Lorentzian Metrics From Relativity
In flat Minkowski space (special relativity), with coordinates the metric is
For a curve with—for example—constant time coordinate, the length formula with this metric reduces to the usual length formula. For a timelike curve, the length formula gives the proper time along the curve.
In this case, the spacetime interval is written as
- .
The Schwarzschild metric describes the spacetime around a spherically symmetric body, such as a planet, or a black hole. With coordinates, we can write the metric as
where G (inside the matrix) is the gravitational constant and M the mass of the body.
Read more about this topic: Metric Tensor, Examples