Re: the black fishing hole
- From: tessel@xxxxxx
- Date: Tue, 19 Sep 2006 08:40:17 +0000 (UTC)
On Sat, 16 Sep 2006, Gerry Quinn wrote:
it's already been pointed out several times in the past few weeks why
this assumption is naive. And the same point must have been made
dozens of times here in the past few years!
I don't recall this being "pointed out",
For example, in a post dated 7 Sep 2006, which appeared in the thread
titled "White Holes are time-reversed black holes?", Steve Carlip wrote:
Thus, I am confused about whether or not an infalling observer would
say that she is *spatially* near the horizon for r just an itty bit
smaller than 2m. Does it make any sense to say that she is *spatially*
closer to the horizon when she arrives at r = 1.999m as compared to
when she is at r = 1.998m? Or is it not valid to ask about the spatial
nearness of events that are inside the horizon?
Spatial closeness is a bit tricky. To define it, you need a "time slice"
In addition to the choice of spatial hyperslice, there are also many
coordinates one could use to define a notion of "radial coordinate". In
any static spherically symmetric spacetime, the Schwarzschild coordinate
has the virtue of having a simple and natural interpretation (surface
area of sphere r=r0 is 4 Pi r0^2). And as I have been trying to
explain, there are competing notions of distance even after you have
chosen a family of observers such as static exterior observers or
Painleve observers. (For the purpose of defining a pedometer distance,
these two choices happen to correspond to choosing the hyperslice
corresponding to the Flamm paraboloid and a hyperslice which is locally
isometric to E^3, respectively.)
but if I had I would have dismissed it as obfuscation.
Not at all. This is a fundamental point about curved spacetimes.
Failure to recognize this is one source of many common misconceptions.
For an observer at a distance - such as Zef, who is after all the active
party in this thought experiment
Will someone please give the citation? (Ed Green omitted to do so in his
original post--- in happier days, his submission would probably have been
returned with a demand that he give a citation.)
- the position of the event horizon is certainly well-defined.
I don't know what you mean by "position" and I don't know why you think it
is ill-defined.
If you are referring to my point about the multiplicity of notions of
distance, I was not saying that any of these are ill-defined. I was
saying that with sufficient care they can be operationally and
unambiguously defined, but they disagree. The disagreement is small for
small h, or small m, or large R; verifying this was one point of taking
a multivariable Taylor expansion; verifying the stated inequalities for
small h, small m, and large R was the other.
And the Schwarzschild metric is defined in terms of a radius. If the
radial coordinate becomes timelike, the length of string will be
redefined along with it, and will remain finite.
You wrote "if the radial coordinate becomes timelike, the length of
string will be redefined". I don't think this makes sense as stated,
probably because it is clearly based on several fundamental
misconceptions. As I and others tried to explain to you, a static
observer hovering outside the horizon cannot lower a weight on a string
under the horizon, because the string will break (because the tension in
the string diverges as the weight nears the horizon).
The use of the word "distance" for the difference between two radial
coordinates is not particularly problematic.
Here you may be thinking of the pedometer distance from
r=r2, theta=theta0, phi=phi0
down to
r=r1, theta=theta0, phi=phi0
where r2 > r1 > 2m, as measured by a quasistatic ant crawling slowly
along the string. It turns out that for a radially oriented taut string
held by a static observer, this is given by an integral, which will
-not- agree with r2-r1, except when the hyperslice has euclidean
geometry, which is -not- the case for any of the hyperslices orthogonal
to static observers in the exterior region of the Schwarzschild vacuum
solution.
Contrary to what you imply, the multiplicity of notions of "distance in
the large" is a -fundamental feature- of curved spacetimes. So is the
fact that in general these notions of distance will not even be
symmetric (given two observers A, B, even if they agree that they are
mutually stationary, and even if they both use the same notion of
distance, in general the distance to B as measured by A will not agree
with the distance to A as measured by B!)
Furthermore, under any interpretation, the distance to a point
arbitrarily close to the event horizon, where there is no need for
ambiguity in distinguishing space from time, is well-defined and tends
toward a finite limit.
"Under any interpretation"? Not true. There are multiple competing
spatial hyperslices, one can integrate along a geodesic in a given
hyperslice to obtain a "distance"; see Steve Carlip's comment. Using
the operational viewpoint I advocated, which is based upon families of
observers, is an alternative and more physically motivated way of seeing
that there exist multiple competing notions of "distance". If you
appeal to notion A in one place and notion B later on, without realizing
that A =/= B, of course you will become confused!
There is no good reason to think of the distance to the event horizon as
other than this limit. So *of course* Zef can pay out more string than
this limit.
No, the string will break before the weight reaches the horizon.
Remember, this thought experiment concerns the observations of Zef.
It seems that you have not read the original (which Ed Green failed to
cite), and neither have I (since Ed Green failed to give the citation).
