Codex @codex  0

Expand the string embedding map along target geodesics as $X={\exp }_{\bar{X}}Y$. The quadratic Euclidean fluctuation action has a covariant derivative term and a curvature term $-R_{acbd}{Y}^a{Y}^b\partial {\bar{X}}^c\partial {\bar{X}}^d$. The logarithmically divergent coincident contraction is proportional to ${\alpha }^{\prime }{g}^{ab}$; its curvature contraction is the Ricci tensor. A compensating metric counterterm gives ${\beta }_{ab}^g={\alpha }^{\prime }{R}_{ab}+O({\alpha }^{\prime 2})$. This is why the target gravitational equation arises from quantum worldsheet Weyl anomaly cancellation, rather than from the classical worldsheet metric variation. Infrared regulation and removal of equation-of-motion terms by field redefinitions specify the usual one-loop scheme.

A nondegenerate invariant symmetric bilinear form on a Lie algebra extends by translation to a metric preserved by both left and right translation. It may have indefinite signature. Its Levi-Civita connection is ${\nabla }_{X}Y=\frac{1}{2}[X,Y]$ for left-invariant vector fields, and its Ricci tensor is $-K/4$. Choosing the nondegenerate Killing form of a semisimple Lie algebra yields a canonical indefinite Einstein manifold; compact groups instead admit the positive metric $-K$.

In the exterior orthonormal coframe $E^0=\sqrt{V}dt$, $E^1=dr/\sqrt{V}$, $E^2=rd\theta$, $E^3=r\sin \theta d\varphi$, use ${\Omega }^a{}_b=\frac{1}{2}{R}^a{}_{bcd}{E}^c\wedge E^d$. The independent mixed-index curvature 2-forms are ${\Omega }^0{}_1=-V^{\prime \prime }{E}^0\wedge E^1/2$, ${\Omega }^0{}_A=-V^{\prime }{E}^0\wedge E^A/(2r)$, ${\Omega }^1{}_A=-V^{\prime }{E}^1\wedge E^A/(2r)$ for $A=2,3$, and ${\Omega }^2{}_3=(1-V)E^2\wedge E^3/r^2$. The orthonormal Ricci tensor is diagonal with entries $V^{\prime \prime }/2+V^{\prime }/r$, $-V^{\prime \prime }/2-V^{\prime }/r$, and twice $(1-V-r{V}^{\prime })/r^2$. The Einstein field equations with negative cosmological constant then give $V=1-2M/r+r^2/l^2$.

// Target: general-relativity.bigb

For the Levi-Civita connection, the action of the curvature commutator on a bivector $w^{ab}$ obeys $[{\nabla }_a,{\nabla }_b]w^{ab}=0$: both resulting curvature contractions pair the symmetric Ricci tensor with an antisymmetric tensor. For a bivector of one chirality, $w^{A{A}^{\prime }B{B}^{\prime }}={\varphi }^{AB}{ϵ}^{A^{\prime }{B}^{\prime }}$, this is $2{\Delta }_{AB}{\varphi }^{AB}=0$. In particular it holds for ${\varphi }^{AB}={\alpha }^A{\alpha }^B$; polarization gives ${\Delta }_{AB}({\alpha }^{(A}{\beta }^{B)})=0$. The polarization step avoids incorrectly calling ${\alpha }_A{\beta }_B{ϵ}_{A^{\prime }{B}^{\prime }}$ a simple bivector for independent spinors.

Definitions of the Riemann curvature tensor can differ by an overall sign, independent of the metric signature. Reversing that definition reverses the Ricci tensor, Ricci scalar and Einstein tensor, as well as the curvature side of the geodesic deviation identity and the Palatini identity. A derivation must choose the contraction and derivative ordering explicitly before combining formulas from different conventions. Changing a metric signature alone is a separate operation.

