 An alternate approach would be to use differential forms, but the approach above is computationally more convenient as well as not requiring the reader to understand differential forms.A lengthy, but straightforward (with computer algebra) calculation from our assumed line element shows us that

formula_19

Thus we can write

formula_20

In any vacuum region of space-time, all components of the Ricci tensor must be zero.This demonstrates that enclosing any amount of vacuum will not change our volume integral.