antisymmetrization, ( )=symmetrization,
F
A A


 A ,  A ,  2 A[ ,  ]
x  x
Einstein-Maxwell theory
The fundamental fields of Einstein-Maxwell theory
• The electromagnetic vector potential A is the fundamental field
 A0 

 electromag netic   A1   


A  


  A    ,
 vector potential   2   A 
 A3 
• Electric and magnetic fields (E and B) are defined in terms of A
F
Ex E y Ez 
 0
 E
 A
A
0

B
B

x
z
y



   

E
B
0

B
y
z
x
x

 x
 E  B B

0 
z
y
x

The fundamental fields of Einstein-Maxwell theory
• Metric determines distance between points in space-time
g 00 g 01 g 02 g 03  1 0 0 0 
metric  g 01 g11 g12 g13  0 -1 0 0  for flat space and
  

g  
 
tensor  g 02 g12 g 22 g 23  0 0 -1 0  t, x, y, z coordinates

 

g 03 g13 g 23 g 33  0 0 0 -1
•
dx2

dx1
•
theorem
ds   dx  g  dx generalized 2Pythagorean
1 2
2 2
(ds) =(dx ) +(dx )
• Connection determines how vectors change when moved
  
 1 
 
 connection    
2
 

 3 
  

0


v r 
v    
v 
r

dx
2D radial
coordinates
(x1,x2)=(r,)

v   v   
v dx 
Almost all field theories can be derived from a Lagrangian
function of fields & derivatives,
Lagrangian  
L  
 for example electromagnetic
density

 

vector
potential
A
,
A
/x








• The field equations are derived from the Euler-Lagrange equations
L
0
A


L
 
L

  

 

A x (A /x ) 
which minimizes the “action”
S   L dx 0 dx 1  dx 3

• Guarantees field equations are coordinate independent and self consistent
• Lagrangian is also necessary for quantization via path integral methods.
Einstein-Maxwell theory = General Relativity + Electromagnetism


1
 g g 1 R ()  2 b
16 
1
 g F  F  L M (u , , g  , A ,),
16 
g  metric tensor  ,
L -
  connection  ,


A  vector potential ,


Euler - Lagrange eqs.
L
0
A
L
L

0
,
0

g 

g  det( g  )





 
R ()       
 

x
x
A A
F   g-1 g-1 F , F    
x
x
Maxwell' s equations
 Faraday' s law 
 Ampere' s law 




,
and
Gauss'
s
law


 and   B  0 
Einstein equations
Lorentz-force equation
Early attempts to unify General Relativity and Electromagnetism
 1916 - Einstein  Maxwell theory :
Doesn' t really unify the two long range forces.
 1920s - Kaluza - Klein theo ry : 5D vacuum general relativity
( 5)
, A  g ( 55)
g   g 
 1946 - Einstein - Straus theory : nonsymmetr ic vacuum general relativity
g   N (  ) , F  N[  ] or F     N[ ]
 1947 - Einstein  Schrodinge r theory  Einstein  Straus theory wi th a  b :
Can be derived from a very simple Lagrangian L  - det(R  ()) .
 1953 - Einstein - Straus theory and Einstein - Schrodinge r theory shown
to predict no Lorentz force between charged particles.
-renormalized Einstein-Schrodinger (LRES) theory

LRES theory vs. Einstein-Maxwell theory
 Einstein  Maxwell theory
1
g  g1 R ()  2 b 
16 
1
gF g g  F  LM (g , A ,u ,, ),
16 
where
L -
F  A,  A,
 LRES theory allows nonsymmetric
g  det(g )

N  and   , excludes F g g  F term,
and includes an additional cosmological constant
 z,
1
L N N 1 R ()  2 b
16 
1
g2 z  LM (g , A ,u ,, ),
N  det(N  )
16 
where the " bare"  b  - z so that    b   z matches measurement, and


