The Einstein field equations can be obtained from a small number of postulates about the structure of spacetime together with a variational principle. This article carries out that derivation in full. Sec. 0 fixes conventions and defines the objects used throughout. Sec. 1 states the six postulates. Sec. 2 constructs the connection and the curvature tensors that follow from them, and Sec. 3 determines the admissible form of the action. The variation is performed in Sec. 4 and the coupling constant is fixed by the Newtonian limit in Sec. 5. Sec. 6 gives equivalent forms of the equations and Sec. 7 verifies internal consistency. Sec. 8 collects every assumption made and Sec. 9 surveys the theories obtained by relaxing them.

Equations are numbered by section and referred to by number. Familiarity with special relativity and multivariable calculus is assumed; no differential geometry is presupposed.

0. Conventions and definitions

The following conventions are used without further comment. Roughly half of the signs appearing below depend on them.

• Signature. The metric has signature $(-,+,+,+)$, so that timelike intervals satisfy $ds^2 \lt 0$.
• Indices. Greek indices $\mu,\nu,\rho,\dots$ range over $0,1,2,3$, with $x^0 = ct$. Latin indices $i,j,k$ range over the spatial values $1,2,3$.
• Summation. An index appearing once as a superscript and once as a subscript within a term is summed over its range: $A^\mu B_\mu \equiv \sum_{\mu=0}^{3} A^\mu B_\mu$.
• Curvature sign. The Riemann tensor is defined by $[\nabla_\mu, \nabla_\nu] V^\rho = R^\rho_{\ \sigma\mu\nu} V^\sigma$, the convention of Misner, Thorne and Wheeler.
• Symmetrization. $A_{(\mu\nu)} = \tfrac{1}{2}(A_{\mu\nu} + A_{\nu\mu})$ and $A_{[\mu\nu]} = \tfrac{1}{2}(A_{\mu\nu} - A_{\nu\mu})$.
• Units. The constants $c$ and $G$ are retained explicitly, since the object of Sec. 5 is to determine the combination in which they appear.

A tensor of type $(p,q)$ at a point is an object with $p$ contravariant and $q$ covariant components transforming under a change of coordinates $x^\mu \to x'^\mu$ as

Two consequences of (0.1) are used repeatedly. First, an equality between tensors that holds in one coordinate system holds in all of them. Second, a tensor whose components vanish at a point in one coordinate system vanishes at that point in every coordinate system. The second is applied four times in Sec. 2 and Sec. 4 to promote a relation established in a convenient frame to a covariant statement.

1. The postulates

P1. Spacetime is a four-dimensional smooth manifold $M$. Each point admits a neighborhood homeomorphic to an open subset of $\mathbb{R}^4$, with smooth transition functions between overlapping charts. The dimension is an input to the theory.

P2. $M$ carries a Lorentzian metric $g_{\mu\nu}$, and the metric determines the readings of ideal clocks and rods. The metric is a symmetric non-degenerate tensor field of type $(0,2)$ and signature $(-,+,+,+)$. It determines the line element

and the proper time elapsed along a timelike worldline $\gamma$ is

Equation (1.2) supplies the operational content of the metric. Non-degeneracy guarantees the existence of the inverse metric $g^{\mu\nu}$ satisfying $g^{\mu\alpha}g_{\alpha\nu} = \delta^\mu_\nu$, by means of which indices are raised and lowered.

P3. General covariance. The laws of physics take the same form in all coordinate systems; equivalently, the action is a scalar functional invariant under diffeomorphisms $x^\mu \to x^\mu + \xi^\mu(x)$. Coordinates are labels and carry no physical content.

P4. The Einstein equivalence principle. At every point $p \in M$ there exist coordinates, called locally inertial or normal coordinates at $p$, in which

and in which all non-gravitational physics reduces to special relativity. A freely falling observer therefore cannot detect a gravitational field by any local experiment. Two consequences are used below: matter couples to $g_{\mu\nu}$ and to no other gravitational field (minimal coupling, Sec. 3.4), and the connection is metric-compatible (Sec. 2.2).

