"JanPB" wrote in message
ups.com...
| LEJ Brouwer wrote:
| The Trouble with Physic(ist)s is that they are Not Even Wrong
|
| I have been thinking about this and earlier discussions and wondering
| what to make of it all. How is it that otherwise intelligent
| individuals can agree en masse upon something which, on the face of it,
| is completely and utterly wrong?
|
| We would like to find the metric outside of a static point particle.
| Clearly, before we even start, we know that the solution metric, like
| the problem itself, must be time-independent. Anyway, we start by
| writing the general form for a spherically symmetric metric WITHOUT
| imposing the requirement that the solution be static. Despite this, we
| are still able to make a change of coordinates in which the solution
| metric takes a static form.
|
| No. It should read: "Despite this, we are still able to make a change
| of coordinates in which the solution metric takes either of the two
| forms:
|
| ds^2 = -A^2 dt^2 + B^2 dr^2 + r^2 dOmega^2
| or:
| ds^2 = +A^2 dt^2 - B^2 dr^2 + r^2 dOmega^2".
|
| For some mysterious reason (laziness?) most texts insist on the first
| metric only (which is the wrong thing to do) and only later pull the
| second metric like a rabbit out of hat which naturally looks very fishy
| to the reader - exactly as you describe below.
|
| We solve the Einstein field equations, and discover the usual exterior
| Schwarzschild solution. Now, Mr ****wit, our resident self-proclaimed
| GR 'expert', notices that the if we let r 2m, we get another solution
| of the field equations,
|
| Actually, if we start off correctly by considering both forms of the
| metric then both ranges (r2m and r2m) follow from solving the
| equations. The interior is not obtained by "oh, look, I can plug in
| r2m and it's still a solution" kind of thing. Some time ago the thread
| you initiated made me rederive this solution with some care. I TeX'd it
| and put it at http://www.mastersofcinema.org/jan/t.pdf
| I hope it addresses some of these issues. I used Cartan's moving frames
| as I find Christoffel symbols way too tedious.
|
| even though it is non-static
|
| We've only assumed spherical symmetry (and signature 2, etc.).
|
| and does not
| happen to fit the form of the metric we derived after applying our
| coordinate transformations.
|
| Again, you would be right if it was not that the second form of the
| metric does in fact arise right at the beginning, while diagonalising
| the general form of the metric.
|
| Any reasonably smart first year undergraduate Oxford physicist taking
| his first course in mathematical methods would at this point politely
| point out to Mr ****wit that the solution he has just 'discovered' does
| not solve our problem as (a) it is not static,
|
| It wasn't supposed to be.
|
| and (b)does not fit the
| form of the metric that we have just derived.
|
| And I won't repeat myself here then.
|
| In order not to upset Mr ****wit
| too greatly, he might mention that Mr ****wit's metric may turn out to
| be the solution to some, non-static problem, but certainly not the one
| we are trying to solve. But also that Mr ****wit should take note that
| his metric has a rather nasty singularity right in the middle of it, so
| that the chances of it being the solution to any physically reasonable
| problem are rather slim.
|
| That's a problem of a sort of different scientific magnitude. It is
| thought that this is a consequence of GR being a classical theory and
| incorporating QM in it will probably resolve this.
|
| [snip]
|
| BTW, what happened to your writing style? You sound almost like
| Androcles today.
|
| --
| Jan Bielawski
|

"JanPB" wrote in message
ups.com...
| LEJ Brouwer wrote:
| The Trouble with Physic(ist)s is that they are Not Even Wrong
|
| I have been thinking about this and earlier discussions and wondering
| what to make of it all. How is it that otherwise intelligent
| individuals can agree en masse upon something which, on the face of it,
| is completely and utterly wrong?
|
| We would like to find the metric outside of a static point particle.
| Clearly, before we even start, we know that the solution metric, like
| the problem itself, must be time-independent. Anyway, we start by
| writing the general form for a spherically symmetric metric WITHOUT
| imposing the requirement that the solution be static. Despite this, we
| are still able to make a change of coordinates in which the solution
| metric takes a static form.
|
| No.

YES!!!

You ****ing idiot, you don't even know what a constant velocity is and
the OP is talking about a symmetric metric.
Androcles.


