black hole spacetimes | vanishing angular momentum | positive angular momentum |
---|---|---|
vanishing charge | Schwarzschild spacetime | Kerr spacetime |
positive charge | Reissner-Nordstrom spacetime | Kerr-Newman spacetime |
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What is called Einstein-Yang-Mills theory in physics is the theory/model (in theoretical physics) describing gravity together with Yang-Mills fields such as the electroweak field or the strong nuclear force of quantum chromodynamics. For the special case that the gauge group is the circle group this reproduces Einstein-Maxwell theory.
Einstein-Yang-Mills theory is a local Lagrangian field theory defined by the action functional which is the Einstein-Hilbert action plus the Yang-Mills action functional involving the given metric,
where
$X$ is the compact smooth manifold underlying spacetime,
$e$ is the vielbein field which encodes the field of gravity
$\nabla$ is the $G$-principal connection which encodes the Yang-Mills 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 $e$.
$\langle -,-\rangle$ is the given invariant polynomial on the Lie algebra of the gauge group.
Einstein-Yang-Mills theory
standard model of particle physics and cosmology
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 |
$L =$ | $R(e) vol(e) +$ | $\langle F_\nabla \wedge \star_e F_\nabla\rangle +$ | $(\psi , D_{(e,\nabla)} \psi) vol(e) +$ | $\nabla \bar H \wedge \star_e \nabla H + \left(\lambda {\vert H\vert}^4 - \mu^2 {\vert H\vert}^2 \right) vol(e)$ |
Section Prequantum gauge theory and Gravity in
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