P5. The field equations follow from a stationary action. There exists a functional $S[g,\psi]$ of the metric and the matter fields $\psi$ whose stationary points under variations vanishing on the boundary are the physical configurations. The metric is the only gravitational field.

P6. Locality and second order. The Lagrangian density is a local scalar constructed from $g_{\mu\nu}$ and finitely many of its derivatives, and the field equations contain derivatives of $g_{\mu\nu}$ of order no higher than two.

P6 requires comment, as it is the postulate least often stated explicitly and the one most frequently relaxed. Second-order equations render the initial-value problem well posed in the standard form: the metric and its first time derivative on a spacelike hypersurface determine the solution. Field equations of higher order generically possess an Ostrogradsky instability, that is, a Hamiltonian unbounded from below.¹ The theories of Sec. 9.1 accept this cost in exchange for improved ultraviolet behavior.

2. Consequences of the postulates

2.1 The covariant derivative

The partial derivative of a vector field is not a tensor. Differentiating (0.1) for a type $(1,0)$ tensor produces a term proportional to $\partial^2 x' / \partial x \partial x$, which spoils the transformation law. One therefore introduces the covariant derivative

where the connection coefficients $\Gamma^\lambda_{\mu\nu}$ are defined to transform inhomogeneously in precisely the manner that cancels the unwanted term. They are consequently not the components of a tensor. On scalars $\nabla_\mu f = \partial_\mu f$, and (2.1) extends to tensors of arbitrary type by the Leibniz rule, with one $+\Gamma$ term for each contravariant index and one $-\Gamma$ term for each covariant index.

A vector field $V^\mu$ is parallel-transported along a curve with tangent $u^\mu$ if $u^\nu \nabla_\nu V^\mu = 0$. A geodesic is a curve that parallel-transports its own tangent, $u^\nu \nabla_\nu u^\mu = 0$; in coordinates,

Equation (2.2) also follows from extremizing the proper time (1.2) of a free test particle, whose action is $S_{\rm pt} = -mc^2\!\int d\tau$. It is used in Sec. 5.1 to identify the Newtonian potential.

2.2 Determination of the connection

Two conditions determine $\Gamma^\lambda_{\mu\nu}$ uniquely.

Lemma 2.1 (Metric compatibility). P4 implies $\nabla_\lambda g_{\mu\nu} = 0$.

Proof. In normal coordinates at a point $p$ the first derivatives of the metric vanish by (1.3). The connection coefficients are constructed from those first derivatives, so they too vanish at $p$, and hence $\nabla_\lambda g_{\mu\nu}(p) = \partial_\lambda g_{\mu\nu}(p) = 0$ in this frame. Since $\nabla_\lambda g_{\mu\nu}$ is a tensor, its vanishing at $p$ in one frame implies its vanishing at $p$ in every frame. The point $p$ was arbitrary. $\blacksquare$

Geometrically, Lemma 2.1 states that lengths and angles are preserved under parallel transport, as (1.2) requires if the metric is to be measured by transported clocks and rods.

The second condition is the vanishing of the torsion tensor $T^\lambda_{\ \mu\nu} = 2\Gamma^\lambda_{[\mu\nu]}$, equivalently $\Gamma^\lambda_{\mu\nu} = \Gamma^\lambda_{\nu\mu}$. The antisymmetric part of the connection is a tensor, and P1–P6 do not require it to vanish; its vanishing is an independent assumption, recorded in Sec. 8 and relaxed in Sec. 9.4.

Proposition 2.2 (Levi-Civita). A torsion-free metric-compatible connection exists and is unique, with coefficients

Proof. Expanding $\nabla_\rho g_{\mu\nu} = 0$ and permuting the free indices cyclically gives

Adding the second and third equations of (2.4) and subtracting the first, six of the eight terms on the right cancel in pairs by the symmetry $\Gamma^\lambda_{\mu\nu} = \Gamma^\lambda_{\nu\mu}$, leaving $2\Gamma^\lambda_{\mu\nu}g_{\lambda\rho}$. Contraction with $g^{\rho\sigma}$ gives (2.3). Uniqueness follows because the construction inverts the two conditions with no freedom remaining. $\blacksquare$

The geometry is therefore completely determined by the metric.

