Ben Rudiak-Gould wrote:

Ben Rudiak-Gould wrote:

This is also true of a Rindler horizon. You need to specify what
you mean by "the outside point" -- it sounds coordinate-dependent.

Perhaps. It is another parallel in the back and forth between
analogies in flat spacetime to phenomena in curved spacetime, and one
that happens -- if we are reduced to "well, it could be like this, but
it could also be like this, while on the other hand ..." to fall on
your side of the balance beam.

Perhaps you think that a "static spatial point" is a coordinate
dependent concept?

I disagree vigorously.

The only exception I can think of (I'm cheating -- I though
of this until later, and inserted it), is the special static
geometry of flat spacetime. There indeed a "point" (static
spatial point) has no natural meaning, since other "static"
points may be flying by an arbitrary relative velocities.

I find this objection bizarre. You stated that there is nothing very
special about the horizon -- other than the behavior of the Killing
vector, and some condition on observability -- and that no coordinate
independent infinities lurk nearby. I claim the radar return time
refutes both these arguments.

If there happens to be an event horizon below us, we are going to have
problems placing some finite meter stick (as I now know); I guess we
keep dropping it. So at that point -- oops... I mean "juncture" -- we
place, say, half a meter stick, or a third, or whatever we can manage.
At this same state of affairs, we notice the radar return times to our
fixed mother ship are growing large very quickly with length. In fact,
we will find that by adding some finite additional increment of stick,
we can make the return time grow as large as we like.

If, on the other hand, our initial determinations were that the
spacetime were flat, we could begin unfolding our infinite folding
ruler, as before, and again measure an unbounded increasing series of
return times -- the difference being, in this case, the amount of ruler
we can extend is unbounded also.

Ben Rudiak-Gould wrote:

Edward Green wrote:

Ben Rudiak-Gould wrote:

This is also true of a Rindler horizon. You need to specify what you mean by
"the outside point" -- it sounds coordinate-dependent.

In hindsight I should have left out the second sentence of that paragraph.
The first sentence is the important one.

Perhaps. It is another parallel in the back and forth between
analogies in flat spacetime to phenomena in curved spacetime, and one
that happens -- if we are reduced to "well, it could be like this, but
it could also be like this, while on the other hand ..." to fall on
your side of the balance beam.

But really sir ... you are quite impossible: "the" outside point is
clearly "an outside point" from the first paragraph (manifested by the
reference "sequence of suspended ropes")

It's equally problematic there, but I didn't bring it up because it's
overshadowed by the other problem with that test.

and "an" outside point means
just that -- a completely arbitrary point outside the event horizon. I
presume "horzon" and "outside" are coordinate free concepts.

Yes, but "point" isn't. Unless you mean "spacetime point", but you can't
mean that because you're talking about radar return times.

Many are the times I've accidentally left some side door open in a
post, realized it afterwards, and sure enough found the reader
wandering about in the broom closet. At the same time I'm perplexed by
your perplexity. Surely it's obvious I meant a spatial point -- and
before you lecture me on the meaninglessness of that concept, recall we
are in a static geometry, where we can return to the "same" spatial
point as often as we like along our worldline.

Perhaps you think that a "static spatial point" is a coordinate
dependent concept? I disagree vigorously.

The only exception I can
think of (I'm cheating -- I though of this until later, and inserted
it), is the special static geometry of flat spacetime. There indeed a
"point" (static spatial point) has no natural meaning, since other
"static" points may be flying by an arbitrary relative velocities.

I
guess you were thinking of this, which perhaps is why you otherwise
inexplicably regard my unqualified use of "point" as problematic.

Flat spacetime may be a kind of analytically limiting case -- like the
extension of fourier series over finite intervals to the "limit" of
fourier integrals -- which is both natural and pathogenic. I'm not
sure of this observations full significance to our discussion, but I
think it may be deep.

The real problem is that there's no way to tell what "a point outside the
event horizon" or even "a worldline outside the event horizon" means until
after you know where the event horizon is, but the purpose of your test is
to characterize the event horizon in the first place.

I find this objection bizarre. You stated that there is nothing very
special about the horizon -- other than the behavior of the Killing
vector, and some condition on observability -- and that no coordinate
independent infinities lurk nearby. I claim the radar return time
refutes both these arguments. I take the point of view that the
spacetime geometry is a given and that I can use that given geometry
freely in formulating my reply. You want me to take the view, I think,
that I'm plunked down in an unknown spacetime, and must begin piecing
its geometry together by operational tests.

I'm not sure why this is necessary, but Ok, let's look at that
question.

