Rindler Horizon Subtleties: Advanced Discussion

[ergospherical]

[Mentors' note: thread spun off from https://www.physicsforums.com/threads/equivalence-principle-and-rindler-horizons.1007879/]

[Disclaimer] Definitely not B Level, but possibly of interest:
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.100.084029

discusses the Rindler horizons of an observer in a radial, locally Rindler trajectory (i.e. hyperbolic in a local inertial frame) in the Schwarzschild spacetime. As in the flat case these trajectories are defined by constant norm of the 4-acceleration and zero 4-rotation.

Note that a stationary observer in a Schwarzschild spacetime is such an observer.

Meanwhile the equations of the future (past) Rindler horizon for a given locally Rindler trajectory can be determined from the null geodesics with the same intercept with future (past) null infinity.

 

[PeterDonis]

a stationary observer in a Schwarzschild spacetime is such an observer.

Not any stationary observer; only a stationary observer sufficiently close to the horizon of a black hole. The problem with the more general claim is that, for an observer to have a Rindler horizon, spacetime has to be flat, at least to a good enough approximation, over a large enough region, where "large enough" means "compared to $c^2 / a$, where $a$ is the observer's proper acceleration", and that will simply not be the case for most stationary observers in most Schwarzschild spacetimes.

For example, if you're standing at rest on the surface of the Earth, $c^2 / a$ for you is about one light-year, and the spacetime around you is certainly not flat to a good enough approximation over that large a region. But if you are sufficiently close to the horizon of a black hole, your $a$ will be large enough that the hole's horizon is within $c^2 / a$ of you, and you will indeed be able to treat spacetime as flat over a region that size, and then the hole's horizon will indeed be your Rindler horizon, at least within the region of spacetime that you can treat as flat to a good enough approximation.

What counts as a good enough approximation? Basically, that any tidal accelerations within the appropriate spacetime region are much smaller than the proper acceleration of your worldline. Or, to put it another way, the spacetime curvature (geodesic deviation) over the appropriate spacetime region must be much smaller than the path curvature of your worldline.

 

[ergospherical]

for an observer to have a Rindler horizon, spacetime has to be flat, at least to a good enough approximation, over a large enough region, where "large enough" means "compared to $c^2 / a$, where $a$ is the observer's proper acceleration"

Is this true? In the paper I didn't notice anything about this flatness condition - they define the future (past) Rindler horizon of an observer in a locally Rindler trajectoy as the causal past (future) of the intercept of that trajectory with null (future) infinity. In analogy to the flat case, the locally Rindler trajectories approach appoach the Rindler horizon asymptotically at spatial infinity.

Maybe I have missed some details...

 

[PeterDonis]

In the paper I didn't notice anything about this flatness condition

The paper is behind a paywall so I can't read it. (I haven't looked to see if there is a preprint on arxiv.) But it looks to me like they are either leaving out conditions or redefining terms (see below for one example). (Note that Physical Review D is a more questionable journal than the Physical Review journals with lower letters, like A, so papers in it are not always as well reviewed.)

they define the future (past) Rindler horizon of an observer in a locally Rindler trajectoy as the causal past (future) of the intercept of that trajectory with null (future) infinity.

First, this is not the standard definition of "Rindler horizon"; that definition requires flatness (at least to a good enough approximation, as I said, in a region large enough to contain both the relevant segment of the observer's worldline and the horizon itself). Second, it doesn't make sense, because, for example, the trajectory of a stationary observer in Schwarzschild spacetime never intercepts future null infinity. It eventually reaches future timelike infinity. So either they're leaving something out or they've made a (not valid, as far as I can see) redefinition of the term "Rindler horizon".

 

[PeterDonis]

I haven't looked to see if there is a preprint on arxiv.

Just looked, here it is:

https://arxiv.org/abs/1906.05134

 

[ergospherical]

it doesn't make sense, because, for example, the trajectory of a stationary observer in Schwarzschild spacetime never intercepts future null infinity. It eventually reaches future timelike infinity. So either they're leaving something out or they've made a (not valid) redefinition of the term "Rindler horizon".

This is true: a stationary observer in Schwarzschild spacetime is a locally Rindler observer (clearly, because $|a|$ is a constant), but they do not have a Rindler horizon.

 

[PeterDonis]

In the paper

I just noticed that the abstract of the paper talks about radial "Rindler trajectories". Obviously a stationary observer's trajectory in Schwarzschild spacetime is not such a trajectory. So now I'm even more skeptical that the paper makes sense. Will read through the preprint in more detail before commenting further.

 

[ergospherical]

I just noticed that the abstract of the paper talks about radial "Rindler trajectories". Obviously a stationary observer's trajectory in Schwarzschild spacetime is not such a trajectory. So now I'm even more skeptical that the paper makes sense. Will read through the preprint in more detail before commenting further.