2.3 The curvature tensor

Since $\Gamma$ vanishes at any preassigned point in normal coordinates, no tensor constructed from the first derivatives of the metric alone can represent the gravitational field. The invariant content lies in the second derivatives, and is expressed by the failure of covariant derivatives to commute. Evaluating $[\nabla_\mu,\nabla_\nu]V^\rho$ from (2.1), the terms containing derivatives of $V^\rho$ cancel by the symmetry of $\Gamma$, and the remainder is algebraic in $V^\rho$:

with

The left-hand side of (2.5) is a tensor for arbitrary $V^\sigma$, and therefore so is $R^\rho_{\ \sigma\mu\nu}$, notwithstanding its construction from the non-tensorial quantities (2.3). This is the Riemann curvature tensor. It admits an equivalent characterization as the change in a vector parallel-transported around an infinitesimal closed loop, and $R^\rho_{\ \sigma\mu\nu} = 0$ throughout a region is the necessary and sufficient condition for that region to be flat.

With the first index lowered, $R_{\rho\sigma\mu\nu} = g_{\rho\lambda}R^\lambda_{\ \sigma\mu\nu}$, the tensor obeys

together with the first Bianchi identity $R_{\rho[\sigma\mu\nu]} = 0$. In $n$ dimensions the relations (2.7) leave $n^2(n^2-1)/12$ algebraically independent components, which for $n = 4$ is twenty, as against the ten components of the metric.

2.4 Contractions

The symmetries (2.7) admit essentially one contraction to a tensor of rank two. Contracting the first and third indices defines the Ricci tensor

which is symmetric by the pair-exchange symmetry in (2.7). Contraction of the first with the second, or of the third with the fourth, vanishes by antisymmetry; contraction of the first with the fourth yields $-R_{\mu\nu}$. A further contraction defines the Ricci scalar

By the same argument, $R$ is the only scalar constructible from the metric and its first two derivatives that is linear in the second derivatives. This fact determines the form of the action in Sec. 3.2.

2.5 The Bianchi identities

Differentiating (2.6) in normal coordinates at a point, where $\Gamma$ vanishes and the quadratic terms do not contribute, and antisymmetrizing, gives the second Bianchi identity

Both sides are tensors, so (2.10) holds in every frame and at every point. It is an identity of the geometry, valid for an arbitrary metric, and independent of any field equation.

Contracting $\rho$ with $\mu$ in (2.10) and using $R^\mu_{\ \sigma\nu\mu} = -R_{\sigma\nu}$ in the third term,

Contracting (2.11) with $g^{\sigma\nu}$, the first term gives $\nabla_\lambda R$, the second $-\nabla^\sigma R_{\sigma\lambda}$, and the third, on applying the symmetries (2.7), a further $-\nabla^\mu R_{\mu\lambda}$. Hence the contracted Bianchi identity

Defining the Einstein tensor

identity (2.12) is equivalent to $\nabla^\mu G_{\mu\nu} = 0$. The tensor (2.13) is thus symmetric, constructed from the metric and its first two derivatives, linear in the second derivatives, and identically divergence-free. These four properties characterize it, by the theorem quoted in Sec. 3.2.

3. The action

3.1 The invariant volume element

The coordinate volume $d^4x$ is not invariant, acquiring the Jacobian determinant under a change of coordinates. Writing $g \equiv \det(g_{\mu\nu})$, which transforms with the square of the inverse Jacobian and is negative for Lorentzian signature, the combination

is invariant. By P3 every admissible action is therefore of the form $\int d^4x\,\sqrt{-g}\,\mathcal{L}$ with $\mathcal{L}$ a scalar.

3.2 The admissible scalars

By P6 the scalar $\mathcal{L}$ is constructed from the metric and its derivatives and must yield field equations of second order. The candidates are enumerated as follows.

