# Horizons question

1. Dec 18, 2005

### Vagelford

I have a question on horizons. Actually I would like to make some
things that I have in mind clear. First of all let's give some
definitions.
Trapped surface: Is a surface on which null vectors point inwards.
That means that for example, if we let a photon on that surface it
could only travel inwards and it could at no time be outside that
surface.
Untrapped surface: Is a surface on which there exist null vectors
pointing outwards. That means that for example, if we let a photon
on that surface, there are null geodesics leading out of that
surface.
Marginally trapped surface: Is a surface on which there exist null
vectors that are tangent to that surface but there are no null
vectors pointing outwards.
A first question could be how exact are those definitions. The
question I would actually like to ask is: What exactly is the
difference between an event horizon and an apparent horizon in
terms of trapped surfaces and null geodesics?
As I understand the subject, an apparent horizon is defined as a
trapped surface that is the border between a region that has
outgoing null geodesics that point outwards and a region that
doesn't have outgoing null geodesics pointing outwards. That means
that the apparent horizon is the lust trapped surface between the
two regions and the next infinitesimally near surface is an
untrapped surface.
On the other hand an event horizon is a marginally trapped surface
an thus it is also a null surface, in the sense that it has null
generators, and it is an asymptotic surface to the outgoing null
geodesics from the inside and the outside.
These things can be seen in the following diagram of the formation
of a spherically symmetric black hole, were the cyan lines are the
null ingoing geodesics, the yellow lines are the null outgoing
geodesics, the green line is the apparent horizon and the event
horizon is defined by the convergence of the yellow lines. We can
see that at late times the two horizons coincide.

In the above example, it is easy to identify the horizons and
distinguish between the event horizon and the apparent horizon.
Both horizons are trapped surfaces as it can be seen by the light
cones, but the event horizon is also a null surface while the
apparent horizon isn't at early times. Will these criteria apply
to more general configurations or should I look for a more general
"tool"?
The point is, are these things generic? The case in the diagram is
a special case with a lot of symmetry. Are the above definitions
general or they apply only to especially symmetric settings like
the above? What are the other trademarks that distinguish these
horizons?
Vagelford
PS. The x-axis is distance r and the y-axis is time t.

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2. Dec 19, 2005

### hellfire

I have little knowledge about this, but it would be a pitty if such interesting questions remain without comment. As far as I can tell, your definitions seam to agree with the ones in Wald's book, chapters 9 and 12. The definitions as such are valid for any generic spacetime. However, the singularity theorems and corolaries relating them, are usually valid only for spacetimes in which the strong or weak energy conditions hold. Specifically, the proposition 12.2.4 of Wald's book, stating that the apparent horizon will be located always within the event horizon, applies in globally hyperbolic spacetimes in which the strong energy condition holds.

3. Dec 22, 2005

### Vagelford

my professors and we concluded that I should check the expansion
$$\theta$$ of the null congruence. So I did and I found something
puzzling. Before checking the time-dependent case in the diagram
above, I tested my calculations on a time independent situation.
So I used a metric of the form
$$ds^2=-(1-(\frac{A}{r})^2)dt^2-2\frac{A}{r}dtdr+dr^2+r^2d\Omega^2$$
with A=-1. I calculated the null vectors from the equation
$$k_a=-\partial_au$$ for the outgoing geodesics and
$$k_a=-\partial_a\upsilon$$ for the ingoing geodesics, were
$$u=t-r+Aln(r+A)$$ and $$\upsilon=t+r+Aln(r-A)$$. And finally I
calculated the expansion from the equation $$\theta=\nabla^ak_a$$.
My surprise was that the result was $$\theta=\pm\frac{2}{r}$$ which
is positive for the outgoing geodesics and negative for the
ingoing geodesics. I was expecting that $$\theta$$ for the outgoing
geodesics would change sign on the horizon at r=1 and become
negative in the sense that inside the horizon the geodesics
converge to the singularity at r=0 and on the horizon the null
geodesics are parallel so the expansion should be zero.
Having found these results I proceeded to calculate the expansion of the
outgoing null geodesics for the time-evolving black hole that I
mentioned above and I found that to the duration of the evolution
at the place were the apparent horizon formes the expansion of the
outgoing null geodesics goes to infinity and as the apparent
horizon merges with the event horizon the expansion becomes finite
and at late times takes the form of $$\theta=\frac{2}{r}$$. These
results can be seen in the following pictures. The second picture
shows the behavior of $$\theta$$ and the first is just a diagram
that shows the ingoing and outgoing null geodesics for the first
time independent metric. I would really like to hear some comments
on these things.

