Metric Variation Identities

Varying a curvature action requires a small set of identities used with one metric variable consistently. The determinant supplies the trace term, the Levi-Civita connection supplies derivatives of the variation, and the Palatini identity separates the Ricci variation into a divergence. Retaining that divergence is essential: it is the source of the gravitational boundary term and presymplectic potential.

Required background. We use covariant variational calculus, the curvature tensor, and hypersurface geometry.

Helpful background. The flat limit comes from special relativity; tensor operations and classical field variations supply the algebraic method.

Direct and inverse metric variations

Vary the identity $g^{ac}g_{cb}=\delta^a_b$:

$$\delta g^{ab}=-g^{ac}g^{bd}\delta g_{cd}, \qquad \delta g_{ab}=-g_{ac}g_{bd}\delta g^{cd}.$$

Thus $\delta g^{ab}$ and $\delta g_{ab}$ are not related merely by raising indices; there is a minus sign. Choose one as the independent variable and translate every term together.

Jacobi’s determinant formula gives $\delta g=g\,g^{ab}\delta g_{ab}$. Since $g<0$ for a Lorentzian metric of signature $(-,+,+,+)$,

$$\delta\sqrt{-g} =\frac12\sqrt{-g}\,g^{ab}\delta g_{ab} =-\frac12\sqrt{-g}\,g_{ab}\delta g^{ab}.$$

The two versions are a valuable sign check. Reall derives them before varying the curvature scalar Reall 2013, § 10.2, pp. 116–117, Eqs. (10.8)–(10.13), PDF.

Variation of the Levi-Civita connection

Although the connection is not a tensor, the difference between two connections is. Therefore $\delta\Gamma^a{}_{bc}$ is a tensor under a fixed coordinate identification. Vary metric compatibility or the Christoffel formula to obtain

$$\delta\Gamma^a{}_{bc} =\frac12g^{ad} \left( \nabla_b\delta g_{dc} +\nabla_c\delta g_{db} -\nabla_d\delta g_{bc} \right).$$

If $\delta g^{ab}$ is independent, substitute $\delta g_{ab}=-g_{ac}g_{bd}\delta g^{cd}$ only after taking account of metric compatibility. The result contains one covariant derivative of the metric variation.

This formula can be derived at any chosen event in normal coordinates, where $\Gamma^a{}_{bc}=0$ and $\nabla$ reduces to $\partial$ at the event. Because both sides are tensors, equality in that frame establishes equality in every frame. Normal coordinates simplify the proof; they do not set $\delta\Gamma$ to zero.

The connection variation in this convention is given in Reall 2013, § 10.2, p. 117, Eqs. (10.14)–(10.16), PDF.

The Palatini identity

With

$$R^a{}_{bcd} =\partial_c\Gamma^a{}_{db}-\partial_d\Gamma^a{}_{cb} +\Gamma^a{}_{ce}\Gamma^e{}_{db} -\Gamma^a{}_{de}\Gamma^e{}_{cb},$$

variation and regrouping into covariant derivatives gives

$$\delta R_{ab} =\nabla_c\delta\Gamma^c{}_{ab} -\nabla_b\delta\Gamma^c{}_{ac}.$$

This is the Palatini identity for the site’s curvature convention. It is a divergence before contraction, but it is not zero. The scalar curvature varies as

$$\delta R =R_{ab}\delta g^{ab} +g^{ab}\delta R_{ab}.$$

Define

$$v^c=g^{ab}\delta\Gamma^c{}_{ab} -g^{ac}\delta\Gamma^b{}_{ab}.$$

Metric compatibility turns the last term into $g^{ab}\delta R_{ab}=\nabla_cv^c$. In terms of the inverse metric variation,

$$v^c=\nabla^c(g_{ab}\delta g^{ab})-\nabla_b\delta g^{bc}.$$

The curvature identity, its contraction, and the resulting divergence are developed without discarding the boundary term in Reall 2013, § 10.2, pp. 117–118, Eqs. (10.17)–(10.23), PDF.

Assembling the curvature density

Combine the determinant and scalar-curvature identities:

$$\begin{aligned} \delta(\sqrt{-g}R) &=R\,\delta\sqrt{-g}+\sqrt{-g}\,\delta R\\ &=\sqrt{-g}\,G_{ab}\delta g^{ab} +\sqrt{-g}\,\nabla_cv^c. \end{aligned}$$

After integration, Stokes’ theorem converts the divergence into a boundary integral. For compactly supported variations it vanishes. On a finite region it contains normal derivatives of $\delta g_{ab}$ and cannot be removed by merely fixing the induced metric.

The cosmological density provides a separate check:

$$\delta(-2\Lambda\sqrt{-g}) =\Lambda\sqrt{-g}\,g_{ab}\delta g^{ab}.$$

It contributes $+\Lambda g_{ab}$ to the equation when the action contains $R-2\Lambda$.

The integration step assumes that the underlying manifold and the coordinate location of its boundary are held fixed during variation. If the boundary itself moves, its displacement changes both the integration domain and the induced geometric data, producing additional terms. Distributional metrics likewise require more care than the smooth identities given here. Within the stated smooth fixed-domain problem, however, the divergence theorem is exact and the boundary term is part of the result rather than a negligible correction.

It is also useful to distinguish a tensor from a tensor density. The combination $\sqrt{-g}g^{ab}$ appearing after integration by parts has density weight one; an affine covariant derivative acting on it includes the corresponding trace of the connection. In the metric calculation this subtlety is hidden by Levi-Civita compatibility, but it becomes explicit in an independent-connection variation.

Common sign failures

Three errors tend to travel together: forgetting the minus sign between $\delta g^{ab}$ and $\delta g_{ab}$, using a Palatini identity from the opposite Riemann convention, and discarding $g^{ab}\delta R_{ab}$ before specifying boundary support. A robust check is to redo the determinant and cosmological variations with $g_{ab}$ as the independent field; translating at the end must reproduce the same equation.

A nonconstant conformal variation tests the derivative terms as well. In $D$ dimensions set $\delta g_{ab}=2\omega(x)g_{ab}$. The identities give

$$\delta\sqrt{-g}=D\omega\sqrt{-g}, \qquad v^a=-2(D-1)\nabla^a\omega,$$

and hence

$$\delta R=-2\omega R-2(D-1)\Box\omega.$$

The constant-$\omega$ limit checks curvature’s inverse-length-squared scaling. Allowing $\omega$ to vary is essential: a constant rescaling would never detect a missing surface divergence. In $D=4$ the curvature-density variation is $\sqrt{-g}(2\omega R-6\Box\omega)$, which also follows by inserting $\delta g^{ab}=-2\omega g^{ab}$ into the Einstein-tensor formula.

Exercises

Starting from $g^{ac}g_{cb}=\delta^a_b$, derive the inverse-metric variation and use it to show that the two formulas for $\delta\sqrt{-g}$ agree.

Solution

Variation gives $(\delta g^{ac})g_{cb}+g^{ac}\delta g_{cb}=0$. Multiplying by $g^{bd}$ yields $\delta g^{ad}=-g^{ac}g^{bd}\delta g_{cb}$. Hence

$$g^{ab}\delta g_{ab} =-g_{ab}\delta g^{ab},$$

and substitution into $\delta\sqrt{-g}=(1/2)\sqrt{-g}g^{ab}\delta g_{ab}$ gives the inverse-metric form with a minus sign.

References

• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013). PDF.