1. A constant is admissible and cannot be excluded. It contributes the cosmological term.
2. No non-constant scalar can be built from $g_{\mu\nu}$ and $\partial_\lambda g_{\mu\nu}$ alone, since normal coordinates set the first derivatives to zero at any preassigned point, whereas a scalar cannot vanish at a point in one frame and not in another.
3. At second order in derivatives the only available tensor is the Riemann tensor, and by Sec. 2.4 the only scalar linear in it is $R$. Its variation yields second-order equations.
4. Scalars quadratic in the curvature, such as $R^2$, $R_{\mu\nu}R^{\mu\nu}$ and $R_{\rho\sigma\mu\nu}R^{\rho\sigma\mu\nu}$, are permitted by P1–P5 but yield fourth-order equations and are excluded by P6. The Gauss-Bonnet combination $\mathcal{G} = R^2 - 4R_{\mu\nu}R^{\mu\nu} + R_{\rho\sigma\mu\nu}R^{\rho\sigma\mu\nu}$ is an exception, being in four dimensions a topological density whose variation vanishes identically; it contributes nothing here and is dynamically relevant only in five or more dimensions.

The enumeration is made precise by the following result, which will not be proved here.

Theorem 3.1 (Lovelock, 1971). In four dimensions, the most general symmetric, divergence-free tensor of rank two constructed from the metric and its first two derivatives is

for constants $a$ and $b$.

Given P1–P6, Theorem 3.1 exhausts the possible field equations, leaving two constants to be fixed: $a$ by the Newtonian limit and $b$ by observation.

3.3 The Einstein-Hilbert action

The constant $\kappa$ is left undetermined; no argument internal to the derivation fixes it, and its value is obtained in Sec. 5 by comparison with Newtonian gravity. The constant $\Lambda$ is the cosmological constant.

3.4 The matter action

By P4 matter couples to the metric alone. The corresponding prescription, minimal coupling, is to take the Lagrangian of a field in special relativity and replace $\eta_{\mu\nu} \to g_{\mu\nu}$, $\partial_\mu \to \nabla_\mu$, and $d^4x \to d^4x\sqrt{-g}$:

No term coupling the curvature directly to matter, such as $R\phi^2$, is included. P1–P6 do not strictly forbid such a term; the equivalence principle excludes it only in the limit in which the radius of curvature is large compared with the scale of the experiment. This is an assumption, recorded in Sec. 8.

The total action is

and by P5 the field equations are $\delta S / \delta g^{\mu\nu} = 0$.

4. Variation of the action

The variation is taken with respect to the inverse metric $g^{\mu\nu}$; variation with respect to $g_{\mu\nu}$ yields the same equations with an overall change of sign.

4.1 Variation of the inverse metric

Varying $g^{\mu\alpha}g_{\alpha\nu} = \delta^\mu_\nu$, whose right-hand side is constant,

Contracting (4.1) with the metric gives the relation

used at several points below.

4.2 Variation of the volume element

For an invertible matrix $M$ the identity $\ln\det M = \mathrm{tr}\ln M$ gives on differentiation $\delta \det M = \det M\;\mathrm{tr}(M^{-1}\delta M)$, known as Jacobi's formula. Applied to the metric,

whence

and, on substituting (4.2),

The term $-\tfrac{1}{2}g_{\mu\nu}R$ in the Einstein tensor originates entirely in (4.5).

4.3 Variation of the connection

Although $\Gamma^\lambda_{\mu\nu}$ is not a tensor, the difference of two connections is, the inhomogeneous parts of their transformation laws being identical. The variation $\delta\Gamma^\lambda_{\mu\nu}$ is therefore a tensor and admits a covariant expression. Evaluating it in normal coordinates at a point, where $\Gamma = 0$ and covariant derivatives reduce to partial derivatives,

Both sides of (4.6) are tensors agreeing at an arbitrary point in one frame, and therefore agree everywhere in every frame.

