# Apparent horizon for the eternally accelerated observer

1. Jul 5, 2015

### naima

The problem with the notion of Event Horizon is that it speaks ot events which will never be seen in the future. So it requires to wait eternally.
So apparent Horizon is introduced. It uses trapped surfaces
They are closed spacelike surfaces. when they emit light there is no diverging wave front.
A theorem says that Apparent Horizons lie behind or on the Event Horizon.
My question is about rhe non eternally accelerated observer. Has she an apparent horizons?

2. Jul 5, 2015

### naima

PeterDonis wrote in an old thread:
""In a "real" scenario where the Unruh effect was observed, the accelerated observer would not be accelerating indefinitely; he would start accelerating at some finite time and stop accelerating at some later finite time. He would still observe the effect during the period when he was accelerating (but not when he was inertial)".

In this case all events will be in his past cone at a given moment. There is no Event Horizon in this case. What are the closed trapped spacelike surfaces that make the time depending Apparent Horizon?

3. Jul 5, 2015

### Staff: Mentor

Not if you fall through the event horizon. Then you can see events on the other side.

The concept of an apparent horizon was not introduced because of problems with the concept of an event horizon. In fact, it was the other way around: the concept of an apparent horizon came first (more precisely, the concept of a trapped surface came first, and the concept of an apparent horizon was an obvious corollary of the concept of a trapped surface). Then Hawking introduced the concept of an event horizon because of problems with the concept of an apparent horizon, in the context of proving rigorous theorems about black holes. For those rigorous theorems, he needed the horizon to be a causal boundary, and an apparent horizon is not. An event horizon is.

More precisely, the theorem says this is true if the energy conditions are satisfied. In spacetimes that violate one of the energy conditions (e.g., a black hole emitting Hawking radiation), the apparent horizon can be outside the event horizon.

No; at least, not if you are talking about an observer in flat spacetime. There are no trapped surfaces in flat spacetime. There aren't any event horizons either; the Rindler horizon of an accelerated observer is not an event horizon. (It shares some properties with one, but it does not share the primary property of an event horizon, that it bounds a region which can't send light signals to infinity. The region behind an accelerated observer's Rindler horizon can send light signals to infinity--all regions can in flat spacetime.)

4. Jul 5, 2015

### Staff: Mentor

No, they won't; this is impossible, since "all events" means the entire spacetime, and there is no point on any observer's worldline that has the entire spacetime in its past light cone.

There aren't any; the Rindler horizon of an accelerated observer in flat spacetime is neither an apparent horizon (trapped surface) nor an event horizon. See my previous post.

5. Jul 6, 2015

### naima

My notions of horizons had (have) to be more precise.
Have you a good link on these subjects?

6. Jul 6, 2015

### Staff: Mentor

The definitive text on global properties is Hawking & Ellis, but that text assumes some pretty heavy background in GR. Misner, Thorne, & Wheeler has some discussion of horizons and trapped surfaces, including the singularity theorems. Wald has some discussion as well. Kip Thorne's Black Holes and Time Warps has a good layman's discussion, including some good history of how the concepts developed.

Unfortunately I don't know of a good online source that discusses horizons specifically. Carroll's lecture notes on GR discuss black holes, but don't have much discussion of the two types of horizons and how they compare.

7. Jul 6, 2015

### naima

I have Black Holes and Time Warps
I will look again for the topic.

8. Jul 7, 2015

### naima

I find this sentence:
"It may be possible to find trapped surfaces in a flat spacetime.
By taking a surface being the intersection of two past light cones in Minkowski space, we see that there
are surfaces such that the two families of light rays both converge. But these
will not be closed, and are therefore not trapped by denition"
in this paper.
coul anyone exlpain

9. Jul 7, 2015

### Staff: Mentor

Read the rest of the paragraph:

"However, such locally trapped surfaces may be closed by performing identifications in Minkowski spacetime. The resulting spacetime is called Misner space, which is a flat spacetime containing trapped surfaces."