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4. Dec 22, 2005

### pervect

Staff Emeritus
I gather that the motivation for your choice of metric was to make dr/dt simple. Solving for ds^2 = 0 gives dr/dt = -1/r + 1 or dr/dt = -1/r -1 given that A=-1.

Integrating this differential equation gives geodesics associated with u=constant and v=constant, and explains where these variables came from.

This is about as far as I've gotten, not having done anything previously with geodesic congruences or expansion, unfortunately.

It's unclear to me what the matter distribution associated with this metric is, except that it's not a vacuum solution to Einstein's equation. As far as I know, there is no reason to assume it meets any of the various energy conditions, so many of the theorems that assume these energy conditions wouldn't apply.

5. Dec 22, 2005

### hellfire

This is also my understanding. The expansion scalar for the outgoing geodesics should become negative inside the event horizon, as one finds trapped surfaces there. Are you sure your calculations are right? It is unclear to me whether your expression for $\theta$ is equivalent to eq. (9.2.6) in Wald.

Last edited: Dec 22, 2005
6. Dec 22, 2005

### Vagelford

pervect, the metric comes from a fluid analogue of a black hole,
so no energy condition applies to that spacetime.

hellfire, the expression (9.2.6) in Wald is for timelike
geodesics. The expression (9.2.27) is for null geodesics and is
equivalent with the one I used. For more details there is a
thorough treatment in Poisson's, A relativist's toolkit. The point
is that I would have thought that I made a mistake too, but the
same result applies for the Schwarzschild metric also. Poisson
mentions the result and I've also calculated it myself, so at this
point I'm confused. The only logical explanation that I can give
is that I'm missing something of the meaning of $$\theta$$. Any
ideas?

7. Dec 22, 2005

### pervect

Staff Emeritus
If $\theta$ measures the increase/decrease in area of a cross-section of the bundle of geodesics with time, what determines the direction of "time" used?

Is there any chance that (for instance) a variable such as 'r' in the Schwazschild metric is being used to determine the direction of 'time' which is backwards from the usual convention?

8. Dec 22, 2005

### George Jones

Staff Emeritus
Sorry - I'm too tired to think, but, In Poisson, read 5.1.7 Apparent Horizon, keeping Figure 5.2 in mind.

Regards,
George

9. Dec 23, 2005

### Vagelford

Thanks for the tip.

10. Dec 23, 2005

### Stingray

Probably everything you ever wanted to know about horizons can be found in http://xxx.lanl.gov/abs/gr-qc/0407042
An event horizon cannot be defined locally in terms of trapped surfaces. It can exist even in a region of spacetime which is completely flat. Still, all trapped surfaces will be inside an event horizon.

Apparent horizons also lie inside event horizons. As you say, they are a border between trapped and untrapped regions. Unfortunately, their definition requires a given Cauchy surface (3 dimensional hypersurface). This is inconvenient and sometimes causes problems, so various other horizon concepts have since been defined which do not require this (as explained in the paper I referenced above).

11. Dec 27, 2005

### George Jones

Staff Emeritus
I don't know about your black hole analogue example, but $$\theta$$ changes sign for radially "outgoing" light rays at the apparant horizon of Schwarzschild spacetime, just as expected. I have used scare quotes because, according to Poisson, the standard terminology is somewhat confusing:

I can see Poisson's point, but I don't think the terminology is so bad, and I might elaborate in a future post.