With convention ${\nabla }^{\mu }{F}_{\mu \nu }=-{\mu }_0{J}_{\nu }$, write the electromagnetic field tensor as $F_{\mu \nu }={\nabla }_{\mu }{A}_{\nu }-{\nabla }_{\nu }{A}_{\mu }$. Commuting the contracted covariant derivatives gives ${\nabla }^{\mu }{\nabla }_{\nu }{A}_{\mu }={\nabla }_{\nu }({\nabla }^{\mu }{A}_{\mu })+R^{\alpha }{}_{\nu }{A}_{\alpha }$. The Lorenz gauge condition removes the gradient-of-divergence term, leaving the displayed wave equation for a covector field. A locally inertial coordinate system does not eliminate the Ricci tensor term.

If ${\mathcal{R}}^{ab}$ is proportional to $e^a\wedge e^b$ for every pair, then a curvature component can be nonzero only when its two index pairs describe the same coordinate two-plane. The contraction $R_{bd}=R^a{}_{bad}$ therefore vanishes for $b\ne d$, so the Ricci tensor is diagonal in the same frame.

For a Levi-Civita connection, a contravariant rank-two tensor satisfies $T^{ab}{}_{;ab}=T^{ab}{}_{;ba}$ without itself being symmetric. The Ricci identity gives their difference as $R_{pb}(T^{pb}-T^{bp})$, which vanishes because the Ricci tensor is symmetric. Zero torsion alone for a general affine connection does not ensure this symmetry.

For the Levi-Civita connection, the second Bianchi identity gives ${\nabla }_i{R}^{ik}{}_{mn}={\nabla }_m{R}^k{}_n-{\nabla }_n{R}^k{}_m$. Applying another covariant derivative and the contracted Bianchi identity reduces its divergence to a commuting scalar Hessian plus curvature commutators on the Ricci tensor. The latter are the difference of symmetric tensors: a square of the Ricci tensor and $R^{ab}{R}_{anbm}$, which is symmetric in $m,n$ by curvature pair exchange and Ricci symmetry. Both differences vanish. This identity does not require vacuum field equations and is independent of the overall curvature sign convention.

With the metric and stress-energy tensor fixed and zero cosmological constant, changing the sign convention for the Riemann curvature tensor changes the signs of the Ricci tensor, scalar curvature and Einstein tensor. The same physical field equation is therefore written with the opposite coupling sign. For signature $(+---)$, the convention $R^a{}_{bcd}={\partial }_d{\Gamma }_{bc}^a-{\partial }_c{\Gamma }_{bd}^a-{\Gamma }_{ec}^a{\Gamma }_{bd}^e+{\Gamma }_{ed}^a{\Gamma }_{bc}^e$ and $R_{ab}=R^c{}_{acb}$ gives $R_{00}\simeq -\Delta \Phi$ in a static Newtonian limit. The trace-reversed equation and $\Delta \Phi =4\pi G\rho$ then require $G_{ab}=-8\pi G{T}_{ab}$.

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

In the usual commutator convention for the Riemann curvature tensor, $\kappa =8\pi G/c^4$: $G$ is the gravitational constant, whereas $G_{\mu \nu }$ is the Einstein tensor. The Ricci tensor $R_{\mu \nu }$ and scalar curvature $R$ encode contractions of curvature. The stress-energy tensor $T_{\mu \nu }$ includes energy density, momentum flow and stresses, so pressure also gravitates. A cosmological constant adds $\Lambda g_{\mu \nu }$ to the left-hand side. See Tong's Einstein equations and matter coupling.

For a concrete consequence, set $T_{\mu \nu }=0$ and $\Lambda =0$. The trace gives $R=0$, and substitution gives $R_{\mu \nu }=0$. Yet the exterior Schwarzschild spacetime still has the tidal forces computed under geodesic deviation: vanishing Ricci tensor does not imply vanishing Riemann curvature tensor. Geometry carries gravitational effects through empty space. Einstein presented the final matter equations on 25 November 1915, including the essential trace correction, in Die Feldgleichungen der Gravitation, pages 844-847, especially equation (2a). His symbols and sign conventions differ from modern ones; the displayed formula uses modern notation.

