shows that the metric becomes singular approaching the event horizon (that is, ). The metric singularity is not a physical one (although there is a real physical singularity at ), as can be shown by using a suitable coordinate transformation (e.g. the Kruskal–Szekeres coordinate system). Alternate derivation using known physics in special cases The Schwarzschild metric can also be derived using the known physics for a circular orbit and a temporarily stationary point mass. Start with the metric with coefficients that are unknown coefficients of : Now apply the Euler–Lagrange equation to the arc length integral Since is constant, the integrand can be replaced with because the E–L equation is exactly the same if the integrand is multiplied by any constant. Applying the E–L equation to with the modified integrand yields: where dot denotes differentiation with respect to In a circular orbit so the first E–L equation above is equivalent to Kepler's third law of motion is In a circular orbit, the period equals implying since the point mass is negligible compared to the mass of the central body So and integrating this yields where is an unknown constant of integration. can be determined by setting in which case the spacetime is flat and So and When the point mass is temporarily stationary, and The original metric equation becomes and the first E–L equation above becomes When the point mass is temporarily stationary, is the acceleration of gravity, So Alternative form in isotropic coordinates The original formulation of the metric uses anisotropic coordinates in which the velocity of light is not the same in the radial and transverse directions. Arthur Eddington gave alternative forms in isotropic coordinates. For isotropic spherical coordinates , , , coordinates and are unchanged, and then (provided ) , , and Then for isotropic rectangular coordinates , , , The metric then becomes, in isotropic rectangular coordinates: Dispensing with the static assumption – Birkhoff's theorem 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. See also Karl Schwarzschild Kerr metric Reissner–Nordström metric References Exact solutions