Working in a coordinate chart with coordinates labelled 1 to 4 respectively, we begin with the metric in its most general form. The solution is assumed to be spherically symmetric, static and vacuum. For the purposes of this article, these assumptions may be stated as follows :
A static spacetime is one in which all metric components are independent of the time coordinate and the geometry of the spacetime is unchanged under a time-reversal.
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 , 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 .
Simplifying the components
On each hypersurface of constant, constant and constant , should only depend on . 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 metric components restricted to this hypersurface should be unchanged under rotations through and . Comparing the forms of the metric on this hypersurface gives: which immediately yields: But this is required to hold on each hypersurface; hence, An alternative intuitive way to see that and must be the same as for a flat spacetime is that stretching or compressing an elastic material in a spherically symmetric manner will not change the angular distance between two points. Thus, the metric can be put in the form: with and as yet undetermined functions of. Note that if or is equal to zero at some point, the metric would be singular at that point.
Using the metric above, we find the Christoffel symbols, where the indices are. The sign denotes a total derivative of a function.
Using the field equations to find ''A(r)'' and ''B(r)''
To determine and, the vacuum field equations are employed: Hence: where a comma is used to set off the index that is being used for the derivative. Only three of these equations are nontrivial and upon simplification become: , where the prime means the r derivative of the functions. Subtracting the first and third equations produces: where is a non-zero real constant. Substituting into the second equation and tidying up gives: which has general solution: for some non-zero real constant. Hence, the metric for a static, spherically symmetric vacuum solution is now of the form: Note that the spacetime represented by the above metric is asymptotically flat, i.e. as, the metric approaches that of the Minkowski metric and the spacetime manifold resembles that of Minkowski space.
The geodesics of the metric must, in some limit, agree with the solutions of Newtonian motion. The constants and are fully determined by some variant of this approach; from the weak-field approximation one arrives at the result: where is the gravitational constant, is the mass of the gravitational source and is the speed of light. It is found that: Hence: So, the Schwarzschild metric may finally be written in the form: Note that: is the definition of the Schwarzschild radius for an object of mass, so the Schwarzschild metric may be rewritten in the alternative form: which shows that the metric becomes singular approaching the event horizon. The metric singularity is not a physical one, as can be shown by using a suitable coordinate transformation.
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 stationarypoint 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 space-time 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
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 Then for isotropic rectangular coordinates,,, The metric then becomes, in isotropic rectangular coordinates:
In deriving the Schwarzschild metric, it was assumed that the metric was vacuum, spherically symmetric and static. In fact, the static assumption is stronger than required, as Birkhoff's theorem states that any spherically symmetric vacuum solution of Einstein's field equations is stationary; then one obtains the Schwarzschild solution. Birkhoff's theorem has the consequence that any pulsating star which remains spherically symmetric cannot generate gravitational waves.
