Schwarzschild spacetime

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

For smooth radial infall through a future Schwarzschild event horizon, outgoing retarded time obeys $u\sim -4M\log ({\tau }_H-\tau )$. A distant observer sees the redshift grow as $e^{t_{\mathrm{rec}}/(4M)}$. Its exponent in emission Schwarzschild time is instead $1/(2M)$, because $t_{\mathrm{rec}}\sim 2t_{\mathrm{em}}$. At finite fixed observer radius the wavelength ratio includes $\sqrt{1-2M/r_B}$.

Schwarzschild time $t$ is the exterior time coordinate normalized to the proper time of static observers at infinity. At fixed exterior radius, $d\tau =\sqrt{1-2M/r}dt$. A received signal and its emission occur at different values of this coordinate.

The Kruskal spacetime is the maximal analytic extension of the positive-mass vacuum Schwarzschild spacetime. It contains two exterior regions, a black hole, and a white hole. Kruskal–Szekeres coordinates remove both horizon coordinate singularities, while $r=0$ remains a curvature singularity.

Every maximal radial timelike geodesic in positive-mass Kruskal spacetime is incomplete in at least one direction. Its equations are ${\dot{r}}^2=E^2-1+2M/r$ and $\ddot{r}=-M/r^2$. A finite turning point is a maximum, and each trajectory reaches $r=0$ at one or both ends. The remaining proper time is finite because $d\tau \sim \sqrt{r/(2M)}dr$. Complete circular timelike geodesics exist, but they have nonzero angular momentum.

In units $G=\hslash =c=k_B=1$, a Schwarzschild black hole has Hawking temperature $T=1/(8\pi M)$, so its heat capacity is $dM/dT=-8\pi M^2$. It heats up as it loses mass, preventing stable canonical equilibrium with an unlimited thermal reservoir.

Boost a Schwarzschild mass with speed $v$ and hold $p=m\gamma$ fixed as $v\to 1$. Away from the null plane $x=t$, the isotropic-coordinate expression tends toThe complete distributional limit on the null plane requires the additional coordinate treatment that produces the Aichelburg-Sexl shock wave.

The Schwarzschild-de Sitter spacetime is the spherically symmetric vacuum solution with positive cosmological constant. In static coordinates its metric function is $f(r)=1-2m/r-H^2{r}^2$, with a black-hole horizon and a cosmological horizon when $mH<1/\sqrt{27}$.

Geodetic precession is the rotation of a parallel-transported gyroscope relative to the orbital radial direction in curved spacetime. For a circular Schwarzschild orbit its exact precession per revolution is $2\pi [1-\sqrt{1-3M/R}]$.

Painlevé–Gullstrand coordinates write the Schwarzschild metric asThe constant-$t$ spatial slices are flat, and the coordinates remain regular at the Schwarzschild radius.

An orthonormal coframe is

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

The orthonormal-frame components of Schwarzschild curvature scale as $M/r^3$. The geodesic deviation equation therefore gives radial stretching and transverse compression whose magnitude diverges as $r\to 0$.

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.

A circular null geodesic remains at constant areal radius. It lies at a stationary point of the null radial effective potential and may be stable or unstable according to the sign of the second derivative there.

A photon sphere is a spherical set of circular null geodesics. In Schwarzschild spacetime it occurs at $r=3M$ and is unstable.

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

The tortoise coordinate satisfies

The retarded coordinate $u=t-r_{\ast }$ puts the radial Schwarzschild metric into $d{s}^2=-fd{u}^2-2dudr+r^{2}d{\Omega }^2$, with $f=1-2M/r$. Outgoing radial null geodesics have constant $u$. These coordinates are regular at the past horizon; an infaller crossing the future horizon has $u\to +\infty$.

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

A Finkelstein diagram plots areal radius $r$ against $T=v-r$ in Ingoing Eddington-Finkelstein coordinates. Its radial null directions show explicitly that both future light-ray families decrease $r$ inside a Schwarzschild black hole. It is not a compactified Penrose diagram.

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 two equal-mass inward timelike geodesics of unit specific Killing energy, their invariant center-of-mass energy has this finite horizon limit. If both are captured from infinity, Schwarzschild marginally bound capture gives $E_{\mathrm{com}}<m\sqrt{20}$, with $m\sqrt{20}$ a supremum. No such universal bound follows for particles prepared inside the angular-momentum barrier.

The Schwarzschild singularity is the spacelike curvature singularity at $r=0$. Curvature invariants and orthonormal tidal components diverge there.

Kruskal–Szekeres coordinates extend the Schwarzschild solution smoothly through its future and past horizons. Constant Schwarzschild radius curves are hyperbolae, while the horizons are null lines through the bifurcation sphere.

A radial geodesic orthogonal to the time-symmetric Kruskal slice starts at $r=2M$ with zero Schwarzschild energy and reaches $r=0$ after proper time $\pi M$.

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 $4M/b$.