A   [ ]/ 18 b ,

gg  N N 1(  )
LRES theory is well motivated
• Einstein-Schrödinger theory is non-symmetric generalization of vacuum GR
• LRES theory basically includes a z  g term in the ES theory Lagrangian
- gives the same Lorentz force equation as in Einstein-Maxwell theory
• z  g 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  g term could also result from the minimum of the potential of
some additional scalar field in the theory, like the Weinberg-Salam  field
• z  g modification is a new idea, particularly the non-Abelian version
The field equations
• The electromagnetic field tensor f can be defined by
 g f    N N 1[ ]i1b/ 2 / 2 ,
• Ampere’s law is identical to Einstein-Maxwell theory
(  g f  ) ,  4  g j 
• Other field equations have tiny extra terms
f  2 A[  , ]  ( f 3 )b1  ( f ' ' )b1 
1

G  8T  2 f ( f  )  g f
4



f    g  ( f 4 )b1  ( f ' f ' )b1 

 Extra terms are  10 -13 of usual terms for worst - case | f  |, | f  , |, | f  , , |
accessible to measuremen t (e.g. 10 20 eV, 10 34 Hz gamma rays)
 Possible Proca field ghost with

M/  2 b ~1/LP , but probably not.
Exact Solutions
Exact charged black hole solution of Einstein-Maxwell theory
• Called the Reissner-Nordström solution
a
0
0
0 1/a 0
g  
0
0
r 2

0
0
0
q
A0  ,
r
where
2M q 2
a  1
 2
r
r
0 
0 
0 

r 2sin 2 
Setting a  1 gives
flat space in spherical
coordinate s. Flat space
in txyz coordinate s is
1 0 0 0 


g    00 01 01 00 
 0 0 0  1


• Becomes Schwarzschild solution for q=0

• -2M/r term is what causes gravitational force
Exact charged black hole solution of LRES theory
• The charged solution is very close to the Reissner-Nordström solution,
a

0
0
0
0 1/ab 0

0
g  b 

0
0
r 2
0

2
2 
0
0
0
r
sin
 


q 
M
4q 2
2
63
2
A0  1


O(
)
,

~10
cm

b
b
r   b r 3 5 b r 4

where

2M q 2 
q2
2q 2
2
a  1
 2 1
 O( b ) , b  1
4
r
r  10 b r
b r4

• Extra terms are tiny for worst-case radii accessible to measurement:

| r  q  M  M sun
r  1017 cm,q  e, M  M e

q 2 / b r 4 |
1073
1061
M/ b r 3 |
1073
1067
Charged solution of Einstein-Maxwell theory vs. LRES theory
Einstein-Maxwell
r-
LRES
Event horizon
conceals interior
(disappears for Q>M
as is the case for
elementary particles)
r+
r+
g00 has1/r 2 singularit y,
all other relevant fields
also have singularit ies
r-
g11 has 1/ r singularity,
A , F , N  , -N , -g ,
-g N  ,
-g g  ,
-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33 cm
Exact Electromagnetic Plane Wave Solution of LRES theory
• EM plane wave solution is identical to that of Einstein-Maxwell theory

E  yˆ f ( x  ct )

B  zˆf ( x  ct )
f  (arbitrary function)
g 
1  h  h
  h 1 h
 0
0
 0
0

h  f 2 (y2  z2 )
f (u )
u  x  ct
0
0
1
0
0
0
0
 1
Equations of Motion
Lorentz force equation is identical to that of Einstein-Maxwell theory
• Usual Lorentz force equation results from divergence of Einstein equations


 
dp
 qE  qv  B
dt
+q/r2
-q/r2

B
+q/r2
• Lorentz force equation in 4D form
u
 
 
mg     u  u  qF u 
x

where
u  ( , v )  (4 - velocity)
• Also includes gravitational “force”; it becomes geodesic equation when q=0

