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					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,

          A A
  F             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                           ,
                 vector potential   A2   A 
                                               
                                        A3 

• Electric and magnetic fields (E and B) are defined in terms of A

                  0     Ex E y Ez 
                  E    0  B z B y  A A
          F        x
                                         
                   E y Bz   0  B x  x  x
                  E  B B           
                      z    y  x 0 
                The fundamental fields of Einstein-Maxwell theory

     • Metric determines distance between points in space-time
                         00 g 01 g 02 g 03  1 0 0 0 
                         g
                                                  
              metric g 01 g11 g12 g13  0 -1 0 0  for flat space and
       g                               
              tensor  02 g12 g 22 g 23  0 0 -1 0  t,x,y,z coordinates
                       g
                                                    
                         g
                         03 g13 g 23 g 33  0 0 0 -1
          •
      dx2                                                Pythagorean theorem
                        ds   dx  g  dx generalized 2
                    •
                                                             1 2   2 2
                                                      (ds) =(dx ) +(dx )
            dx1
     • Connection determines how vectors change when moved
                                                    v r 
                                 
                                    0         v                     2D radial
                                1                 v               coordinates
                                 
       
            connection                         dx
                                                                         (x1,x2)=(r,)
                                 
                                   2                 r
                                3                                         
                                                               v   v    v dx 
                                 
                                             
      Almost all field theories can be derived from a Lagrangian
                                function of fields & derivatives, 
                               
                 Lagrangian 
                                                                
            L             example electromagnetic 
                                for
                 density                                   
                               vector potential A , A /x 
 • The field equations are derived from the Euler-Lagrange equations



                    L        L        L       
                 0               
                                     (A /x  ) 
                                                    
                    A       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

                   g  g 1 R ()  2 b 
              1
     L -
            16 
               1
           -        g F  F  L M (u , , g  , A ,),       g  det( g  )
             16 
                           
      g  metric tensor ,
                                                  
                                                       
        connection ,
                                                                    
                                      R ()             
                                                 x        x
                                                                      A A
                           
      A  vector potential,         F   g-1 g-1 F ,   F    
                                                                 
                                                                      x  x

   Euler - Lagrange eqs.               Maxwell' s equations
      L                                                Faraday's law 
           0                     Ampere' s law 
    A                                         ,   and   B  0 
                                                                
                                                                       
      L         L               and Gauss's law                   
        
            0,    
                     0
     g                     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 vacuumgeneral relativity
            g   g ) , A  g 55)
                     (5           (




   1946 - Einstein - Straus theory: nonsymmetr vacuumgeneral relativity
                                                   ic
           g   N (  ) , F  N[  ] or F     N[ ]

   1947 - Einstein  Schrodinger theory  Einstein  Straus theory wi a  b :
                                                                     th
           Can be derived from a very simple Lagrangian L  - det(R  ()).

   1953 - Einstein - Straus theory and Einstein - Schrodinger theoryshown
           topredict no Lorentz forcebetween charged particles.
-renormalized Einstein-Schrodinger (LRES) theory
                        LRES theory vs. Einstein-Maxwell theory
      Einstein  Maxwell theory
             1
       L -       g  g1 R ()  2 b 
           16 
               1
           -       gF g g  F  LM (g , A ,u ,, ),         g  det(g )
             16 
       where
              F  A,  A,
                                                    
      LRES theory allows nonsymmetric N  and   , excludes F g g  F term,
                                                  

      and includes an additional cosmological constant  z,
              1
     L -
            16 
                         
                   N N 1 R ()  2 b
              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 )1  ( f ' ' )1 
                                   b              b

                                   1                         
      G  8T  2 f  f  )  g f
                         (
                                                    
                                                         f    g  ( f 4 )1  ( f ' f ' )1 
                                                                                 b                b
                                   4                         
      Extra terms are  10-13 of usual terms for worst - case | f  |, | f  , |, | f  , , |
       accessible to measurement (e.g.1020 eV, 1034 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

