Questions about the Kretschmann curvature scalar

[student85]

The Kretschmann curvature scalar is defined to be K = R_abcdR^abcd where R_abcd is the Riemann tensor. I believe I heard in class that this scalar can be used to demonstrate the existence of a curvature singularity. Can somebody tell me why this is so? Also, I heard that it is better (easier I suppose) to use this Kretschmann curvature scalar rather than examining the Riemann curvature tensor component by component, to achieve this conclusion.
Can someone help explain why (and how maybe?) we can demonstrate the existence of the singularity in the first place?

Thanks in advance.

 

[tiny-tim]

Hi student85!

See http://en.wikipedia.org/wiki/Schwarzschild_metric#Singularities_and_black_holes:

The case r = 0 is different, however. If one asks that the solution be valid for all r one runs into a true physical singularity, or gravitational singularity, at the origin. To see that this is a true singularity one must look at quantities that are independent of the choice of coordinates. One such important quantity is the Kretschmann invariant, which is given by

48G²M²/c⁴r⁶​

At r = 0 the curvature blows up (becomes infinite) indicating the presence of a singularity.

 

[student85]

Hey tiny-tim, thanks!
I think you have helped me with several questions in the past, and you're always so kind. You deserve a prize! Oh wait, you do have some awards there I see :p. Anyway, thanks again...