Initial conditions for BH in GR

[unih]

HI all
There is smth I don't understand in building initial data for BH
The equations everybody uses uses spacetime or timelike hypersurfaces but for example horizon is a null surface!

 

[atyy]

I think the basic idea is to find coordinates which are "nice" and penetrate the horizon.

These guys use Painleve-Gullstrand coordinates: http://arxiv.org/abs/0812.0993.

Cook has many references in his review: http://relativity.livingreviews.org/Articles/lrr-2000-5/ . He starts with ones that get stuck at the horizon, then describes a bunch to can see through it.

 

[PeterDonis]

The equations everybody uses uses spacetime or timelike hypersurfaces but for example horizon is a null surface!

The horizon is a null surface but there are still plenty of timelike and spacelike hypersurfaces that pass through it. As atyy says, you just have to pick a coordinate chart that is nonsingular at the horizon to be able to specify spacelike or timelike surfaces that pass through it.

Also, I'm not sure whether by "initial conditions" you mean initial conditions to specify an "eternal" BH spacetime (one that exists forever) or initial conditions to specify a spacetime in which a BH is formed by the collapse of a massive object (such as the Oppenheimer-Snyder solution for perfectly spherical collapse). For the latter type of spacetime, you can specify initial conditions on a spacelike hypersurface that doesn't include the horizon (because the hypersurface is taken at a time before the horizon forms.

 

[unih]

Thank you veryu much for your answer
Yes, I saw review by Cook, and still doesn't understand it.
Lets take for example the classical article of Shapiro and co
http://prd.aps.org/abstract/PRD/v38/i10/p2972_1

They are looking for apperent horizon, that they define as a marginal surface witha null expansion. Ok. Thats I understand. But when they write Gauss- Codazzi equations they write it for three surface (spacelike) and not for null surface. Thats I don't understand

 

[unih]

Thank you veryu much for your answer
Yes, I saw review by Cook, and still doesn't understand it.
Lets take for example the classical article of Shapiro and co
http://prd.aps.org/abstract/PRD/v38/i10/p2972_1

They are looking for apperent horizon, that they define as a marginal surface witha null expansion. Ok. Thats I understand. But when they write Gauss- Codazzi equations they write it for three surface (spacelike) and not for null surface. Thats I don't understand

 

[atyy]

A spacelike 3-surface can contain both spacelike and null 2-surfaces.

 

[PeterDonis]

A spacelike 3-surface can contain both spacelike and null 2-surfaces.

Is this true? I understand that one can "cut" a 2-surface out of a 3-surface at different "angles". But if a 3-surface is spacelike, then all tangent vectors in the surface are spacelike. And the tangent vectors of any 2-surface cut out of the 3-surface must be expressible as linear combinations of tangent vectors in the 3-surface. There is no way to make a null vector out of linear combinations of spacelike vectors. So I don't think you can cut a null 2-surface out of a spacelike 3-surface; the 3-surface has to have at least one timelike dimension.

 

[PeterDonis]

Lets take for example the classical article of Shapiro and co
http://prd.aps.org/abstract/PRD/v38/i10/p2972_1

They are looking for apperent horizon, that they define as a marginal surface witha null expansion. Ok. Thats I understand. But when they write Gauss- Codazzi equations they write it for three surface (spacelike) and not for null surface. Thats I don't understand

This paper is behind a paywall that I don't have access to (probably many others here don't either). Can you be more specific about their definition of apparent horizon? As I understand it, an apparent horizon is a congruence of null geodesics with zero expansion; but such a congruence can be described using a chart with one timelike and one or more spacelike dimensions (depending on whether the angular spacelike coordinates are included or not). So by writing equations in such a chart you could still describe the congruence in terms of the "time evolution" of its intersection with a foliation of spacelike surfaces, even though it's a congruence of null geodesics. For example, you can describe a black hole horizon using the Kruskal (T, X) chart.

 

[atyy]

Is this true? I understand that one can "cut" a 2-surface out of a 3-surface at different "angles". But if a 3-surface is spacelike, then all tangent vectors in the surface are spacelike. And the tangent vectors of any 2-surface cut out of the 3-surface must be expressible as linear combinations of tangent vectors in the 3-surface. There is no way to make a null vector out of linear combinations of spacelike vectors. So I don't think you can cut a null 2-surface out of a spacelike 3-surface; the 3-surface has to have at least one timelike dimension.

