Gravitation and Electromagnetism in Purely Affine Gravity

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18 Οκτ 2013 (πριν από 3 χρόνια και 11 μήνες)

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Gravitation and Electromagnetism
in Purely Affine Gravity
Nikodem J. Popławski
Department of Physics, Indiana University, Bloomington, IN
10th
Eastern Gravity Meeting, 1 June, 2007
Cornell University, Ithaca, NY
3FORMULATIONSOFGR PURELYAFFINE(Einstein–Eddington)
•LagrangiandensityLdependsontorsionlessaffineconnectionΓand
symmetricpartoftheRiccitensoroftheconnectionR(Γ)
•ThemetrictensorgisthederivativeofLwithrespecttoR
•Thefieldequationsarederivedfromvaryingthetotalactionwith
respecttoΓ
3FORMULATIONSOFGR PURELYAFFINE(Einstein–Eddington)
•LagrangiandensityLdependsontorsionlessaffineconnectionΓand
symmetricpartoftheRiccitensoroftheconnectionR(Γ)
•ThemetrictensorgisthederivativeofLwithrespecttoR
•Thefieldequationsarederivedfromvaryingthetotalactionwith
respecttoΓ
METRIC–AFFINE(Einstein–Palatini)
•BothgandΓareindependentvariables
•LislinearinthesymmetricpartofR(Γ)
3FORMULATIONSOFGR PURELYAFFINE(Einstein–Eddington)
•LagrangiandensityLdependsontorsionlessaffineconnectionΓand
symmetricpartoftheRiccitensoroftheconnectionR(Γ)
•ThemetrictensorgisthederivativeofLwithrespecttoR
•Thefieldequationsarederivedfromvaryingthetotalactionwith
respecttoΓ
METRIC–AFFINE(Einstein–Palatini)
•BothgandΓareindependentvariables
•LislinearinthesymmetricpartofR(Γ)
PURELYMETRIC(Einstein–Hilbert)
•gisthedynamicalvariable
•ΓistheLevi-Civitaconnectionofg
•LislinearinR
All3formulationsofgravitationaredynamicallyequivalentM.FerrarisandJ.Kijowski,Gen.Relativ.Gravit.14,165(1982)
‘UNIFIED’THEORYOFGRAVITATIONANDEM •EnoughdegreesoffreedominГtodescribegravitationaland
electromagneticfields(classically)
•Purelyaffineformulationallowsforelegantunificationofbothinthe
absenceofsourcesM.FerrarisandJ.Kijowski,Gen.Relativ.Gravit.14,37(1982)
‘UNIFIED’THEORYOFGRAVITATIONANDEM •EnoughdegreesoffreedominГtodescribegravitationaland
electromagneticfields(classically)
•Purelyaffineformulationallowsforelegantunificationofbothinthe
absenceofsourcesM.FerrarisandJ.Kijowski,Gen.Relativ.Gravit.14,37(1982)
Gravitationalfield–symmetricpartoftheRiccitensorofthe
connection
EMfield–thesecondRiccitensor(whichisacurl)
WeconsiderpurelyaffineLagrangiansthatdependoncurvaturevia
contractedcurvaturetensorsandconnection
ArXiv: 0705.0351 [gr-qc]
FIELDEQUATIONS Metric density of purely affine gravity
Metric tensor algebraicrelation R(g)
If is independent of then is symmetric
FIELDEQUATIONS Metric density of purely affine gravity
Metric tensor algebraicrelation R(g)
If is independent of then is symmetric
Define
Field equations from affine variational principle (variation of action
with respect to )
–differentialrelation Г(g) differential eqs. for g
FIELDEQUATIONS
yield
(generalization of Schrödinger pure-affine gravity with nonsymmetric g)
FIELDEQUATIONS
yield
(generalization of Schrödinger pure-affine gravity with nonsymmetric g)
or
looks like Maxwelleqs.
•The current density is conserved even if depends on
•If is independent of Q then FE constrain how depends on Г
ANALOGYWITHCLASSICALMECHANICS J. Kijowski, Gen. Relativ. Gravit. 9, 857 (1978)
Generalized coordinate
Generalized velocity
Lagrangian
Canonical momentum
Generalized force
Affine Hamiltonianvia Legendre transformation (Ferraris & Kijowski)
EQUIVALENCEOFAFFINEANDMETRICGRAVITY
Affine Hamiltonianvia Legendre transformation (Ferraris & Kijowski)
1st
Hamiltonequation
Einsteinequations
is metric Lagrangian for matter
EQUIVALENCEOFAFFINEANDMETRICGRAVITY
Affine Hamiltonianvia Legendre transformation (Ferraris & Kijowski)
1st
Hamiltonequation
Einsteinequations
is metric Lagrangian for matter
2nd
Hamiltonequation FE
Metric Lagrangian for the gravitational field is a Legendreterm pv
L is automatically linear in R –no f(R) models!
Metric–Affine Metric: Legendre transformation wrt Γ
EQUIVALENCEOFAFFINEANDMETRICGRAVITY
SOLUTIONOFFIELDEQUATIONS •The algebraic relation R(g) depends on how R enters L
•The differential relation Г(g) can be solved
•
ArXiv: 0705.0351 [gr-qc]
SOLUTIONOFFIELDEQUATIONS •The algebraic relation R(g) depends on how R enters L
•The differential relation Г(g) can be solved
•
R quadratic in Π
V linear in Π
ArXiv: 0705.0351 [gr-qc]
Metric Maxwell Lagrangian
Affine Ferraris–KijowskiLagrangian (dynamically equivalent to Maxwell)
This equivalence is valid so long as there are no other terms in L
depending on
ELECTROMAGNETICFIELD
Metric Maxwell Lagrangian
Affine Ferraris–KijowskiLagrangian (dynamically equivalent to Maxwell)
This equivalence is valid so long as there are no other terms in L
depending on
‘Unified’ Ferraris–Kijowski Lagrangian (associate F with eQ)
Without sources the torsion vector represents the EM potential
ELECTROMAGNETICFIELD
Take