Androcles

LEJ Brouwer wrote:

[...]
We would like to find the metric outside of a static point particle.
Clearly, before we even start, we know that the solution metric, like
the problem itself, must be time-independent. Anyway, we start by
writing the general form for a spherically symmetric metric WITHOUT
imposing the requirement that the solution be static. Despite this, we
are still able to make a change of coordinates in which the solution
metric takes a static form.

in a region of spacetime; we know enough elementary differential
geometry to realize that a change of coordinates typically does not
cover the whole manifold.

[...]

Anyway, we notice that an infalling particle appears to take an
infinite amount of time to reach the 'event horizon' at r = 2m, where
some of the metric components either vanishing or become divergent.
Unfortunately the infalling particle reaches the horizon in finite
proper time, so that it is not clear where it goes after this - so our
solution must be incomplete.

Exactly. Since we are not flat-earthers, we reject the possibility
that the infalling particle falls off the edge of spacetime. (This
would actually be worse than the usual flat earth stereotype -- there
would be no place for the particle to fall to, so it would have to
just poof out of existence.) We also note that the divergence of some
metric components is an indication that our coordinate system is not
valid at r=2m, much as the divergence of the component g^{\theta\theta}
of the metric in ordinary polar coordinates on the plane indicates
that the coordinate system breaks down at the origin. We check this
by noting that the coordinate-independent quantities that we can build
-- the curvature invariants -- are perfectly well-behaved at the
horizon.

[...]
We take a more rational approach and note that our initial formulation
of the problem was not sufficiently general to take into account all
possible solutions, which could be multivalued in r. Using the method
of Synge, we are able to derive the complete solution, containing both
exterior patches of the 'maximal' (sic) Kruskal extension.

We then remember that our coordinate system broke down at r=2m, so we
switch to a set of coordinates that is well-behaved there. We immediately
find that the solution is then the whole Kruskal extension, including
the interior regions.

Since we were (apparently) worried about the adjective "maximal," we do
a bit more research, and find that the Kruskal solution is the unique
maximal analytic extension of the solution we originally found for r2m.

Mr ****wit
again jumps up with excitement, and proclaims vociferously that the
entire plane, including both the black and white hole interior
solutions must be included. We tell him once again to calm down, and
that the interior solutions are still non-static and are still not
valid solutions to our original problem.

Since we are not flat-earthers, and are not prepared to accept a solution
in which test particles fall off the edge of space, we conclude that there
must have been something wrong with our original formulation -- that the
assumption of a static *point* particle must have been inconsistent.

Or, if we are the original poster, we decide that:

Indeed the only two valid
solutions are the two (static) exterior solutions, labelled I and II:
_
\ / /.\
\ / / . \
I \/ II --- I( x )II
/\ \ . /
/ \ \./
/ \ -
Two (spatially superimposed) An infinite cone with regions I and
II
quadrants with light cones --- patched along the EH (dotted line)
with
pointing upwards in region I lightcones rotating clockwise around
and downwards in region II. the cone. (We are looking down into
the
cone here - note that regions I and
II
are still spatially superimposed)

We then note that the spacetime still has an "edge," and that an infalling
particle can still fall out of space and vanish. Oops! So we realize that
we must have been wrong -- we really should have included regions III and IV.

We note that the acceleration on a particle at the event horizon
diverges, so that something unusual must be happening at the event
horizon.

We quickly realize, however, that this observation depended on our choice
of coordinates. Having already realized that the original coordinate
system broke down, we change to any one of an enormous number of known
coordinate systems that are well-behaved at the horizon, and find that
there is nothing at all strange happening to the acceleration of a freely
falling particle. We *do*, on the other hand, find that it would take
infinite acceleration for a particle to remain at rest at the horizon.
We feel gratified that our original reaction that "something unusual must
be happening" was correct, with the added benefit that we now understand
what the unusual something is.

We also perform some calculations showing that the area of the horizon
is 16m^2,

Yes

but that it is at distance zero from the central mass,

Apparently "we" have made a rather bad miscalculation here, since the
actual calculation shows no such thing.

which
must therefore be at (or, rather, just inside) the event horizon.

