Schwarzschildtype

Schwarzschild-type, in general relativity, refers to a class of static, spherically symmetric spacetime metrics that generalize the Schwarzschild solution. These spacetimes describe non-rotating, gravitating systems such as black holes or stellar interiors and are characterized by a line element that, in standard spherical coordinates, can be written as ds^2 = -f(r) dt^2 + g(r) dr^2 + r^2(dθ^2 + sin^2θ dφ^2), where f(r) and g(r) are positive functions of the radial coordinate r outside horizons. An equivalent form introduces a mass function m(r): ds^2 = -e^{2Φ(r)} dt^2 + [1-2m(r)/r]^{-1} dr^2 + r^2(dθ^2 + sin^2θ dφ^2), with Φ(r) and m(r) determined by the matter content via the Einstein field equations.

Common examples include the Schwarzschild solution (f(r)=1-2M/r in geometric units), the Reissner–Nordström solution for charged bodies, and Schwarzschild–de Sitter or anti-de Sitter spacetimes with a cosmological constant. More generally, any static, spherically symmetric solution of the vacuum or nonvacuum Einstein equations that can be cast in these forms is described as Schwarzschild-type.

Key properties include the presence or absence of horizons where f(r)=0, potential physical singularities at r=0, and asymptotic behavior determining whether the spacetime is asymptotically flat, de Sitter, or anti-de Sitter. These metrics are used to model non-rotating black holes, relativistic stars, and as backgrounds in tests of alternative theories of gravity.

See also: Schwarzschild solution, horizons, Einstein field equations, static spacetimes.