Einstein field equations

With signature $(-,+,+,+)$ and vanishing cosmological constant, the Einstein field equations areIn four spacetime dimensions, contraction with $g^{\mu \nu }$ gives the useful trace relation $R=-\kappa T$.

This metric has $R_{ab}=-3g_{ab}$ and cosmological constant $\Lambda =-3$ in unit curvature-radius normalization. Its regular static coframe is real when $F>0$ and $z\ne 0$. Setting $q=\alpha z^3$, its six independent curvature 2-form coefficients are $-(1-q)$ in the $03$ and $12$ planes and $-(1+q/2)$ in the other four planes. Their Ricci contractions cancel $q$. The Weyl tensor squared invariant is $12{\alpha }^2{z}^6$, so nonzero $\alpha$ does not give constant sectional curvature despite the Einstein equation.

In four dimensions, a local natural symmetric divergence-free metric tensor involving the metric and its derivatives of at most second order has the Lovelock form $a{G}_{ab}+b{g}_{ab}$ with constant coefficients. Generic nonlinear metric f(R) gravity instead has fourth-order terms and does not meet that derivative-order hypothesis. For affine $f$, the metric field equation is precisely of Lovelock form.

In a region with vanishing stress-energy tensor, the Einstein field equations reduce to $R_{\mu \nu }=0$ when the cosmological constant vanishes.

A vacuum spacetime has vanishing matter stress-energy tensor. In $D>2$ dimensions, tracing the Vacuum Einstein equations gives $R=2D\Lambda /(D-2)$ and hence $R_{ab}=2\Lambda g_{ab}/(D-2)$; in four dimensions this reduces to $R_{ab}=\Lambda g_{ab}$. With zero cosmological constant it is Ricci flat, although its full Riemann curvature tensor can be nonzero: a vacuum Kerr black hole and vacuum gravitational waves illustrate that distinction.

The Einstein-Hilbert action is the integral of the Ricci scalar against the metric volume form. Varying it with respect to the metric gives the vacuum Einstein field equations, up to a boundary term.

Einstein-Proca theory couples general relativity to a massive vector field governed by the Proca action. Its matter stress-energy tensor isThe contracted Bianchi identity makes the Einstein equation compatible with the covariant Proca equation.

The Proca stress-energy tensor is the metric variation of the matter part of the Proca action. It consists of the electromagnetic stress-energy tensor plus the quadratic mass contribution $m^2(A_a{A}_b-g_{ab}{A}^2/2)$.

Einstein-Maxwell-dilaton theory couples the metric to a scalar field and a Maxwell field with scalar-dependent gauge coupling, typically through $e^{\alpha \Phi }{F}_{ab}{F}^{ab}$. Its Maxwell equation is ${\nabla }_a(e^{\alpha \Phi }{F}^{ab})=0$.

The unit-coupling magnetic solution of Einstein-Maxwell-dilaton theory has $d{s}^2=-Vd{t}^2+V^{-1}d{r}^2+R^{2}d{\Omega }_2^2$, magnetic potential $A=P\cos \theta d\varphi$, and dilaton $\varphi =-\frac{1}{2}\log (1-r_{-}/r)$, where $r_{-}=2P^2/r_{+}$ and $r_{+}>0$. Its physical asymptotic component has $r>r_{-}$. For $0\le r_{-}<r_{+}$ it has a regular Killing horizon at $r_{+}$, area $4\pi r_{+}(r_{+}-r_{-})$ and surface gravity $1/(2r_{+})$. Its asymptotic mass is $r_{+}/2$. For $r_{-}>r_{+}$ the zero-area singularity is naked. The limiting case $r_{-}=r_{+}$ is singular and must not be treated as a regular nondegenerate horizon.

For $r_{-}<r_{+}$, an ingoing chart with $v=t+r_{\ast }$ has metric $-Vd{v}^2+2dvdr+R^{2}d{\Omega }_2^2$. In its future interior, $\nabla r={\partial }_v+V{\partial }_r$ is future timelike, so every future causal curve decreases $r$ and cannot reach future null infinity. The singularity $r=r_{-}$ is spacelike. A Kruskal–Szekeres coordinates extension has the same qualitative Penrose diagram as Schwarzschild spacetime, with two exterior regions, a future black-hole interior and a past white hole interior. The past interior has the same radial interval but is not a black hole region; specifying the radial interval alone does not identify the future component. For $r_{-}>r_{+}$, the physical exterior $r>r_{-}$ has $V>0$ everywhere and outgoing null rays escape from arbitrarily near a timelike naked singularity.