Having gotten this far, we realize that our original assumption of a
static point mass was, indeed, inconsistent, as we had suspected earlier,
since a point can't have a finite surface area. To investigate further,
we try replacing our point mass by a static sphere of fluid, with an
arbitrary equation of state. Sure enough, we find that if we try to
shrink the sphere to one that has a surface area of less than 16 pi m^2,
no static solution exists -- as long as we have an equation of state
for which the speed of sound is less than the speed of light, such a
sphere of fluid inevitably collapses (a rather nonstatic process!).

Noting that there is no curvature singularity at the horizon, and that
the infalling particle must go somewhere, we realise that the only
physically consistent scenario is that a particle beginning in region
I, which has light cones pointing upwards, must cross to the other side
of the 'wormhole' on reaching the horizon at which point it enters into
region II (which is spatially superimposed upon region I), but now
travelling backwards in time relative to region I, so that forward
light cones in region II point downwards.

Having made this guess, we go back and compute the geodesics that
describe the particle's motion, and discover that it does no such
thing. Oops! So we go back to our earlier observation that the
omission of regions III and IV left an "edge" at which particles
could vanish, and we find that the geodesics do, in fact, reach this
"edge." We thus confirm our earlier suspicion that we really needed
regions III and IV after all.

We realize that we have been rather impolite to a number of people who,
as it turns out, know quite a bit more physics and mathematics than we
do. We politely ask their pardon.

[Or, if we are the original poster, we stick our fingers in our ears
and yell, "Nyah, nyah, I'm smarter than you!"]

Steve Carlip

wrote:
LEJ Brouwer wrote:

[...]
We would like to find the metric outside of a static point particle.
Clearly, before we even start, we know that the solution metric, like
the problem itself, must be time-independent. Anyway, we start by
writing the general form for a spherically symmetric metric WITHOUT
imposing the requirement that the solution be static. Despite this, we
are still able to make a change of coordinates in which the solution
metric takes a static form.

in a region of spacetime; we know enough elementary differential
geometry to realize that a change of coordinates typically does not
cover the whole manifold.

[...]

Anyway, we notice that an infalling particle appears to take an
infinite amount of time to reach the 'event horizon' at r = 2m, where
some of the metric components either vanishing or become divergent.
Unfortunately the infalling particle reaches the horizon in finite
proper time, so that it is not clear where it goes after this - so our
solution must be incomplete.

Exactly. Since we are not flat-earthers, we reject the possibility
that the infalling particle falls off the edge of spacetime. (This
would actually be worse than the usual flat earth stereotype -- there
would be no place for the particle to fall to, so it would have to
just poof out of existence.) We also note that the divergence of some
metric components is an indication that our coordinate system is not
valid at r=2m, much as the divergence of the component g^{\theta\theta}
of the metric in ordinary polar coordinates on the plane indicates
that the coordinate system breaks down at the origin. We check this
by noting that the coordinate-independent quantities that we can build
-- the curvature invariants -- are perfectly well-behaved at the
horizon.

[...]
We take a more rational approach and note that our initial formulation
of the problem was not sufficiently general to take into account all
possible solutions, which could be multivalued in r. Using the method
of Synge, we are able to derive the complete solution, containing both
exterior patches of the 'maximal' (sic) Kruskal extension.

We then remember that our coordinate system broke down at r=2m, so we
switch to a set of coordinates that is well-behaved there. We immediately
find that the solution is then the whole Kruskal extension, including
the interior regions.

Since we were (apparently) worried about the adjective "maximal," we do
a bit more research, and find that the Kruskal solution is the unique
maximal analytic extension of the solution we originally found for r2m.

Mr ****wit
again jumps up with excitement, and proclaims vociferously that the
entire plane, including both the black and white hole interior
solutions must be included. We tell him once again to calm down, and
that the interior solutions are still non-static and are still not
valid solutions to our original problem.

Since we are not flat-earthers, and are not prepared to accept a solution
in which test particles fall off the edge of space, we conclude that there
must have been something wrong with our original formulation -- that the
assumption of a static *point* particle must have been inconsistent.