Lorentz force also results from Einstein-Infeld-Hoffman (EIH) method
• Requires no sources (no L M 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
q/r2
q/r2
• Motion of the particles agrees with the Lorentz force equation
Observable Consequences

Pericenter Advance
Kepler’s third law
2
 orbital

o  


frequency

 period

r3
M1, Q1
  M1  Q1Q2 /M2
This ignores
radiation reaction
M2, Q2
 pericenter 

frequency


p  
LRES theory modification
Einstein-Maxwell theory
p
Q12 Q22
3M 1 Q12  1 6 
Q12
    3



2
 o 2M 2 r
r
2r   r  r  b

6Q M

 3  2 1  
Q1 M 2
r

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
Deflection of Light
photon

 impact 
b

parameter


Einstein-Maxwell
theory
M, Q
LRES theory
modification
4M 3Q 2 3Q 2
 


2
b
4b
8 b b 4
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
satellite
–(
t=0
radio signal
d
)–
 impact 
b

parameter


M, Q
Einstein-Maxwell
theory
t=d/c+t
LRES theory
modification
2
Q 2
 d  3Q
t  4M ln   

3
b
2
b

b
 
b
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
Shift in Hydrogen Atom Energy Levels
1
N N 1 R ()  2 b 
16
1

g2 z  LM (u ,,g ,A
16
L
 Charged fluid : LM  
q

),
u A  u g u ,
m
2
N  det(N  )
u  ( , v)  (4  velocity)

• LM may contain all of the Standard Model (excluding FFterm)

 

iq
1


 QED : L M   g  (  D   D  )  m  , D    A

x
2

- Dirac equation is unchanged
- Charged solution gives fractional change of 10 - 49 in H atom energy levels
Application of Newman Penrose Methods
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
0th order  1st order 1 2nd order  1
0  
 
  
 2
equations
equations
equations

 
r 
r
 One of the field equations is f   2A[, ]  (f 3 )-1b  (f ")-1b .
Taking the curl of this gives something similar to the Proca equation
   (   ;;   apparently negligible terms) /2 b where     f [, ] / 4.
 This suggests that Proca waves with mass M/  2 b ~1/LP might exist.
If they exist, a rough calculation suggests they might have negative energy.
• 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?
Non-Abelian LRES theory
Non-Abelian LRES theory vs. Einstein-Weinberg-Salam theory
 Einstein - Weinberg - Salam theory
1
L
g  g1 R ()  2 b 
16
1

g tr(F g g  F )  LM (g , A , e ,  ),
g  det(g )
32
where A  Ia   ib i is composed of 2x2 Hermitian matrix components and

i 
̂
F  A,  A, 
[A , A ],
[A,B]  AB-BA
2LP sin  w


 Non - Abelian LRES theory allows nonsymmetr ic N and ˆ 
with 2x2
Hermitian matrix components , excludes tr( F g  g  F ), and includes a  z ,


1
1/ 4 
1
L
N  tr( N R ())  4 b 


16 
1 1/ 4

g  z  LM (g , A , e , , ),
N  det(N  )
4
where the " bare"  b  - z so that    b   z matches measurement, and

A   [ ]/ 18 b ,
g1/ 4 g   N 1/ 4 N 1(  ),
(assume g  Ig  )
The non-Abelian field equations
• The electro-weak field tensor f is defined by
g 1 / 4 f   N 1 / 4 N 1[ ]i1b/ 2 / 2 ,
• Ampere’s law is identical to Weinberg-Salam theory
(g1/ 4 f  ),  2 b g1/ 4 [ f  , A ]  4g1/ 4 j  ,
 - z   b 

8L2P sin 2 w
~ 1063 cm2
consistent with a  z caused 


by zero - point fluctuations 
• Other field equations have tiny extra terms

f  2 A[  , ]   2 b [ A , A ]  ( f 2 )b1 / 2  ( f ' ' )b1 
1

G  8T  tr f ( f  )  g f
4



f    g  ( f 3 )b1 / 2  ( f ' f ' )b1 

 Extra terms are  10 -13 of usual terms for worst - case | f  |, | f  , |, | f  , , |
accessible to measuremen t (e.g. 10 20 eV, 10 34 Hz gamma rays)


