Dr. Photon wrote:
JB wrote
Hovering above the BH is a system
of stationary "calibrated" spaceship
markers. Each is hovering 1m apart
according to some "standard ruler"
that everyone accepts.
First, I should point out that no
such ruler exists. Even in ordinary
special relativity, a distance
measurement is only meaningful if you
specify the frame of reference in which it is made.
But you have said previously that the "distance to the BH is finite".
What did this refer to?
This was as it would be measured by hovering rulers. I was merely
pointing out that the distance would not be the same in every frame of
reference.
If you have made an integral along a path in
Schwartzschild metric, then mark out 1m intervals on this path. Have
a
marker hover at each point. Have the mothership hover at a fixed
distance from the BH - I hope this means something? In the metric
(maths description), choose the coordinate of the mothership to be
constant with time, and mark everything from there? The purpose of
the
winch line with 1m marks on it was so I could define everything in
terms of "mothership metres".
Does your comment imply that it is possible that the answer "what is
the distance to a BH?" is "finite" if you do it one way, a different
"finite" if you do it another way,
Yes, you can get a different finite answer for the distance from the
event horizon to the ship if you do it a different way.
or even "infinite" if you do it
another way?
I don't think you can get infinity as an answer, but it's hard to check
every conceivable way of doing it.
What conditions did you and TR assume when you said it
was finite?
and the first ship passes
through the EH in a finite amount of mothership time,
Once again, this is a meaningless statement, unless you carefully
specify a definition of simultaneity.
How about this: If the mothership can bounce a light pulse off the
infalling object, and receive that pulse again, then the object had
not fallen in at the halfway time between sending and receiving. The
mothership can send 10 ps pulses at a rep rate of 1 ns quite easily,
to have a good "realtime" update. The mothership can count the
transmitted and received pulses in order to know which pulse it is
receiving.
This is a useful definition of simultaneity outside the event horizon,
and along a vertical line passing through the mothership, it is
equivalent to saying two events are simultaneous if their Schwarzschild
t coordinates are the same. It's not so good for understanding what
happens at the event horizon because it puts everything that happens on
the event horizon, in the infinitely distant future. That's because if
you bounce a light pulse off an object that's at the event horizon, the
reflected light pulse can never get back to you. There's nothing wrong
with thinking about the event horizon as being in the infinitely
distant future; it satisfies the consistency requirement that time only
goes forward for every object. But it makes it difficult to analyze
anything that happens on the event horizon.
The definition is also almost what Einstein used to define simultaneity
in special relativity. The crucial difference is that the observers in
special relativity weren't accelerating.
For a mothership very close to the event horizon, space-time curvature
can be safely ignored in the analysis, and we can safely use the rules
of special relativity. A freely falling observer using a special
relativistic definition of simultaneity would say that the reason we
thought the event horizon was in the infinitely distant future was
because, although we used the special relativistic definition of
simultaneity for the frame the spaceship was at rest in at each
instant, that frame was changing. From the free-falling observer's
perspective, we are declaring events that happened increasing longer
ago in what he would call our past to be happening right now.
Close up to the event horizon, this definition of simultaneity, and the
time dilation that occurs in it seem silly. By choosing another
definition, the time dilation goes away. But over long distances,
where space-time curvature is significant, it isn't possible to pick a
definition of simultaneity without dilation somewhere.
It still seems to me that this time will have a very large dilation,
and the mothership will receive return signals at longer and longer
intervals, for a considerable length of time. Hence the object will
take a "long" time to fall the finite distance. As there will be some
"last bounce", and after that silence, once the mothership does not
receive a signal in two days, then we can call it quits.
There is no last time at which a signal could be received. With a
powerful enough transmission from the probe just before it crossed the
event horizon, the mothership could receive signals at an arbitrarily
late time.
This gives a
lower bound measure of the time of fall to the nearest day, (although
if we had waited longer, there may well have been another pulse). I
expect that the dilation proceeds over many days, possibly years. So
again, the infalling mass "really" looks like it is going that finite
distance at slower and slower speed.
From the point of view of the first probe, which is freely falling,
the
markers are moving upward at some velocity v. The distance between
adjacent markers is 1m * sqrt(1-v^2/c^2), less than 1m.
If the mothership is hovering, take its coordinates as fixed. I
wanted
to get away from the picture according to the falling objects. I
think
I am right in saying that even though the mothership is accelerating,
its coordinates can be fixed???
Its coordinates can easily be fixed. Coordinates are just labels and
can be whatever you want them to be. The mothership is still
accelerating.
I have here a problem with "real" and
"apparent" again. If you have a helicopter on earth, hovering at a
particular height, it is producing a force upwards to maintain
position. But it is not "moving" relative to the Earth centre (an
inertial frame (not counting rotation, or orbit around Sun)). The
"force" of the upthrust, counteracts the "curvature" due to gravity,
but the *distances* remain the same.
The force accelerates the helicopter upward. The space-time curvature
of the space between the earth's center and the helicopter counteracts
the change in relative velocity and distance this acceleration would
otherwise cause.
Similarly, according to another
inertial observer floating in free-space, the distance between the
observer and the helicopter can be constant. So the helicopter may as
well be given a fixed coordinate? I found Section 2.3 in the
following
link
http://www.vallis.org/publications/tesidott.pdf
titled "The equivalence principal paradox" (p17) very interesting,
I'll post again once I crystallise what my point is here. Something
like a "supported frame in a gravitational field" can have different
properties to an "accelerating frame in a flat spacetime". But I'll
have to read it closely first.
It looks interesting even in and of itself. I'm going to want to read
it closely too soon.
Anyway, relative to the hovering ship, the EH is not moving at all,
although relative to the infalling object, it is moving at c. When I
spoke of the EH expanding, I referred it to the spaceship. So after
the mass fell in, the distance from the ship to the EH is smaller
than
it used be.
This seems like it should be right, or close to right, although it
could take a nasty calculation to check it.
If no further mass falls in, then the distance from the
ship to the EH is again constant. If the EH does not move towards the
spaceship at all (an ideal stationary hole), then the time to cross
the EH is measured as infinite according to the "bouncing signal"
described above (remove the two day limit).
It would be infinite in either case.
Over long distances, such as what
would happen if the spaceship waited
far away from the black hole, space-
time curvature can play tricks on
you and give you various speeds for
light if you attempt to measure it
this way.
I think(!) we are agreed. Does TR agree?
Thanks,
BR
.
