- #1

PAllen

Science Advisor

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*[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.

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