Feeling out the local spacetime geometry is a complicated business, but
I take it if we are plopped into a static geometry, we can, given the
ability to carry out measurements over a local, but not too local
region (not pointlike), discover that we are in a static spacetime.
Assume for now the spacetime is not flat. We will simultaneously
establish, perhaps, that we are moving wrt the static background. Then
we may being firing our rockets sufficiently to maintain a fixed
position in the time invariant spatial geometry. Once we are in a
static position, we may identify "down" as the opposite of the
direction we are accelerating in to hold ourselves in place.
Identifying "down", we may then begin placing meter sticks end to end
in that direction, with reflectors attached at certain lengths.

If there happens to be an event horizon below us, we are going to have
problems placing some finite meter stick (as I now know); I guess we
keep dropping it. So at that point -- oops... I mean "juncture" -- we
place, say, half a meter stick, or a third, or whatever we can manage.
At this same state of affairs, we notice the radar return times to our
fixed mother ship are growing large very quickly with length. In fact,
we will find that by adding some finite additional increment of stick,
we can make the return time grow as large as we like.

This entire construction sounds to me like a coordinate independent
operation characterization of an infinity associated with something
that begins to look like an event horizon.

If, on the other hand, our initial determinations were that the
spacetime were flat, we could begin unfolding our infinite folding
ruler, as before, and again measure an unbounded increasing series of
return times -- the difference being, in this case, the amount of ruler
we can extend is unbounded also. I guess in this case I would like to
say we have a "coordinate independent infinity" also -- the usual one.

An objection that can be made:

"We never measure 'infinity', but only an increasing sequence of
positive numbers."

This is more a kind of statement of the halting problem, or the
impossibility of "measuring an infinity" in general, than a specific
objection to characterizing this infinity. Of course we can never be
sure that, measuring a sequence of ever increasing numbers which do not
seem to approach an asymptote, that the sequence will not stop growing
with the next number or the one after it. At best we can say that "the
postulate of an infinity has not yet been falsified".

Need I say I would find this objection silly?

Let's fire up our spaceship and go event-horizon hunting. Event horizons
always move toward you at the speed of light (until you fall through, then
they move away at the speed of light). So our first order of business had
better be to fire the rockets to avoid falling in. Which direction? Well, we
don't know yet, but let's try thataway. Now as soon as we start
accelerating, we'll find that there's an event horizon below us which passes
your radar-time test, *whether or not there's a black hole nearby*.
Accelerating makes a Rindler horizon, and Rindler horizons have the same
properties as black hole horizons. In fact you can choose any approaching
null surface whatsoever, and as long as you avoid falling through it, it'll
pass your test.

Now that's an interesting argument. Essentially you are saying that my
first assertion is false -- that we _can't_ survey the local spacetime
geometry by kind of poking around with scout ships and lasers and dust
clouds. Maybe, worse, we can't even survey it when it is static,
finding out in the process that it is static, and hold ourselves in
(now meaningful) spatial position. If we could carry out this survey,
and if we just turned off the rocket for a minute, we could discover
that not only is the spacetime static but flat, and we wouldn't be
fooled by the dumb old Rindler horizon for a minute, recognizing it for
what it is.

I think we can infer the local geometry, at least when it is static.

Some years back, when giants roamed sci.physics, there was reference to
a published paper, or at least a respectable pre-print, asserting that
(static?) gravitational fields were in fact distinguishable from
effects associated with an accelerating observer in flat spacetime
_even_ in the limiting case of a gravitational field with zero spatial
curvature.

You will agree this assertion is of some interest here.
(I've tried to search the archive, but I've been unable to find the
reference again.)

You in effect assert the contrary -- that the
uniformly accelerating observer in flat spacetime _cannot_ tell the
difference between his situation and a gravitational field of an
"infinitely large" black hole geometry

-- that's why he can't tell the
difference (you imply) between his own Rindler horizon and a black hole
in the ground.

I remember that the paper was over my head, but even now I can think of
some supporting arguments based on low brow reasoning. For example...
if we were outside the event horizon of a black hole so large that we
couldn't detect the spatial curvature, then we would find we had
manuvering room in how far (as measured by a finite extension of meter
sticks -- I bet you're sorry you showed me that) we were from the
horizon at a constant proper acceleration g: if the horizon is
essentially flat, gravitational acceleration is essentially constant.
But we would have no such freedom with the Rindler horizon, since its
distance behind us depends only on g: what else could it possibly
depend on?

[...] so is the finite proper length
filament somehow a mapping of an infinitely long filament whose
"spatial acceleration" (variation of velocity with position) is strong
enough that the integral of the relativistically contracted length is
finite, again into a curved spacetime which hides the effective
relative velocity in the curvature -- and also contrives to allow a
static filament whose segments are effectively moving at different
velocities.

