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 Extensions of the Einstein-Schrodinger Non-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 Some conventions Einstein summation convention: paired indices imply summation comma=derivative, [ ]=antisymmetrization, ( )=symmetrization,

Einstein-Maxwell theory

The fundamental fields of Einstein-Maxwell theory The electromagnetic vector potential A  is the fundamental field 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 dx 1 dx 2 Connection determines how vectors change when moved  dx  r 2D radial coordinates (x 1,x 2 )=(r,  ) generalized Pythagorean theorem (ds) 2 =(dx 1 ) 2 +(dx 2 ) 2

Almost all field theories can be derived from a Lagrangian 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 Lorentz-force equation Einstein equations

Early attempts to unify General Relativity and Electromagnetism

 -renormalized Einstein-Schrodinger (LRES) theory

LRES theory vs. Einstein-Maxwell theory

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 ≈ 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

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

Exact Solutions

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

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:

Charged solution of Einstein-Maxwell theory vs. LRES theory LRES Einstein-Maxwell Event horizon conceals interior (disappears for Q>M as is the case for elementary particles) r+r+ r-r- r+r+ r-r-

EM plane wave solution is identical to that of Einstein-Maxwell theory Exact Electromagnetic Plane Wave Solution of LRES theory

Equations of Motion

Lorentz force equation is identical to that of Einstein-Maxwell theory Usual Lorentz force equation results from divergence of Einstein equations +q/r 2 -q/r 2 +q/r 2 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/r 2 Lorentz force also results from Einstein-Infeld-Hoffman (EIH) method

Observable Consequences

M 1, Q 1 M 2, Q 2 Pericenter Advance Kepler’s third law This ignores radiation reaction Einstein-Maxwell theory LRES theory modification Comparison to Einstein-Maxwell theory extremal charged black hole Q=M=M sun, r=4M atomic parameters Q 1 =-Q 2 =e, M=M P, r=a 0 fractional difference

Deflection of Light photon M, Q  Einstein-Maxwell theory LRES theory modification Comparison to Einstein-Maxwell theory extremal charged black hole Q=M=M sun, r=4M atomic parameters Q=e, M=M P, r=a 0 fractional difference

Time Delay of Light radio signal M, Q t=d/c+  t t=0 satellite –(–( )–)– Einstein-Maxwell theory LRES theory modification d Comparison to Einstein-Maxwell theory extremal charged black hole Q=M=M sun, r=4M atomic parameters Q=e, M=M P, r=a 0 fractional difference

may contain all of the Standard Model (excluding F  F  term) Shift in Hydrogen Atom Energy Levels

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/r 2 ) for k=  propagation 1/r expansion may not necessarily rule out wave-packet Proca solutions. Perhaps a Proca field with M/ħ~1/L P could be a built-in Pauli-Villars field? Asympotically flat 1/r expansion of the field equations

Non-Abelian LRES theory

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

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

L  L under SU(2) gauge transformation, with 2x2 matrix U L  L under U(1) gauge transformation, with scalar  L*=L when A and f  are Hermitian

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/ Additional Archived Papers “A modification of Einstein-Schrodinger theory which closely approximates Einstein-Weinberg-Salam theory”, Apr. 2008, gr-qc/ “Lambda-renormalized Einstein-Schrodinger theory with spin-0 and spin-1/2 sources”, Apr. 2007, gr-qc/ “Einstein-Schrodinger theory in the presence of zero-point fluctuations”, Apr. 2007, gr-qc/ “Einstein-Schrodinger theory using Newman-Penrose tetrad formalism”, Jul. 2005, gr-qc/ Other material on 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: for extremal charged black hole; for atomic charges/masses/radii. Standard tests extremal charged black holeatomic charges/masses/radii pericenter advance deflection of light time delay of light Other Standard Model fields included like Einstein-Weinberg-Salam theory: - Energy levels of Hydrogen atom have fractional difference of < fractional difference from Einstein-Maxwell result Extra terms in the field equations are < 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.

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

Proca waves as Pauli-Villars ghosts? 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 4πsin 2  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)? If wave-packet Proca waves exist and if they have negative energy, perhaps the Proca field functions as a built-in Pauli-Villars ghost

Electron Self Energy  mass renormalization m = m b - m b ·ln(ћω c /mc 2 )3  /2  Photon Self Energy (vacuum polarization)  charge renormalization e = e b - e b ·ln(M/m)  /3  Zero-Point Energy (vacuum energy density)  cosmological constant renormalization  =  b - L P 2  c 4 (fermions-bosons)/2   c = (cutoff frequency) L P = (Planck length) M= (Pauli-Villars cutoff mass)  = (fine structure constant)  e-e-  e+e+ e-e-   e-e- e-e-  =  b +  z is similar to mass/charge renormalization in QED