Deriving The Schwarzschild Solution - Using The Weak-Field Approximation To Find And

Using The Weak-Field Approximation To Find And

The geodesics of the metric (obtained where is extremised) must, in some limit (e.g., toward infinite speed of light), agree with the solutions of Newtonian motion (e.g., obtained by Lagrange equations). (The metric must also limit to Minkowski space when the mass it represents vanishes.)

(where KE is the kinetic energy and PE_g is the Potential Energy due to gravity) 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:

and

Hence:

and

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 approching 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).