Lagrangian has U(1)  SU(2) invariance like Wienberg - Salam theory
• LL under SU(2) gauge transformation, with 2x2 matrix U
A  UA U 1 


  U  U 1  2[ U, ]U 1,



i
U, U 1,
2 b

R  U R U 1,
N  UNU 1, g  UgU 1, f   UfU 1 .
• LL under U(1) gauge transformation, with scalar 
A  A 


     2iI[ , ],



1
, ,
2 b

R  R , N  N , g  g , f   f  .
• L*=L when A and f are Hermitian
T
*
T
*
T
ˆ *  ˆ 
, Rˆ
 Rˆ 
, N
 N 
, N *  N,
*
T
A*  AT , f*  fT , g
 g
, g*  g.
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
• Extra terms in the field equations are <10-13 of usual terms.
 Lagrangian has U(1)  SU(2) invariance .
• EM plane-wave solution is identical to that of Einstein-Maxwell theory.
• Charged solution and Reissner-Nordström sol. have tiny fractional difference:
10-73 for extremal charged black hole; 10-61 for atomic charges/masses/radii.
• Lorentz force equation is identical to that of Einstein-Maxwell theory
• Standard tests
pericenter advance
deflection of light
time delay of light
fractional difference from Einstein-Maxwell result
extremal charged black hole
10-75
10-76
10-75
atomic charges/masses/radii
10-85
10-54
10-55
• Other Standard Model fields included like Einstein-Weinberg-Salam theory:
- Energy levels of Hydrogen atom have fractional difference of <10-49.
 Possible Proca field ghost with
M/  2 b ~1/LP , but probably not.
Backup charts
The non-Abelian/non-symmetric Ricci tensor
• We use one of many non-symmetric generalizations of the Ricci tensor



,
R  


1  
1     1   
   ( )  ( )       [ ] [  ]
2
2
3

( ( ), )
• Because it has special transformation properties



R  (

T




)  R  (  )


(almost SU(2) invariant)
R  (   2iI[ , ] )  R  (  )
(U(1) invariant)

1

[
(transposition symmetric)
R  (U  U  2 U, ]U )  U R  (  )U 1



T
1


• For Abelian fields the third and fourth terms are the same

 Reduces to the ordinary Ricci tensor for ˆ [ ]  0 and ˆ [ , ]  0,
as occurs in ordinary general relativity .

Proca waves as Pauli-Villars ghosts?
• If wave-packet Proca waves exist and if they have negative energy,
perhaps the Proca field functions as a built-in Pauli-Villars ghost



  Pr oca  2 b ,
 c  cutoff
- z   b  2
frequency 
8LP sin 2  w
 c4 L2P  fermion
boson 


z  

spin
states
spin
states

2 
 fermion
 4  sin 2  w
boson

 

 412.8  2
spin states spin states 

• For the Standard Model this difference is about 60
• Non-Abelian LRES theory works for dd matrices as well as 22 matrices
• Maybe 4πsin2w/ or its “bare” value at c works out correctly for some “d”
• SU(5) almost unifies Standard Model, how about U(1)xSU(5)?
 = b+ z is similar to mass/charge renormalization in QED
Electron Self Energy
 mass renormalization
m = mb- mb·ln(ћωc/mc2)3/2

e-


ee+

e-
e-
Photon Self Energy (vacuum polarization)
 charge renormalization
e = eb - eb·ln(M/m)/3
Zero-Point Energy (vacuum energy density)
 cosmological constant renormalization
 = b - LP2c4(fermions-bosons)/2
c= (cutoff frequency)
M= (Pauli-Villars cutoff mass)
LP = (Planck length)
 = (fine structure constant)