Minkowski Metric

In Cartesian Coordinates,

$$ds^2 = dx^2+dy^2+dz^2$$ (1)

$$d\tau^2 = -c^2\,dt^2+dx^2+dy^2+dz^2,$$ (2)

and

$$g_{\alpha\beta}\equiv \eta_{\alpha\beta} =\left[{\matrix{ -... ...0 & 1 & 0 & 0\cr 0 & 0 & 1 & 0\cr 0 & 0 & 0 & 1\cr}}\right].$$ (3)

In Spherical Coordinates,

$$ds^2 = dr^2+r^2\,d\theta^2+r^2\sin^2\theta\,d\phi^2$$ (4)

$$d\tau^2 = -c^2\,dt^2+dr^2+r^2\,d\theta^2+r^2\sin^2\theta\,d\phi^2,$$ (5)

and

$$g = \left[{\matrix{ -1 & 0 & 0 & 0\cr 0 & 1 & 0 & 0\cr 0 & 0 & r^2 & 0\cr 0 & 0 & 0 & r^2\sin^2\theta}}\right].$$ (6)

See also Lorentz Transformation, Minkowski Space