James A. Shifflett Dissertation Presentation For Degree of Doctor of Philosophy in Physics Washington University in St.

1. 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

2. 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

3. 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,

4. Einstein-Maxwell theory

5. 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

6. 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 

7. 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.

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

9. Early attempts to unify General Relativity and Electromagnetism

10. -renormalized Einstein-Schrodinger (LRES) theory

11. LRES theory vs. Einstein-Maxwell theory

12. 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

13. The field equations The electromagnetic field tensor fcan be defined by Ampere’s law is identical to Einstein-Maxwell theory Other field equations have tiny extra terms

14. Exact Solutions

15. 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

16. 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:

17. 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+

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

19. Equations of Motion

20. 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

21. 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

22. Observable Consequences

23. M1, Q1 M2, Q2 Pericenter Advance Kepler’s third law This ignores radiation reaction LRES theory modification Einstein-Maxwell theory

24. Deflection of Light photon  M, Q Einstein-Maxwell theory LRES theory modification

25. Time Delay of Light satellite radio signal –( d t=0 )– t=d/c+t M, Q LRES theory modification Einstein-Maxwell theory

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

27. Application of Newman Penrose Methods

28. 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?

29. Non-Abelian LRES theory

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

31. The non-Abelian field equations The electro-weak field tensor fis defined by Ampere’s law is identical to Weinberg-Salam theory Other field equations have tiny extra terms

32. 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

33. 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)

34. 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)?

35. 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.

36. Backup charts

37. 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

38. 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 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)?

39.  = 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 - LP2c4(fermions-bosons)/2 e-  e- c= (cutoff frequency) LP = (Planck length) M= (Pauli-Villars cutoff mass)  = (fine structure constant)