Definition of Rindler Horizon: A Global View

[PAllen]

[Moderator's note: Thread spun off from previous one due to closure of the previous thread.]

I have been thinking about this off and on, and though late to the thread, want to propose another way of looking at this that can be presented both at B level or A level. I post here at B level, and will post in the other thread more technically.

To answer the OP at all, one must clarify what is meant by the equivalence principle and a Rindler Horizon.

The generally accepted definitions of the equivalence principle are all local, the gist of them being that in a small region of space and for a short time, all of physics in the presence of gravity is indistinguishable from special relativity (without gravity). This implies first, that a 'lab' free falling in gravity is indistinguishable over short time scales from a 'lab' far away from everything with no acceleration (measured by an accelerometer). It also implies that a 'lab' sitting on a planet is indistinguishable from a 'lab' carried by an accelerating rocket, over short time scales.

Any type of horizon is global phenomenon, in that there are statements about 'never', so, at minimum, the short time scale part of the equivalence principle is violated. Thus, it can be immediately stated that the equivalence principle is wholly irrelevant to any discussion of horizons in special or general relativity - full stop. This is irrespective of whether there may be analogous horizon situations - even if there are, the equivalence principle is not involved.

To talk more generally about comparing Rindler horizons in special versus general relativity (now ignoring the irrelevant equivalence principle) one needs to accept a definition that works in the general case. In my opinion, there is a unique such definition that covers all world lines (i.e. eternal observers), in all spacetimes, with a physical criterion that has nothing to do with coordinates, and coincides with the standard Rindler case in special relativity. My proposed definition (at the B level - I will use more technical language in the other thread) is the boundary of events such that the world line can both receive a signal from some event (at some event on the world line), and also send a signal to it (from some earlier event on the world line). Note, this definition is defined by the observer, not by coordinates, and is totally different from general relativity definitions of event horizon (which are global features of the the spacetime, independent of any observer).

By this definition, it is obvious that inertial observers in special relativity have no Rindler horizon because they can 'communicate' with all of spacetime. It is also true that any observer in special relativity for which there is a lower bound on proper acceleration for all time, has Rindler horizon. (This does not cover all cases with Rindler horizons in special relativity, but it is not relevant to try to describe all cases).

With this preparation, it is trivially true that an observer sitting on a planet has no Rindler horizon - there are no events they can't communicate with (at least if you consider the planet in isolation, and don't bring in cosmological horizons). IMO, this fully answers the OP question.

More interesting, is that for a stationary observer anywhere in a black hole spacetime, the event horizon is, in fact, their Rindler horizon. This is different from other statements in this thread, but is clearly correct by the definition above.

Finally, as @Ibix noted much earlier, there are Rindler horizon cases in a planet or black hole spacetime that are essentially similar to the special relativity case, having nothing to do with the BH event horizon. An observer eternally accelerating at 1 g, with closest approach to a BH or planet being 10 light years, will have a Rindler horizon at around 9 ly from the planet or BH, essentially indistinguishable from the special relativity case. The BH or planet will be completely 'behind' this Rindler horizon.

 

[TSny]

It is also true that any observer in special relativity for which there is a lower bound on proper acceleration for all time, has Rindler horizon.

Would the following be a counterexample of this statement?

Suppose the observer is in Minkowski spacetime and the observer is forever sitting on the rim of a merry-go-round that rotates at constant speed. Thus, the observer experiences a constant magnitude of proper acceleration. But, there are no events that can't communicate with this observer.

I haven't thought much about this, so I am probably overlooking something or misinterpreting something.

 

[PeterDonis]

Moderator's note: "A" level discussion of Rindler Horizons is here:

https://www.physicsforums.com/threads/rindler-horizon-subtleties-advanced-discussion.1007892/

 

[PAllen]

Would the following be a counterexample of this statement?