Yes, that seems right. Perhaps it is only the union of 2-surfaces on several spacelike 3-surfaces that can be null, and that is what is referred to as a null apparent horizon? I was thinking of a statement in Poisson's http://www.physics.uoguelph.ca/poisson/research/agr.pdf in which he says an apparent horizon can be null (p133, section 5.1.8), but now I see he uses apparent horizon to describe both apparent horizons and trapping horizons as "apparent horizons" (p132, section 5.1.7).

There's a similar comment about terminology in section 7.2, footnote 21 of Thornburg's http://relativity.livingreviews.org/Articles/lrr-2007-3/ . There he says that it is the union of apparent horizons (defined as 2-surfaces within spacelike 3-surfaces) that can be spacelike or null.

Looking at http://arxiv.org/abs/gr-qc/9606010v1 together with the Poisson and Thornburg reviews, it looks like an apparent horizon is a spacelike 2-surface, not a null 2-surface. It is defined by light rays orthogonal to it, not tangent to it. Then it is only the union of 2-surfaces from many spacelike 3-surfaces that can be described as a conguence of spacelike or null geodesics.

 

[PeterDonis]

I was thinking of a statement in Poisson's http://www.physics.uoguelph.ca/poisson/research/agr.pdf in which he says an apparent horizon can be null (p133, section 5.1.8), but now I see he uses apparent horizon to describe both apparent horizons and trapping horizons as "apparent horizons" (p132, section 5.1.7).

Yes, he uses "apparent horizon" in different ways: to describe just the 2-surface which is the boundary of the "trapping region" within a single spacelike hypersurface (as on p. 132), and also to describe the 3-surface formed by "stacking together" all the apparent horizon 2-surfaces from each spacelike 3-surface in the spacetime. (On p. 132 Poisson says that this "sloppiness of language is fairly standard", so the dual usage of the term is not inadvertent.) The latter 3-surface is null (at least, it is when the spacetime is stationary; he points out that it will have a short spacelike portion when matter is falling into the black hole and increasing its mass), but the individual 2-surfaces are spacelike (they are 2-spheres at r = 2M for a stationary black hole).

In my previous post I was using "apparent horizon" to describe the null 3-surface (in the case of a stationary black hole), which can also be described, as I did, as a congruence of null geodesics whose expansion is zero.

There's a similar comment about terminology in section 7.2, footnote 21 of Thornburg's http://relativity.livingreviews.org/Articles/lrr-2007-3/ . There he says that it is the union of apparent horizons (defined as 2-surfaces within spacelike 3-surfaces) that can be spacelike or null.

This appears to fit with Poisson's comment that I mentioned above, where the 3-surface formed by the union of all the 2-surfaces will be null if the spacetime is stationary, but spacelike if there is additional matter falling into the hole.

Looking at http://arxiv.org/abs/gr-qc/9606010v1 together with the Poisson and Thornburg reviews, it looks like an apparent horizon is a spacelike 2-surface, not a null 2-surface.

This would be the first usage of "apparent horizon" from above (a single 2-surface cut out of a single spacelike 3-surface, not the union of all of them). I don't think there's ever a case where there is a null 2-surface involved; the union of all the spacelike 2-surfaces is a null 3-surface when the spacetime is stationary. As I noted above, Poisson, at least, says that both usages of the term "apparent horizon" are "fairly standard".

It is defined by light rays orthogonal to it, not tangent to it.

Agreed.

Then it is only the union of 2-surfaces from many spacelike 3-surfaces that can be described as a conguence of spacelike or null geodesics.

Agreed.

 

[unih]

Can you be more specific about their definition of apparent horizon?.

Thats smthh with the same idea
arXiv:gr-qc/9904054v2

 

[PeterDonis]

Thats smthh with the same idea
arXiv:gr-qc/9904054v2

It looks like their definition of "apparent horizon" is consistent with atyy's and my previous posts. And it looks like the way they are handling initial data and time evolution is as I conjectured in my post #8. The Gauss-Codazzi equation does not appear in this paper.