4.4 The Palatini identity

Varying (2.6) in normal coordinates, where the quadratic terms do not contribute at the point, gives $\delta R^\rho_{\ \sigma\mu\nu} = \partial_\mu \delta\Gamma^\rho_{\nu\sigma} - \partial_\nu \delta\Gamma^\rho_{\mu\sigma}$. Every object in this relation is now a tensor, so it may be written covariantly:

Contracting $\rho$ with $\mu$,

Equation (4.8) expresses the variation of the Ricci tensor, which contains second derivatives of $\delta g_{\mu\nu}$, as a difference of covariant divergences.

4.5 The boundary term

From (2.9),

In the second term $g^{\mu\nu}$ may be taken through the covariant derivatives in (4.8), by Lemma 2.1, giving

or equivalently $v^\lambda = g_{\mu\nu}\nabla^\lambda \delta g^{\mu\nu} - \nabla_\mu \delta g^{\lambda\mu}$. For any vector field, $\sqrt{-g}\,\nabla_\lambda v^\lambda = \partial_\lambda(\sqrt{-g}\,v^\lambda)$, so by the divergence theorem the contribution of (4.10) to the variation is a surface integral,

The vanishing of (4.11) requires both $\delta g^{\mu\nu}$ and its normal derivative to vanish on $\partial M$, which over-determines the variational problem. The standard remedy is to add to (3.3) the Gibbons-Hawking-York surface term

in which $h$ is the induced metric on $\partial M$ and $K$ the trace of its extrinsic curvature; the variation of (4.12) cancels the normal-derivative contribution exactly.² For variations of compact support the surface term is absent, and it is omitted in what follows.

4.6 The gravitational contribution

Combining (4.5), (4.9) and (4.10), and varying the cosmological term with (4.5),

The cosmological term enters with $+g_{\mu\nu}$ because $\delta\!\left(\sqrt{-g}\,(-2\Lambda)\right) = +\Lambda\sqrt{-g}\,g_{\mu\nu}\,\delta g^{\mu\nu}$.

4.7 The energy-momentum tensor

The energy-momentum tensor is defined as the response of the matter action to a variation of the metric,

and is symmetric by construction. Definition (4.14) is justified by its agreement with the expressions obtained in special relativity. As an illustration, consider a real scalar field with

whose only metric dependence is the explicit inverse metric and the volume element. Then

so that $T_{\mu\nu} = \partial_\mu\phi\,\partial_\nu\phi + g_{\mu\nu}\mathcal{L}_M$. In flat space with $c = 1$ its time-time component is $\dot\phi^2 - \left(-\tfrac{1}{2}\dot\phi^2 + V\right) = \tfrac{1}{2}\dot\phi^2 + V$, the sum of the kinetic and potential energy densities, in agreement with the special-relativistic result.

4.8 The field equations

Requiring $\delta S = \delta S_{EH} + \delta S_M = 0$ for arbitrary $\delta g^{\mu\nu}$ annihilates the integrand of (4.13) and (4.14) pointwise,

that is,

These are ten coupled nonlinear partial differential equations of second order for the ten independent components of $g_{\mu\nu}$.

5. The Newtonian limit and the value of $\kappa$

The constant $\kappa$ is determined by requiring that (4.18) reproduce Newtonian gravitation in the regime in which the latter is valid. That regime is defined by three approximations: the field is weak, $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ with $|h_{\mu\nu}| \ll 1$ and all terms beyond first order in $h_{\mu\nu}$ neglected; the field is static, $\partial_0 h_{\mu\nu} = 0$; and the sources move slowly, $|dx^i/dt| \ll c$. The cosmological term is negligible at the relevant scales and is set to zero.

Step 1. Identification of the potential. For a slowly moving particle $dx^0/d\tau \approx c$ while the spatial components of $dx^\mu/d\tau$ are negligible, so that (2.2) reduces to

For a static field, (2.3) gives to first order

Comparison of (5.1) and (5.2) with the Newtonian equation of motion $d^2x^i/dt^2 = -\partial_i\Phi$ yields

The Newtonian potential is thus a component of the metric; gravitational time dilation is the same statement applied to (1.2).