In other words, the paper is not really talking about trapped surfaces in Minkowski spacetime; it's talking about trapped surfaces in Misner space, which is a flat spacetime, true, but with non-trivial topology. The bit about "performing identifications" means, basically, taking a flat spacetime with the spatial topology of a circle (or in 3 dimensions, a torus) instead of an infinite line--so, for example, in a particular inertial frame, the spatial point at $x = + L$ is identified with (the same as) the spatial point at $x = - L$ (so the total "circumference" of the spatial circle is $2L$). In such a case, one can no longer unambiguously say that one side of a closed 2-surface is the "outside" and the other is the "inside"; light emitted in both directions is "trapped" by the global topology of the spacetime.

None of this is relevant to the discussion we've been having in this thread, because the only flat spacetime we've been considering is flat Minkowski spacetime. The key feature that flat Minkowski spacetime has, that Misner space does not have, is a well-defined notion of "infinity": in Misner space, each spacelike slice is finite in extent. Since we're discussing horizons, and since the definition of an event horizon requires the spacetime to have a well-defined notion of infinity, the only flat spacetime that is relevant is Minkowski spacetime.

10. Jul 7, 2015

### Staff: Mentor

It's worth noting, btw, that this is a thesis, not a peer-reviewed paper. Theses are generally supposed to be reviewed, but on average they do not necessarily meet the same standards as peer-reviewed papers. This one doesn't; just in skimming I've already found several fairly basic errors. So I would not recommend using this as a source of information about black holes.

11. Jul 7, 2015

### naima

Can we read what she says step by step.
My question here is basic.
I read that she finds (not closed) trapped surfaces in the intersection of two past light cones. Is it true? What are these surfaces?
I am not yet interested in Misner surfaces.
Thank you for the time you spend with my questions!

12. Jul 7, 2015

### Staff: Mentor

Her terminology here is confused. If it's not closed, it's not a trapped surface; her use of that term is inaccurate in this case. I'm actually not quite sure what she's referring to, but since she says the surfaces are not closed, they're not trapped.

Once again, I do not recommend using this paper as a source; I have found enough errors in it that I don't think it can be trusted for the understanding you're trying to build.

13. Jul 7, 2015

### naima

trapped surfaces are defined as 2d spacelike manifolds such that the scalar expansion functions are negative on its two sides. I understand that you do not like this paper but is what she says about trapped open surfaces in Minkowsky false?

14. Jul 7, 2015

### Staff: Mentor

2d closed spacelike manifolds. (Note that the paper itself admits that the "closed" is part of the definition.)

Yes, because they don't meet the definition, as corrected above. (This is why I said the paper's use of terminology here is inaccurate.)

15. Jul 7, 2015

### Staff: Mentor

It's not a question of my not "liking" it. It's a question of whether it's an acceptable source for discussion on PF. In my opinion, it isn't. I've asked for input from other Mentors.

16. Jul 7, 2015

### PAllen

I question her analogy of intersection of two light cones as 'sort of a trapped surface'. The closure part of the definitions isn't arbitrary, as I understand it. It is essential to being able deduce that non-orthogonal (to the surface) null rays are also effectively trapped. In the intersection of two past null cones, the fact that the surface is open correlates to the fact that non-orthogonal null rays are not trapped. The Misner space trick then just says: "I'll trap the rest of you topologically by giving you no place to go" (being anthropomorphic about it).

17. Jul 7, 2015

### Staff: Mentor

Yes; in a sense, if the spatial topology is closed, every 2-surface could be viewed as a "trapped surface", simply because there is no way to distinguish either side of it as the "outer" side, the side from which light rays "should" be able to escape to infinity, because there is no infinity to escape to.

18. Jul 8, 2015

### naima

Could you give me an example of such an intersection in Minkowski spacetime and the coordinates of a point in it where there is an open "locally trapped"surface (its equation)?

19. Jul 8, 2015

### Staff: Mentor

No, because, as I said, I don't quite understand what the thesis is talking about here or why it is included. Once more: if you're trying to build an understanding of horizons and trapped surfaces based on something you read in that thesis, don't. It's not worth the trouble. There are better sources. Try Carroll's online lecture notes for a start, and Black Holes and Time Warps. Don't waste your time trying to dig into this particular question just because you read it in the thesis.

20. Jul 8, 2015

### naima

BH and Time warps is a good book but the difference between absolute and apparent horizons is not clear to me. Why do apparent horizons appear suddenly and absolute horizons begin at one point and increase?