Why Scalar Quantities Matter in Singularity Tests

[binbagsss]

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.

 

[binbagsss]

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?

 

[Dale]

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

 

[George Jones]

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?

 

[martinbn]

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.

 

[binbagsss]

You need some norm on the tensors, /QUOTE]

Makes sense