Jim Shifflett shifflethbar wustl edu WUGRAV group Advisor

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

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

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

The fundamental fields of Einstein-Maxwell theory • Metric determines distance between points in space-time • dx 2 • generalized Pythagorean theorem (ds)2=(dx 1)2+(dx 2)2 dx 1 • Connection determines how vectors change when moved r θ dxα 2 D radial coordinates (x 1, x 2)=(r, θ)

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 Einstein equations Lorentz-force equation

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.

Λ-renormalized Einstein-Schrödinger theory (LRES theory) • Einstein-Maxwell Lagrangian again • LRES theory excludes Fμ ע F ע μ part, uses non-symmetric and , and uses “bare” Λb≈ -Λz so that “physical” Λ =Λb+Λz matches measurement where gμ ע and Aμ are defined by

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 where ωc=(cutoff frequency)~1/Lp Lp=(Planck length) using “geometrized” units c=G=1 • We will assume ωc~1/LP and Λz~1/LP 2 for the rest of this presentation

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

Λ = Λb+ Λz is similar to mass/charge renormalization in QED Electron Self Energy → mass renormalization m = mb- mb·ln(ћωc/mc 2)3α/2π γ e- γ γ ee+ γ e- e- Photon Self Energy (vacuum polarization) → charge renormalization e = eb - eb·ln(M/m)α/3π Zero-Point Energy (vacuum energy density) → cosmological constant renormalization Λ = Λb - LP 2ωc 4(fermions-bosons)/2π ωc= (cutoff frequency) M= (Pauli-Villars cutoff mass) LP = (Planck length) α = (fine structure constant)

Exact charged black hole solution of Einstein-Maxwell theory • Called the Reissner-Nordström solution • Becomes Schwarzschild solution for q=0 • -2 M/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 r- LRES Event horizon conceals interior (disappears for Q>M as is the case for elementary particles) r+ r+ r-

LRES theory allows other fields to be included in the Lagrangian • Additional fields couple to gμ ע and • A ע just as in Einstein-Maxwell theory may contain all of the Standard Model (excluding Fμ ע term) - gives fractional changes of <10 -50 in Hydrogen atom energy levels

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 4 D 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 gμ ע , Γαμ ע and Aμ q/r 2 • Motion of the particles agrees with the Lorentz force equation

LRES theory avoids ghosts

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

Periastron Advance – LRES theory vs. Einstein-Maxwell theory test case → extremal charged black hole Q 1=Msun, Q 2=0, r=4 M 1=5. 9 x 105 cm ωo = 2. 6 x 104 rad/s Bohr atom Q 2= -Q 2=e, M 1=MP, M 2=Me, r=a 0=. 53 x 10 -8 cm ωo = 4. 1 x 1016 rad/s periastron frequency (ωp) Einstein-Maxwell 1. 6 x 104 rad/s theory 1. 1 x 1012 rad/s LRES theory modification 5. 5 x 10 -74 rad/s 2. 1 x 10 -76 rad/s fractional difference 10 -78 10 -88

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?

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

Deflection of Light – LRES theory vs. Einstein-Maxwell theory test case → extremal charged black hole Q=M=Msun, b=4 M=5. 9 x 105 cm atomic charges/masses/radii Q=e, M=MP, b=a 0=. 53 x 10 -8 cm deflection of light (ΔΦ) Einstein-Maxwell. 85 rad theory 9. 4 x 10 -44 rad LRES theory modification 2. 1 x 10 -79 rad 2. 9 x 10 -101 rad fractional difference 10 -79 10 -57

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

Time Delay of Light – LRES theory vs. Einstein-Maxwell theory test case → extremal charged black hole atomic charges/masses/radii Q=M=Msun, Q=e, M=MP, b=4 M=5. 9 x 105 cm b=a 0=. 53 x 10 -8 cm time delay of light (Δt) Einstein-Maxwell 2. 1 x 10 -5 sec theory 2. 7 x 10 -62 sec LRES theory modification 4. 2 x 10 -83 sec 6. 3 x 10 -120 sec fractional difference 10 -78 10 -58

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.

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.

Backup charts

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 • Avoids ghosts

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. • Standard tests fractional difference from Einstein-Maxwell result extremal charged black hole atomic 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.

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

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

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