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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

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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

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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,

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Einstein-Maxwell theory

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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

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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

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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

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Einstein-Maxwell theory = General Relativity + Electromagnetism Lorentz-force equation Einstein equations

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Early attempts to unify General Relativity and Electromagnetism

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-renormalized Einstein-Schrodinger (LRES) theory

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LRES theory vs. Einstein-Maxwell theory

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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 ≈ 10 63 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

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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

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Exact Solutions

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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

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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:

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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-

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

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Equations of Motion

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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

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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

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Observable Consequences

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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 difference10 -75 10 -85

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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 difference10 -76 10 -54

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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 difference10 -75 10 -55

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

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Application of Newman Penrose Methods

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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

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Non-Abelian LRES theory

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

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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

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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

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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)

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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)?

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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 holeatomic 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. 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.

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Backup charts

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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

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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

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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

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