Apparent horizon for the eternally accelerated observer

[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 http://www.fysik.su.se/~ingemar/relteori/Emmaslic.pdf
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?

 

[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?

 

[PeterDonis]

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.

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

So apparent Horizon is introduced.

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.

A theorem says that Apparent Horizons lie behind or on the Event Horizon.

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.

My question is about rhe non eternally accelerated observer. Has she an apparent horizons?

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.)

 

[PeterDonis]

In this case all events will be in his past cone at a given moment.

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.

What are the closed trapped spacelike surfaces that make the time depending Apparent Horizon?

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.

 

[naima]

Thank you very much for your "no go" answers!
My notions of horizons had (have) to be more precise.
Have you a good link on these subjects?

 

[PeterDonis]

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.

 

[naima]

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

 

[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 http://www.fysik.su.se/~ingemar/relteori/Emmaslic.pdf.
coul anyone exlpain

 

[PeterDonis]

coul anyone exlpain

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.

 

[PeterDonis]

this http://www.fysik.su.se/~ingemar/relteori/Emmaslic.pdf.

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.

 

[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!

 

[PeterDonis]

I read that she finds (not closed) trapped surfaces in the intersection of two past light cones.

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.

 

[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?

 

[PeterDonis]

trapped surfaces are defined as 2d spacelike manifolds

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

I understand that you do not like this paper but is what she says about trapped open surfaces in Minkowsky false?

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.)

 

[PeterDonis]

I understand that you do not like this paper

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.

 

[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).

 

[PeterDonis]

The Misner space trick then just says: "I'll trap the rest of you topologically by giving you no place to go"

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.

 

[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)?

 

[PeterDonis]

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)?

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.

 

[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?

 

[PeterDonis]

Why do apparent horizons appear suddenly and absolute horizons begin at one point and increase?

Let's take the example of a perfectly spherically symmetric collapse of dust (matter with zero pressure); this is the idealized case that Oppenheimer and Snyder modeled.

Here are the (heuristic, but sufficient, I hope, for this discussion) definitions of the two types of horizons:

(1) An apparent horizon is a surface at which radially outgoing light does not move outward, but stays at the same radius.

(2) An absolute horizon is a surface at which light rays just fail to escape to infinity.

Consider a piece of matter at the exact center of the collapsing dust, i.e., at $r = 0$. This piece of matter emits light rays outward; we'll suppose that the light rays don't interact with the matter as they go, to eliminate any complications from that. So, before the collapse, and during the first part of the collapse, those light rays will escape to infinity.

However, at some instant, the piece of matter at the exact center will emit an outgoing light ray that happens to reach the surface of the dust at the exact same instant that that surface is collapsing through radius $r = 2M$. That is the radius of the absolute horizon, and that outgoing light ray will stay at $r = 2M$ forever, even after it emerges from the collapsing dust (which will continue collapsing to smaller and smaller radius until it hits $r = 0$ and a singularity forms). So in fact the entire path of that light ray lies on the absolute horizon, and a family of light rays emitted at the same instant from the exact center in all possible directions will all lie on the absolute horizon (because none of those light rays will ever escape to infinity, but light rays emitted just an infinitesimal time before will, so those light rays are the ones that just fail to escape to infinity). Since those light rays start at $r = 0$ and gradually diverge outward until they stop diverging at $r = 2M$, we can say that the absolute horizon begins at one point and increases in radius until it reaches $r = 2M$, when it stops increasing and stays there forever after.

Note, though, that these light rays, until they reach $r = 2M$, are still diverging, i.e., going to larger radius. So even though there is an absolute horizon from the instant that family of light rays is emitted from $r = 0$, there is no apparent horizon until they stop diverging, i.e., until they reach $r = 2M$. So the apparent horizon, for these light rays, appears suddenly at $r = 2M$. (Here we have only considered that one family of light rays; if we consider other families, that are emitted from $r = 0$ later, we find that they also stop diverging at some point inside $r = 2M$--how far inside depends on how late they were emitted. So in fact the apparent horizon "appears suddenly" at a slightly different radius for different families of light rays, which means that, considering the spacetime as a whole, the apparent horizon appears "suddenly" on some spacelike surface going from $r = 0$ to $r = 2M$. Only at $r = 2M$ does the apparent horizon become a null surface and coincide with the absolute horizon.)

