The Einstein-Hilbert action

Plain-language version: Gravity from an action

The Einstein-Hilbert action produces the field equations of general relativity from a variational principle. It is the last application in this volume and it is where three threads left open earlier converge: the surface term that Several independent variables could not discard, Noether's second theorem, and her classification of energy conservation as proper or improper.

The action

Two stationary light paths around a mass.

The dynamical variable is the metric $g_{\alpha \beta }$. Requiring the action to be a scalar under arbitrary coordinate transformations, built from the metric and its derivatives, leaves very little freedom. The simplest choice is

$S_{\text{EH}}=\frac{1}{2\kappa }\int R\sqrt{-g}{d}^{4}x\mathrm{(}\mathrm{1}\mathrm{)}$

with $R$ the Ricci scalar, $g$ the determinant of the metric, and $\kappa =8\pi G/c^4$. The factor $\sqrt{-g}{d}^{4}x$ is the invariant volume element; without it the integral would not be a scalar.

(1) contains second derivatives of the metric. $R$ is built from the Christoffel symbols of Geodesics and their derivatives, so $ℒ$ depends on $g$, $\partial g$ and ${\partial }^{2}g$. This is the exception flagged on Classical field theory, where first-order Lagrangians were said to be the rule.

It escapes the usual instability because the second derivatives enter linearly and can be moved into a total divergence. The field equations come out second order, not fourth. That is a special property of the Ricci scalar and not a general feature of higher-derivative actions.

Varying

Free fall as a straight path in a curved geometry.

Write the variation of (1) using $\delta \sqrt{-g}=-\frac{1}{2}\sqrt{-g}{g}_{\alpha \beta }\delta g^{\alpha \beta }$ and $R=g^{\alpha \beta }{R}_{\alpha \beta }$:

$\delta S_{\text{EH}}=\frac{1}{2\kappa }\int [R_{\alpha \beta }-\frac{1}{2}R{g}_{\alpha \beta }]\delta g^{\alpha \beta }\sqrt{-g}{d}^{4}x+\frac{1}{2\kappa }\int g^{\alpha \beta }\delta R_{\alpha \beta }\sqrt{-g}{d}^{4}x\mathrm{(}\mathrm{2}\mathrm{)}$

The second integral is the one that matters. The variation of the Ricci tensor is a total divergence,

$g^{\alpha \beta }\delta R_{\alpha \beta }={\mathrm{\nabla }}_{\gamma }{v}^{\gamma }\mathrm{(}\mathrm{3}\mathrm{)}$

for a vector $v^{\gamma }$ built from $\delta g$ and $\mathrm{\nabla }\delta g$, so by the divergence theorem it becomes a surface integral over $\partial \Omega$.

That surface integral does not vanish when only $\delta g$ is required to vanish on the boundary, because $v^{\gamma }$ also contains the normal derivative of $\delta g$, which is unconstrained. This is precisely the second case named on Several independent variables: with second derivatives in the Lagrangian, two integrations by parts are needed and the surface term involves the derivative of the variation.

Setting the first integral to zero gives the Einstein field equations in vacuum:

$R_{\alpha \beta }-\frac{1}{2}R{g}_{\alpha \beta }=0\mathrm{(}\mathrm{4}\mathrm{)}$

and adding a matter action with stress-energy tensor $T_{\alpha \beta }$ gives the full equations, $G_{\alpha \beta }=\kappa T_{\alpha \beta }$.

The surface term

The boundary term in one dimension and in a field.

The standard repair is to add a compensating boundary term to (1),

$S=S_{\text{EH}}+\frac{1}{\kappa }{\oint }_{\partial \Omega }K\sqrt{|h|}{d}^{3}x\mathrm{(}\mathrm{5}\mathrm{)}$

with $K$ the trace of the extrinsic curvature of the boundary and $h$ the induced metric. Its variation cancels the unwanted term, and the variational problem is then well posed with $\delta g=0$ on the boundary alone.

