Conditions for curvature singularity

[FunkyDwarf]

Hi All,

I was wondering if it is correct to say that a vanishing metric determinant is a necessary (but probably not sufficient) condition for a curvature singularity to exist at some point(s), or is one forced to construct the full Kretschmann scalar?

Cheers!
FD

 

[bcrowell]

It's neither necessary nor sufficient.

It can't be necessary, because if there is a curvature singularity, then the singularity isn't even considered to be a point in the spacetime. Therefore there is no way to talk about the metric's determinant at that "point." The equivalence principle guarantees that even at points that lie at an arbitrarily small affine distance from the singularity, the metric's determinant is perfectly normal.

It's not sufficient, because for example it's possible to have a flat spacetime and have its metric determinant become degenerate at any arbitrarily chosen point, if you make a perverse choice of coordinates. Here's an example: http://lightandmatter.com/html_books/genrel/ch06/ch06.html#Section6.4

If the Kretschmann scalar blows up, that's sufficient but not necessary to prove a curvature singularity.

The standard formulation of GR can't even describe changes of signature, although some others, like the Ashtekar formulation, apparently can. It's not obvious to me that this can have any physical significance, because it's not obvious to me that the signature is measurable. E.g., if the signature was not +--- but ---- somewhere, then there would be no timelike dimension there, but I don't think you can measure anything if there's no time. Measurement requires collecting and processing information, so that at some later time you have more information than you had before.

 

[FunkyDwarf]

Hi bcrowell, thanks for the reply/info.

What would then be the minimal condition to prove a curvature singularity (as opposed ofc to a coordinate singularity).

 

[bcrowell]

Hi bcrowell, thanks for the reply/info.

What would then be the minimal condition to prove a curvature singularity (as opposed ofc to a coordinate singularity).

If you're asking for a technique for testing for a singularity, I don't think there's any simple answer that works in all cases. Carroll has a good discussion of this, IIRC. There is a free online version of his book.

If you're asking for a definition, then I think the standard definition is that a singularity exists when you have geodesic incompleteness, meaning that some geodesics can't be extended past a certain finite affine parameter.

 

[FunkyDwarf]

Ok, I have his notes somewhere and can have a dig around, thanks!