Step 2. Linearized curvature. To first order in $h_{\mu\nu}$ the quadratic terms of (2.6) are of second order and may be dropped, so that $R_{\mu\nu} \approx \partial_\lambda\Gamma^\lambda_{\mu\nu} - \partial_\nu\Gamma^\lambda_{\lambda\mu}$. For a static field the second term contributes nothing to the time-time component, and with (5.2),

Step 3. The source. For pressureless dust of rest-mass density $\rho$ at rest, $T_{\mu\nu} = \rho\,u_\mu u_\nu$ with $u^\mu \approx (c,0,0,0)$, giving $T_{00} \approx \rho c^2$ and $T = g^{\mu\nu}T_{\mu\nu} \approx -\rho c^2$.

Step 4. Comparison. The time-time component of the trace-reversed equation (6.1) below, with $\Lambda = 0$, reads

so that $\nabla^2\Phi = \tfrac{1}{2}\kappa\rho c^4$. Comparison with Poisson's equation $\nabla^2 \Phi = 4\pi G\rho$ gives

The factor $8\pi$ is a consequence of the factor $4\pi$ in Poisson's equation, which in turn follows from the normalization of Newton's law of gravitation. With (5.6), the field equations (4.18) take their standard form

6. Equivalent forms

6.1 Trace-reversed form

Contracting (5.7) with $g^{\mu\nu}$ and using $g^{\mu\nu}g_{\mu\nu} = \delta^\mu_\mu = 4$ gives $-R + 4\Lambda = \kappa T$, whence $R = 4\Lambda - \kappa T$. Substituting back,

For $\Lambda = 0$ in the absence of matter this reduces to the vacuum field equations

A solution of (6.2) is Ricci-flat but need not be flat. Of the twenty independent components of the Riemann tensor, (6.2) constrains only the ten contained in the Ricci tensor; the remaining ten, comprising the Weyl tensor, are unconstrained and propagate. Gravitational radiation in vacuum is a solution of (6.2).

6.2 The cosmological term as vacuum energy

Transposing the cosmological term in (5.7) to the right-hand side,

which is the stress tensor of a perfect fluid of energy density $\rho_\Lambda c^2 = \Lambda c^4/8\pi G$ and pressure $p = -\rho_\Lambda c^2$. The field equations do not distinguish between a geometrical constant and a vacuum energy density. The discrepancy between the value of $\Lambda$ inferred from cosmological observation and the vacuum energy estimated from quantum field theory, which exceeds it by many tens of orders of magnitude, is the cosmological constant problem and remains unresolved.

7. Consistency conditions

7.1 Conservation of energy and momentum

Taking the divergence of (5.7) and applying the contracted Bianchi identity (2.12), which annihilates the left-hand side identically, gives

Local conservation of energy and momentum is therefore not an independent postulate but a consequence of the field equations; a matter model violating (7.1) is inconsistent with them.

The same conclusion follows from P3 alone, independently of the field equations. Under an infinitesimal diffeomorphism generated by $\xi^\mu$ the metric changes by its Lie derivative,

General covariance requires $S_M$ to be invariant, so that for matter fields satisfying their own equations of motion, using (4.14), (7.2) and the symmetry of $T_{\mu\nu}$, and integrating by parts,

Since $\xi^\nu$ is arbitrary, (7.1) follows. Equation (7.1) is thus the Noether identity associated with general covariance, of which the contracted Bianchi identity is the gravitational counterpart.

7.2 Counting of degrees of freedom

Equations (5.7) number ten, as do the components of the metric, but the four contracted Bianchi identities imply that four combinations of them contain no second time derivatives. These four are constraints on the initial data rather than evolution equations. Independently, P3 implies that any solution may be transformed by a diffeomorphism, corresponding to four arbitrary functions of gauge. Subtracting both from the ten components leaves two propagating degrees of freedom, corresponding to the two polarization states of a gravitational wave.

8. Summary of assumptions

The following table lists every assumption employed above, the point at which it enters, and the theories obtained by relaxing it. The last column is expanded in Sec. 9.

