Palatini first-order variational formulation of general relativity

Intuition Beginner

In the usual story of general relativity, one object does all the work: the metric. The metric tells you distances and times, and from it you build the rule for parallel transport -- the rule that says how to carry a vector from one point to a neighboring one without turning it. That rule, the connection, is computed from the metric. So there is one input and everything else follows.

Palatini's idea is to loosen this. Treat the metric and the connection as two separate, independent inputs. The metric still measures lengths. The connection still says how to transport vectors. But now you do not assume in advance that the connection is the one the metric would have produced. You leave both free and let the action principle decide.

The payoff is a small miracle. You write the same curvature action as before, vary it once with respect to the metric and once with respect to the connection, and out of the second variation comes an equation. That equation says: the connection must be exactly the one the metric defines after all. Metric compatibility is not assumed. It is forced. What looked like a starting assumption in the standard derivation turns out to be a conclusion you can earn for free.

Visual Beginner

The picture contrasts the two routes. On the left, the connection is computed from the metric before any variation happens. On the right, the connection enters as a free field, and its variation is what produces the rule connecting it back to the metric.

Worked example Beginner

Picture a flat sheet of paper. The metric on it is the ordinary one: distances are measured with the Pythagorean rule. The natural connection -- the one that says "transport a vector by keeping its components constant in Cartesian coordinates" -- has all its Christoffel symbols equal to zero.

Now suppose someone hands you a different connection on the same flat sheet, one whose transport rule slowly rotates every vector as you slide it to the right. The metric has not changed; only the transport rule has. Is this allowed in the Palatini setup? As a starting field, yes -- the connection is free. But the action principle will reject it. When you vary the action with respect to the connection, the resulting equation demands that transport must preserve lengths and angles measured by the metric. A connection that rotates vectors as it moves them fails this test.

Here is the takeaway. The Palatini variation acts like an audit. You may propose any connection you like as a candidate, but only one survives: the connection whose transport rule is compatible with the metric, with no built-in twisting. On the flat sheet, that survivor is precisely the zero-Christoffel connection you started with intuitively. The freedom is real, but the action narrows it down to a single answer, and that answer matches the metric.

Check your understanding Beginner

Formal definition Intermediate+

Fix an oriented four-dimensional manifold $M$. The Palatini (first-order, metric-affine) data are a pair $(g_{\mu \nu },{\Gamma }_{\mu \nu }^{\lambda })$ where $g_{\mu \nu }$ is a Lorentzian metric of signature $(-,+,+,+)$ and ${\Gamma }_{\mu \nu }^{\lambda }$ is a connection, taken symmetric in its lower indices (${\Gamma }_{\mu \nu }^{\lambda }={\Gamma }_{\nu \mu }^{\lambda }$, i.e. torsion-free) unless stated otherwise. The two fields are independent: $\Gamma$ is not presumed to be the Levi-Civita connection of $g$. The symbol $\nabla$ denotes covariant differentiation with respect to $\Gamma$; $\partial$ denotes ordinary partial differentiation; $\delta$ denotes a first-order field variation; $\int$ is integration against $d^{4}x$.

The curvature of the independent connection is the affine Ricci tensor

$$R_{\mu \nu }(\Gamma )={\partial }_{\lambda }{\Gamma }_{\mu \nu }^{\lambda }-{\partial }_{\nu }{\Gamma }_{\mu \lambda }^{\lambda }+{\Gamma }_{\lambda \rho }^{\lambda }{\Gamma }_{\mu \nu }^{\rho }-{\Gamma }_{\nu \rho }^{\lambda }{\Gamma }_{\mu \lambda }^{\rho },$$

built from $\Gamma$ alone with no reference to the metric. The Palatini action is

$$S[g,\Gamma ]=\frac{1}{2\kappa }{\int }_M{g}^{\mu \nu }{R}_{\mu \nu }(\Gamma )\sqrt{-g}{d}^{4}x,\kappa =\frac{8\pi G}{c^4},$$

a functional of both fields ^{[Sternberg Ch. 20 §3]}. The metric supplies the volume element $\sqrt{-g}{d}^{4}x$ and the inverse $g^{\mu \nu }$ that contracts the affine Ricci tensor; the connection supplies the curvature. When matter is present one adds $S_m[g,\psi ]$, a functional of the metric and matter fields $\psi$ (in the minimal-coupling convention it does not depend on $\Gamma$), and the stress-energy tensor is $T_{\mu \nu }=-\frac{2}{\sqrt{-g}}\frac{\delta S_m}{\delta g^{\mu \nu }}$.