                                                   Setting a  1 gives
                a    0     0         0         flat space in spherical
                 1/a 0
                  0                    0         coordinates. Flat space
          g  
                  0    0    r 2       0         in txyz coordinates is
                                            
                0    0     0     r 2sin 2           1 0 0 0 
                                                                       
           A0  ,
                 q                                 g    0  1  1 0 
                                                            0 0
                                                                  0
                                                                      0
                 r                                         0 0 0  1
          where                                                        
                    2M q 2
             a 1        2
                     r    r

      • 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
                                                
                   0    0      0 r 2 sin 2  
                q     M      4q 2         2
                                               
          A0    1                 O( b ) ,      b ~1063 cm2
                r   b r 3 5 b r 4          
         where
                    2M q 2         q2           2
                                                                2q 2
            a  1       2 1           O( b ) , b  1
                      r   r  10 b r  4
                                                               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                  Event horizon                   LRES
                                  conceals interior
                              (disappears for Q>M
                               as is the case for
                               elementary particles)
                 r-                                           r-
                      r+                                 r+



g 00 has1/r 2 singularit y,              g11 has 1/ r singularity,
all other relevant fields                 A , F , N , -N , -g , -g g ,
also have singularit ies
                                          -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33 cm
  Exact Electromagnetic Plane Wave Solution of LRES theory

• EM plane wave solution is identical to that of Einstein-Maxwell theory

E  yf ( x  ct )
    ˆ                                     1  h  h      0    0
                                            h 1 h     0    0
                                  g     0     0       1   0
                                          0                   1
B  zf ( x  ct )
    ˆ                                            0       0       

 f  (arbitrary function)            h  f 2 (y2  z2 )

   f (u)




                                              u  x  ct
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          B
                               dt
   +q/r2           -q/r2                                              +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 LM 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
           orbital    2      
     o                    3
           frequency period   r                          M1, Q1

       M1  Q1Q2 /M2                                              M2 , Q 2

     This ignores                         pericenter 
                                    p              
   radiation reaction                   frequency 

               Einstein-Maxwell theory               LRES theory modification
     p   Q12 Q22
                    3M 1 Q12  1 6   Q12    6Q M   
                              3 3  2 1  
     o 2M 2 r
              2
                     r    2r   r  r  b 
                                           Q1 M 2 r
                                                      
         Comparison to           extremal charged black hole     atomic parameters
     Einstein-Maxwell theory           Q=M=Msun,r=4M            Q1=-Q2=e, M=MP, r=a0
       fractional difference                 10-75                      10-85
                          Deflection of Light
 photon
                                                                       
             impact 
          b                                M, Q
             parameter 


        Einstein-Maxwell     LRES theory
             theory          modification
        4 M 3Q 2 3Q 2
              
         b   4b 2
                    8 b b 4

    Comparison to         extremal charged black hole   atomic parameters
Einstein-Maxwell theory         Q=M=Msun,r=4M            Q=e, M=MP, r=a0
 fractional difference               10-76                    10-54
                            Time Delay of Light
   satellite
                    radio signal
        –(
                                                        d
        t=0
                                                                )–
          impact 
       b                                      M, Q          t=d/c+t
          parameter 


                         Einstein-Maxwell       LRES theory
                              theory            modification
                            d  3Q     Q 2
                                     2
               t  4 M ln          
                           b    2b      bb3
    Comparison to            extremal charged black hole    atomic parameters
Einstein-Maxwell theory            Q=M=Msun,r=4M             Q=e, M=MP, r=a0
 fractional difference                  10-75                     10-55
                  Shift in Hydrogen Atom Energy Levels

             1
      L        N N 1 R ()  2 b 
           16
              1
                 g2 z  LM (u ,,g ,A       ),              N  det(N  )
             16
                               q        
      Charged fluid: LM       u A  u g u ,     
                                                        u  (,v)  (4  velocity)
                               m        2
   • LM may contain all of the Standard Model (excluding FFterm)
                      1    
                                                                 iq
      QED : L M   g  (  D   D  )  m  , D    A
                        2                                     x    
      - Dirac equationis unchanged
      - Chargedsolution gives fractionalchange 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 onu  t-kr/, not on r or t separately
      Expand the fields and field equations in a NewmanPenrose frame as
                                                         -
           order  order 1 
           0th           1st            2nd order 1
       0                             
           equations 
                       equationsr equationsr 2

