Einstein-Hilbert action

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.

An independent vierbein and metric-compatible spin connection allow spin density to source a torsion tensor. The connection equation from the Einstein-Hilbert action is algebraic in torsion. In the absence of connection-dependent matter it fixes the Levi-Civita connection. For a minimally coupled Dirac field, eliminating the connection generates a local four-fermion interaction. Thus its equivalent torsion-free second-order action includes this contact term; simply dropping it defines a different matter theory.

Solve the algebraic connection equation for $\hat{\omega }(e,\chi )$, but perform the remaining variations as though the connection were independent. The chain-rule contribution $(\delta S/\delta \omega )\delta \hat{\omega }$ vanishes on that solution. The resulting equations are those of the connection-eliminated second-order action. This is especially useful for local supersymmetry calculations, where the gravitino-induced contorsion is quadratic in fermions and the omitted induced connection variation would otherwise be lengthy.

Use $K_{abc}=e_a{}^{\mu }(\omega -\omega (e){)}_{\mu bc}$ and $T_{abc}=K_{bac}-K_{cab}$. For the symmetrized Dirac action $-e\bar{\chi }{\gamma }^{\mu }{\overleftrightarrow{\nabla }}_{\mu }\chi /2-em\bar{\chi }\chi$, varying the spin connection gives $-e\bar{\chi }{\gamma }^{\mu ab}\chi \delta {\omega }_{\mu ab}/4$. Only totally antisymmetric contorsion is sourced. Its action is $-e{K}^2/(2{\kappa }^2)-e{K}_{abc}\bar{\chi }{\gamma }^{abc}\chi /4$, hence the stated solution. Substitution generates $e{\kappa }^2(\bar{\chi }{\gamma }^{abc}\chi )(\bar{\chi }{\gamma }_{abc}\chi )/32$. The torsion is an auxiliary field, not an independent propagating degree of freedom.

For a nondegenerate orthonormal coframe in spacetime $E^a$ and independent antisymmetric Lorentz connection 1-form, the action $\int {ϵ}_{abc}(R^{ab}\wedge E^c+\lambda E^a\wedge E^b\wedge E^c)$ has independent coframe and connection variations. Since $\delta R^{ab}=D\delta {\omega }^{ab}$, integration by parts yields a connection equation ${ϵ}_{abc}{T}^c=0$. The three variations of the cubic coframe term give the coframe equation ${ϵ}_{abc}(R^{ab}+3\lambda E^a\wedge E^b)=0$. In three dimensions these are exactly the equations displayed above. The solution has constant sectional curvature $K=-3\lambda$. Omitting the factor of three changes the cosmological normalization.

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 $f=1-2M/r$ and squared areal radius $C=r(r-b)$, an equatorial null geodesic obeys ${\dot{r}}^2=E^2[1-B^{2}f/C]$, with impact parameter $B=L/E$. At $b=2M$, $f/C=1/r^2$ in the exterior. Its supremum is $1/(4M^2)$ at the null singular boundary, rather than at a regular exterior photon sphere. Rays with $B>2M$ turn at $r=B$; rays with $B<2M$ reach the boundary. Assuming those rays are absorbed gives the displayed geometric-optics cross-section despite zero limiting horizon area.

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 $M_K(r_{+})=(r_{+}-r_{-})/2=\kappa A_H/(4\pi )$.