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

Let $X$ be a compact smooth manifold with boundary.

For $(X,g)$ the structure of a Riemannian manifold or pseudo-Riemannian manifold on $X$, hence for $g$ a (pseudo-)Riemannian metric on $X$, its vacuum Einstein-Hilbert action is the number

$$g \;\mapsto\; \int_X R(g) dvol(g) \,,$$

where

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

• $dvol(g)$ 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

$$\nabla \;\mapsto\; \int_X \langle R \wedge e \wedge e \rangle \,,$$

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);

• $\langle -\rangle$ is the invariant polynomial which in terms of the canonical basis has components $(\epsilon_{a b c d})$.

In either case this assignment defines a function

$$S_{EH} \;\colon\; Met(X) \longrightarrow \mathbb{R}$$

on the space of (pseudo-)Riemannian metrics on $X$. This is the Einstein-Hilbert action functional.

This functional is invariant under the action of diffeomorphisms on $X$ (e.g. Bertschinger 02, section 4.2). Therefore more properly the EH-action functional is the function descended to the moduli stack $Met(X)//Diff(X)$, see at general covariance.

More generally, 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

$$(g,\phi) \;\mapsto\; \int_X (L_{EH}(g) + L_\phi(g)) d vol(g) \,.$$

The critical points of the Einstein-Hilbert action define the physically realized metrics (and matter fields) on $X$, hence the physically realized spacetimes with underlying manifold $X$. This are Einstein's equations (the Euler-Lagrange equations of this action)

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

To derive the Einstein Field equations from this, we will look at the Einstein-Hilbert action action in its usual form ($c$, the speed of light, is set to unity), in spacetime with no matter fields (i.e., matter Lagrangian $\mathcal{L}_M=0$):

$$S = \frac{1}{16\pi G} \int \operatorname{tr}_g(R) \sqrt{-g} \, \mathrm{d}^4x \;,$$

The variation of this action can be performed:

$$\frac{\delta S}{\delta g}=0$$

The Wikipedia Page shows how to vary the action (with matter fields) through components, giving us the famed Einstein field equations:

$$R-\frac{1}{2}g\,\operatorname{tr}_g(R)=T$$

Here, $T$ is the energy-momentum tensor.

(…)

History

Historically, Einstein had been working towards what today are called Einstein's equations for some time. What is today called the Einstein tensor in the equation had been missing the trace term in one of his first publications, and other things remained unclear. Einstein was working with the 10 coordinate components of these tensors and trying to make the theory match classical Newtonian physics in the right limit. The “general covariance” of the equations made this a subtle problem, at least back in the times when these issues were just beginning to be appreciated.

Hilbert had been following these developments and had listened to Einstein talk about them. Apparently at some point Hilbert, who believed in the axiomatic foundation of physics (as is evident from Hilbert's sixth problem), decided that whatever the correct equations of motion are, they ought to follow from an action principle, hence that they characterize the critical points of an action functional on the space of pseudo-Riemannian metrics on some smooth manifold. Furthermore he observed that the most obvious such functional with the right symmetry properties is the integral over the Riemann scalar curvature times the volume form of the given metric (hence the “Einstein-Hilbert action”). He computed the variational derivative of this and thereby found Einstein's equations apparently around the same time if not a little bit before Einstein published the final version of his theory.

In (Sauer-Majer, from p. 402) a talk of Hilbert is reproduced that reviews Einstein’s work, from the point of view of Hilbert’s having known the right answer all along from the variational principle (hence from the axioms) as Einstein struggled (until a few days before the lecture was given!) with finding the right equations of gravity. This comment culminates (Sauer-Majer, from p. 405) with the curious remark, paraphrasing just a little: that the fact that Einstein after his “colossal detour” ends up with the same equations of motion as Hilbert already had is a “nice consistency check” (schöne Gewähr)

p. 402:

Das von mir aufgestellte Hamiltonsche Prinzip lautet

$\delta \int \int \int \int I \sqrt{g} \, d x^1 d x^2 d x^3 d x^4 = 0$

(this defines the critical point of the action functional being the integral of the Lagrangian $I$ over spacetime, i.e the principle of extremal action)

p. 404:

Indess mündet das Schlussresultat der letzten Einsteinschen Untersuchung wieder auf ein Hamiltonsches Prinzip, das dem ursprünglich von mir aufgestellten gleicht; ja es könnte sein, dass diese Einsteinsche Theorie inhaltlich sich mit der von mir ursprünglich aufgestellten Theorie völlig deckt.

p. 405:

Als Entgelt dafür können wir gegenwärtig als sicher ansehen, dass die oben von mir aufgestellten Gl. gewiss den Grundstock für die Weltgl. ausmachen. In einer letzten vor wenigen Tagen erschienenen Note gelangt Einstein zu dem Hamiltonschen Prinzip:

$\delta \int \int \int \int \{K + \alpha \Phi + \beta \phi\} \sqrt{g} d x^1 d x^2 d x^3 d x^4 = 0$

$[...]$

Und wenn auf dem kollossalen Umweg über Levi Civita, Weyl, Schouten, Eddington Einst. zu dem Resultat zurückgelangt, so liegt darin sicher eine schöne Gewähr.

In Constance Reid’s biography of Hilbert, one finds in Chapter XVII the following:

In Göttingen, in the weekly Hilbert-Debye seminars, it seemed to those few students who were left $[$ due to WWI $]$ that the “living pulse” of physical research was at their finger tips. The work of Einstein as he pressed forward toward a general theory of relativity was followed with great interest. Also followed was the work of others who were trying to reach the same goal. Hilbert was especially fascinated by the ideas of Gustav Mie, then in Greifswald, who was attempting to develop a theory of matter on the fundamentals of the relativity principle; and in his own investigations he was able to bring together Mie’s program of pure field theory and Einstein’s theory of gravitation. At the same time, while Einstein was attempting in a rather roundabout way to develop the binding laws for the 10 coefficients of the differential form which determines gravitation, Hilbert independantly solved the problem in a different, more direct way. Both men arrived at almost the same time at the same goal. As the western front settled down for the winter, Einstein presented his two papers, “On general relativity theory” to the Berlin Academy on November 11 and 25; Hilbert presented his first note on “The foundations of physics” to the Royal Society of Science in Gottingen on November 20, 1915. It was a remarkable coincidence … but even more remarkable, according to $[$Max$]$ Born (who was now in Berlin with Einstein), was the fact that it led, not to a controversy over priority, but to a series of friendly encounters and letters. Hilbert freely admitted, and frequently stated in lectures, that the great idea was Einstein’s.

More citations on the priority issue are listed in this Physics.SE comment.

The success of the Einstein-Hilbert action principle in the formulation of gravity probably had a considerable influence on the theoretical physics to follow and led some physcist to explore more systematically the space of action functionals invariant under prescribed symmetries. This way notably the supergravity variant of Einstein-gravity in the context of Riemannian supergeometry was found, a theory which eventually leads to superstring theory. These variants of course remain speculative as theories of phenomenological physics, but to some extent their development does follow Hilbert’s attitude in the discovery of general relativity.

Related concepts

• Einstein-Maxwell theory

• Einstein-Yang-Mills theory

• Einstein-Yang-Mills-Dirac theory

• Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory

• higher order curvature corrections

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
$L =$	$R(e) vol(e) +$	$\langle F_\nabla \wedge \star_e F_\nabla\rangle +$	$(\psi , D_{(e,\nabla)} \psi) vol(e) +$	$\nabla \bar H \wedge \star_e \nabla H + \left(\lambda {\vert H\vert}^4 - \mu^2 {\vert H\vert}^2 \right) vol(e)$

References

General

Lecture notes include

• Edmund Bertschinger, Symmetry transformations, the Einstein-Hilbert action, and gauge invariance 2002 (pdf)

Section Prequantum gauge theory and Gravity at

• geometry of physics

History

• Tilman Sauer, Ulrich Majer with Arne Schirrmacher, Heinz-Jürgen Schmidt (eds.): David Hilbert’s Lectures on the foundations of physics 1915-1927 — Relativity, Quantum Theory and Epistemology, Springer (2009) [doi:10.1007/b12915]