Singularity of spacetime/singularity of the riemanian tensor

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Discussion Overview

The discussion revolves around the nature of singularities in spacetime and their relationship with the Riemannian tensor, specifically questioning whether every singularity of spacetime corresponds to a singularity in the Riemannian scalar RαβμνRαβμν. The scope includes theoretical aspects of general relativity and the definitions of singularities.

Discussion Character

  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant states that a singularity of the Riemannian tensor scalar RαβμνRαβμν leads to a singularity of spacetime, but questions the reverse implication.
  • Another participant suggests that a true spacetime singularity should be evident in the Kretschmann scalar due to geodesic incompleteness.
  • A different participant challenges the definition of singularity, asserting that geodesic incompleteness is the key criterion, rather than the behavior of the metric tensor in a coordinate system.
  • This participant also claims that not every singularity of spacetime corresponds to a singularity in RαβμνRαβμν, providing the example of a non-curvature singularity, such as a conical singularity.
  • There is a mention of the possibility of a curvature singularity where the Kretschmann invariant does not blow up, although no specific examples are provided.
  • Several participants express uncertainty and seek clarification on the conditions under which a curvature singularity may exist without a corresponding blowup in the Kretschmann scalar.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between spacetime singularities and the Riemannian tensor. There is no consensus on whether every singularity of spacetime must correspond to a singularity in RαβμνRαβμν, and the discussion remains unresolved.

Contextual Notes

Participants highlight limitations in definitions and examples, indicating that the understanding of singularities in general relativity is complex and may depend on specific conditions or interpretations.

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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