Consider a standard eternal (maximally extended) Schwarzschild black hole, which has apparent and event horizons that coincide. In Kruskal coordinates $$u[$$ and $$v$$, the metric, with $$M=1$$, is given by

$$ds^2 = - \frac{32}{r} e^{-\frac{r}{2}} du dv + r^{2} d\Omega^{2},$$

where $$r$$ is defined implicitly by

$$e^{-\frac{r}{2M} \left( \frac{r}{2} - 1 \right) = -uv.$$

I have attached a (rather too busy!) gif illustrating the situation.

Red lines: event and apparant horizons.
Green lines: outgoing radial null geodesics.
Blue line: ingoing radial null geodesic.
Black lines: curves of contant $$r$$.

The red horizons are the axes for the Kruskal coodinate system. The line going from lower left to upper right is the $$v$$ axis, i.e, the line $$u = 0$$. Points above this line have $$u > 0$$ axis, while points below have $$u < 0$$. The line going from lower lright to upper left is the $$u$$, i.e, the line $$v = 0$$. Points above this line have $$v > 0$$, while points below have $$v < 0$$.

According to Poisson's calculation, the expansion $$\theta$$ is given by

$$\theta = - \frac{u}{2r}.$$

The bottom (outside the horizon) outgoing geodesic is the line $$u = -0.2$$, so, by the expression above, the expansion $$\theta$$ is positive. Contrast this with the top (inside the horizon) outgoing geodesic $$u = +0.2$$, which has negative expansion $$\theta$$.

The apparant horizon is a maginally trapped surface that has $$u = 0$$ and thus zero expansion, i.e., the apparant (and, in this case, the event) horizon is a marginally trapped surface. This is true in the generic case (Wald, Theorem 12.2.5), since, as both you and Stingray note, the apparant horizon is the boundary of the positve expansion region.

As Stingray also noted, when the apparant and event horizons do not coincide, the apparant horizon lies inside the event horizon (Wald, Propostion 12.2.2), which is the boundary of the region for which light rays can reach future null infinity. In the region between the apparant and event horizons, the expansion is positive, but light cannot reach future null infinity.

Poisson's book is excellent! This book nicely fills a gap in the literature.
It has a nice discussion of the apparant and event horizon's for Vaidya spacetime (a growing black hole), which is very similar to the original situation that you gave.

Regards,
George

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12. Dec 28, 2005

### Vagelford

Thanks George. I’ve seen Poisson’s analysis in Kruskal coordinates and I can see that the behaviour of U is what makes the expansion positive, negative or zero. What I don’t see is the relation of the analysis in Kruskal coordinates and the one given in the example 2.4.7 at page 52.

13. Dec 28, 2005

### George Jones

Staff Emeritus
The expansion $$\theta$$ depends on which affine paramterization is chosen for the null geodesics. The sign of the of the expansion, however, does not depend on choice of affine parameterization.

Example 2.4.7 deals with null geodesics *outside* the event horizon. Here, the outgoing geodesics are affinely parameterized by $$+r$$, and the ingoing geodesics are affinely parameterized by $$-r$$.

From my diagram, it is clear that inside the event horizon, outgoing future-directed null geodesics cannot be parameterized by $$+r$$, as $$r$$ becomes smaller as the geodesic proceeds. Inside the diagram, note the symmetry with respect to $$r$$ between outgoing and ingoing geodesics. This leads me to believe that $$-r$$ is the affine paramterization for both outgoing and ingoing geodesics inside the horizon. This might also be the solution to the problem with your example.

The expressions for expansion of congruences of null geodesics in Schwarzschild spacetime given on pages 52 and 171 differ because different affine parameterizations have been used for the null geodesics.

Regards,
George

14. Jan 11, 2006

### Vagelford

Happy New Year to everyone. I 'm returning with a few questions and thoughts on the subject.