What is wrong with this? A radial trajectory is one with $\theta, \varphi$ constant and a stationary trajectory is one with $\theta, \varphi, r$ constant (and stationary trajectories have constant $|a| = \dfrac{r_s}{2r^2}\dfrac{1}{\sqrt{1 - \dfrac{r_s}{r}}}$ so are locally Rindler...)

 

[PeterDonis]

A radial trajectory is one with $\theta, \varphi$ constant

And $r$ not constant.

and a stationary trajectory is one with $\theta, \varphi, r$ constant.

And $r$ constant means it is not radial. See above. If the paper is using "radial" to mean "stationary", my level of skepticism just went up another notch since that is an obvious abuse of standard terminology.

 

[ergospherical]

And $r$ not constant.

I don't think that's right; stationary trajectories ought to be a subset of radial trajectories, because the only condition on a radial trajectory is that the angular coordinates are constant - it doesn't matter if $r$ is constant or not!

 

[PeterDonis]

I don't think that's right; stationary trajectories ought to be a subset of radial trajectories

I have never seen "radial" used to include "stationary" when describing trajectories in any GR textbook or any other paper. The two are always treated in practice as mutually exclusive when describing trajectories.

The term "radial" is of course commonly used to describe vectors pointing in the radial direction, such as the proper acceleration of a stationary observer. But that is not the same as using the term to describe the trajectory. The tangent vector of the stationary observer's worldline is not radial; it points in the $t$ direction, not the $r$ direction.

 

[PeterDonis]

the only condition on a radial trajectory is that the angular coordinates are constant

As I have always seen the term used, the condition on a radial trajectory is that its tangent vector has a nonzero $r$ component and zero angular components.

 

[Dale]

I don't think that's right; stationary trajectories ought to be a subset of radial trajectories, because the only condition on a radial trajectory is that the angular coordinates are constant - it doesn't matter if $r$ is constant or not!

I agree. It is a “degenerate” type of radial trajectory.

 

[PeterDonis]

a stationary observer in a Schwarzschild spacetime is such an observer.

Ok, having looked at the preprint in some more detail, this statement is not correct. The trajectories that the paper is calling "radial Rindler trajectories" are, heuristically, trajectories that "look like Rindler hyperbolas" in Schwarzschild coordinates (in the $t$-$r$ plane of such coordinates). In other words, they are trajectories which fall in from infinity, have a turning point (although this condition sets limits on the proper acceleration, as the paper discusses), and then go back out to infinity. But, unlike the case usually considered, where the hyperbolic trajectory is a geodesic, and has nonzero angular velocity, here it is not: it has constant nonzero proper acceleration, radially outward, and has zero angular velocity throughout. The constant outward proper acceleration is what makes the trajectory intersect past and future null infinity instead of past and future timelike infinity (the way a geodesic hyperbolic trajectory would).

The paper does discuss the trajectories of stationary observers, but only to compare them to the class of trajectories described above. Stationary observers to belong to the more general class of "LUA" trajectories discussed in the paper, but they do not have "Rindler horizons" in the sense the paper is using the term. The "Rindler horizon" of a trajectory in the class described above is not the event horizon of the black hole (and does not even require a black hole to be present--trajectories such as those described in the paper can exist in the vacuum region outside any spherically symmetric mass); the Rindler horizon of such a trajectory is in the exterior region of the spacetime (i.e., outside the event horizon, if there is one). Heuristically, the Rindler horizon is the asymptote of the hyperbola describing the trajectory as it would appear in exterior Schwarzschild coordinates. As the paper discusses in Section 3.3, this horizon will always be outside the horizon of the black hole for any trajectory that meets the conditions described above. There are trajectories which come in from past null infinity (instead of past timelike infinity, because of the outward proper acceleration, as above), and do not have a Rindler horizon outside the black hole event horizon--but these trajectories have no turning point; they fall into the black hole.

So I am no longer skeptical that the paper makes sense; it does. But what it calls a "Rindler horizon" is not the same as a standard Rindler horizon and what it is discussing has nothing to do with the local equivalence of a black hole's event horizon to a Rindler horizon for a stationary observer that is close enough to the event horizon.

 

[ergospherical]

Ok, having looked at the preprint in some more detail, this statement is not correct.

Stationary observers to belong to the more general class of "LUA" trajectories discussed in the paper, but they do not have "Rindler horizons" in the sense the paper is using the term.

But this is exactly what I meant when I wrote that statement, that "a stationary observer in a Schwarzschild spacetime is such an observer". I used the phrase "locally Rindler" to describe what they call "LUA".

 

[PeterDonis]

But this is exactly what I meant when I wrote that statement, that "a stationary observer in a Schwarzschild spacetime is such an observer". I used the phrase "locally Rindler" to describe that they call "LUA".