Or, if we are the original poster, we decide that:

Indeed the only two valid
solutions are the two (static) exterior solutions, labelled I and II:
_
\ / /.\
\ / / . \
I \/ II --- I( x )II
/\ \ . /
/ \ \./
/ \ -
Two (spatially superimposed) An infinite cone with regions I and
II
quadrants with light cones --- patched along the EH (dotted line)
with
pointing upwards in region I lightcones rotating clockwise around
and downwards in region II. the cone. (We are looking down into
the
cone here - note that regions I and
II
are still spatially superimposed)

We then note that the spacetime still has an "edge," and that an infalling
particle can still fall out of space and vanish. Oops! So we realize that
we must have been wrong -- we really should have included regions III and IV.

We note that the acceleration on a particle at the event horizon
diverges, so that something unusual must be happening at the event
horizon.

We quickly realize, however, that this observation depended on our choice
of coordinates. Having already realized that the original coordinate
system broke down, we change to any one of an enormous number of known
coordinate systems that are well-behaved at the horizon, and find that
there is nothing at all strange happening to the acceleration of a freely
falling particle. We *do*, on the other hand, find that it would take
infinite acceleration for a particle to remain at rest at the horizon.
We feel gratified that our original reaction that "something unusual must
be happening" was correct, with the added benefit that we now understand
what the unusual something is.

We also perform some calculations showing that the area of the horizon
is 16m^2,

Yes

but that it is at distance zero from the central mass,

Apparently "we" have made a rather bad miscalculation here, since the
actual calculation shows no such thing.

which
must therefore be at (or, rather, just inside) the event horizon.

Having gotten this far, we realize that our original assumption of a
static point mass was, indeed, inconsistent, as we had suspected earlier,
since a point can't have a finite surface area. To investigate further,
we try replacing our point mass by a static sphere of fluid, with an
arbitrary equation of state. Sure enough, we find that if we try to
shrink the sphere to one that has a surface area of less than 16 pi m^2,
no static solution exists -- as long as we have an equation of state
for which the speed of sound is less than the speed of light, such a
sphere of fluid inevitably collapses (a rather nonstatic process!).

Noting that there is no curvature singularity at the horizon, and that
the infalling particle must go somewhere, we realise that the only
physically consistent scenario is that a particle beginning in region
I, which has light cones pointing upwards, must cross to the other side
of the 'wormhole' on reaching the horizon at which point it enters into
region II (which is spatially superimposed upon region I), but now
travelling backwards in time relative to region I, so that forward
light cones in region II point downwards.

Having made this guess, we go back and compute the geodesics that
describe the particle's motion, and discover that it does no such
thing. Oops! So we go back to our earlier observation that the
omission of regions III and IV left an "edge" at which particles
could vanish, and we find that the geodesics do, in fact, reach this
"edge." We thus confirm our earlier suspicion that we really needed
regions III and IV after all.

We realize that we have been rather impolite to a number of people who,
as it turns out, know quite a bit more physics and mathematics than we
do. We politely ask their pardon.

[Or, if we are the original poster, we stick our fingers in our ears
and yell, "Nyah, nyah, I'm smarter than you!"]

Steve Carlip

Dear Steve,

First of all, I apologise sincerely to you if you felt at all offended
by what I wrote. My message was certainly not directed towards to you
in any way - it was not actually directed to anyone at all in
particular - not even Tom Roberts or "T. Essel", both of whom have been
quite rude to me in the past. If anything, 'Mr ****wit and his
Sheep-like companions' merely represents my own imperfect mental
abstraction of the scientific establishment and some of its inherent
flaws. The message was really just a record of the current state of
affairs as I see it with relation to this problem (admittedly you may
not think that there is a problem). I 'know' that the Schwarzschild
solution is wrong, and I also 'know' that my proposal must be either
correct, or if not completely correct at least on the right path. I
can't tell you precisely how I know - it is just a very strong gut
feeling, and when I feel like this, I am usually right. That's all. I
could be completely wrong, though the results I have managed to come up
with so far do suggest to me otherwise, which is why I am continuing to
press the issue.

