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 . Assumptions and notation 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 (+,+,+,−). Diagonalising the metric The first simplification to be made is to diagonalise the metric. Under the coordinate transformation, , all metric components should remain the same. The metric components () change under this transformation as: () But, as we expect (metric components remain the same), this means that: () Similarly, the coordinate transformations and respectively give: () () Putting all these together gives: () and hence the metric must be of the form: where the four metric components are independent of the time coordinate (by the static assumption). Simplifying the components On each hypersurface of constant , constant and constant (i.e., on each radial line), should only depend on (by spherical symmetry). Hence is a function of a single variable: A similar argument applied to shows that: On the hypersurfaces of constant and constant , it is required that the metric be that of a 2-sphere: Choosing one of these hypersurfaces (the one with radius , say), the metric components restricted to this hypersurface (which we denote by and ) should be unchanged under rotations through and (again, by spherical symmetry). Comparing the forms of the metric on this hypersurface gives: which immediately yields: and But this is required to hold on each hypersurface; hence, and An alternative intuitive way to see that and must be