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shinobi20

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- TL;DR Summary
- I'm reading a paper about the volume inside a black hole. Reading the abstract and the early part of the introduction, I already have questions regarding the event horizon as viewed by different observers and the uniqueness of the black hole area.

The paper is The Volume Inside a Black Hole (0801.1734)

Looking at the abstract, I have a question already.

It is stated:

I don't understand this statement, as I know the horizon ##r_H## is found by calculating the norm of the null normal vector to the event horizon (a null hypersurface) which then leads to the radial component of the metric to be ##g^{rr} = 0 \rightarrow r_H = 2GM##.

Looking at the first paragraph of the Introduction,

It is stated:

Looking at the abstract, I have a question already.

It is stated:

*Because the light rays are orthogonal to the spatial 2-dimensional surface at one instant of time, the surface of the black hole is the same for all observers (i.e. the same for all coordinate definitions of "instant of time")*I don't understand this statement, as I know the horizon ##r_H## is found by calculating the norm of the null normal vector to the event horizon (a null hypersurface) which then leads to the radial component of the metric to be ##g^{rr} = 0 \rightarrow r_H = 2GM##.

- How can light rays (null) be orthogonal to a spatial surface (spacelike tangent)?
- What is this spatial 2-dimensional surface being talked about?
- What does a light ray being orthogonal to the spatial surface have to do with the black hole surface being the same for all observers?

Looking at the first paragraph of the Introduction,

It is stated:

*Because the black hole is spherical, we simply need to measure the area in a transverse direction. This produces the unique result (Area ##= 16 \pi m^2##). The uniqueness follows because if we consider a different definition of the 3-space in which we measure the area, we just shift our points in null directions along the (null) horizon. Null directions have zero length and cannot contribute to (or change) the area.*- What does "
*the**black hole is spherical, we simply need to measure the area in a transverse direction"*mean? - Can anyone explain more about the statement on the uniqueness of the derived black hole area? If there's a different 3-space definition, why should we just shift in the null direction?

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