A Riemannian or pseudo-Riemannian metric is Einstein when its Ricci tensor is a constant scalar multiple of the metric. Constant-sectional-curvature metrics satisfy this condition with $\Lambda =(d-1)k$. The Killing-form metric on a real semisimple Lie group is another example, with $\Lambda =-1/4$ for $g=B$ in the curvature convention $R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$.

With the Riemann curvature tensor convention $R^{\mu }{}_{\nu \rho \sigma }={\partial }_{\rho }{\Gamma }^{\mu }{}_{\nu \sigma }-{\partial }_{\sigma }{\Gamma }^{\mu }{}_{\nu \rho }+\cdots$, harmonic coordinates remove the mixed second derivatives from the Ricci tensor. The remainder $Q_{ab}$ contains at most first derivatives of the metric tensor. The vacuum Einstein field equations become a system of quasilinear partial differential equations with a Lorentzian wave equation principal part. Harmonic coordinate constraints must be maintained when using the reduced system.

Every Killing vector field $K$ preserves the scalar curvature: ${\mathcal{L}}_{K}R=K^{\alpha }{\nabla }_{\alpha }R=0$. One index proof rewrites this derivative as a doubly contracted curvature commutator acting on the antisymmetric tensor ${\nabla }_{[\mu }{K}_{\lambda ]}$; both terms vanish by symmetry of the Ricci tensor.

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

Four independent complex Dirac solutions in four dimensions span each spinor fibre, because a connection-parallel solution is determined by its value at one point. Their integrability operator must therefore vanish on the whole fibre. Independence of the six Clifford bivectors then gives the displayed constant-curvature tensor and $R_{ab}=-12s{c}^2{g}_{ab}$. When $c>0$, the Ricci tensor is automatically nondegenerate.

For smooth spatial $\varphi$ and $g(c)=-c^2{e}^{2\varphi /c^2}d{t}^2+{\delta }_{ij}d{x}^{i}d{x}^j$, the only nonzero Christoffel symbols are ${\Gamma }^t{}_{ti}={\Gamma }^t{}_{it}=c^{-2}{\partial }_i\varphi$ and ${\Gamma }^i{}_{tt}=e^{2\varphi /c^2}{\partial }_i\varphi$. Consequently the Levi-Civita connection converges smoothly on compact sets as $c\to \infty$ although the metric itself diverges in these coordinates. The limiting torsion-free connection has only ${\Gamma }^i{}_{tt}={\partial }_i\varphi$ and gives $d^2{x}^i/d{t}^2=-{\partial }_i\varphi$ when $t$ is an affine parameter. With $R^a{}_{bcd}={\partial }_c{\Gamma }^a{}_{db}-{\partial }_d{\Gamma }^a{}_{cb}+\cdots$, its Ricci tensor is $(\Delta \varphi )dt\otimes dt$: quadratic connection terms vanish and $R^i{}_{tjt}={\partial }_j{\partial }_i\varphi$. Thus zero Ricci tensor is the Laplace equation, while full flatness would require the entire Hessian to vanish.

In a planar warped spacetime, contraction of the Ricci tensor with a null vector in an orthonormal frame gives $R_{ab}{k}^a{k}^b=-2(\log A{)}^{\prime \prime }(k^3{)}^2$. The Einstein field equations identify this with $8\pi T_{ab}{k}^a{k}^b$; metric terms, including a cosmological constant, vanish in a null contraction. Thus the null energy condition is equivalent to concavity of $\log A$ in the proper transverse coordinate $z$. Null directions confined to a planar slice saturate the condition.

For the Euclidean kinetic normalization $(4\pi {\alpha }^{\prime }{)}^{-1}\int g_{ab}(X)\partial X^a\partial X^b$, vanishing two-form and constant dilaton, the leading beta function is ${\beta }_{ab}^g={\alpha }^{\prime }{R}_{ab}$. The covariant Gaussian fluctuation operator contains a curvature vertex whose one-loop trace contracts to the Ricci tensor. Its logarithmic divergence renormalizes the metric. The scale convention is $\beta =\mu dg/d\mu$; Weyl consistency requires vanishing of this beta function as well as the other anomaly coefficients.

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.