Singularity of spacetime/singularity of the riemanian tensor

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SUMMARY

This discussion centers on the relationship between spacetime singularities and the Riemannian tensor scalar RαβμνRαβμν. It is established that while a singularity of the Riemannian tensor scalar leads to a singularity in spacetime, the reverse is not necessarily true. Specifically, a counterexample is provided with non-curvature singularities, such as conical singularities, which do not imply a singularity in the Riemannian tensor. Additionally, the Kretschmann scalar's behavior is discussed, indicating that curvature singularities can exist without a blowup in this invariant.

PREREQUISITES
  • Understanding of General Relativity (GR) concepts, particularly singularities.
  • Familiarity with the Riemannian tensor and its scalar RαβμνRαβμν.
  • Knowledge of the Kretschmann scalar and its significance in GR.
  • Concept of geodesic incompleteness in the context of spacetime.
NEXT STEPS
  • Research the definition and implications of geodesic incompleteness in General Relativity.
  • Study the properties and examples of non-curvature singularities, such as conical singularities.
  • Examine the relationship between curvature singularities and the behavior of the Kretschmann scalar.
  • Explore advanced texts on General Relativity, particularly sections discussing singularities and curvature.
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This discussion is beneficial for theoretical physicists, mathematicians specializing in differential geometry, and students of General Relativity seeking a deeper understanding of singularities and their implications in spacetime.

Neutrinos02
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Hello,

I have a question to the singularities of spacetime (where the metric tensor is infinite, but not the coordinate singularities which can be removed be a change of coordinate)
It's easy to show that a singularity of the riemanian tensor scalar RαβμνRαβμν leads to a singularity of the spacetime. But what about the other way round? Is it possible to proof that each singularity of spacetime is also a singularity of RαβμνRαβμν ?

Thanks for help.
Neutrinos
 
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Neutrinos02 said:
Hello,

I have a question to the singularities of spacetime (where the metric tensor is infinite, but not the coordinate singularities which can be removed be a change of coordinate)
It's easy to show that a singularity of the riemanian tensor scalar RαβμνRαβμν leads to a singularity of the spacetime. But what about the other way round? Is it possible to proof that each singularity of spacetime is also a singularity of RαβμνRαβμν ?

Thanks for help.
Neutrinos
I'm not 100% sure, but I think so. A true spacetime singularity (a curvature singularity actually) should pop up on the Kretschmann scalar as that point will not be well-behaved (with the geodesic incompleteness and all).
 
Neutrinos02 said:
I have a question to the singularities of spacetime (where the metric tensor is infinite, but not the coordinate singularities which can be removed be a change of coordinate)
This is actually not a correct definition of a singularity. A singularity is defined as geodesic incompleteness, meaning that there are geodesics that can't be extended past a finite affine parameter. It is neither necessary nor sufficient for the metric tensor to have a component that blows up, when expressed in some coordinate system, at some set of coordinates.

Neutrinos02 said:
It's easy to show that a singularity of the riemanian tensor scalar RαβμνRαβμν leads to a singularity of the spacetime.
This is not quite true. A blowup in a curvature scalar only indicates a curvature singularity if it can be reached along a geodesic in a finite affine parameter.

Neutrinos02 said:
But what about the other way round? Is it possible to proof that each singularity of spacetime is also a singularity of RαβμνRαβμν ?

No, this is false. A counterexample would be a non-curvature singularity such as a conical singularity. I'm pretty sure it's also possible to have a curvature singularity in which the Kretschmann invariant doesn't blow up, but I don't know of an example.

We seem to have had multiple discussions and a lot of confusion recently about the definition of singularities in GR. I have a discussion of this in section 6.3.6 of my GR book: http://www.lightandmatter.com/genrel/ .
 
bcrowell said:
I'm pretty sure it's also possible to have a curvature singularity in which the Kretschmann invariant doesn't blow up, but I don't know of an example.
How would this be possible?
 

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