I actually admire you a great deal. You are like a walking
encyclopaedia on gravity, yet you do not appear to be at all
pretentious or arrogant about it. You even take the time to patiently
correct simple conceptual mistakes of laymen posting on unmoderated
newsgroups who are not at all professional physicists and probably
never will be. This is particularly admirable as you must have so many
other things you could be doing instead. And you do not go around
insulting crackpots.

I will be the first to admit that I know very little mathematics or
physics. But I don't stick my fingers in my ears, and I certainly don't
go around thinking that I am smarter than everyone else. Besides,
different people are smart in very different ways. There are people who
post on these newsgroups who do perhaps think that they are smarter
than everyone else, but I am not one of them. For one thing, Daryl and
Jan appear to be smarter than me, which is good, as I have learnt a lot
from them. Daryl is very methodical, and Jan has a very sympathetic
nature. I like them both very much. Many people (yourself included)
clearly know so much about GR than me that it is not even funny, but
amongst them are individuals that have an air of pomposity about them
which makes them seem extraordinarily foolish.

But that's also not to say that I am not smart at all. I am the kind of
person that can know next to nothing about maths and physics but given
a little effort, can still come up with a consistent theory of
everything on demand and at fairly short notice. I don't really know
how I do this. It just 'happens'. I am often surprised by some of the
things I have done and the ideas I have come up with as they are often
beyond my own perceived ability.

BTW, could you please explain what you mean when you say that my
infinite cone has an 'edge'? I thought I had patched it up pretty well
with my infinite piece of sticky tape.

Best wishes,

Sabbir.

Daryl McCullough wrote:
Tom Roberts says...
Daryl McCullough wrote:
In any case, it seems mistaken to say that a white hole
is *repulsive*. dr/ds 0 for the white hole interior
region,
Remember that r is TIMELIKE there, and increasing r does NOT mean
"coming out".

Yes, it certainly does mean that.

No, in the white hole region increasing r means "moving into the
future", not "coming out" -- the latter phrase means a change in spatial
position, not merely a movement to the future. IOW: in this region +d/dr
is future pointing, not outside pointing.

Now it is also true that any future-directed timelike path in this
region is coming out, but that requires more than merely r increasing.


If we have a timelike geodesic
such that initially r 2m and dr/ds 0, then r will increase with
increasing s until eventually r 2m. At that point, r is spacelike,
and it is certainly appropriate to say that the test particle "came out"
of the event horizon.

Sure.


I don't think we actually disagree on anything here.


Tom Roberts

JanPB wrote:
No. It should read: "Despite this, we are still able to make a change
of coordinates in which the solution metric takes either of the two
forms:

ds^2 = -A^2 dt^2 + B^2 dr^2 + r^2 dOmega^2
or:
ds^2 = +A^2 dt^2 - B^2 dr^2 + r^2 dOmega^2".

For some mysterious reason (laziness?) most texts insist on the first
metric only (which is the wrong thing to do) and only later pull the
second metric like a rabbit out of hat which naturally looks very fishy
to the reader - exactly as you describe below.

It's probably more likely to do with the fact that there should not be
an explicit time-dependence in the solution of a manifestly
time-independent problem.

We solve the Einstein field equations, and discover the usual exterior
Schwarzschild solution. Now, Mr ****wit, our resident self-proclaimed
GR 'expert', notices that the if we let r 2m, we get another solution
of the field equations,

Actually, if we start off correctly by considering both forms of the
metric then both ranges (r2m and r2m) follow from solving the
equations. The interior is not obtained by "oh, look, I can plug in
r2m and it's still a solution" kind of thing. Some time ago the thread
you initiated made me rederive this solution with some care. I TeX'd it
and put it at http://www.mastersofcinema.org/jan/t.pdf
I hope it addresses some of these issues. I used Cartan's moving frames
as I find Christoffel symbols way too tedious.

I am sure there is nothing wrong with your derivation - the problematic
issue for me is the second of the two equations you set out to solve.

even though it is non-static

We've only assumed spherical symmetry (and signature 2, etc.).

Yes, but coordinate transformations can be made to bring the general
solution to a static form.

and does not
happen to fit the form of the metric we derived after applying our
coordinate transformations.

Again, you would be right if it was not that the second form of the
metric does in fact arise right at the beginning, while diagonalising
the general form of the metric.