      One of the field equations isf   2A[, ]  (f 3 )-1  (f ")-1 .
                                                              b          b

     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  2 b ~1/LP might exist.
                                                 M/
                                                      might have negative energy.
       If they exist, a rough calculation suggests they

     • 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                                   ˆ
                                                                          [A,B] AB-BA
            F  A,  A,                [A , A ],
                                  2LP sin  w
                                                                        ˆ
      Non - Abelian LRES theory allows nonsymmetric N and  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   Ig  )
                                                                              g
                                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  ,
                                                                    consistent with a  z caused
       - z   b                       ~ 10 63 cm2                                            
                          8L2 sin 2  w
                            P
                                                                     by zero - point fluctuations 

     • Other field equations have tiny extra terms

     f  2 A[  , ]   2 b [ A , A ]  ( f 2 )1 / 2  ( f ' ' )1 
                                                        b                  b

                                      1                               
       G  8T  tr  f  f  )  g f
                            (
                                                             
                                                                  f    g  ( f 3 )1 / 2  ( f ' f ' )1 
                                                                                          b                    b
                                      4                               
      Extra terms are  10-13 of usual terms for worst - case | f  |, | f  , |, | f  , , |
       accessible to measurement (e.g.1020 eV, 1034 Hz gamma rays)
     Lagrangianhas U(1)  SU(2) invariancelike Wienberg- Salam theory

     • LL under SU(2) gauge transformation, with 2x2 matrix U
                           i
      A  UA U 1           U, U 1,
                          2 b
                  
          
        U   U 1  2[ U, ]U 1,
                      
                  
      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 
                 1
      A  A       , ,
                2 b
            
          
           2iI[ , ],
                 
              
      R  R , N  N , g  g , f  f  .
     • L*=L when A and f are Hermitian
      ˆ     ˆ       ˆ*    ˆT
      *  T , R  R , N  N  , N *  N ,
                                   *     T


    A*  AT , f  f , g  g , g*  g.
                    *     T     *    T
                          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.
 Lagrangianhas 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         fractional difference from Einstein-Maxwell result
                        extremal charged black hole   atomic charges/masses/radii
  pericenter 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.
 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
                
                      
                                  
                                                 1        1     1   
       R          ,                        ( )  ( )       [ ] [  ]
                                       ( ( ), )
                                                  2            2                       3
     • Because it has special transformation properties
                                     
                 T
       R  (        )  R  (  )
                                  T
                                                                                (transposition symmetric)
     
                 
                 
                                  
                                                                     
                                           
       R  (U  U  2 U, ]U )  U R  (  )U 1
                      
                             1
                                            [
                                             
                                                         1
                                                                            (almost SU(2) invariant)
                                                           
                                 
       R  (  2iI , ] )  R  (  )
                                [                                         (U(1) invariant)

     • For Abelian fields the third and fourth terms are the same
                                                ˆ         ˆ
      Reduces to the ordinary Ricci tensor for [ ]  0 and  [ , ]  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

                                                                 
       c  cutoff   Pr oca  2 b ,
              frequency                              - z   b  2
                                                             8LP sin 2  w
                 c L2  fermion
                  4
                                               
      z           P
                                  boson 
                 2   spin states spin states
           fermion          boson  4 sin  w
                                               2
                                 
           spin states spin states             412.8  2
                                          

     • 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
          
e-                           mass renormalization
                            m = mb- mb·ln(ћωc/mc2)3/2

          e-              Photon Self Energy (vacuum polarization)
                             charge renormalization
           e+               e = eb - eb·ln(M/m)/3


               e-           Zero-Point Energy (vacuum energy density)
                              cosmological constant renormalization
e-                            = b - LP2c4(fermions-bosons)/2


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

				
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