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 Einstein summation convention: paired indices imply summation comma=derivative, [ ]=antisymmetrization, ( )=symmetrization,
The fundamental fields of Einstein-Maxwell theory The electromagnetic vector potential Ais 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 generalized Pythagorean theorem (ds)2=(dx1)2+(dx2)2 dx2 dx1 Connection determines how vectors change when moved 2D radial coordinates (x1,x2)=(r,) dx r
Almost all field theories can be derived from a Lagrangian The field equations are derived from the Euler-Lagrange equations which minimizes the “action” 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 Lorentz-force equation Einstein equations
Early attempts to unify General Relativity and Electromagnetism
LRES theory is well motivated • 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+zresembles 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
The field equations The electromagnetic field tensor fcan be defined by Ampere’s law is identical to Einstein-Maxwell theory Other field equations have tiny extra terms
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 Einstein-Maxwell LRES Event horizon conceals interior (disappears for Q>M as is the case for elementary particles) r- r- r+ r+
Exact Electromagnetic Plane Wave Solution of LRES theory EM plane wave solution is identical to that of Einstein-Maxwell theory
Lorentz force equation is identical to that of Einstein-Maxwell theory Usual Lorentz force equation results from divergence of Einstein equations +q/r2 -q/r2 +q/r2 Lorentz force equation in 4D form 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 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
M1, Q1 M2, Q2 Pericenter Advance Kepler’s third law This ignores radiation reaction LRES theory modification Einstein-Maxwell theory
Deflection of Light photon M, Q Einstein-Maxwell theory LRES theory modification
Time Delay of Light satellite radio signal –( d t=0 )– t=d/c+t M, Q LRES theory modification Einstein-Maxwell theory
Shift in Hydrogen Atom Energy Levels may contain all of the Standard Model (excluding FFterm)
Asympotically flat 1/r expansion of the field equations • 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?
The non-Abelian field equations The electro-weak field tensor fis defined by Ampere’s law is identical to Weinberg-Salam theory Other field equations have tiny extra terms
LL under SU(2) gauge transformation, with 2x2 matrix U LL under U(1) gauge transformation, with scalar L*=L when Aand fare 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/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. 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 fractional difference from Einstein-Maxwell result • Other Standard Model fields included like Einstein-Weinberg-Salam theory: - Energy levels of Hydrogen atom have fractional difference of <10-49.
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? • If wave-packet Proca waves exist and if they have negative energy, perhaps the Proca field functions as a built-in Pauli-Villars ghost • 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)?
= b+ zis similar to mass/charge renormalization in QED Electron Self Energy mass renormalization m = mb- mb·ln(ћωc/mc2)3/2 e- e- Photon Self Energy (vacuum polarization) charge renormalization e = eb - eb·ln(M/m)/3 e+ Zero-Point Energy (vacuum energy density) cosmological constant renormalization = b - LP2c4(fermions-bosons)/2 e- e- c= (cutoff frequency) LP = (Planck length) M= (Pauli-Villars cutoff mass) = (fine structure constant)