Suppose the observer is in Minkowski spacetime and the observer is forever sitting on the rim of a merry-go-round that rotates at constant speed. Thus, the observer experiences a constant magnitude of proper acceleration. But, there are no events that can't communicate with this observer.

I haven't thought much about this, so I am probably overlooking something or misinterpreting something.

You are correct. I was only considering proper acceleration in one direction for that particular statement, but failed to state this. A merry go round observer does not have a Rindler horizon.

 

[PeterDonis]

I post here at B level

Moderator's note: I have made this an "I" level thread, not "B" level. The original thread was "B" level, but the discussion here really goes beyond that level.

 

[PAllen]

Moderator's note: I have made this an "I" level thread, not "B" level. The original thread was "B" level, but the discussion here really goes beyond that level.

Fine, but I plan to continue most discussion in the already existing A level thread.

 

[PeterDonis]

My proposed definition (at the B level - I will use more technical language in the other thread) is the boundary of events such that the world line can both receive a signal from some event (at some event on the world line), and also send a signal to it (from some earlier event on the world line).

This "two-sided" definition actually combines two standard definitions--but most discussions of Rindler horizons don't make it clear that there are, in fact, two of them, not one. (The same is true of most discussions of event horizons.)

The "Rindler horizon" that most people think of in the SR case is the future Rindler horizon--the boundary of the region of spacetime that cannot send light signals to the accelerating observer. But there is also a past Rindler horizon--the boundary of the region of spacetime that cannot receive light signals from the accelerating observer. The key point is that these are two different null surfaces in spacetime. (They do intersect at a "bifurcation surface"--so called because the two horizons together form a "bifurcate Killing horizon" for the congruence of Rindler observers. This surface appears as the "origin" point in a 1 x 1 Minkowski diagram.)

Your definition combines these two different null surfaces into one "horizon", conceptually. For the SR case, which is manifestly time symmetric, this works fine. But it doesn't necessarily generalize to cases which are not time symmetric--for example, the spacetime of a black hole formed by gravitational collapse of a massive body, such as the Oppenheimer-Snyder model. In that spacetime, there is no past event horizon; there is only the future event horizon. So the analogy with the Rindler case in SR is incomplete. It is true that the black hole region satisfies the letter of your definition; but it also satisfies the usual definition that only considers the "future" part of the horizon--because, of course, it is perfectly possible for a stationary observer to send light signals into the black hole.

The paper that is being discussed in the "A" level thread uses a somewhat similar "two-sided" definition, but its definition is more restrictive, so that a stationary observer in a black hole spacetime does not have a "Rindler horizon" (and the event horizon of the hole is not a "Rindler horizon" for a stationary observer), but observers with appropriate profiles of radial acceleration (so that they come in from infinity, turn around, and accelerate back out to infinity) do have "Rindler horizons" by their definition.

 

[PeterDonis]

I plan to continue most discussion in the already existing A level thread.

Fair enough. You might want to consider formulating your definition in more technical language and comparing it with the definition given in the paper being discussed in that thread, which makes use of the intersections of the observer's worldline with past and future null infinity.

 

[PAllen]

I’ll add that a coordinate formulation of my definition is simply the boundary of the region of spacetime that can be covered by radar coordinates based on a given world line. This also matches the boundary of coverage of standard Rindler coordinates.

 

[PeterDonis]

I’ll add that a coordinate formulation of my definition is simply the boundary of the region of spacetime that can be covered by radar coordinates based on a given world line.

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.

 

[cianfa72]

1) Spherically symmetric refers to a physical system which depends only on some radial parameter and is the same in all directions. To a first approximation the solar system is spherically symmetric about the Sun: the spacetime/gravity around the Sun depends only the distance from the Sun, not on the direction.

Quoted from other thread.

I believe a point to be highlighted is that in a spherically symmetric spacetime the radial parameter does not represent in general a 'distance' (e.g. the length of a spacelike geodesic). It is actually just a parameter (coordinate) describing the radial symmetry w.r.t. a given point (e.g. the Sun).