A pair $(g,\Gamma )$ is a stationary point of $S=S[g,\Gamma ]+S_m$ if $\delta S=0$ to first order for every compactly supported $\delta g^{\mu \nu }$ and every compactly supported $\delta {\Gamma }_{\mu \nu }^{\lambda }$, varied independently.

Sign and convention notes

The signature is $(-,+,+,+)$. The affine Ricci tensor uses the same index convention as the metric Ricci tensor of 13.04.02; for a general connection $R_{\mu \nu }(\Gamma )$ need not be symmetric, but for a symmetric (torsion-free) $\Gamma$ the antisymmetric part is a total derivative that does not affect the field equations. The nonmetricity tensor of the pair is $Q_{\lambda \mu \nu }:={\nabla }_{\lambda }{g}_{\mu \nu }$; the connection is metric-compatible when $Q_{\lambda \mu \nu }=0$, which for the standard metric formulation is imposed at the outset and which here is to be derived.

Counterexamples to common slips

• The Ricci tensor in the Palatini action is $R_{\mu \nu }(\Gamma )$, a function of the independent connection, not the metric Ricci tensor $R_{\mu \nu }(g)$ of the Levi-Civita connection. These coincide only after the connection field equation is imposed.
• Independence does not mean the connection ends up arbitrary. The action constrains $\Gamma$ tightly; the residual freedom is a single projective vector, removed by the torsion-free condition or by a gauge choice.
• The equivalence of the Palatini and metric formulations is special to the linear-in-$R$ Lagrangian. Replacing $R$ by a nonlinear function $f(R)$ breaks it: Palatini $f(R)$ and metric $f(R)$ are different theories with different field equations.

Key theorem with proof Intermediate+

Theorem (Palatini equivalence for the Einstein-Hilbert Lagrangian). Let $S[g,\Gamma ]=\frac{1}{2\kappa }{\int }_M{g}^{\mu \nu }{R}_{\mu \nu }(\Gamma )\sqrt{-g}{d}^{4}x+S_m[g,\psi ]$ on a four-dimensional Lorentzian manifold, with $\Gamma$ a symmetric (torsion-free) connection varied independently of $g$. Then $(g,\Gamma )$ is a stationary point under independent compactly supported variations $\delta g^{\mu \nu }$, $\delta {\Gamma }_{\mu \nu }^{\lambda }$ if and only if (i) $\Gamma$ is the Levi-Civita connection of $g$, and (ii) $g$ satisfies the Einstein field equations $R_{\mu \nu }(g)-\frac{1}{2}R(g)g_{\mu \nu }=\kappa T_{\mu \nu }$.

Proof. The two variations are independent, so $\delta S=0$ for all variations is equivalent to the vanishing of the coefficient of $\delta g^{\mu \nu }$ and of $\delta {\Gamma }_{\mu \nu }^{\lambda }$ separately.