This makes it impossible for us to discuss what "Zef" (I suspect he may
have meant "Zeh") might have said, particularly since I suspect Ed Green
may have misunderstood the thought experiment he was putatively quoting.
What goes on at or below the event horizon has no relevance to him, as
he will never observe it.
No, the string will snap while the weight is -still outside the horizon-.
I thought that's precisely the thought experiment which we've been
discussing, except that of course that in a quasistatic thought
experiment the tension increases -monotonically-.
Actually the original poster referred to "lowering an object below the
event horizon",
Which was promptly corrected: according to gtr (recall that the
mainstream view is firmly that gtr should hold true to an excellent
approximation in some neighborhood of the horizon of any astrophysical
hole--- e.g. including the region just -inside- the horizon), an
exterior static observer cannot lower a weight below the horizon; the
string will break before the weight actually reaches the horizon.
Even if he maintains constant tension for, as you say, a controlled
fall, the amount of string to be paid out per second will eventually
fall to zero.
No, to maintain constant tension, he needs to pay out string -faster-.
Since this is obviously true in Newtonian theory (agreed?) and since gtr
has a Newtonian limit, if you claim otherwise you have some explaining
to do.
But I suggest that you first learn how to compute the tension in a
radially oriented string, held by a static observer hovering somewhere
outside the horizon and using the taut string to hold up a quasistatic
weight at the end nearer to the horizon.
If you find that the use of curved spacetime confuses the issue, I
suggest you consider transforming the problem into Zef's coordinates, in
which spacetime is flat.
If you are claiming that some coordinate tranformation can turn a curved
spacetime into a flat spacetime, this is obviously incorrect. I suggest
you reread Darryl's post to see if you can spot why this fundamental
point is -not- incompatible with what he wrote there!
Am I to infer from the above paragraph that you think it will increase
without bound so long as the string resists breaking?
Of course not. If you had worked the exercises I gave you would know
that.
Certainly that's the impression that the OP seems to have got.
Huh? Who is this "OP" who allegedly says that he/she has the
(incorrect) impression that I am saying that if a static observer uses a
rocket engine to hover outside the horizon of a Schwarzschild and lowers
a weight at the end of a radially oriented string toward the horizon,
then he can pay out an infinite length of string before the string
breaks? Are you talking about Ed Green? If so, where did he say that
he has gotten this impression from something I wrote?
In both Newtonian gravitation and in gtr, the tension in the string
should monotonically increase as the weight is slowly lowered. The
computation works the same way in both theories: consider the situation
at a given time and work out the total tension in the string at that
time.
When I say 'restored' I mean that the fact of tension is restored; that
the dangling object is supported by a tensile force in the string. There
seems no need to concern ourselves with its value.
To the contrary, verifying that the tension diverges as the weight nears
the horizon is the easiest way to see that the string must in fact break
(according to gtr, but since we are discussing an exact solution of the
EFE, if you have some other theory in mind you have a lot of explaining
to do).
As for "length of the string", this obviously means its proper length -
as measured by the number of knots along it or in some such manner.
Exercise: does this "reel distance" correspond to "pedometer distance"?
To put it another way, it is the length paid out by Zef. We assume that
under the conditions of the experiment it is so strong as to be
effectively inextensible until such time as it breaks and the gedanken
is no longer valid. In flat or curved spacetime, this proper length is
a well-defined value, admitting no ambiguity.
No! There are various well-defined but -distinct- notions. Part of the
problem here may be that in places you seem to be using "radar distance"
in some places and in other places you seem to be using "pedometer
distance", or some related notion, without realizing that these give two
methods of defining distance differ.
It is unfortunate that you have chosen to dismiss my comments without even
attempting to understand the points I was making.
The problem for Zef is gravitational time dilation.
Citation? (Second request.)
I don't recall a previous request.
I was under the impression that you knew the citation Ed Green failed to
give in the original post. Ed G, if you are reading this, you can see
what trouble a little laziness costs, so please provide the missing
citation forthwith.
I can give no citation, except the post to which you are responding.
OK, check, I now recognize that you do not in fact know the citation
either. Ed Green, can you please just give the darned citation?
But your assertion that paying out the string faster and faster will
maintain constant tension is simply wrong. The effects of gravitational
time dilation will eventually overcome the effects of increased weight.
I suggest you try again, this time considering the simpler thought
experiment involving quasistatic weight on the end of a radially
oriented taut string held by a static observer using his rocket engine
to hover outside the horizon of a Schwarschild object. This will enable
you to verify the claim that the string will snap before the slowly
lowered weight reaches the horizon.
Is there anything wrong with the above?
That is precisely what I have been trying to explain. But if you are
unwilling to follow along with my discussion by working the exercises I
gave, there is no point in my continuing this discussion.
"T. Essel"
.
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