# Interpreting the Kretschmann scalar

How do you interpret the Kretschmann scalar (in general relativity)? What can you tell from it?

The Kretschmann scalar is defined as

$$K = R_{abcd} R^{abcd}$$

where $R_{abcd}$ is the Riemann curvature tensor.

## 2 Answers

The Kretschmann scalar can be used as an indicator of curvature singularities in the manifold. For instance, in the Schwarzschild black hole (given in the Wikipedia link in your post), $$K\propto\frac1{r^6}$$ so as $r\to0$, $K\to\infty$.

• Is there a reasonable interpretation for other values of $K$? Such as when it vanishes, $K = 0$? – XYZT Dec 2 '14 at 19:15
• Sure: there are two other cases: (1) $K=0$, then there's no curvature (2) if $K>0$ (but finite), then the manifold is not flat. – Kyle Kanos Dec 2 '14 at 19:20
• Is (1) necessarily true? – XYZT Dec 2 '14 at 19:20
• Actually, I'm not 100% sure on that note. I presume yes, but I'm willing to be corrected. – Kyle Kanos Dec 2 '14 at 19:26
• @KyleKanos For pseudo-Riemannian manifolds, you can have non-zero null curvature tensors, just like you can have nonzero, null vectors. – asperanz Dec 3 '14 at 4:11

For vacuum solutions, since the Ricci tensor $R_{ab}$ vanishes, the Kretschmann scalar is equal to the norm of the Weyl tensor, $K = C_{abcd}C^{abcd}$. This means it is telling you something about the tidal forces at a given point. I might use $K^{1/2}$ to characterize the strength of the tidal forces. This can be used in Schwarzschild or Kerr spacetimes to see that the tidal forces go like $M/r^3$ (at least in the equatorial plane for Kerr).

• This is much better than the accepted answer. – Ben Crowell Feb 22 '18 at 1:08