The metric variation. Holding $\Gamma$ fixed, only $g^{\mu \nu }$ and $\sqrt{-g}$ depend on $g$; the affine Ricci tensor $R_{\mu \nu }(\Gamma )$ is metric-independent, so ${\delta }_g{R}_{\mu \nu }(\Gamma )=0$. Using $\delta \sqrt{-g}=-\frac{1}{2}\sqrt{-g}{g}_{\mu \nu }\delta g^{\mu \nu }$ from 13.04.02,

$${\delta }_g{S}_{\mathrm{grav}}=\frac{1}{2\kappa }{\int }_M(R_{\mu \nu }(\Gamma )-\frac{1}{2}{g}_{\mu \nu }{g}^{\alpha \beta }{R}_{\alpha \beta }(\Gamma ))\delta g^{\mu \nu }\sqrt{-g}{d}^{4}x.$$

Adding ${\delta }_g{S}_m=-\frac{1}{2}\int T_{\mu \nu }\delta g^{\mu \nu }\sqrt{-g}{d}^{4}x$ and setting the coefficient to zero gives

$$R_{\mu \nu }(\Gamma )-\frac{1}{2}{g}_{\mu \nu }R(\Gamma )=\kappa T_{\mu \nu },R(\Gamma ):=g^{\alpha \beta }{R}_{\alpha \beta }(\Gamma ).\text{\hspace{1em}(∗)}$$

This is the Einstein equation written for the affine Ricci tensor of the still-undetermined $\Gamma$.

The connection variation. Holding $g$ fixed, only $R_{\mu \nu }(\Gamma )$ depends on $\Gamma$. The variation of the affine Ricci tensor is the Palatini identity for the independent connection,

$$\delta R_{\mu \nu }(\Gamma )={\nabla }_{\lambda }(\delta {\Gamma }_{\mu \nu }^{\lambda })-{\nabla }_{\nu }(\delta {\Gamma }_{\mu \lambda }^{\lambda }),$$

where $\nabla$ is the covariant derivative of $\Gamma$ and $\delta {\Gamma }_{\mu \nu }^{\lambda }$ is a $(1,2)$-tensor. Hence

$${\delta }_{\Gamma }{S}_{\mathrm{grav}}=\frac{1}{2\kappa }{\int }_M\sqrt{-g}{g}^{\mu \nu }({\nabla }_{\lambda }\delta {\Gamma }_{\mu \nu }^{\lambda }-{\nabla }_{\nu }\delta {\Gamma }_{\mu \lambda }^{\lambda })d^{4}x.$$

Integrate by parts. For a general connection $\nabla$ the covariant divergence does not reduce to $\frac{1}{\sqrt{-g}}{\partial }_{\lambda }(\sqrt{-g}\cdots )$, because ${\nabla }_{\lambda }\sqrt{-g}=-\frac{1}{2}\sqrt{-g}{g}^{\mu \nu }{\nabla }_{\lambda }{g}_{\mu \nu }$ need not vanish. Moving the derivatives onto $\sqrt{-g}{g}^{\mu \nu }$ and discarding the boundary terms (the variations are compactly supported) yields, after relabeling,

$${\delta }_{\Gamma }{S}_{\mathrm{grav}}=-\frac{1}{2\kappa }{\int }_M[{\nabla }_{\lambda }(\sqrt{-g}{g}^{\mu \nu })-{\delta }_{\lambda }^{\nu }{\nabla }_{\rho }(\sqrt{-g}{g}^{\mu \rho })]\delta {\Gamma }_{\mu \nu }^{\lambda }{d}^{4}x.$$

Stationarity for all $\delta {\Gamma }_{\mu \nu }^{\lambda }$ symmetric in $(\mu ,\nu )$ forces the symmetric part of the bracket to vanish. Tracing the bracket over $\lambda =\nu$ shows the second term is fixed by the first, and the resulting condition reduces to

$${\nabla }_{\lambda }\left(\sqrt{-g}{g}^{\mu \nu }\right)=0.\text{\hspace{1em}(∗∗)}$$

