Flat Minkowski space-time

This solution is so simple, that it was known even before A. Einstein came up with the equations it is a solution for! It serves as a mathematical implementation of Special Relativity principles and as such is required to be ``locally present'' in any other solution for the metric of space-time.

One can describe metric $g_{\mu\nu}(x)$ by listing all its 16 components (say, in matrix form). But more conveniently one can just write down interval $ds^2$ from which all the components can be easily read off. For flat space-time we have

Both in rectangular and in spherical coordinates the metric is diagonal (terms like $dxdy$ or $drd\theta$ come in $ds^2$ with zero coefficients). But notice that it is only in rectangular coordinates (or, more correctly, ``galilean'', because time coordinate is included) the metric is coordinate independent.

This solution has the greatest possible number of symmetries: ten.

They are

4 translations	(3 in space and 1 in time coordinate),
3 rotations	(in space),
3 ``boosts''	(analogs of rotations, but involving time).

It has no curvature singularities. And more than that, the curvature tensor identically vanishes: $R_{\mu\nu\alpha\beta}(x)=0$ (easy to see in galilean coordinates, because curvature tensor depends only on derivatives of the metric; and it is zero in any other coordinates, because it transforms as a tensor).