In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including Earth and the Sun. It was found by Karl Schwarzschild in 1916.

According to Birkhoff's theorem, the Schwarzschild metric is the most general spherically symmetric vacuum solution of the Einstein field equations. A Schwarzschild black hole or static black hole is a black hole that has neither electric charge nor angular momentum (non-rotating). A Schwarzschild black hole is described by the Schwarzschild metric, and cannot be distinguished from any other Schwarzschild black hole except by its mass.

The Schwarzschild black hole is characterized by a surrounding spherical boundary, called the event horizon, which is situated at the Schwarzschild radius (⁠${\displaystyle r_{\text{s}}}$⁠), often called the radius of a black hole. The boundary is not a physical surface, and a person who fell through the event horizon (before being torn apart by tidal forces) would not notice any physical surface at that position; it is a mathematical surface which is significant in determining the black hole's properties. Any non-rotating and non-charged mass that is smaller than its Schwarzschild radius forms a black hole. The solution of the Einstein field equations is valid for any mass M, so in principle (within the theory of general relativity) a Schwarzschild black hole of any mass could exist if conditions became sufficiently favorable to allow for its formation.

In the vicinity of a Schwarzschild black hole, space curves so much that even light rays are deflected, and very nearby light can be deflected so much that it travels several times around the black hole.

The Schwarzschild metric is a spherically symmetric Lorentzian metric (here, with signature convention (+ − − −)), defined on (a subset of) $${\displaystyle \mathbb {R} \times \left(E^{3}-O\right)\cong \mathbb {R} \times (0,\infty )\times S^{2}}$$ where ${\displaystyle E^{3}}$ is 3 dimensional Euclidean space, and ${\displaystyle S^{2}\subset E^{3}}$ is the two-sphere. The rotation group ${\displaystyle \mathrm {SO} (3)=\mathrm {SO} (E^{3})}$ acts on the ${\displaystyle E^{3}-O}$ or ${\displaystyle S^{2}}$ factor as rotations around the center ${\displaystyle O}$, while leaving the first ${\displaystyle \mathbb {R} }$ factor unchanged. The Schwarzschild metric is a solution of Einstein's field equations in empty space, meaning that it is valid only outside the gravitating body. That is, for a spherical body of radius ${\displaystyle R}$ the solution is valid for ${\displaystyle r>R}$. To describe the gravitational field both inside and outside the gravitating body the Schwarzschild solution must be matched with some suitable interior solution at ⁠${\displaystyle r=R}$⁠, such as the interior Schwarzschild metric.

In Schwarzschild coordinates ${\displaystyle (t,r,\theta ,\phi )}$ the Schwarzschild metric (or equivalently, the line element for proper time) has the form $${\displaystyle {ds}^{2}=c^{2}\,{d\tau }^{2}=\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)c^{2}\,dt^{2}-\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)^{-1}\,dr^{2}-r^{2}{d\Omega }^{2},}$$ where ${\displaystyle {d\Omega }^{2}}$ is the metric on the two-sphere, i.e. ⁠${\displaystyle {d\Omega }^{2}=\left(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\right)}$⁠. Furthermore,

The Schwarzschild metric has a singularity for r = 0, which is an intrinsic curvature singularity. It also seems to have a singularity on the event horizon r = r_s. Depending on the point of view, the metric is therefore defined only on the exterior region ${\displaystyle r>r_{\text{s}}}$, only on the interior region ${\displaystyle r<r_{\text{s}}}$ or their disjoint union. However, the metric is actually non-singular across the event horizon, as one sees in suitable coordinates (see below). For ⁠${\displaystyle r\gg r_{\text{s}}}$⁠, the Schwarzschild metric is asymptotic to the standard Lorentz metric on Minkowski space. For almost all astrophysical objects, the ratio ${\displaystyle {\frac {r_{\text{s}}}{R}}}$ is extremely small. For example, the Schwarzschild radius ${\displaystyle r_{\text{s}}^{({\text{Earth}})}}$ of the Earth is roughly 8.9 mm, while the Sun, which is 3.3×10⁵ times as massive has a Schwarzschild radius ${\displaystyle r_{\text{s}}^{({\text{Sun}})}}$ of approximately 3.0 km. The ratio becomes large only in close proximity to black holes and other ultra-dense objects such as neutron stars.

The radial coordinate turns out to have physical significance as the "proper distance between two events that occur simultaneously relative to the radially moving geodesic clocks, the two events lying on the same radial coordinate line".