From ($\ast \ast$) to Levi-Civita. Expanding the weight, ${\nabla }_{\lambda }(\sqrt{-g}{g}^{\mu \nu })=\sqrt{-g}({\nabla }_{\lambda }{g}^{\mu \nu }+\frac{1}{2}{g}^{\mu \nu }{g}^{\alpha \beta }{\nabla }_{\lambda }{g}_{\alpha \beta })$ using ${\nabla }_{\lambda }\sqrt{-g}=\frac{1}{2}\sqrt{-g}{g}^{\alpha \beta }{\nabla }_{\lambda }{g}_{\alpha \beta }$. Setting this to zero and tracing over $\mu ,\nu$ gives $g_{\mu \nu }{\nabla }_{\lambda }{g}^{\mu \nu }\cdot (1+\frac{n}{2})=0$, which in $n=4$ forces $g^{\alpha \beta }{\nabla }_{\lambda }{g}_{\alpha \beta }=0$; substituting back leaves ${\nabla }_{\lambda }{g}^{\mu \nu }=0$, equivalently ${\nabla }_{\lambda }{g}_{\mu \nu }=0$. A torsion-free connection that annihilates the metric is, by uniqueness of the Levi-Civita connection 13.02.02, the Levi-Civita connection of $g$:

$${\Gamma }_{\mu \nu }^{\lambda }=\frac{1}{2}{g}^{\lambda \rho }({\partial }_{\mu }{g}_{\nu \rho }+{\partial }_{\nu }{g}_{\mu \rho }-{\partial }_{\rho }{g}_{\mu \nu }).$$

Closing the loop. With $\Gamma$ now the Levi-Civita connection, $R_{\mu \nu }(\Gamma )=R_{\mu \nu }(g)$ is the metric Ricci tensor, and ($\ast$) becomes the Einstein field equation $R_{\mu \nu }(g)-\frac{1}{2}R(g)g_{\mu \nu }=\kappa T_{\mu \nu }$. Conversely, if (i) and (ii) hold, both variations vanish and $(g,\Gamma )$ is stationary. $■$

Bridge. This result builds toward 13.05.01, where the Schwarzschild metric -- a vacuum solution of the Einstein equation reproduced here -- can be obtained by either variation, and it appears again in 13.07.01, where first-order (vielbein-and-connection) variables are the natural setting for linearizing gravity and for coupling spinor fields whose action sees the connection directly. The first-order viewpoint connects to 05.05.08 through the role of the connection as a gauge field whose own field equation, not an external postulate, fixes its relation to the metric. It also threads into 13.08.01, because the Palatini route is the standard arena for $f(R)$ modifications proposed as dark-energy models; the structural fact that metric compatibility is an output for the linear Lagrangian but not for $f(R)$ is exactly why those modifications carry distinct cosmological dynamics.

Exercises Intermediate+

Lean formalization Intermediate+

The first-order formulation sits one layer above the metric variation of 13.04.02 in formalization difficulty, and Mathlib does not reach either layer, so this unit ships with lean_status: none. The missing tower is:

1. Lorentzian metrics and the space of connections. Mathlib has positive-definite Riemannian metrics and affine connections on vector bundles, but not indefinite Lorentzian metrics as a first-class structure, and not the space of all connections on $TM$ as an affine space modeled on $(1,2)$-tensor fields. The Palatini variation lives precisely on the product of these two infinite-dimensional spaces.

2. Curvature of a general connection. The affine Ricci tensor $R_{\mu \nu }(\Gamma )$ of a connection that is not assumed metric-compatible, together with its nonmetricity tensor $Q_{\lambda \mu \nu }={\nabla }_{\lambda }{g}_{\mu \nu }$ and its torsion. Mathlib has neither nonmetricity nor the torsion of a general affine connection as named objects.

3. Independent variational derivatives. The functional $S[g,\Gamma ]$ has two independent Gateaux derivatives, $\delta S/\delta g^{\mu \nu }$ and $\delta S/\delta {\Gamma }_{\mu \nu }^{\lambda }$. Formalizing the connection variation requires the projective equivalence relation $\Gamma \sim \Gamma +{\delta }_{\mu }^{\lambda }{A}_{\nu }$ on connections, which has no Mathlib analogue.

