Metric and Palatini Variations Compared

Metric and Palatini variations begin from similar-looking Einstein–Hilbert integrals but treat different fields as independent. Metric variation assumes the Levi-Civita connection from the outset. The torsion-free Palatini problem studied first below varies a nondegenerate metric and a symmetric affine connection independently. For the ordinary Einstein–Hilbert action in dimension greater than two, with matter independent of that connection, its connection equation returns Levi-Civita geometry and the two formulations agree. If the connection is instead completely arbitrary, the solution is Levi-Civita only up to a projective gauge field; that gauge freedom does not by itself change Einstein observables. Connection-dependent matter and modified gravitational actions require separate consistency checks.

Required background. We use variational geometry, affine curvature, and boundary geometry.

Helpful background. Special-relativistic fields, tensor operations, and classical actions help identify connection-dependent matter.

Two variational problems

In the metric formulation,

$$S_{\mathrm{metric}}[g] =\frac{1}{16\pi G}\int\sqrt{-g}\, \bigl(R[g]-2\Lambda\bigr)d^Dx,$$

where $R[g]$ uses the Levi-Civita connection of $g_{ab}$. Varying the metric therefore varies the connection and produces the familiar Palatini divergence.

In the independent-connection formulation,

$$S_{\mathrm P}[g,\Gamma] =\frac{1}{16\pi G}\int\sqrt{-g} \left(g^{ab}R_{ab}(\Gamma)-2\Lambda\right)d^Dx +S_{\mathrm m}[g,\psi],$$

with $\Gamma^a{}_{bc}=\Gamma^a{}_{cb}$ assumed torsion free. The Ricci tensor need not initially be symmetric, but only $R_{(ab)}$ couples to symmetric $g^{ab}$. Reall sketches this comparison after the metric derivation Reall 2013, § 10.2, p. 119, PDF.

The connection equation

For a torsion-free independent connection,

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

Integrating by parts, using compact support or appropriate connection boundary data, and remembering that $\delta\Gamma^c{}_{ab}$ is symmetric in $a,b$ gives

$$\nabla_c\!\left(\sqrt{-g}\,g^{ab}\right) -\delta_c{}^{(a} \nabla_d\!\left(\sqrt{-g}\,g^{b)d}\right)=0.$$

Writing $Q^b=\nabla_d(\sqrt{-g}g^{bd})$ and contracting $c$ with $a$ gives $(1-D)Q^b/2=0$. Thus, for $D\ne1$,

$$\nabla_d(\sqrt{-g}g^{bd})=0,$$

and hence

$$\nabla_c(\sqrt{-g}g^{ab})=0.$$

To see exactly where dimension enters, define $q_c=g^{ab}\nabla_cg_{ab}$. The weight-one density derivative obeys

$$\nabla_c\sqrt{-g}=\frac12\sqrt{-g}\,q_c, \qquad g_{ab}\nabla_cg^{ab}=-q_c.$$

Contracting densitized compatibility with $g_{ab}$ therefore gives

$$0=g_{ab}\nabla_c(\sqrt{-g}g^{ab}) =\frac{D-2}{2}\sqrt{-g}\,q_c.$$

For $D>2$, first $q_c=0$ and then $\nabla_cg^{ab}=0$. Together with zero torsion, metric compatibility selects the unique Levi-Civita connection. Substitution into the metric equation then produces ordinary Einstein gravity. The exceptional $D=2$ case fails at this trace step, not at the preceding contraction of the connection equation.

Starting with a symmetric connection is a consistent truncation of the unrestricted Einstein–Palatini problem: it removes the projective gauge mode without removing physical degrees of freedom Dadhich and Pons 2012, §§ 1–2 and 4.1, pp. 2–7, Eqs. (2.1)–(2.9) and (4.1), Open PDF.

The metric equation and equivalence

Holding $\Gamma$ fixed while varying $g^{ab}$ gives

$$R_{(ab)}(\Gamma) -\frac12g_{ab}R(\Gamma) +\Lambda g_{ab} =8\pi G\,T_{ab},$$

where the assumed matter action defines $T_{ab}$ but has no explicit $\Gamma$ dependence. Once the connection equation yields $\Gamma=\{g\}$, the Ricci tensor becomes the metric Ricci tensor and the equation is exactly

$$G_{ab}[g]+\Lambda g_{ab}=8\pi G\,T_{ab}.$$

The two independent equations together establish equivalence. Solving only the metric equation while leaving $\Gamma$ arbitrary does not.

