comparison of einstein-maxwell theory to an alternative theory
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DESCRIPTIONComparison of Einstein-Maxwell theory to an alternative theory. Jim Shifflett firstname.lastname@example.org WUGRAV group Advisor: Clifford Will. Overview: Einstein-Maxwell theory Lambda-renormalized Einstein-Schr ö dinger theory (LRES theory) Motivation for LRES theory - PowerPoint PPT Presentation
Jim Shifflettshifflet@hbar.wustl.eduWUGRAV groupAdvisor: Clifford WillGraduate student seminar, 16 Feb. 2007Overview: Einstein-Maxwell theory Lambda-renormalized Einstein-Schrdinger theory (LRES theory) Motivation for LRES theory Comparison of LRES theory to Einstein-Maxwell theory
Greek indices , ,, etc. always go from 03 Geometrized units: c=G=1Some conventions: Einstein summation convention: paired indices imply summation
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
The fundamental fields of Einstein-Maxwell theory Metric determines distance between points in space-timedx1dx2 Connection determines how vectors change when moveddxr2D radial coordinates(x1,x2)=(r,)generalized Pythagorean theorem (ds)2=(dx1)2+(dx2)2
Almost all field theories can be derived from a Lagrangian The field equations are derived from the Euler-Lagrange equationswhich minimizes the action Lagrangian is also necessary for quantization via path integral methods. Guarantees field equations are coordinate independent and self consistent
Einstein-Maxwell theory = General Relativity + ElectromagnetismLorentz-force equationEinstein equations
Some Nitpicking of Einstein-Maxwell theory Measurement gives ~10-56 cm-2. QED indicates z~-1066 cm-2 from zero-point fluctuations. This suggests bare b cancels z so that physical =b+z. Gravitation and electromagnetism are the two long range forces. They are not unified in Einstein-Maxwell theory.
-renormalized Einstein-Schrdinger theory (LRES theory) Einstein-Maxwell Lagrangian again LRES theory excludes FF part, uses non-symmetric and , and uses bare b -z so that physical =b+z matches measurementwhere g and A are defined by
LRES theory can be used to derive Einstein-Maxwell theory But because of quantum gravity effects we probably have instead We will assume c~1/LP and z~1/LP2 for the rest of this presentationwhere z = (cosmological constant from zero-point fluctuations)where c=(cutoff frequency)~1/Lp Lp=(Planck length)usinggeometrizedunitsc=G=1
Einstein-Schrdinger 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 z term should be expected to occur as a quantization effect
Zero-point fluctuations are essential to QED - demonstrated by Casimir effect, etc.
= b+z is like mass/charge/field-strength renormalization in QED - physical mass of an electron is sum of bare mass and self energy - its fine tuning but this is not new; its already in Standard Model + GR
z modification of ES theory has never been considered beforeLRES theory is well motivated
Electron Self Energy mass renormalizationm = mb- mbln(c/mc2)3/2Photon Self Energy (vacuum polarization) charge renormalizatione = eb - ebln(M/m)/3Zero-Point Energy (vacuum energy density) cosmological constant renormalization = b - LP2c4(fermions-bosons)/2c= (cutoff frequency) LP = (Planck length) M= (Pauli-Villars cutoff mass) = (fine structure constant)e-e+e-e-e- = b+ z is similar to mass/charge renormalization in QED
Exact charged black hole solution of Einstein-Maxwell theory Called the Reissner-Nordstrm 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-Nordstrm solution, Extra terms are tiny for worst-case radii accessible to measurement:
Charged solution of Einstein-Maxwell theory vs. LRES theoryLRESEinstein-Maxwell Event horizon conceals interior(disappears for Q>M as is the case for elementary particles)r+r-r+r-
- Additional fields couple to g and A just as in Einstein-Maxwell theory may contain all of the Standard Model (excluding FFterm)LRES theory allows other fields to be included in the Lagrangian - gives fractional changes of
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
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 Motion of the particles agrees with the Lorentz force equationq/r2q/r2Lorentz force also results from Einstein-Infeld-Hoffman (EIH) method
LRES theory avoids ghosts
Periastron AdvanceEinstein-Maxwell theoryLRES theory modificationKeplers third lawThis ignoresradiation reaction
Periastron Advance LRES theory vs. Einstein-Maxwell theory
test case extremal charged black holeQ1=M1=Msun,Q2=0,r=4M1=5.9x105 cmo = 2.6x104 rad/sBohr atomQ2= -Q2=e,M1=MP, M2=Me,r=a0=.53x10-8 cmo = 4.1x1016 rad/speriastron frequency (p)periastron frequency (p)Einstein-Maxwelltheory1.6x104 rad/s1.1x1012 rad/sLRES theorymodification5.5x10-74 rad/s2.1x10-76 rad/s fractionaldifference10-7810-88
EM plane wave solution is identical to that of Einstein-Maxwell theoryExact Electromagnetic Plane Wave Solution of LRES theorywhos profile is this?
Deflection of LightphotonM, QEinstein-Maxwell theoryLRES theorymodification
Deflection of Light LRES theory vs. Einstein-Maxwell theory
test case extremal charged black holeQ=M=Msun,b=4M=5.9x105 cmatomic charges/masses/radiiQ=e, M=MP,b=a0=.53x10-8 cmdeflection of light ()deflection of light ()Einstein-Maxwelltheory.85 rad9.4x10-44 radLRES theorymodification2.1x10-79 rad2.9x10-101 rad fractionaldifference10-7910-57
Time Delay of Lightradio signalM, Qt=d/c+tt=0satellite()Einstein-Maxwell theoryLRES theorymodificationd
Time Delay of Light LRES theory vs. Einstein-Maxwell theory
test case extremal charged black holeQ=M=Msun,b=4M=5.9x105 cmatomic charges/masses/radiiQ=e, M=MP,b=a0=.53x10-8 cmtime delay of light (t)time delay of light (t)Einstein-Maxwelltheory2.1x10-5 sec2.7x10-62 secLRES theorymodification4.2x10-83 sec6.3x10-120 sec fractionaldifference10-7810-58
Why pursue LRES theory if its 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 k1/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 dont 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 >0 and broken symmetry, little predictive ability, read The Trouble With Physics by Lee Smolin.
Summary of -renormalized Einstein-Schrdinger 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-Schrdinger theory.
Well motivated its 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.
LRES theory avoids the problems of Einstein-Schrdinger 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
- 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-Nordstrm sol. have tiny fractional difference: 10-76 for extremal charged black hole; 10-64 for atomic charges/masses/radii. Other Standard Model fields can be added just like Einstein-Maxwell theory:- Energy levels of Hydrogen atom have fractional difference of
Electromagnetism in 4-dimensional formGreek indices,,, = 03implies
Derivation of Electromagnetism from a LagrangianMaxwells equationsGreek indices,,, = 03
Derivation of Vacuum General Relativity from a LagrangianEinstein equationsGreek indices,,, = 03