A realistic first target, purely tensor-algebraic, is the implication ($\ast \ast$) $\Rightarrow$ metric compatibility: given a torsion-free connection with ${\nabla }_{\lambda }(\sqrt{-g}{g}^{\mu \nu })=0$, prove ${\nabla }_{\lambda }{g}_{\mu \nu }=0$ and hence that $\Gamma$ equals the Levi-Civita connection. This is a finite linear-algebra identity once the connection and metric are available as data, and it would be the clean entry point for a Mathlib contribution.

Advanced results Master

The first-order action needs no boundary term. The second-order metric action of 13.04.02 produced the Einstein tensor only after discarding a surface integral whose integrand contained normal derivatives of $\delta g_{\mu \nu }$; cancelling it required the Gibbons-Hawking-York term. The Palatini action behaves differently. Its Lagrangian density $g^{\mu \nu }{R}_{\mu \nu }(\Gamma )\sqrt{-g}$ is linear in first derivatives of the connection and contains no second derivatives of the metric. The boundary term produced by the connection variation is proportional to $\delta {\Gamma }_{\mu \nu }^{\lambda }$ on $\partial M$, and the natural Dirichlet datum of the first-order problem fixes $\Gamma$ on the boundary, so the surface term vanishes with no added counterterm. The variational problem is well-posed as it stands. This is one of the structural advantages of first-order gravity and is why the tetrad-connection (Palatini) action is the preferred starting point for canonical and loop quantizations, where the connection is the configuration variable.

Coupling to spinors. A second advantage is essential rather than cosmetic. Dirac fields cannot be coupled to the metric alone; the Dirac operator requires a connection acting on a spin bundle, and the spin connection is built from a tetrad and an independent connection. In the second-order formulation one must vary a metric and then reconstruct the spin connection from it, which generates four-fermion contact terms only after the fact. In the first-order (Einstein-Cartan-Palatini) formulation the connection is already present as an independent field, the spinor action depends on it directly, and varying the connection yields an algebraic equation whose solution feeds the torsion sourced by the spin current. The first-order action is the natural home for fermionic matter precisely because the object the spinor couples to is a field, not a derived quantity.

The projective structure. When the torsion-free restriction is lifted, the connection variation of the linear action determines $\Gamma$ only up to the projective transformation ${\Gamma }_{\mu \nu }^{\lambda }\mapsto {\Gamma }_{\mu \nu }^{\lambda }+{\delta }_{\mu }^{\lambda }{A}_{\nu }$, under which the contracted curvature $g^{\mu \nu }{R}_{\mu \nu }(\Gamma )$ is invariant. The residual covector $A_{\nu }$ is pure gauge for the Einstein-Hilbert Lagrangian; fixing $A_{\nu }=0$, or equivalently imposing vanishing torsion, selects the Levi-Civita representative. The projective invariance is a genuine symmetry of the first-order action and is the metric-affine shadow of the diffeomorphism-Noether identity of 05.05.08; it explains why the connection equation, which is forty equations for forty components, nonetheless leaves a four-parameter family of solutions before the gauge is fixed.

The f(R) inequivalence as a theorem. For a nonlinear Lagrangian $f(R(\Gamma ))$ the connection equation becomes ${\nabla }_{\lambda }(\sqrt{-g}{f}^{\prime }(R)g^{\mu \nu })=0$, so $\Gamma$ is the Levi-Civita connection not of $g$ but of the conformal metric $\tilde{g}=f^{\prime }(R)g$. The metric field equation, traced, fixes $R$ algebraically through $f^{\prime }(R)R-2f(R)=\kappa T$. The resulting dynamics differ from metric $f(R)$ gravity: there is no propagating scalar, the field equations are second-order in $g$, and the vacuum theory ($T=0$) collapses to Einstein gravity with a cosmological constant determined by the roots of the structural equation. The two $f(R)$ theories share a Lagrangian symbol and almost nothing else ^{[Sotiriou-Faraoni 2010]}.

Synthesis. The Palatini formulation reorganizes the variational derivation of gravity around the independence of metric and connection, and four facts follow from that single move. First, metric compatibility is demoted from postulate to field equation: the connection variation produces ${\nabla }_{\lambda }(\sqrt{-g}{g}^{\mu \nu })=0$, whose unique torsion-free solution is the Levi-Civita connection, so the structure that the metric formulation assumes is here derived. Second, the metric variation reproduces the Einstein field equations of 13.04.01, and once the connection is reconstructed the two formulations coincide on vacuum and on minimally coupled matter, which is the equivalence theorem. Third, the first-order action is well-posed without a boundary term and couples spinors directly, making it the natural starting point for fermionic matter and for canonical quantization. Fourth, the equivalence is fragile under nonlinearity: Palatini $f(R)$ and metric $f(R)$ are distinct theories, the difference traceable to the conformal factor $f^{\prime }(R)$ that relabels which metric the connection is compatible with. These four facts -- emergent metricity, on-shell equivalence, boundary-term economy, and $f(R)$ inequivalence -- are the content of treating the connection as a field.

Full proof set Master

Proposition (metricity from the connection equation). Let $\nabla$ be a symmetric connection on a four-dimensional manifold with metric $g$, and suppose ${\nabla }_{\lambda }(\sqrt{-g}{g}^{\mu \nu })=0$ for all indices. Then ${\nabla }_{\lambda }{g}_{\mu \nu }=0$, and consequently $\nabla$ is the Levi-Civita connection of $g$.

Proof. Write the weight derivative via Jacobi's formula for the derivation $\nabla$: ${\nabla }_{\lambda }\sqrt{-g}=\frac{1}{2}\sqrt{-g}{g}^{\alpha \beta }{\nabla }_{\lambda }{g}_{\alpha \beta }$. Expanding the hypothesis,

$$0={\nabla }_{\lambda }(\sqrt{-g}{g}^{\mu \nu })=\sqrt{-g}({\nabla }_{\lambda }{g}^{\mu \nu }+\frac{1}{2}{g}^{\mu \nu }{g}^{\alpha \beta }{\nabla }_{\lambda }{g}_{\alpha \beta }).$$

Since $\sqrt{-g}\ne 0$, the bracket vanishes. Contract it with $g_{\mu \nu }$ and use $g_{\mu \nu }{\nabla }_{\lambda }{g}^{\mu \nu }=-g^{\alpha \beta }{\nabla }_{\lambda }{g}_{\alpha \beta }$ (from ${\nabla }_{\lambda }(g^{\mu \nu }{g}_{\mu \nu })={\nabla }_{\lambda }(n)=0$):

$$0=-g^{\alpha \beta }{\nabla }_{\lambda }{g}_{\alpha \beta }+\frac{n}{2}{g}^{\alpha \beta }{\nabla }_{\lambda }{g}_{\alpha \beta }=(\frac{n}{2}-1)g^{\alpha \beta }{\nabla }_{\lambda }{g}_{\alpha \beta }.$$

For $n=4$ the coefficient is $1\ne 0$, so $g^{\alpha \beta }{\nabla }_{\lambda }{g}_{\alpha \beta }=0$. Substituting back into the vanishing bracket leaves ${\nabla }_{\lambda }{g}^{\mu \nu }=0$, equivalently ${\nabla }_{\lambda }{g}_{\mu \nu }=0$ (lower the indices with the now-parallel metric). A connection that is both torsion-free and metric-compatible is unique and equals the Levi-Civita connection of $g$ by the fundamental theorem of (pseudo-)Riemannian geometry 13.02.02. $■$

Proposition (projective invariance of the linear action). Under ${\Gamma }_{\mu \nu }^{\lambda }\mapsto {\Gamma }_{\mu \nu }^{\lambda }+{\delta }_{\mu }^{\lambda }{A}_{\nu }$ for any covector field $A_{\nu }$, the Palatini action $S[g,\Gamma ]=\frac{1}{2\kappa }\int g^{\mu \nu }{R}_{\mu \nu }(\Gamma )\sqrt{-g}{d}^{4}x$ is invariant.

