# nLab Einstein-Maxwell theory

### Context

#### Gravity

gravity, supergravity

## Quantum theory

#### Differential cohomology

differential cohomology

# Contents

## 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,

${S}_{G+\mathrm{EM}}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\left(e,\nabla \right)↦{\int }_{X}R\left(e\right)\mathrm{vol}\left(e\right)+{\int }_{X}{F}_{\nabla }\wedge {\star }_{e}{F}_{\nabla }\phantom{\rule{thinmathspace}{0ex}},$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 $U\left(1\right)$-principal connection which encodes the electromagnetic field,

• $\mathrm{vol}\left(e\right)$ is the volume form induced by $e$;

• $R\left(e\right)$ 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 $e$.

This is the special case of Einstein-Yang-Mills theory for the gauge group being the circle group.

standard model of particle physics and cosmology

theory:Einstein-Yang-Mills-Dirac-Higgs
gravityelectroweak and strong nuclear forcefermionic matterscalar field
field content:vielbein field $e$principal connection $\nabla$spinor $\psi$scalar field $H$
Lagrangian:scalar curvature densityfield strength squaredDirac operator component densityfield strength squared + potential density
$L=$$R\left(e\right)\mathrm{vol}\left(e\right)+$$⟨{F}_{\nabla }\wedge {\star }_{e}{F}_{\nabla }⟩+$$\left(\psi ,{D}_{\left(e,\nabla \right)}\psi \right)\mathrm{vol}\left(e\right)+$$\nabla \overline{H}\wedge {\star }_{e}\nabla H+\left(\lambda {\mid H\mid }^{4}-{\mu }^{2}{\mid H\mid }^{2}\right)\mathrm{vol}\left(e\right)$

## References

Section Prequantum gauge theory and Gravity in