How close could a tachyon get to a black hole and still escape?
This question can't be answered because we don't have any tested or accepted theory of tachyons to apply in answering it. Without such a theory, if we assume that tachyons exist at all (which, to the best of our current knowledge, they don't), we have no way of ruling out the possibility of their escaping from anywhere inside a black hole.
For example, if you assume a tachyon can follow any spacelike geodesic, then (for SC geometry), then some tachyon can escape from any event in the black hole interior (the singularity itself not being an event but an open 'edge' of the manifold).
I don't see why #2 and #3 have to be worded so cautiously. I don't think there's that much wiggle room in the properties of tachyons. If they exist, and if Lorentz invariance (LI) holds, then their kinematic properties are fully determined by LI.
It's true that during the OPERA neutrino debacle, there were a lot of goofy theories floating around. This was because the experimental claims were seemingly inconsistent with LI. But now that we know the whole thing was due to a loose cable, there is no motivation to take non-LI theories seriously.
Define how close according to what observer.
I agree, but "kinematic properties" isn't enough by itself. That just tells you that tachyons travel on spacelike curves whose tangent vectors transform appropriately. But there are plenty of such curves from any event in the interior that do *not* escape into the exterior region, as well as plenty that do. *Which* spacelike curve will a particular tachyon, emitted at a particular event in a black hole's interior, follow? Kinematics can't tell you that. So kinematics can't answer the OP's question.
Why can't kinematics tell you that? A test particle moves along a geodesic. Given the initial spacelike velocity four-vector, there is a unique geodesic tangent to it.
I suppose we could worry about Cerenkov radiation...? Is that what you had in mind as an uncertainty about the dynamics? I have seem claims that even a sterile tachyon, which doesn't emit Cerenkov radiation through electromagnetic or weak-force interactions, emits gravitational Cerenkov radiation.
*Which* initial spacelike vector? That's what kinematics can't tell you.
For example: I point my tachyon pistol at you. The pointing of the pistol does define a spatial direction. But it doesn't define a unique spacelike vector, because I don't know which surface of simultaneity I should use. Should I use mine? Yours? The simultaneity of the CMBR rest frame? That's what kinematics can't tell you; you need a dynamical theory of tachyons.
The generic dynamical theory of tachyons simply says: for a given local frame, the total energy and invariant (imaginary) mass determine the 4-vector, which then determines a geodesic. Given this, from any point inside the BH horizon, there will exist choices such that the tachyon will escape the horizon and proceed to spatial infinity (assuming SC geometry).
The total energy and invariant mass together don't determine a unique 4-vector. They only determine a particular "mass shell" hyperbola (I put "mass shell" in scare-quotes because the invariant mass is imaginary, as you say, but it works the same as an ordinary mass shell for a timelike object). *Any* spacelike 4-vector whose endpoint lies on the hyperbola will satisfy the equation specified by the total energy and invariant mass, and there are an infinite number of them.
I agree that it's true that at least *some* of those spacelike 4-vectors, at any event in the BH interior, will determine spacelike geodesics that escape into the exterior. So as long as the dynamical theory of tachyons allows tachyons that follow any 4-vector on the "mass shell" to exist, then as you said in your earlier post, there will be some tachyon that escapes from any event in the BH interior.
I don't see any reason why a dynamical theory of tachyons *wouldn't* allow the above, but we don't actually have such a theory, so we don't know for sure.
I'm speaking of 'classical' tachyons. No such thing as mass shell. Spatial direction, total energy, and (imaginary) invariant mass then uniquely determine a spacelike 4-vector, which uniquely determines a geodesic.
However, this discussion proves your point: you must specify some theory of tachyons.
Actually, the mass shell can be defined classically (it's just the hyperbola defined by all 4-vectors with the same invariant length). However, I was misreading "total energy" to mean the same thing as "length of the energy-momentum 4-vector", which of course is what "invariant mass" means; "total energy" means the 0 component of the 4-vector in a given frame. So your specifications *do* determine a unique spacelike vector. Sorry for the mixup.
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