Proof. Insert the shifted connection into $R_{\mu \nu }(\Gamma )={\partial }_{\lambda }{\Gamma }_{\mu \nu }^{\lambda }-{\partial }_{\nu }{\Gamma }_{\mu \lambda }^{\lambda }+{\Gamma }_{\lambda \rho }^{\lambda }{\Gamma }_{\mu \nu }^{\rho }-{\Gamma }_{\nu \rho }^{\lambda }{\Gamma }_{\mu \lambda }^{\rho }$. The shift of each term, keeping first order in $A$ and then verifying the second order cancels: from ${\partial }_{\lambda }{\Gamma }_{\mu \nu }^{\lambda }$ comes ${\partial }_{\lambda }({\delta }_{\mu }^{\lambda }{A}_{\nu })={\partial }_{\mu }{A}_{\nu }$; from $-{\partial }_{\nu }{\Gamma }_{\mu \lambda }^{\lambda }$ comes $-{\partial }_{\nu }({\delta }_{\mu }^{\lambda }{A}_{\lambda })=-{\partial }_{\nu }{A}_{\mu }$. The quadratic terms contribute $A$-dependent pieces that, using ${\delta }_{\lambda }^{\lambda }=n$ and the contraction patterns, cancel in pairs, and the $A\cdot A$ terms cancel by symmetry. The net change is $\Delta R_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$, antisymmetric in $\mu ,\nu$. Contracting with the symmetric inverse metric, $g^{\mu \nu }\Delta R_{\mu \nu }=0$. Hence the integrand, and so the action, is unchanged. $■$

Proposition (structural equation of Palatini f(R)). For the action $\frac{1}{2\kappa }\int f(R(\Gamma ))\sqrt{-g}{d}^{4}x+S_m$, with $R(\Gamma )=g^{\mu \nu }{R}_{\mu \nu }(\Gamma )$, the metric field equation is $f^{\prime }(R)R_{\mu \nu }(\Gamma )-\frac{1}{2}f(R)g_{\mu \nu }=\kappa T_{\mu \nu }$, whose trace gives the algebraic relation $f^{\prime }(R)R-2f(R)=\kappa T$ with $T=g^{\mu \nu }{T}_{\mu \nu }$.

Proof. Varying with respect to $g^{\mu \nu }$ at fixed $\Gamma$: ${\delta }_g(f(R)\sqrt{-g})=f^{\prime }(R){\delta }_{g}R\sqrt{-g}+f(R)\delta \sqrt{-g}$. Here ${\delta }_{g}R={\delta }_g(g^{\mu \nu })R_{\mu \nu }(\Gamma )=R_{\mu \nu }(\Gamma )\delta g^{\mu \nu }$, since $R_{\mu \nu }(\Gamma )$ is metric-independent. With $\delta \sqrt{-g}=-\frac{1}{2}\sqrt{-g}{g}_{\mu \nu }\delta g^{\mu \nu }$,

$${\delta }_g(f(R)\sqrt{-g})=(f^{\prime }(R)R_{\mu \nu }(\Gamma )-\frac{1}{2}f(R)g_{\mu \nu })\sqrt{-g}\delta g^{\mu \nu }.$$

Adding ${\delta }_g{S}_m=-\frac{1}{2}\int T_{\mu \nu }\delta g^{\mu \nu }\sqrt{-g}{d}^{4}x$ and requiring stationarity yields $f^{\prime }(R)R_{\mu \nu }(\Gamma )-\frac{1}{2}f(R)g_{\mu \nu }=\kappa T_{\mu \nu }$. Contracting with $g^{\mu \nu }$ and using $g^{\mu \nu }{R}_{\mu \nu }(\Gamma )=R$ and $g^{\mu \nu }{g}_{\mu \nu }=4$ gives $f^{\prime }(R)R-2f(R)=\kappa T$. The left side is an algebraic function of $R$, so $R$ is determined pointwise by $T$ with no derivatives, which is the absence of a propagating scalar. $■$

Connections Master

The metric field equation reproduced by the Palatini variation is identical to the Einstein field equations posited in 13.04.01; the first-order route derives them without assuming metric compatibility, so the postulate that the connection is Levi-Civita is replaced by a field equation that yields it.

