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

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

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

4. 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 θ dx α r 2D radial coordinates (x 1,x 2 )=(r,θ) generalized Pythagorean theorem (ds) 2 =(dx 1 ) 2 +(dx 2 ) 2

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” Lagrangian is also necessary for quantization via path integral methods. Guarantees field equations are coordinate independent and self consistent

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

7. Some Nitpicking of Einstein-Maxwell theory Measurement gives Λ~10 -56 cm -2. QED indicates Λ z ~-10 66 cm -2 from zero-point fluctuations. This suggests “bare” Λ b cancels Λ z so that “physical” Λ=Λ b +Λ z. Gravitation and electromagnetism are the two long range forces. They are not unified in Einstein-Maxwell theory.

8. Λ-renormalized Einstein-Schrödinger theory (LRES theory) Einstein-Maxwell Lagrangian again LRES theory excludes F μע F עμ part, uses non-symmetric and, and uses “bare” Λ b ≈ -Λ z so that “physical” Λ =Λ b +Λ z matches measurement where g μע and A μ are defined by

9. LRES theory can be used to derive Einstein-Maxwell theory But because of quantum gravity effects we probably have instead We will assume ω c ~1/L P and Λ z ~1/L P 2 for the rest of this presentation where Λ z = (cosmological constant from zero-point fluctuations) where ω c =(cutoff frequency)~1/L p L p =(Planck length) using “geometrized” units c=G=1

10. 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 Λ z term should be expected to occur as a quantization effect Zero-point fluctuations are essential to QED - demonstrated by Casimir effect, etc. Λ = Λ b +Λ z 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 Λ z modification of ES theory has never been considered before LRES theory is well motivated

11. Electron Self Energy → mass renormalization m = m b - m b ·ln(ћω c /mc 2 )3α/2π Photon Self Energy (vacuum polarization) → charge renormalization e = e b - e b ·ln(M/m)α/3π Zero-Point Energy (vacuum energy density) → cosmological constant renormalization Λ = Λ b - L P 2 ω c 4 (fermions-bosons)/2π ω c = (cutoff frequency) L P = (Planck length) M= (Pauli-Villars cutoff mass) α = (fine structure constant) γ e-e- γ e+e+ e-e- γ γ e-e- e-e- Λ = Λ b + Λ z is similar to mass/charge renormalization in QED

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 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-

15. Additional fields couple to g μע and A ע just as in Einstein-Maxwell theory may contain all of the Standard Model (excluding F μע F μע term) LRES theory allows other fields to be included in the Lagrangian - 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/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

17. 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 μע, Γ α μע and A μ Motion of the particles agrees with the Lorentz force equation q/r 2 Lorentz force also results from Einstein-Infeld-Hoffman (EIH) method

18. LRES theory avoids ghosts

19. M 1, Q 1 M 2, Q 2 Periastron Advance Einstein-Maxwell theory LRES theory modification Kepler’s third law This ignores radiation reaction

20. test case →extremal charged black hole Q 1 =M 1 =M sun, Q 2 =0, r=4M 1 =5.9x10 5 cm ω o = 2.6x10 4 rad/s Bohr atom Q 2 = -Q 2 =e, M 1 =M P, M 2 =M e, r=a 0 =.53x10 -8 cm ω o = 4.1x10 16 rad/s periastron frequency (ω p ) Einstein-Maxwell theory 1.6x10 4 rad/s1.1x10 12 rad/s LRES theory modification 5.5x10 -74 rad/s2.1x10 -76 rad/s fractional difference 10 -78 10 -88 Periastron Advance – LRES theory vs. Einstein-Maxwell theory

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

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

23. Deflection of Light – LRES theory vs. Einstein-Maxwell theory test case →extremal charged black hole Q=M=M sun, b=4M=5.9x10 5 cm atomic charges/masses/radii Q=e, M=M P, b=a 0 =.53x10 -8 cm deflection of light (ΔΦ) Einstein-Maxwell theory.85 rad9.4x10 -44 rad LRES theory modification 2.1x10 -79 rad2.9x10 -101 rad fractional difference 10 -79 10 -57

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

25. Time Delay of Light – LRES theory vs. Einstein-Maxwell theory test case →extremal charged black hole Q=M=M sun, b=4M=5.9x10 5 cm atomic charges/masses/radii Q=e, M=M P, b=a 0 =.53x10 -8 cm time delay of light (Δt) Einstein-Maxwell theory 2.1x10 -5 sec2.7x10 -62 sec LRES theory modification 4.2x10 -83 sec6.3x10 -120 sec fractional difference 10 -78 10 -58

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/L P - 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? ~ 10 500 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/0310124 www.arxiv.org/abs/gr-qc/0403052, www.arxiv.org/abs/gr-qc/0411016.www.arxiv.org/abs/gr-qc/0403052www.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 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. Standard tests extremal charged black holeatomic charges/masses/radii periastron advance 10 -78 10 -88 deflection of light 10 -79 10 -57 time delay of light 10 -78 10 -58 Other Standard Model fields can be added just like Einstein-Maxwell theory: - Energy levels of Hydrogen atom have fractional difference of <10 -50. fractional difference from Einstein-Maxwell result 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.

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

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

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