Deriving The Schwarzschild Solution - Diagonalising The Metric

Diagonalising The Metric

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 (metric components remain the same), this means that:

Similarly, the coordinate transformations and respectively give:

Putting all these together gives:

and hence the metric (line element) must be of the form:

where the four metric components are independent of the time coordinate (by the static assumption).