No, it's just a plain, ordinary finite distance. Really truly it is.

That was a completely unedited flight of speculative fancy. However,
while I don't present it as a close argument, I'm unconvinced that it
might not correspond to a close argument.

Physics
near the event horizon is just like physics everywhere else.

Except of course for that pesky Killing vector thing.

You're intent
on making it look weird by viewing it in a weird way, but as you've pointed
out yourself, even good old Minkowski space is weird if you admit
sufficiently weird worldlines.

Well, the longer we discuss this, the more I see many things hinging on
a series of parallels between behavior in Minkowski space, as you
prefer to call it, and analogous behaviors in more general spacetimes.
Not all the parallels seem to urge us in the same direction: we can
either focus on the Rindler horizon, and conclude that that looks
plenty weird -- though we know its a kind of observational artifact in
flat spacetime -- and hence convince ourselves that maybe the
Schwarzschild horizon isn't all that weird either, or else we can focus
on the trajectories which "reach infinity" in finite proper time in
Minkowski space -- though we would tend to think they never get there
-- and convince ourselves that maybe crossing the event horizon never
happens either. I've mentioned that the good old (and relevant) Zeno's
paradox can be looked at from two ends also (Zeno and Onez?), though
the tradition apparently only includes one reading.

That's a kind of ethically relativistic "I'm ok, you're ok" kind of
version, though I still think I'm just a little more ok here ;-) I
deny I'm looking at things in a "weird way" -- perhaps I'm just looking
at things in a fresh way: I am in fact a tyro, but not a completely
unsophisticated one. The Rindler horizon thing is a standard analogy
-- I see -- known by many -- but the aberrant trajectories I happened
to discover independently don't seem to be all that well known, nor
traditionally discussed. Hence they don't seem to carry equal weight
in the philosophically tinged discussions touching on this topic --
though I don't see why not.

There's nothing weird in the vicinity of a
black hole event horizon that you couldn't find pretty much anywhere else in
the universe.

Still disagree.

If you expand your viewpoint to take in the entire black hole, or even just
an equatorial slice through it, then you can probably find a working
characterization of the event horizon. But that's no longer a local test,
because the size of the black hole sets a lower bound on the size of your
laboratory.

I indeed took it as a given that we could simply take the entire black
hole geometry as a given in looking for interesting operational tests.
You apparently are interested in a slightly different question.

However, adding your requirements, it still seems to me that the
operational characterization can be carried out. Here's the meta-code:

Survey local spacetime geometry
If geometry is flat, then stop: no nearby horizon

If geometry is nonstatic, stop: we are not outside a pure
Schwarzschild horizon

Else, stop motion wrt static spatial geometry by proper acceleration
Find "down"
Extend array of meter sticks with reflectors
If return time and proper length both diverge, we are facing a
"normal" infinity
If return time diverges while proper length remains finite, we are
facing a presumptive bh
If proper length diverges while return time remains finite... ???

That would be weird. ;-)

Surely it's obvious I meant a spatial point -- and
before you lecture me on the meaninglessness of that concept, recall we
are in a static geometry, where we can return to the "same" spatial
point as often as we like along our worldline.

Perhaps you think that a "static spatial point" is a coordinate
dependent concept?

You want me to take the view, I think,
that I'm plunked down in an unknown spacetime, and must begin piecing
its geometry together by operational tests.

Feeling out the local spacetime geometry is a complicated business, but
I take it if we are plopped into a static geometry, we can, given the
ability to carry out measurements over a local, but not too local
region (not pointlike), discover that we are in a static spacetime.

Some years back, when giants roamed sci.physics, there was reference to
a published paper, or at least a respectable pre-print, asserting that
(static?) gravitational fields were in fact distinguishable from
effects associated with an accelerating observer in flat spacetime
_even_ in the limiting case of a gravitational field with zero spatial
curvature.

You in effect assert the contrary -- that the
uniformly accelerating observer in flat spacetime _cannot_ tell the
difference between his situation and a gravitational field of an
"infinitely large" black hole geometry --

if the horizon is
essentially flat, gravitational acceleration is essentially constant.
But we would have no such freedom with the Rindler horizon, since its
distance behind us depends only on g: what else could it possibly
depend on?

Physics near the event horizon is just like physics everywhere else.

Except of course for that pesky Killing vector thing.

The Rindler horizon thing is a standard analogy
-- I see -- known by many -- but the aberrant trajectories I happened
to discover independently don't seem to be all that well known, nor
traditionally discussed.

Hence they don't seem to carry equal weight
in the philosophically tinged discussions touching on this topic --
though I don't see why not.