The action boundary problems are also formulated differently. In Palatini form the curvature is first order in derivatives of $\Gamma$, so one declares connection data or a matching boundary term; this is not automatically the same Dirichlet problem as fixing the induced metric in the metric formulation.

When equivalence fails

If matter depends explicitly on the independent connection, its variation supplies a hypermomentum source to the connection equation. Spinor matter naturally couples through a tetrad and spin connection; theories with torsion can then produce algebraic spin–torsion interactions. One cannot infer Levi-Civita geometry by setting the matter variation to zero when it is not zero.

For a completely arbitrary connection, pure Einstein–Hilbert gravity determines

$$\Gamma=\{g\}+\text{projective gauge field},$$

not an additional physical connection mode. The Einstein–Hilbert scalar is projectively invariant, and the unfixed vector is gauge; imposing zero torsion consistently removes it Dadhich and Pons 2012, §§ 2 and 4.1, pp. 3–7, Eqs. (2.3)–(2.9) and (4.1), Open PDF.

Generic connection-dependent matter cannot simply be said to “break and fix” this freedom. In the unrestricted metric-affine problem, the projective trace of the gravitational connection equation vanishes identically, so the same trace of the hypermomentum source must vanish. Matter that breaks projective symmetry need not satisfy that condition, and the unmodified field equations can then be inconsistent Sotiriou and Liberati 2007, § 4.2, pp. 18–20, Eqs. (40)–(47), Open PDF. A consistent theory must instead restrict matter to projectively invariant couplings, impose an explicit projective gauge constraint, or add gravitational terms that break the symmetry consistently; these are distinct theories, not automatic consequences of the Einstein–Hilbert action.

If the gravitational Lagrangian is changed to $f(R)$ while matter remains independent of a torsion-free connection, independent-connection variation generally makes $\Gamma$ compatible with an auxiliary metric conformally related to $g_{ab}$ rather than with $g_{ab}$ itself Sotiriou and Faraoni 2010, § II B, pp. 6–9, Eqs. (13)–(30), Open PDF. Metric and Palatini $f(R)$ theories are dynamically distinct.

Dimension also matters. In $D=2$, the Einstein–Hilbert bulk integral is topological for Levi-Civita metrics and the densitized compatibility equation is degenerate; the $D>2$ inference must not be imported. Degenerate metrics lie outside the inverse-metric action entirely.

Independent consistency check

Write a torsion-free connection as $\Gamma^a{}_{bc}=\{^a{}_{bc}\}+C^a{}_{bc}$, where $C^a{}_{bc}$ is a tensor symmetric in its lower indices. Inserting this difference into $\nabla_c(\sqrt{-g}g^{ab})=0$ gives a homogeneous algebraic system for $C$. For $D>2$ its traces and trace-free parts vanish, confirming $C=0$. This avoids assuming the desired result through a coordinate choice.

Exercises

Why does connection-independent scalar matter preserve metric–Palatini equivalence, whereas a matter term containing $\nabla_aV^b$ can spoil it?

Solution

A scalar derivative satisfies $\nabla_a\phi=\partial_a\phi$, so a standard scalar matter action has no independent $\Gamma$ dependence and does not source the connection equation. For a vector, $\nabla_aV^b=\partial_aV^b+\Gamma^b{}_{ac}V^c$; varying the matter action with respect to $\Gamma$ generally produces nonzero hypermomentum. The connection equation is then sourced and need not imply the Levi-Civita connection of $g_{ab}$. If the connection is unrestricted, the source must also respect the projective trace identity or the action must be completed by one of the consistent mechanisms described above.

References

• Dadhich, Naresh, and Josep M. Pons. “On the Equivalence of the Einstein–Hilbert and the Einstein–Palatini Formulations of General Relativity for an Arbitrary Connection.” General Relativity and Gravitation 44 (2012): 2337–2352. DOI; arXiv.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013). PDF.
• Sotiriou, Thomas P., and Stefano Liberati. “Metric-Affine $f(R)$ Theories of Gravity.” Annals of Physics 322 (2007): 935–966. DOI; arXiv.
• Sotiriou, Thomas P., and Valerio Faraoni. “$f(R)$ Theories of Gravity.” Reviews of Modern Physics 82 (2010): 451–497. DOI; arXiv.