Any reasonably smart first year undergraduate Oxford physicist taking
his first course in mathematical methods would at this point politely
point out to Mr ****wit that the solution he has just 'discovered' does
not solve our problem as (a) it is not static,

It wasn't supposed to be.

and (b)does not fit the
form of the metric that we have just derived.

And I won't repeat myself here then.

In order not to upset Mr ****wit
too greatly, he might mention that Mr ****wit's metric may turn out to
be the solution to some, non-static problem, but certainly not the one
we are trying to solve. But also that Mr ****wit should take note that
his metric has a rather nasty singularity right in the middle of it, so
that the chances of it being the solution to any physically reasonable
problem are rather slim.

That's a problem of a sort of different scientific magnitude. It is
thought that this is a consequence of GR being a classical theory and
incorporating QM in it will probably resolve this.

I don't think QM is relevant if we are considering GR on its own. I am
slightly confused that Steve Carlip and others worry (understandably)
about the original Schwarzschild solution having an edge, and therefore
disappearing off into nothingness, but in the same breath advocate an
extension to the original solution which contains a singularity into
which all infalling particles disappear into who-knows where. Why is
the same objection not raised about the presence of this singularity,
and why is the extension any better than the original?

[snip]

You snipped my piece de resistance!

BTW, what happened to your writing style? You sound almost like
Androcles today.

Well it was very late and I was in a funny mood. I actually hadn't
meant it to turn out quite as rude as it did, but I did enjoy reading
it and I still think the "No black holes. No white holes..." quote is
very funny, even if it is totally disrespectful and unfair. I guess
you're not likely to find it funny if you are at the receiving end. One
thing at is annoying is that my messages are censored from s.p.research
yet they continue to discuss the matter (and insult me and my friend
Abhas Mitra) without giving me a right to reply on the same group. The
same topic then generates 500+ messages on s.p.rel, and the topic
somehow comes back into the frame of discussion on s.p.research, to
which I am still not going to be allowed to contribute. How's that for
inconsistency? I have to agree with T. Essel that s.p.research is
pointless and pathetic, with everyone just wanting to pat each other on
the back for being so bloody smart. At least in s.p.rel you can speak
freely, and simply ignore the background noise.

Anyway, sorry - the tone of my messahe was a bit out of order, but I
don't think I have quite managed to match Androcles yet. He is much
more succinct and to the point.

--
Jan Bielawski

- Sabbir.

Henning Makholm wrote:

Scripsit "Edward Green"
Henning Makholm wrote:

Other black-hole metrics, such as Kruskal coordinates, make clear how
the outer part of the Schwarzschild metric is actually connected to
two copies of the inner part; one being a black hole and one being a
white hole. Timelike geodesics outside r=2m may approach the event
horizon asymtotically (by coordinates) either in the far future or
in the far past of the Schwarzschild t coordinate.

I take it "far future" means "infinite", in t?

Yes. Of course things don't look any different for any finite t
coordinate.

Of course. Infinity wouldn't be infinity if we could get any closer to
it. :-)

Conversely, geodesics that approach r=2m for large negative t join
with geodesics in the _white_ hole; these are the worldlines of
test particles that the white hole spits out unpredictably.

That last part seems a bit fanciful. The geodesics merely describe the
trajectory of test particles _if_ they should somehow happen to appear,
it says nothing about whether there _are_ any test particles, still
less requiring the white hole to randomly spit them out as if it were
some kind of quantum process.

I didn't say they were "randomly" spit out, just "unpredictably". It
is fine for the white hole not to spit out anything; the theory just
cannot _predict_ that it won't.

Hmm... If you told me a certain person flew into rages unpredictably,
and I countered there was no evidence he did fly into rages, and _you_
countered that you never implied he did fly into rages but merely
stated that you could not make any such predictions... one might say
either your language was a trifle misleading, or that you had a bright
future at the bar. ;-)

But I take your meaning.

(This is another way of saying that the theory will not *reject*
histories where something comes out of the white hole, simply because
the theory is time-symmetric and does allow things to fall into the
black hole).