This is not cosmetic. The boundary term carries the gravitational energy in several standard constructions, and it is what makes the action finite for asymptotically flat spacetimes after a subtraction. A treatment that discards the surface term silently gets (4) right and loses everything that depends on the value of the action.

The general lesson, stated on Boundary terms and natural boundary conditions and now realised: a boundary term must be argued away, not assumed away. Here it cannot be argued away and must be cancelled by hand.

Noether's second theorem, realised

The action (1) is invariant under all coordinate transformations $y^{\alpha }=p^{\alpha }(x)$ — a group depending on four arbitrary functions, with their first derivatives entering. That is a $G_{\mathrm{\infty }4}$, so Noether's second theorem applies and there must be four identities among the field equations.

They are the contracted Bianchi identities:

${\mathrm{\nabla }}_{\alpha }(R^{\alpha \beta }-\frac{1}{2}R{g}^{\alpha \beta })=0\mathrm{(}\mathrm{6}\mathrm{)}$

holding identically, for every metric, before any field equation is imposed. Noether names general relativity as the second of her two examples of Theorem II, and identifies exactly these $n$ dependencies among the Lagrange expressions and their first derivatives.

Three consequences, each an instance of a general statement from that page.

Four of the ten field equations are redundant. Ten components, four identities, four gauge functions — leaving two propagating polarisations, the same count as electromagnetism on the electromagnetic Lagrangian reached by the same arithmetic.

The evolution is not determined. Four functions remain free for all time. That is the coordinate freedom, and the initial-value problem for (4) is constrained rather than free.

Conservation of matter energy is forced. Applying ${\mathrm{\nabla }}_{\alpha }$ to $G_{\alpha \beta }=\kappa T_{\alpha \beta }$ and using (6) gives ${\mathrm{\nabla }}_{\alpha }{T}^{\alpha \beta }=0$. As with charge conservation in electromagnetism, the symmetry of the action requires the conservation law rather than permitting it.

Improper energy conservation

The sharpest consequence is the one Noether wrote the paper for.

Her §6 result, stated on Noether's second theorem, is that the energy relations of a theory are improper exactly when the theory is invariant under an infinite group containing the displacement group. General relativity is such a theory: translations are a special case of arbitrary coordinate changes.

Therefore gravitational energy conservation is improper. The conserved quantity is a combination of the field equations and their derivatives — an identity wearing the costume of a conservation law — rather than a dynamical statement of the kind that holds in classical mechanics or special relativity.

This is why there is no local gravitational stress-energy tensor. The various pseudotensors in the literature are coordinate dependent, and their coordinate dependence is not a defect of the constructions: it is what Noether's theorem predicts. Global energy can still be defined for spacetimes with suitable asymptotic structure, and the boundary term (5) is where that definition lives.

Noether adds that for the Hilbert assertion to hold literally, general relativity must be read as any theory invariant under such a group. The property is group-theoretic, and gravity is one instance rather than the reason.

Geodesics recovered

The matter action for a free test particle is its proper time,

$S=-mc\int ds\mathrm{(}\mathrm{7}\mathrm{)}$

which is the arc-length functional of Geodesics in the spacetime metric. Its extremals are geodesics, so free fall is a geodesic and (6) of that page is the equation of motion under gravity.

Conjugate points then acquire a physical reading. Neighbouring geodesics refocused by curvature are light rays refocused by a gravitational lens, and the multiple images a lens produces are the several stationary paths between source and observer. Only the first-arriving image corresponds to a minimal path; the others are stationary and are not minima, in the sense established on the Jacobi condition and conjugate points.

That is the volume's central distinction appearing as an observation rather than an argument.

Provenance: every equation on this page is standard and is derived or stated here; none is transcribed from a primary source. The characterisations of Noether 1918 §6 and of her general-relativity example are paraphrased from the ESTABLISHED source under the translation rule on Noether's first theorem. Hilbert's 1915 paper and Einstein's field-equation papers are not sources of this volume and are not cited. See Sources.

Next

the Feynman path integral closes the applications track.