General relativity

Forthe only potentially nonzero Ricci component isThe vacuum equation is therefore the transverse Laplace equation.

A connection defines covariant differentiation by adding one connection term for each contravariant index and subtracting one for each covariant index.

A tensor of type $(r,s)$ has $r$ contravariant and $s$ covariant indices. Taking a covariant derivative adds one covariant index, while an outer tensor product adds the corresponding contravariant and covariant index counts.

A vector is parallel transported along a curve when its covariant derivative along the curve vanishes.

For $d{s}^2=d{\theta }^2+{\sin }^2\theta d{\varphi }^2$, a vector initially having coordinate components $(V^{\theta },V^{\varphi })=(1,0)$ returns after one constant-$\theta$ circuit with

For $W_{\mu \nu }$,

The Levi-Civita connection is the unique torsion-free connection compatible with the metric; its coefficients are the Christoffel symbols formed from first metric derivatives.

Metric compatibility is ${\nabla }_{\alpha }{g}_{\mu \nu }=0$, allowing the metric to pass through covariant derivatives and index operations.

The Riemann tensor measures the failure of covariant derivatives to commute.

For a rank-two covariant tensor, the derivative commutator contributes one curvature action on each index:

For a torsion-free connection, $U_{\mu }={\partial }_{\mu }\varphi$ satisfies ${\nabla }_{\alpha }{U}_{\beta }={\nabla }_{\beta }{U}_{\alpha }$.

For a torsion-free connection, $R^{\mu }{}_{\alpha \beta \gamma }+R^{\mu }{}_{\beta \gamma \alpha }+R^{\mu }{}_{\gamma \alpha \beta }=0$.

The Weyl tensor is the completely trace-free part of the Riemann curvature tensor. In four dimensions,

Every contraction of the Weyl tensor vanishes. Contracting its first and third indices directly cancels the Ricci and scalar-curvature terms; antisymmetry in each pair and symmetry under exchanging the two pairs reduce all other contractions to this one or make them vanish immediately.

Every conformally flat metric in dimension at least four has identically zero Weyl tensor. The Weyl tensor is the obstruction to local conformal flatness in dimensions at least four.

Applying the curvature commutator to the parallel metric gives $R_{\mu \alpha \beta \gamma }=-R_{\alpha \mu \beta \gamma }$.

The Ricci tensor is the contraction $R_{\alpha \beta }=R^{\mu }{}_{\alpha \mu \beta }$ and is symmetric for the Levi-Civita connection.

The algebraic symmetries of the Riemann curvature tensor leave only one independent curvature component in two dimensions and forceConsequently the Einstein tensor vanishes identically.

With no cosmological term, the two-dimensional Einstein field equations reduce identically on the geometric side to $G_{\alpha \beta }=0$. They impose $T_{\alpha \beta }=0$ and supply no local metric dynamics; the Einstein-Hilbert action is topological.

The Ricci tensor and scalar satisfy ${\nabla }^{\alpha }{R}_{\alpha \beta }=\frac{1}{2}{\nabla }_{\beta }R$.

The Einstein tensor isThe contracted Bianchi identity and metric compatibility give
${\nabla }^{\nu }{G}_{\mu \nu }=0$, ensuring compatibility of the Einstein equations with covariant stress-energy conservation.

For $d{s}^2=-d{t}^2+a(t{)}^{2}d{\mathbf{x}}^2$, the nonzero connection coefficients areWith $G_t^t=-3H^2$ and
$G_i^j=-(2\ddot{a}/a+H^2){\delta }_i^j$, direct substitution into the mixed-tensor covariant derivative gives zero.

The vacuum equation with positive cosmological constant gives
$H^2=\Lambda /3$. Its expanding flat-slicing solution is

In dimension greater than two, if sectional curvature is pointwise independent of the two-plane, the contracted Bianchi identity forces that curvature to be constant on each connected component.

At any spacetime point, normal coordinates make the metric equal to the Minkowski metric and all Christoffel symbols vanish at that point.

The stress-energy tensor describes local energy density, momentum density, and stress. Compatibility with the Einstein field equations requires covariant conservation ${\nabla }^{\mu }{T}_{\mu \nu }=0$.

For ${\nabla }_{\alpha }{\nabla }^{\alpha }\varphi =m^2\varphi$, considerCovariant conservation for all solutions holds exactly when

The weak energy condition requiresfor every timelike vector $X$, so every observer measures nonnegative energy density.

For the general scalar ansatz in Conserved Klein-Gordon scalar stress-energy ansatz, work in a unit timelike frame. ThenThe weak energy condition for arbitrary local field data is therefore equivalent to

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

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

For the conserved scalar stress tensor above,

Linearized gravity writes $g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$ with $|h_{\mu \nu }|\ll 1$ and discards terms quadratic in $h$ and its derivatives.

In four dimensions, the trace-reversed perturbation isTrace reversal simplifies the linearized Einstein equations.

The Lorenz condition ${\partial }^{\mu }{\bar{h}}_{\mu \nu }=0$ reduces the vacuum field equation to $\square {\bar{h}}_{\mu \nu }=0$. A plane-wave amplitude therefore satisfies $k^{\mu }{H}_{\mu \nu }=0$ and $k^{\mu }{k}_{\mu }=0$.

An infinitesimal coordinate change generated by ${\xi }_{\mu }$ changesIt preserves Lorenz gauge exactly when $\square {\xi }_{\mu }=0$.

A vacuum mode ${\bar{h}}_{\mu \nu }=H_{\mu \nu }{e}^{ik\cdot x}$ has a null wavevector and a transverse symmetric amplitude. Residual gauge freedom can further reduce it to two physical transverse-traceless polarizations.

