nLab
Einstein-Hilbert action

Idea

The Einstein-Hilbert action is the action functional that defines the dynamics of gravity in general relativity. It is a canonical invariant of pseudo-Riemannian manifolds.

Definition

For $(X,g)$ a Riemannian manifold or pseudo-Riemannian manifold, its vacuum Einstein-Hilbert action is the number

$$(X,g)\mapsto {\int }_{X}R(g)\mathrm{dvol}(g),$$(X,g) \mapsto \int_X R(g) dvol(g) \,,

where

• $R(g):X\to ℝ$ is the scalar curvature function of the metric $g$;

• $\mathrm{dvol}(g)$ is the volume form defined by the metric

• ${\int }_X$ is integration of differential forms over $X$.

If the metric is instead encoded in terms of an Poincare group-connection $\nabla$ (the first-order formulation of gravity) then (for $\dim X=4$ and assuming for simplicity that the underlying bundle is trivial) this is equivalently

$$(X,\nabla )\mapsto {\int }_X⟨R\wedge e\wedge e⟩,$$(X,\nabla) \mapsto \int_X \langle R \wedge e \wedge e \rangle \,,

where now

• $R$ denotes the curvature 2-form of the orthogonal group-connection part of $\nabla$;

• $e$ denotes the vielbein, (the translation part of the connection 1-form);

• $⟨-⟩$ is the invariant polynomial which in terms of the canonical basis has components $({ϵ}_{abcd})$.

This extends to an action functional for gravity coupled to “matter” (which in the context of general relativity conventionally means every other field, for instance the electromagnetic field) by simply adding the matter action on a given gravitational background

$$(X,g,\varphi )\mapsto {\int }_X(L_{\mathrm{EH}}(g)+L_{\varphi }(g))d\mathrm{vol}(g).$$(X,g,\phi) \mapsto \int_X (L_{EH}(g) + L_\phi(g)) d vol(g) \,.

The critical points of the Einstein-Hilbert action define the physically realized spacetimes. This are Einstein's equations (the Euler-Lagrange equations of this action)

$$\delta S_{\mathrm{EH}}(g)=0.$$\delta S_{EH}(g) = 0 \,.

(…)

Related concepts

• Einstein-Maxwell theory

• Einstein-Yang-Mills theory

• Einstein-Yang-Mills-Dirac theory

• Einstein-Maxwell-Yang-Mills-Dirac-Higgs 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
$L=$	$R(e)\mathrm{vol}(e)+$	$⟨F_{\nabla }\wedge {\star }_e{F}_{\nabla }⟩+$	$(\psi ,D_{(e,\nabla )}\psi )\mathrm{vol}(e)+$	$\nabla \overline{H}\wedge {\star }_e\nabla H+(\lambda {\mid H\mid }^4-{\mu }^2{\mid H\mid }^2)\mathrm{vol}(e)$

References

Section Prequantum gauge theory and Gravity at

• geometry of physics