Understand Degeneracy of Metric at Schwarzschild Singularity

[ChrisVer]

I am reading this paper
http://arxiv.org/pdf/1111.4837.pdf
and I came across under eq12 that the new metric is degenerate...
How can someone see that from the metric's form?
Degeneracy for a metric means that it has at least 2 same eigenvalues (but isn't that the same for the Minkowski metric since it's diagonal with 3 times -1 eigenvalues)? Or that you can define two different metric tensors $g$ and $\bar{g}$ which keep $ds^{2}$ invariant?

Thanks

 

[Bill_K]

Degeneracy for a metric means that det(g_ab) = 0. In this case, g_ττg_ξξ - g_τξ² = 0.

 

[ChrisVer]

Oh well then that means that my first choice was the correct one about Eigenvalues...
and why is the degenerate metric singular? or put in another manner, what does singular metric mean?

 

[Bill_K]

Oh well then that means that my first choice was the correct one about Eigenvalues...
and why is the degenerate metric singular? or put in another manner, what does singular metric mean?

For the Minkowski metric, the eigenvalues are -1, +1, +1, +1 and det(g) = -1. Det(g) is a scalar density, transforming under a coordinate transformation as Det(g') = |∂x/∂x'|² Det(g) where |∂x/∂x'| is the Jacobian of the transformation. Consequently there exists no nonsingular transformation that can take a singular metric to Minkowski. Wherever Det(g) = 0, the spacetime is not locally Minkowskian.

 

[sardar]

for bringing this paper to my attention. The degeneracy of the metric at the Schwarzschild singularity is an interesting and important concept in understanding the behavior of spacetime in the vicinity of a black hole.

Firstly, let's define what we mean by degeneracy in this context. A degenerate metric means that there are multiple ways to construct the metric tensor, or multiple sets of coordinates, that result in the same spacetime geometry. In other words, there are multiple ways to describe the same physical phenomenon using different mathematical formulations.

In the context of the Schwarzschild singularity, the degeneracy of the metric arises because the singularity itself is a coordinate singularity. This means that the equations used to describe the spacetime geometry break down at the singularity, and we are left with a point where the metric is not well-defined. This is similar to the singularity at the center of a black hole, where the equations of general relativity break down and we are unable to describe the spacetime geometry.

Now, let's take a look at equation 12 in the paper you linked. This equation shows the new metric that is proposed for describing the geometry at the Schwarzschild singularity. It is important to note that this metric is only valid in a specific coordinate system, known as the Painlevé-Gullstrand coordinates. This metric is degenerate in the sense that there are multiple sets of coordinates that can be used to describe the same geometry at the singularity. This is different from the Minkowski metric, which is not degenerate because there is only one set of coordinates that can be used to describe flat spacetime.

In conclusion, the degeneracy of the metric at the Schwarzschild singularity is a result of the breakdown of the equations used to describe the spacetime geometry. This degeneracy highlights the importance of using different coordinate systems and mathematical formulations to fully understand the behavior of spacetime in extreme situations such as black holes.