To first order around flat spacetime,and

For a weak, static field produced by nonrelativistic matter, $T_{00}=\rho$ is the dominant stress-energy component. The field equations and the slow-particle geodesic equation implywhere the Poisson equation is ${\nabla }^2\Phi =4\pi G\rho$.

For a point mass and $r>0$, writeThe vacuum scalar-curvature equation gives $r{f}^{\prime }+3f=0$, so $f=C/r^3$. The remaining vacuum equations fix $C=2GM$ in the corresponding Schwarzschild gauge.

In units $G=c=1$, the Schwarzschild exterior metric is

With the curvature conventionthe vacuum Schwarzschild metric satisfiesThe opposite Riemann-sign convention reverses this sign.

Weyl curvature produces relative acceleration between neighbouring freely falling trajectories and therefore tidal deformation. A freely falling frame removes gravity at one event, as required by the equivalence principle, but cannot remove curvature over an extended region.

For inverse radius $y=1/r$ and azimuth $\varphi$, a Schwarzschild null geodesic obeyswhere $b=L/E$ is the impact parameter.

The impact parameter is the perpendicular miss distance of the corresponding asymptotic straight trajectory. For a null Schwarzschild orbit it is the ratio $b=L/E$ in geometrized units.

The Schwarzschild radius is $r_s=2GM/c^2$, or $2M$ in geometrized units.

The tortoise coordinate satisfies

The advanced coordinate $v=t+r_{\ast }$ puts the radial Schwarzschild metric into the formwhich is regular at the future horizon.

Radial null curves comprise the ingoing family $v=\text{constant}$ and the outgoing familyOutside the horizon the outgoing family moves to larger $r$; inside it both future-directed families move to smaller $r$.

The surface $r=2M$ is the future event horizon of Schwarzschild spacetime. Its singularity in Schwarzschild coordinates is removable, whereas $r=0$ is a curvature singularity.

For static observers at radii $r_A$ and $r_B$, equal Schwarzschild-coordinate time intervals correspond to proper times $d\tau =\sqrt{1-2M/r}dt$. Hence

A static observer at Schwarzschild radius $r>2M$ must maintain the outward proper accelerationIt approaches the Newtonian value $M/r^2$ at large radius and diverges at the horizon.

Every spherically symmetric vacuum solution of the Einstein field equation is locally a portion of Schwarzschild spacetime. In particular, a spherically symmetric body's exterior field depends only on its total mass.

A thin spherical shell of proper surface mass density $\rho$ held at Schwarzschild radius $r_{\mathrm{st}}$ around mass $M$ contributes, to first order in its mass,to the exterior Schwarzschild mass parameter.

A null Schwarzschild geodesic with $b\gg r_s$ is deflected through $2r_s/b=4GM/(b{c}^2)$ to first order.

Writing $y=b^{-1}\sin \varphi +\Delta y$ in $y^{\prime \prime }+y=3M{y}^2$ and retaining first order in $M$ givesThe displacement of the outgoing zero is $4M/b$.

Nordstrom's scalar theory restricts the spacetime metric to the conformally flat form $g_{\mu \nu }={\phi }^2{\eta }_{\mu \nu }$. Its null geodesics have the same unparametrized paths as those of the Minkowski metric, so the theory predicts no gravitational bending of light.

A metric is conformally flat when it is locally a positive scalar multiple of a flat metric. Multiplication by the conformal factor preserves null cones and unparametrized null geodesics.

For $g_{\mu \nu }={\phi }^2{\eta }_{\mu \nu }$,For a null curve, the last term drops from the geodesic equation; the remaining acceleration is parallel to the tangent and can be removed by reparametrization.

Anti-de Sitter spacetime is the maximally symmetric Lorentzian spacetime of constant negative curvature.

In four dimensions, global static coordinates have

A radial timelike geodesic through the origin obeys a harmonic radial equation and returns to the origin after proper time $\pi a$, independently of its energy.

At every radius $r_0>0$, global anti-de Sitter spacetime has stable circular timelike geodesics with $|\dot{\varphi }|=1/a$ and $\dot{t}=1$.

Rotational symmetry conserves the angular-momentum direction, so every nonradial geodesic lies in a plane through the symmetry centre; coordinates may place that plane at $\theta =\pi /2$.

After using Killing constants and four-velocity normalization, radial timelike geodesic motion can often be written as kinetic energy plus an effective potential equal to a constant.

A circular orbit at a strict local minimum of the radial effective potential is stable under small radial perturbations.

For equatorial timelike motion with specific angular momentum $h$,A circular orbit has $h^2=m{r}^2/(r-3m)$ and therefore requires $r>3m$.

The nearly circular orbit equationhas radial angular frequency $1-3m^2/h^2$ to leading order. The perihelion advances by $6\pi m^2/h^2$ per orbit.

A metric tensor is a nondegenerate symmetric covariant tensor that defines squared line elements by $d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$.

The inverse metric satisfies $g^{\mu \rho }{g}_{\rho \nu }={\delta }_{\nu }^{\mu }$ and raises covariant tensor indices.

This metric is the hyperbolic plane with curvature scale $a$. Its nonzero connection coefficients are

Every radial coordinate line $\varphi =\text{constant}$ is a geodesic. No nonconstant coordinate circle $r=r_0>0$ is a geodesic.

For the curvature conventionthe independent nonzero mixed components areand the Ricci scalar is $R=-2/a^2$.

The Ricci scalar is the full metric contraction of the Ricci tensor: