Big Bang: A True Singularity That is Coordinate Independent

[victorvmotti]

Consider a flat Robertson-Walker metric.

When we say that there is a singularity at

$$t=0$$

Clearly it is a coordinate dependent statement. So it is a "candidate" singularity.

In principle there is "another coordinate system" in which the corresponding metric has no singularity as we approach that point in the manifold.

However, we know that Big Bang is "a true" singularity, but how should we test that?

Is it intuitively self-evident, or should we check rigorously all scalars based on the Ricci tensor? If so "which order of scalar" goes to infinity at that point called Big Bang?

 

[bapowell]

The idea is to examine a curvature invariant: a quantity built out of the various curvature tensors that is diffeomorphism invariant (unchanged by coordinate transformation). These should be infinite at true curvature singularities. One such invariant, the Kretschmann scalar, is found from the Riemann tensor, $K = R_{\mu \nu \rho \sigma}R^{\mu \nu \rho \sigma}$.