Ok, but not all "LUA"s have Rindler horizons in the sense the paper is using the term. Only the ones that do are relevant to any discussion of Rindler horizons. So since this thread is about Rindler horizons, naturally I assumed that whatever category of observers you were referring to were the ones that had Rindler horizons. Stationary observers do not (except for the one case I mentioned before, where they are close enough to a black hole event horizon--but the paper does not discuss that case).

 

[ergospherical]

Ok, but not all "LUA"s have Rindler horizons in the sense the paper is using the term. Only the ones that do are relevant to any discussion of Rindler horizons. So since this thread is about Rindler horizons, naturally I assumed that whatever category of observers you were referring to were the ones that had Rindler horizons. Stationary observers do not (except for the one case I mentioned before, where they are close enough to a black hole event horizon).

I already mentioned this in my post #27 [moderator's note--now post #15 since the thread spin-off] though!

 

[PeterDonis]

I already mentioned this in my post #27 [moderator's note--now post #15 since the thread spin-off] though!

Ah, yes, sorry, I didn't see that clarification from you.

 

[pervect]

I wrote something a while back about the relationship between Rindler and Schwarzschild coordinates close to a black hole. See https://www.physicsforums.com/threads/jetpacking-above-a-black-hole.914198/post-5762877

To summarize, if we set $\theta = \phi = 0$ for the Schwarzschild metric, we can use the line element
$$ds^2 = -\left(1-\frac{r_s}{r} \right) dt^2 + \frac{dr^2}{1-\frac{r_s}{r}}$$

where $r_s$ is the Schwarzschild radius (2M in geometric units).

By making the transformation

$$r = r_s + \frac{R^2}{4r_s} $$

which I won't motivate here, but is motivated in the original thread, we transform the (t,r) line element of the Schwarzschild metric in terms of (t,R) as

$$ds^2 = -\frac{R^2}{R^2 + 4\,r_s^2}\,dt^2 + \left(1 + \frac{R^2}{4\,r_s^2} \right) dR^2$$

and if R << r_s this becomes a version of the Rindler metric

$$ds^2 = -\frac{1}{4\,r_s^2} \,R^2 \, dt^2 + dR^2$$

From the form of this line element, we can conclude that, near the event horizon, R plays the role of the "distance away from the event horizon", which occurs at R=0 and r=$r_s$. We can also conclude that the product of proper acceleration and distance away from the event horizon for a static observer is constant and equal to 1 in geometric units, or c^2 in non-geometric units.

 

[PAllen]

I would like to propose what seems to me the uniquely natural definition of Rindler horizon for SR and GR. The main idea is that while event horizons are global constructs related to null infinity, Rindler horizons (if they exist) are defined by a particular observer. The motivation for the following is simply a geometric definition of the boundary of coverage of Rindler coordinates in SR, that works for any observer in any spacetime.

definition: The Rindler horizon for a past and future inextensible timelike world line $\lambda$ is the boundary of $J^+(\lambda) \cap J^-(\lambda)$

Some features of this definition are:

1) In all cases, it is the boundary of coverage by radar coordinates
2) In the case of uniform acceleration in SR, it is the combination of past and future Rindler horizon that bounds the coordinate coverage
3) In a BH geometry, for a stationary observer (but not other observers), it is the event horizon
4) In Kruskal geometry, for an observer that starts from the white hole singularity and terminates on the black hole singularity, it is different from both past and future event horizon.
5) For a uniform (proper) accelerating trajectory in Kruskal geometry whose smallest radial coordinate is 'large', it is essentially indistinguishable from an SR Rindler horizon, and the whole BH is well "inside" the Rindler horizon.
6) For a typical exterior Schwarzschild geometry combined with a material ball solution, stationary observers have no Rindler horizon. But trajectories like (5) do, with the matter ball inside the horizon.

In a later post I will try to discuss differences from the referenced paper's definitions. However, IMO, this definition is by far the most physically natural one.

[I am using definitions and notation from Wald's GR text].

 

[PAllen]

This is similar to the definition in the paper being discussed in the "A" level thread, but the latter has one key extra condition: the worldline must intersect past and future null infinity. The latter property is also possessed by Rindler worldlines in flat spacetime, but is not possessed by the worldline of a stationary observer in Schwarzschild spacetime. So this is an interesting separation in the curved spacetime case of two properties that go together in the flat spacetime case.

(Quoted from other thread)

I find the requirement you describe from the paper restrictive and unnatural. Note, that with my definition, the absence of Rindler horizons in many cases in SR flows naturally, without having to 'define them away'. Further, it merges naturally with the case you have extensively analyzed of stationary observer near a supermassive BH being quasi-locally indistinguishable from the SR Rindler case (whereas their definition excludes this case).

 

[PeterDonis]

I find the requirement you describe from the paper restrictive and unnatural.

As far as I can tell, the reason the paper's authors include that requirement is to "emulate" the corresponding property of Rindler worldlines in flat spacetime, but they don't really give any explanation of why they are concerned with that particular property.