First of all, the problem with the null congruences is that there exists an arbitrariness in the choice of the null tangent vector. That is, since the norm of the null vector is $$k^ak_a=0$$ you can't normalize your vector in any way (as you would if you had a timelike vector were you could normalize to $$u^au_a=-1$$). This arbitrariness makes it "difficult" to find the appropriate null vector for the calculation of the expansion. We discussed in previews posts the calculation of the expansion for the Schwarzschild metric

$$ds^2=-(1-\frac{2M}{r})dt^2+(1-\frac{2M}{r})dr^2+r^2d\Omega^2$$

were we have found it to be $$\theta=\frac{2}{r}$$ for the outgoing null geodesics. I should point out that this result is valid both inside and outside the horizon. George argued that the above expression for the expansion should not be parameterized with the same affine parameter inside and outside the horizon. That could be expected bye the fact that the null outgoing rays $$u=const.$$ (were $$u=t-[r+2Mln|\frac{r}{2M}-1|]$$) are singular on the horizon and the behavior of r changes on crossing the horizon. Let's make a few calculations regarding the affine parameter. In order to calculate the above expansion we have used as a null vector the vector $$k_a=-\partial_au=(-1,\frac{r}{r-2M},0,0)$$. I note here that this is the null vector both inside and outside with the difference that inside $$r<2M$$ and the expression $$\frac{r}{r-2M}$$ is negative. From $$k_a$$ we can calculate $$k^a=(\frac{r}{r-2M},1,0,0)$$. We can see that outside the horizon the vector $$k^a$$ is in fact the vector $$\frac{dx^{\mu}}{d\lambda}$$ were the affine parameter $$\lambda$$ is r. Inside the horizon the vector $$k^a$$ doesn't change form (assuming that it is $$k_a=-\partial_au$$) but the expression $$\frac{r}{r-2M}$$ changes sign. This introduces a flip of the sign of the time component of the vector $$k^a$$. So in order for the vector $$k^a$$ to point "forward in time" I am assuming that the correct $$k_a$$ is given by the expression $$k_a=\partial_au$$ which gives $$k^a=(\frac{r}{2M-r},-1,0,0)$$ that points forward in time (it is also pointing inwards instead of outwards but that is expected). We can see that this $$k^a$$ is what we will get if we evaluate $$\frac{dx^{\mu}}{d\lambda}$$ with affine parameter the parameter $$\lambda=-r$$ on the trajectory $$u=t-[r+2Mln(1-\frac{r}{2M})]$$ (which is inside the horizon). Finally the expansion evaluated from $$k_a=\partial_au$$ is $$\theta=-\frac{2}{r}$$ which is negative.

Now in the case of the Kruskal coordinates the situation is a little different. The outgoing null vector is given by the expression $$k_a=-\partial_aU$$ where $$U=\mp e^{-\frac{u}{4M}}$$. We can see here the arbitrariness that we talked about in the beginning. Since $$\partial_au$$ is a null direction, any expression of the form $$\partial_af(u)$$ will be a null direction. So we have $$k_a=-\partial_aU=(-\frac{1}{4M})(\mp e^{-\frac{u}{4M}})(-\partial_au)$$. We can see from this expression that when we cross the horizon $$(-\partial_au)$$ has the same behavior as before, but in this case the expression $$(-\frac{1}{4M})(\mp e^{-\frac{u}{4M}})$$ on crossing the horizon changes sign from positive to negative and so it gives the right description for the whole spacetime using r as the affine
parameter. The expression for the expansion is the one mentioned earlier $$\theta=-\frac{U}{2Mr}$$ and can be calculated by the above expression for $$k_a$$.

This presentation is my attempt to rationalize the whole situation and to understand maybe the criteria for selecting $$U$$ for the calculation of $$\theta$$ instead of $$u$$ and for the choice of the affine parameter(i.e. the criteria being the good behavior of the null vector on crossing the horizon).