The second-order metric derivation of 13.04.02 is the companion formulation; the two agree on the vacuum and minimally coupled equations, but the first-order action dispenses with the Gibbons-Hawking-York boundary term that the metric action requires, which is the cleanest practical contrast between the two routes.

The projective invariance of the connection equation is the metric-affine instance of the diffeomorphism-Noether identity of 05.05.08; both express a local gauge redundancy that forces the field equations to leave a residual gauge family before a condition such as vanishing torsion fixes the representative.

Palatini $f(R)$ gravity, shown here to differ from metric $f(R)$ gravity, supplies modified-gravity models for the accelerated expansion analyzed through the Friedmann equations in 13.08.01; the algebraic structural equation for $R$ controls whether such a model passes solar-system and cosmological tests.

The first-order, connection-as-field viewpoint is the setting in which the linearized and tetrad formulations of 13.07.01 are most natural, since spinor and gauge matter couple to the connection directly rather than through a metric-derived object.

Historical & philosophical context Master

The first-order variational method is conventionally named after Attilio Palatini, whose 1919 paper in the Rendiconti del Circolo Matematico di Palermo derived the gravitational field equations by an invariant variational argument ^{[Palatini 1919]}. The attribution is, on closer reading, imprecise. Palatini's 1919 work varied the metric and used an identity for the variation of the Ricci tensor -- the identity now bearing his name within the metric derivation of 13.04.02 -- rather than treating the connection as a wholly independent field. The genuinely metric-affine version, in which metric and connection are varied independently and metric compatibility emerges as a field equation, was introduced by Einstein in 1925 in the course of his unified-field-theory program ^{[Einstein 1925]}. The careful historical reconstruction by Ferraris, Francaviglia, and Reina traced this lineage and showed that the method universally called "Palatini's" was in fact first deployed by Einstein, six years after Palatini's paper ^{[Ferraris-Francaviglia-Reina 1982]}.

Sternberg's treatment situates the first-order principle within the geometry of connections on principal bundles, where the independence of metric and connection is the natural structure rather than a device, and where the projective ambiguity appears as the kernel of the contraction map on curvature ^{[Sternberg Ch. 20 §3]}. The modern revival of the formulation is driven by two developments: the recognition that fermionic matter requires an independent connection, which makes the first-order action the physically primary one; and the rise of Palatini $f(R)$ gravity as a modified-gravity candidate, whose inequivalence to metric $f(R)$ gravity was systematized by Sotiriou and Faraoni ^{[Sotiriou-Faraoni 2010]}.

Bibliography Master

@article{Palatini1919,
  author  = {Palatini, Attilio},
  title   = {Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton},
  journal = {Rendiconti del Circolo Matematico di Palermo},
  volume  = {43},
  year    = {1919},
  pages   = {203--212}
}

@article{Einstein1925,
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  year    = {1925},
  pages   = {414--419}
}

@article{FerrarisFrancavigliaReina1982,
  author  = {Ferraris, Marco and Francaviglia, Mauro and Reina, Cesare},
  title   = {Variational formulation of general relativity from 1915 to 1925: ``Palatini's method'' discovered by Einstein in 1925},
  journal = {General Relativity and Gravitation},
  volume  = {14},
  number  = {3},
  year    = {1982},
  pages   = {243--254}
}

@article{SotiriouFaraoni2010,
  author  = {Sotiriou, Thomas P. and Faraoni, Valerio},
  title   = {f(R) theories of gravity},
  journal = {Reviews of Modern Physics},
  volume  = {82},
  number  = {1},
  year    = {2010},
  pages   = {451--497}
}

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  author    = {Wald, Robert M.},
  title     = {General Relativity},
  publisher = {University of Chicago Press},
  year      = {1984}
}

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}

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}