Rotate any geodesic into an equatorial plane of the magnetically charged dilaton black hole. Conserved energy and angular momentum are $E=V\dot{t}$ and $L=R^2\dot{\varphi }$. Normalizing its tangent to $g(\dot{x},\dot{x})=-ϵ$, with $ϵ=1,0,-1$ for timelike, null and spacelike geodesics, gives ${\dot{r}}^2=E^2-V(ϵ+L^2/R^2)$. Thus its radial equation is ${\dot{r}}^2/2+\tilde{V}=0$. Radial null rays have $\dot{r}=\pm E$ and use the same Schwarzschild tortoise coordinate with $2M$ replaced by $r_{+}$.

For the magnetically charged dilaton black hole, the metric-dual one-form of $k={\partial }_t$ is $k^{♭}=-Vdt$. With orientation $dt\wedge dr\wedge d\theta \wedge d\varphi$, its Hodge star operator gives $\star d{k}^{♭}=-V^{\prime }{R}^2\sin \theta d\theta \wedge d\varphi$. Therefore the Komar mass is $R^2{V}^{\prime }/2=r_{+}/2-P^2/r$. Its radial dependence is allowed because the Ricci tensor is nonzero; the divergence of $d{k}^{♭}$ is sourced by $R_{ab}{k}^b$. At the horizon $M_K(r_{+})=(r_{+}-r_{-})/2=\kappa A_H/(4\pi )$.

For $D>2$, tracing and substituting back gives

For the conserved scalar stress tensor above,

For $h_{ab}=\delta g_{ab}$, variation of the inverse metric and metric volume form givesThe Levi-Civita connection varies by

For the Levi-Civita connection, the difference of two connections is a tensor. Varying the coordinate expression for the Ricci tensor then gives the displayed covariant identity. It is the starting point for the metric variation of scalar curvature.

For $h_{ab}=\delta g_{ab}$, the Ricci scalar varies asThe last two terms form a covariant divergence and hence contribute only a boundary term to a compactly supported variation of the Einstein-Hilbert action.

Modified gravity changes the Einstein-Hilbert action or its field equations in order to produce gravitational dynamics different from general relativity.

An f(R) theory replaces the Ricci scalar in the gravitational action by a smooth function $f(R)$. Its metric field equation is

Every four-dimensional Einstein metric with $R_{ab}=\Lambda g_{ab}$ has constant Ricci scalar $R=4\Lambda$. The vacuum metric f(R) gravity tensor reduces to $(\Lambda f^{\prime }(4\Lambda )-f(4\Lambda )/2)g_{ab}$, giving the displayed necessary and sufficient condition for a fixed cosmological constant. Both function and derivative may vanish, so division by $f^{\prime }$ is unnecessary. Requiring this for all real $\Lambda$ and a smooth $f$ instead yields $f(R)=C{R}^2$.

For a metric action $S_g=A\int \sqrt{-g}f(R)d^{4}x$ and the usual stress-energy tensor convention, covariant metric variation gives $A{E}_{ab}=T_{ab}/2$, where $E_{ab}=f^{\prime }{R}_{ab}-f{g}_{ab}/2+(g_{ab}\square -{\nabla }_a{\nabla }_b)f^{\prime }$. With $A=1/(16\pi G)$ this becomes $E_{ab}=8\pi G{T}_{ab}$; with $A=1$ it becomes $E_{ab}=T_{ab}/2$. The chosen normalization does not alter the vacuum equation.

If $f(0)=0$, $f^{\prime }(0)=1$, and $f^{\prime \prime }(0)=0$, then $f(R)=R+O(R^3)$ near zero. The field equations linearized about Minkowski spacetime are therefore the Linearized Einstein equations; deviations first appear beyond linear order.