 

[naima]

(1) An apparent horizon is a surface at which radially outgoing light does not move outward, but stays at the same radius.

(2) An absolute horizon is a surface at which light rays just fail to escape to infinity.

If apparent horizon is inside absolute horizon can i say that light between them:
1) moves outward and travels toward increasing radius.
2) fails to escape to infinity.

could it escape to twice the BH radius?

 

[PeterDonis]

If apparent horizon is inside absolute horizon

Then the black hole must be gaining mass, which means the absolute horizon is growing.

can i say that light between them:
1) moves outward and travels toward increasing radius.

As long as the hole continues to gain mass, because the absolute horizon is itself moving outward, so there is "room" for light inside it to move outward. But if the hole stops gaining mass, the absolute horizon will stop growing, the "gap" between the apparent and absolute horizon will disappear, and there will no longer be any region inside the absolute horizon where light can move outward.

2) fails to escape to infinity.

Yes, this is always true.

could it escape to twice the BH radius?

No; it can never escape beyond the absolute horizon, which is the "BH radius". But the BH radius itself could grow to twice its original radius, if the hole's mass increased to twice its original mass. (More precisely, the area of the horizon would grow to four times its original area if the hole's mass increased to twice its original mass. The horizon doesn't have a physical "radius"; we just quote its size in terms of the $r$ coordinate instead of the physical area of the horizon because that's how it's usually quoted.)

 

[naima]

All this seems logically coherent.
What about the future light cones between the 2 horizons when the BH grows? is light with increasing radius permitted by the metric?

 

[PeterDonis]

What about the future light cones between the 2 horizons when the BH grows? is light with increasing radius permitted by the metric?

Yes, but it will not increase in radius as fast as the absolute horizon does.

 

[naima]

This point is interesting.
Where could i find a non static metric of a BH in which i could compute these 2 horizons

 

[PeterDonis]

Where could i find a non static metric of a BH in which i could compute these 2 horizons

The Vaidya metric is the simplest non-static metric I'm aware of with these horizons in it. It describes a black hole that is either emitting or absorbing radiation:

https://en.wikipedia.org/wiki/Vaidya_metric

The discussion of the ingoing Vaidya metric talks about the horizons.

 

[naima]

According to https://en.wikipedia.org/wiki/Vaidya_metric#Ingoing_Vaidya_with_pure_absorbing_field
when r < 2M(v) in the ingoing case the (l) and (n) expansion rates are negative
So it is the apparent horizon.
In this paper I find the formula for the event radius:
2M(v) / [1 - dr/dv] without the proof.
Can you explain it?
thanks

 

[PeterDonis]

Can you explain it?

Do you mean derive it, or explain what it means?

I think the derivation is straightforward: the event horizon is a null surface which means $ds^2 = 0$. If we assume that $d\Omega^2 = 0$ (i.e., we are considering a purely radial line element, with no angular motion), then setting $ds^2 = 0$ in equation (1) of the paper gives an equation for $r_H$:

$$
\left( 1 - \frac{2M(v)}{r_H} \right) dv^2 = 2 dv dr_H
$$

$$
1 - \frac{2M(v)}{r_H} = 2 \frac{dr_H}{dv}
$$

$$
1 - 2 \frac{dr_H}{dv} = \frac{2M(v)}{r_H}
$$

$$
r_H = \frac{2M(v)}{1 - 2 \frac{dr_H}{dv}}
$$

Physically, this is telling you that if $dr_H / dv$ is positive (because the hole is growing, i.e., absorbing radiation), the event horizon $r = r_H$ is outside the apparent horizon $r = 2M(v)$, and how far outside depends on how fast the hole is growing. If $dr_H / dv$ is negative (because the hole is shrinking, i.e., emitting radiation), the event horizon $r_H$ is inside the apparent horizon $r = 2M(v)$, and how far inside depends on how fast the hole is shrinking.

 

[naima]

Great!
That is what was lacking in Black Holes and Time wraps.