Example of curvature scalar diverging at infinity?

[jinawee]

Reading Geroch's "What is a Singularity in General Relativity?", it seems that polynomial scalar invariants constructed from the Riemann tensor can diverge if we are at infinite distance, and not in a true singularity.

Can someone give an example of space-time whose scalar invariant diverges at infinity?

 

[Ambitwistor]

Sure, here is an example of a space-time whose scalar invariant diverges at infinity:

Consider the Schwarzschild metric, which describes the space-time around a non-rotating, spherically symmetric mass. The scalar invariant in this case is the Kretschmann scalar, given by:

K = R_abcdR^abcd

where R_abcd is the Riemann tensor. In this metric, as we approach infinite distance (r → ∞), the Kretschmann scalar diverges, indicating a singularity at infinity. This can be seen by calculating the Kretschmann scalar at different values of r:

At r = 2M (where M is the mass of the object), K = 48M^2/r^6
As r → ∞, K → ∞

This shows that the scalar invariant diverges at infinity in the Schwarzschild metric, indicating the presence of a singularity at infinite distance.