# Questions about the Kretschmann curvature scalar

## Main Question or Discussion Point

The Kretschmann curvature scalar is defined to be K = RabcdRabcd where Rabcd 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?

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tiny-tim
Homework Helper
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.

• gustavo_vasques
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...