Comparison of Einstein-Maxwell theory to an alternative theory

1. Comparison of Einstein-Maxwell theory to an alternative theory Jim Shifflett shifflet@hbar.wustl.edu WUGRAV group Advisor: Clifford Will Overview: • Einstein-Maxwell theory • Lambda-renormalized Einstein-Schrödinger theory (LRES theory) • Motivation for LRES theory • Comparison of LRES theory to Einstein-Maxwell theory Graduate student seminar, 16 Feb. 2007

2. Some conventions: Geometrized units: c=G=1 Greek indices μ, ע,α,β etc. always go from 0…3 Einstein summation convention: paired indices imply summation

3. The fundamental fields of Einstein-Maxwell theory The electromagnetic vector potential Aμis the fundamental field Electric and magnetic fields (E and B) are defined in terms of Aμ

4. 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,θ) dxα r θ

5. 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.

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

7. Some Nitpicking of Einstein-Maxwell theory • Gravitation and electromagnetism are the two long range forces. They are not unified in Einstein-Maxwell theory. • Measurement gives Λ~10-56 cm-2. QED indicates Λz~-1066 cm-2 from zero-point fluctuations. This suggests “bare” Λb cancels Λz so that “physical” Λ=Λb+Λz.

8. Λ-renormalized Einstein-Schrödinger theory (LRES theory) Einstein-Maxwell Lagrangian again • LRES theory excludes FμעFעμ part, uses non-symmetric and , and uses “bare” Λb≈ -Λz so that “physical” Λ =Λb+Λz matches measurement where gμעand Aμare defined by

9. LRES theory can be used to derive Einstein-Maxwell theory where Λz = (cosmological constant from zero-point fluctuations) But because of quantum gravity effects we probably have instead using “geometrized” units c=G=1 where ωc=(cutoff frequency)~1/Lp Lp=(Planck length) We will assume ωc~1/LP and Λz~1/LP2 for the rest of this presentation

10. LRES theory is well motivated • Einstein-Schrödinger theory is non-symmetric generalization of GR - developed by two Nobel prize winning physicists • LRES theory matches measurement as well as Einstein-Maxwell theory • LRES theory reduces to ordinary GR without E&M for symmetric fields • LRES theory is a vacuum energy renormalization of ES theory • Λz term should be expected to occur as a quantization effect • Zero-point fluctuations are essential to QED - demonstrated by Casimir effect, etc. • Λ = Λb+Λz is like mass/charge/field-strength renormalization in QED - “physical” mass of an electron is sum of “bare” mass and “self energy” - it’s fine tuning but this is not new; its already in Standard Model + GR • Λz modification of ES theory has never been considered before

11. Λ = Λb+ Λz is similar to mass/charge renormalization in QED Electron Self Energy → mass renormalization m = mb- mb·ln(ћωc/mc2)3α/2π γ e- γ γ e- Photon Self Energy (vacuum polarization) → charge renormalization e = eb - eb·ln(M/m)α/3π e+ Zero-Point Energy (vacuum energy density) → cosmological constant renormalization Λ = Λb - LP2ωc4(fermions-bosons)/2π e- γ e- ωc= (cutoff frequency) LP = (Planck length) M= (Pauli-Villars cutoff mass) α = (fine structure constant)

12. 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

13. 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:

14. 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+

15. LRES theory allows other fields to be included in the Lagrangian Additional fields couple to gμעand Aע just as in Einstein-Maxwell theory may contain all of the Standard Model (excluding FμעFμעterm) - gives fractional changes of <10-50 in Hydrogen atom energy levels

16. 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

17. 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 gμע, Γαμעand Aμ q/r2 q/r2 Motion of the particles agrees with the Lorentz force equation

18. LRES theory avoids ghosts

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

20. Periastron Advance – LRES theory vs. Einstein-Maxwell theory

21. Exact Electromagnetic Plane Wave Solution of LRES theory EM plane wave solution is identical to that of Einstein-Maxwell theory who’s profile is this?

22. Deflection of Light photon ΔΦ M, Q LRES theory modification Einstein-Maxwell theory

23. Deflection of Light – LRES theory vs. Einstein-Maxwell theory

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

25. Time Delay of Light – LRES theory vs. Einstein-Maxwell theory

26. Why pursue LRES theory if it’s so close to Einstein-Maxwell theory • It unifies gravitation and electromagnetism ! • Quantization of LRES theory is untried approach to quantization of gravity - LRES theory gets much different than Einstein-Maxwell theory as k→1/LP - This could possibly fix some infinities which spoil the quantization of GR • LRES theory suggests untried approaches to a complete unified field theory - Higher dimensions, but with LRES theory instead of vacuum GR? - Non-abelian fields, but with LRES theory instead of Einstein-Maxwell? • We still don’t have a unified field theory, 50 years after Einstein: - Standard Model: excludes gravity, 25 parameters, not very “beautiful” - String theory: background dependent, spin-2 particle → GR? ~ 10500 versions, none of which have been shown to give Standard Model, problems accounting for Λ>0 and broken symmetry, little predictive ability, read “The Trouble With Physics” by Lee Smolin.

27. Summary of Λ-renormalized Einstein-Schrödinger theory • • Matches measurement as well as Einstein-Maxwell theory. • Reduces to ordinary GR without electromagnetism for symmetric fields. • Other Standard Model fields can be added just like Einstein-Maxwell theory. • Avoids the problems of the original Einstein-Schrödinger theory. • Well motivated – it’s the ES theory but with a quantization effect. • Unifies gravitation and electromagnetism in a classical sense. • Suggests untried approaches to a complete quantized unified field theory. • For the details see my papers: www.arxiv.org/abs/gr-qc/0310124, www.arxiv.org/abs/gr-qc/0403052, www.arxiv.org/abs/gr-qc/0411016.

28. Backup charts

29. LRES theory avoids the problems of Einstein-Schrödinger theory Matches measurement as well as Einstein-Maxwell theory Definitely predicts a Lorentz force Allows other fields in the Lagrangian, which couple to symmetric metric Avoids ghosts

30. LRES theory matches measurement as well as Einstein-Maxwell theory Reduces to ordinary GR without electromagnetism for symmetric fields Lorentz force equation is identical to that of Einstein-Maxwell theory Extra terms in Einstein and Maxwell equations are <10-16 of usual terms. • Exact solutions: - EM plane-wave solution is identical to that of Einstein-Maxwell theory - Charged solution and Reissner-Nordström sol. have tiny fractional difference: 10-76 for extremal charged black hole; 10-64 for atomic charges/masses/radii. fractional difference from Einstein-Maxwell result • Other Standard Model fields can be added just like Einstein-Maxwell theory: - Energy levels of Hydrogen atom have fractional difference of <10-50.

31. Electromagnetism in 4-dimensional form implies Greek indices μ,ע,α,β = 0…3

32. Derivation of Electromagnetism from a Lagrangian Greek indices μ,ע,α,β = 0…3 Maxwell’s equations

33. Derivation of Vacuum General Relativity from a Lagrangian Greek indices μ,ע,α,β = 0…3 Einstein equations