In any case, as far as I understand, "test particles" in GR are
supposed to be present wherever and whenever we stipulate they are,
and come into existence at the whim of the calculator :-)

Certainly it appears we can rule out the idea that, as written, the
white hole is a brightly glowing region in space: the solution as
written contains no finite radiation density.

* I have wondered if black holes may not be better described as models
of processes which never go to completion, rather than static objects,
sub species aeternis.

Well, the outer region of a black hole is (in the metrics I know of)
static, but that is just because it makes for a nice idealized
solution.

Ok... I'm shortly going to fall forward and suffer an oral
pedalization, but let's say for the sake of argument I am referring to
the Schwarzschild metric -- regarded I suppose as a black hole already
in progress -- and an infalling test particle. You go on to give a
well-known reply, that...

However, charts such as Eddington-Finkelstein coordinates show that
the process certainly go to completion from the POV of particles
falling into the black hole: they always reach the central singularity
after a finite amount of proper time. ...

The
sometimes-told story that infalling particles will sit on the event
horizon for the remainder of the universe and only cross it after
infinite time has passed seems to be wrong - it's an artifact of the
choice of Schwarzchild coordinates rather than an intrinsic feature of
the model.

At this point, in rebuttal, I wish to beg the indulgence of the
moderators to express something involving a phrase which might
ordinarily raise red flags, and the phrase I wish to use is: "Inverse
Zeno's paradox".

In the ordinary version of this paradox a process reaching completion
in finite time via local clocks is implicitly reparameterized via a
coordinate system inflating the remaining coordinate increment to the
event to infinity: a runner starting at event A and crossing the finish
at event B is implicitly described via coordinates mapping unit
increments to ever smaller increments of our natural metric. The
protagonist points to the infinity and claims it shows the runner never
reaches B.

In modern language, we dismiss this infinity as a coordinate
singularity.

In the inverse version, we have a process which fails to go to
completion in finite time via local clocks, reparameterized via a
coordinate system which _compresses_ the remaining infinity in our
preferred coordinates into a finite increment, and keeps going. The
protagonist then claims the opposite of what was claimed in the first
case: in this case he is the one who rejects the infinity as a
coordinate singularity, and now claims the runner will reach the finish
line, although our local clocks tell us otherwise.

Individuals who in both cases reject the infinity as a "coordinate
singularity" at least appear to be consistent, and acquaintance with
relativity, particularly general relativity, encourages us in an
egalitarian view of coordinate systems, every one as good as the
others.

On the other hand, a case can be made that our local clocks should have
a special role in deciding whether an event will "ever happen". We
might want to regard a claim wherein our local time tells us an event
won't happen, but a remapping tells us it does, with as much skepticism
as a claim that the runner never crosses the finish line, though local
time tells us he does.

By a slight extension, though this is speculation, we might guess that
metrics describing the initial collapse of a mass to a black hole share
a similar feature, and that the singularity never forms: the black hole
itself might be taken to describe a incomplete process rather than an
object.

As far as I can reason out, the infalling particle is even going to
reach the singularity _even_ if the black hole evaporates away by
Hawking radiation later in the history of the universe.

It boils down to how much "reality" we are willing to ascribe to events
happening beyond infinity in some possibly privileged (cosmological?)
time. Who knows... maybe there are time compressed mites living on
Achilles' heel who regarded the idea that he would even cross the
finish line as equally fanciful: their universe evaporated into void
before he ever got there.

wrote:
in a region of spacetime; we know enough elementary differential
geometry to realize that a change of coordinates typically does not
cover the whole manifold.

This is true, but your statement on its own is a bit too vague if you
expect me to read this and then immediately come to the realisation
that the interior must also a correct solution. Clearly there are a few
steps missing in the chain of logic and it would help if you could be a
bit more explicit.

Exactly. Since we are not flat-earthers, we reject the possibility
that the infalling particle falls off the edge of spacetime. (This
would actually be worse than the usual flat earth stereotype -- there
would be no place for the particle to fall to, so it would have to
just poof out of existence.) We also note that the divergence of some
metric components is an indication that our coordinate system is not
valid at r=2m, much as the divergence of the component g^{\theta\theta}
of the metric in ordinary polar coordinates on the plane indicates
that the coordinate system breaks down at the origin. We check this
by noting that the coordinate-independent quantities that we can build
-- the curvature invariants -- are perfectly well-behaved at the
horizon.

