Usefulness of Kretschmann scalar

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In summary, the divergence of the curvature scalar at a point proves the existence of a true gravitational singularity. It is not necessary, however, for the divergence to become unbounded.
  • #1

As I'm sure you are aware the Kretschmann scalar (formed by contracting the contravariant and covariant Riemann tensors) has some use in the identification of gravitational singularities. Specifically, because K is essentially the sum of all permutations of R's components, but is itself coordinate invariant, its divergence at a point is sufficient to prove the existence of a true gravitational singularity at that point.

I am wondering whether this is a necessary condition as well. It seems to me that it is not, since one could imagine a situation in which two terms in the sum diverge in opposite directions. Perhaps I have missed something, though?
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  • #2
In fact, singularities are extremely hard to define generally. The divergence of curvature scalars is not always sufficient, nor necessary conditions for us to want to classify a specific space-time as "singular".

An example of the "insufficient" case, it could be that the divergence of the curvature only blows up "at infinity", so that no observer could ever get to this singularity. On the other hand, there are some spacetimes which have vanishing curvature everywhere but is still singular.

Whether the curvature becomes unbounded or not is only one of the "indicators" we use to figure out if a singularity exists or not.
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  • #3
I've only used it in sense of a rough measure of 'amount of curvature' (no possible scalar is adequate, otherwise we wouldn't need a tensor), that is more useful than R in GR because it is typically non-zero in vacuum regions of GR solutions, while R is identically zero in vacuum regions. I've never run across any theorems using it to draw significant non-obvious conclusions (example of obvious: non-vanishing K implies manifold not flat). (Don't take my not having seen any as a very strong statement).
  • #4
Okay, I can see that K diverging at infinity would not be so unexpected. So what you're saying is that if K blows up at some ordinary point in your coordinate system (which I guess might map to infinity in some other coordinate system), that point isn't necessarily a gravitational singularity, but it is a point you might want to investigate in more detail, correct?

Would the following be sufficient to demonstrate the existence of a singularity:
1) K diverges at a point p in a coordinate system Q
2) It is possible to map Q onto some new coordinate system S, with p going to a set of points P.
3) S is maximally extended.
4) In S, P is timelike separated from some class of observers.

PAllen: You mean either Rab (the Ricci tensor) or R (the Ricci scalar), I'm assuming (my fault for not subscripting before). Rabcd (the Riemann tensor) need not be zero in vacuum, I don't think?
  • #5
HolyCats said:
PAllen: You mean either Rab (the Ricci tensor) or R (the Ricci scalar), I'm assuming (my fault for not subscripting before). Rabcd (the Riemann tensor) need not be zero in vacuum, I don't think?

Since we were discussing scalars, I assumed Ricci scalar was obvious by context.
  • #6
Yeah, my bad. Anyways thanks.

1. What is the Kretschmann scalar?

The Kretschmann scalar is a mathematical quantity used in general relativity to measure the curvature of spacetime at a specific point. It is named after German mathematician Erich Kretschmann.

2. How is the Kretschmann scalar calculated?

The Kretschmann scalar is calculated by taking the square of the Riemann curvature tensor and contracting it with itself using the metric tensor. This results in a single value that represents the amount of curvature at a specific point in spacetime.

3. What does the Kretschmann scalar tell us about spacetime?

The Kretschmann scalar provides information about the strength of the gravitational field at a specific point in spacetime. It can also indicate the presence of singularities, which are points of infinite curvature and are associated with black holes.

4. How is the Kretschmann scalar useful in physics?

The Kretschmann scalar is useful in general relativity as it allows us to calculate the curvature of spacetime at a specific point, which is essential in understanding the behavior of gravitational fields. It also helps us identify regions of extreme curvature, such as black holes.

5. Can the Kretschmann scalar be used in other fields of science?

Yes, the Kretschmann scalar has applications in other fields of science, such as cosmology and astrophysics. It is also used in theoretical physics to study the properties of spacetime and the behavior of matter and energy in extreme conditions.