9. Theories relaxing the postulates

9.1 Quadratic gravity

Adding $\alpha R^2 + \beta R_{\mu\nu}R^{\mu\nu}$ to (3.3) relaxes P6. Stelle showed that the resulting theory is renormalizable in perturbation theory, which general relativity is not. The fourth-order field equations, however, propagate a massive spin-2 mode of negative kinetic energy, so that unitarity is not manifest; whether this ghost is physical or an artifact of the perturbative treatment remains a matter of debate. Independently of that question, quadratic terms are unavoidable as corrections: treated as an effective field theory, general relativity generates them at one loop with coefficients suppressed by the Planck scale, where they are innocuous.

9.2 $f(R)$ gravity

Replacing $R$ in (3.3) by a function $f(R)$ likewise yields fourth-order equations, but the additional propagating mode is a scalar of positive kinetic energy rather than a ghost, and the theory is dynamically equivalent to a scalar-tensor theory. It has been studied extensively as a means of producing accelerated cosmological expansion geometrically rather than by a dark energy component, and is strongly constrained by solar-system tests.

9.3 Lovelock and Gauss-Bonnet gravity

Retaining second-order field equations but relaxing the restriction to four dimensions, the Gauss-Bonnet term of Sec. 3.2 ceases to be topological and contributes to the equations of motion, and Theorem 3.1 admits an infinite series of higher-order invariants. The four-dimensional case is thus a degenerate member of a larger family.

9.4 Einstein-Cartan theory

Relaxing the vanishing of torsion while retaining P1–P6 gives a theory in which intrinsic spin acts as a source for torsion. The torsion field is non-propagating and algebraically determined by the spin density, so the theory coincides with general relativity except at densities at which spin alignment is appreciable.

9.5 Scalar-tensor and Horndeski theories

Relaxing the requirement in P5 that the metric be the only gravitational field, one introduces a scalar $\phi$ coupled to the curvature. The Brans-Dicke theory, motivated by Mach's principle, is the original example, and the Horndeski construction gives the most general scalar-tensor theory with second-order field equations. The near-simultaneous arrival of gravitational waves and light from the neutron star merger GW170817, after a propagation time of order $10^8$ years, constrains the speed of gravitational waves to equal $c$ to high precision and excludes those members of the family predicting otherwise.

9.6 Massive gravity

Assigning a mass to the graviton breaks diffeomorphism invariance, or requires its restoration by the introduction of Stückelberg fields. Generic mass terms propagate a sixth mode, the Boulware-Deser ghost; the construction of de Rham, Gabadadze and Tolley eliminates it by a particular tuning of the potential.

9.7 Unimodular gravity

Restricting the variation of the action to those variations preserving $\sqrt{-g}$ removes the trace of the field equations from the variational principle. The cosmological constant then arises as a constant of integration rather than as a coupling in the action. The classical predictions are otherwise identical to those of general relativity, so the theory reformulates the cosmological constant problem without resolving it.

9.8 Thermodynamic derivations

Jacobson showed that imposing the Clausius relation $\delta Q = T\,dS$ on every local Rindler horizon, with entropy proportional to horizon area, yields (5.7) as an equation of state. This construction does not relax a single postulate but reinterprets the entire framework, suggesting that the metric is a coarse-grained thermodynamic variable and that the quantization of $g_{\mu\nu}$ is analogous to the quantization of sound waves in a solid.

10. Further development

The derivation above yields the field equations and nothing further. Their content is obtained by solving them in particular cases.

The natural first application is the vacuum equation (6.2) under the assumption of spherical symmetry and staticity. The metric then contains two unknown functions of the radial coordinate, the field equations reduce to ordinary differential equations, and integration yields the Schwarzschild solution together with Birkhoff's theorem, which states that it is the unique spherically symmetric vacuum solution. The classical tests, namely the precession of the perihelion of Mercury and the deflection of light by the Sun, are then perturbative calculations on that background, and the horizon at $r = 2GM/c^2$ appears as a property of the solution.

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