No metric necessary to define SIMPLESTMATTERLAGRANGIAN
J. Kijowski and G. Magli, Class. Quantum Grav. 15, 3891 (1998)
ArXiv: 0705.0351 [gr-qc]
Take

No metric necessary to define
We find:
Effective energy density and pressure for matter:
SIMPLESTMATTERLAGRANGIAN
J. Kijowski and G. Magli, Class. Quantum Grav. 15, 3891 (1998)
ArXiv: 0705.0351 [gr-qc]
Take

No metric necessary to define
We find: equivalent to
Maxwelleqs. if

Effective energy density and pressure for matter:
SIMPLESTMATTERLAGRANGIAN
J. Kijowski and G. Magli, Class. Quantum Grav. 15, 3891 (1998)
ArXiv: 0705.0351 [gr-qc]
SIMPLESTMATTERLAGRANGIAN The Bianchi identities and yield
the continuum Lorentz equation
SIMPLESTMATTERLAGRANGIAN The Bianchi identities and yield
the continuum Lorentz equation
•Integrating it over space with p=0 leads to the Lorentz equation for
a particle with mass and charge
•yields
SIMPLESTMATTERLAGRANGIAN The Bianchi identities and yield
the continuum Lorentz equation
•Integrating it over space with p=0 leads to the Lorentz equation for
a particle with mass and charge
•yields
The simplest purely affine Lagrangian for matter generates the
Einstein and Lorentz equations if the electromagnetic field
contributes to mass
On larger scales, electric charge averages to zero and mass does not
Lorentz eq. is geodesic –WEP
SUMMARY&OUTLOOK
•Mathematical exercise. Classical theory.
•DoF of affine connection allow to ‘unify’ gravity and electromagnetism.
It works without sources –we extended it to sources.
•Simplest matter Lagrangian yields Maxwell–Einstein–Lorentz eqs. If
electric charge contributes to mass (EM origin of mass).
•Mass is of not purely EM origin need to explore other DoF of Г.
SUMMARY&OUTLOOK
•Mathematical exercise. Classical theory.
•DoF of affine connection allow to ‘unify’ gravity and electromagnetism.
It works without sources –we extended it to sources.
•Simplest matter Lagrangian yields Maxwell–Einstein–Lorentz eqs. If
electric charge contributes to mass (EM origin of mass).
•Mass is of not purely EM origin need to explore other DoF of Г.
•Use geometrical quantities to describe matter? Try with
(Eddington) matter L depending on R.
•Legendre transformation wrt Q?
EM Lagrangian could be a Legendreterm pv like gravitational
Lagrangian was.