Why Scalar Quantities Matter in Singularity Tests

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Discussion Overview

The discussion revolves around the significance of scalar quantities in the context of singularity tests within General Relativity (GR), particularly in Schwarzschild space-time. Participants explore the necessity of using scalars over tensors when identifying quantities that exhibit singular behavior.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants propose that scalar quantities are preferred because they remain invariant across all coordinate systems, making them reliable indicators of singularities.
  • Others argue that while tensors are used in GR for their physical invariance, the specific nature of singularities may necessitate the use of scalars to clearly identify when a quantity diverges.
  • A participant questions the reasoning behind the necessity of scalars, suggesting that tensors could also suffice, but acknowledges the challenge in defining what it means for a tensor to "blow up."
  • There is a discussion about the components of the metric tensor near the event horizon and the curvature scalar, indicating a search for specific values in that context.
  • One participant notes that for a tensor to exhibit divergence, one would need to establish a norm, which leads back to scalar quantities.

Areas of Agreement / Disagreement

Participants express differing views on whether scalars are strictly necessary for identifying singularities, with some supporting the use of scalars while others suggest that tensors could also be applicable. The discussion remains unresolved regarding the necessity and sufficiency of scalars versus tensors.

Contextual Notes

Participants highlight the complexity of defining divergence in tensor quantities, which may depend on the choice of coordinate systems and norms. There is also an exploration of specific metrics and curvature scalars, but no consensus is reached on these aspects.

binbagsss
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Apologies if this is a stupid question, so for e.g, in a Schwarzschild space-time we look at ##R^{abcd}R_{abcd} ## (we seek some scalar quantity that blows-up and can not use ##R##as we are looking at vacuum solutions so we know this is zero)

The reason we seek a scalar is because it is the same in all coordinate systems, if it blows up in one, it blows up in all.

But, in GR we aim to write the laws of physics as tensor expressions? so that it's physical meaning is invariant under coordinate transformations-so doesn't this say a tensor would suffice?

thanks.
 
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binbagsss said:
But, in GR we aim to write the laws of physics as tensor expressions? so that it's physical meaning is invariant under coordinate transformations-so doesn't this say a tensor would suffice?

the physical meaning is the same, but not apparent in all coordinate systems?
whereas vector components will vary coordinate system to coordinate system, a scalar will not, and so makes the easiest check or?
 
binbagsss said:
so doesn't this say a tensor would suffice?
Hmm, that is a good point! I know that it is a scalar which blows up at the singularity, but I don’t understand the reasoning that it must be a scalar
 
binbagsss said:
Apologies if this is a stupid question, so for e.g, in a Schwarzschild space-time we look at ##R^{abcd}R_{abcd} ## (we seek some scalar quantity that blows-up and can not use ##R##as we are looking at vacuum solutions so we know this is zero)

The reason we seek a scalar is because it is the same in all coordinate systems, if it blows up in one, it blows up in all.

But, in GR we aim to write the laws of physics as tensor expressions? so that it's physical meaning is invariant under coordinate transformations-so doesn't this say a tensor would suffice?.

In standard spherical coordinates coordinates, what are the components of the metric tensor "near" the event horizon? What is the curvature scalar there?
 
binbagsss said:
doesn't this say a tensor would suffice?
What would it mean for a tensor to blow up? Some of the components? That would be coordinate dependent. You need some norm on the tensors, that would lead to scalars. In any case you can say that a real valued quantity gets larger and larger, but for a tensor valued one it makes no sense.
 
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martinbn said:
You need some norm on the tensors, /QUOTE]

Makes sense
 

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