Шварцшильд шешімі: Сфералық симметриялы кеңістік-уақыт

The Schwarzschild solution describes spacetime under the influence of a massive, non rotating, spherically symmetric object. It is considered by some to be one of the simplest and most useful solutions to the Einstein field equations .

Working in a coordinate chart with coordinates labelled 1 to 4 respectively, we begin with the metric in its most general form (10 independent components, each of which is a smooth function of 4 variables). The solution is assumed to be spherically symmetric, static and vacuum. For the purposes of this article, these assumptions may be stated as follows (see the relevant links for precise definitions):

A spherically symmetric spacetime is one that is invariant under rotations and taking the mirror image. A static spacetime is one in which all metric components are independent of the time coordinate (so that ) and the geometry of the spacetime is unchanged under a time reversal A vacuum solution is one that satisfies the equation From the Einstein field equations (with zero cosmological constant), this implies that since contracting yields Metric signature used here is (+,+,+,−).

Using the metric above, we find the Christoffel symbols, where the indices are The sign denotes a total derivative of a function.

In deriving the Schwarzschild metric, it was assumed that the metric was vacuum, spherically symmetric and static. The static assumption is unneeded, as Birkhoff's theorem states that any spherically symmetric vacuum solution of Einstein's field equations is stationary; the Schwarzschild solution thus follows. Birkhoff's theorem has the consequence that any pulsating star that remains spherically symmetric does not generate gravitational waves, as the region exterior to the star remains static.