nLab Einstein-Maxwell theory

Contents

Formalism

Definition

Spacetime configurations

Properties

Spacetimes

black hole spacetimes	vanishing angular momentum	positive angular momentum
vanishing charge	Schwarzschild spacetime	Kerr spacetime
positive charge	Reissner-Nordstrom spacetime	Kerr-Newman spacetime

Quantum theory

Idea

What is called Einstein-Maxwell theory in physics is the theory/model (in theoretical physics) describing gravity together with electromagnetism.

It is a local Lagrangian field theory defined by the action functional which is the Einstein-Hilbert action plus the Maxwell action functional? involving the given metric,

S_{G+EM} \; \colon \; (e, \nabla) \mapsto \int_{X} R(e) vol(e) + \int_X F_\nabla \wedge \star_e F_\nabla \,,

where

$X$ is the compact smooth manifold underlying spacetime,
$e$ is the vielbein field which encodes the field of gravity
$\nabla$ is the $U(1)$ -principal connection which encodes the electromagnetic field,
$vol(e)$ is the volume form induced by $e$ ;
$R(e)$ is the scalar curvature of $e$ ;
$F_\nabla$ is the field strength/curvature differential 2-form of $\nabla$ ;
$\star_e$ is the Hodge star operator induced by $e$ .

theory:	Einstein-	Yang-Mills-	Dirac-	Higgs
	gravity	electroweak and strong nuclear force	fermionic matter	scalar field
field content:	vielbein field $e$	principal connection $\nabla$	spinor $\psi$	scalar field $H$
Lagrangian:	scalar curvature density	field strength squared	Dirac operator component density	field strength squared + potential density
$L =$	$R(e) vol(e) +$	$\langle F_\nabla \wedge \star_e F_\nabla\rangle +$	$(\psi , D_{(e,\nabla)} \psi) vol(e) +$	$\nabla \bar H \wedge \star_e \nabla H + \left(\lambda {\vert H\vert}^4 - \mu^2 {\vert H\vert}^2 \right) vol(e)$