Deriving The Schwarzschild Solution - Using The Field Equations To Find And

Using The Field Equations To Find And

To determine and, the vacuum field equations are employed:

Only four of these equations are nontrivial and upon simplification become:

(The fourth equation is just times the second equation)

Here dot 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.