Why is falling into a singularity of spacetime any better than falling
off the edge of spacetime?

We then remember that our coordinate system broke down at r=2m, so we
switch to a set of coordinates that is well-behaved there. We immediately
find that the solution is then the whole Kruskal extension, including
the interior regions.

Since we were (apparently) worried about the adjective "maximal," we do
a bit more research, and find that the Kruskal solution is the unique
maximal analytic extension of the solution we originally found for r2m.

As I recall, when someone asked earlier, you were not able to give a
reference to the original proof of this statement. Are you taking this
on blind faith? I don't doubt that it's true, but it's just curious
that physicists have a habit of taking for granted statements made by
other physicists merely on the basis of their authority rather than on
the basis of a mathematical proof.

Since we are not flat-earthers, and are not prepared to accept a solution
in which test particles fall off the edge of space, we conclude that there
must have been something wrong with our original formulation -- that the
assumption of a static *point* particle must have been inconsistent.

I don't see why the incompleteness of the original solution at all
implies that the assumption of a static point particle is inconsistent.
Sure, the solution you propose is non-static, but that does not
necessarily imply that static solutions do not exist. Unless there is a
uniqueness theorem which states otherwise, like "only _maximal_
extensions" are valid. I prefer "only singularity-free solutions are
valid".

Or, if we are the original poster, we decide that:

Indeed the only two valid
solutions are the two (static) exterior solutions, labelled I and II:
_
\ / /.\
\ / / . \
I \/ II --- I( x )II
/\ \ . /
/ \ \./
/ \ -
Two (spatially superimposed) An infinite cone with regions I and
II
quadrants with light cones --- patched along the EH (dotted line)
with
pointing upwards in region I lightcones rotating clockwise around
and downwards in region II. the cone. (We are looking down into
the
cone here - note that regions I and
II
are still spatially superimposed)

Well, at least it's interesting and original.

We then note that the spacetime still has an "edge," and that an infalling
particle can still fall out of space and vanish. Oops! So we realize that
we must have been wrong -- we really should have included regions III and IV.

What edge? It is an infinite cone. There is a pointy bit in the middle,
but I can't say that bothers me at all.

We note that the acceleration on a particle at the event horizon
diverges, so that something unusual must be happening at the event
horizon.

[snip]

Apparently "we" have made a rather bad miscalculation here, since the
actual calculation shows no such thing.

That depends on which solution you are using for the calculation,
doesn't it?

which
must therefore be at (or, rather, just inside) the event horizon.

Having gotten this far, we realize that our original assumption of a
static point mass was, indeed, inconsistent, as we had suspected earlier,
since a point can't have a finite surface area.

Again, this appears to be a logical non-sequitur.

To investigate further,
we try replacing our point mass by a static sphere of fluid, with an
arbitrary equation of state. Sure enough, we find that if we try to
shrink the sphere to one that has a surface area of less than 16 pi m^2,
no static solution exists -- as long as we have an equation of state
for which the speed of sound is less than the speed of light, such a
sphere of fluid inevitably collapses (a rather nonstatic process!).

Unless you have very large hands, you can't physically shrink the
sphere to such a small size. The Helmholtz-Klein mechanism prevents
matter from collapsing to form a black hole:

http://arxiv.org/abs/gr-qc/0605066

Having made this guess, we go back and compute the geodesics that
describe the particle's motion, and discover that it does no such
thing.

Depends what you think the solution manifold looks like, doesn't it?

Oops! So we go back to our earlier observation that the
omission of regions III and IV left an "edge" at which particles
could vanish, and we find that the geodesics do, in fact, reach this
"edge." We thus confirm our earlier suspicion that we really needed
regions III and IV after all.

We realize that we have been rather impolite to a number of people who,
as it turns out, know quite a bit more physics and mathematics than we
do. We politely ask their pardon.

Sorry sir, we won't do it again.

[Or, if we are the original poster, we stick our fingers in our ears
and yell, "Nyah, nyah, I'm smarter than you!"]

I think you may be confusing me with Chris Hillman, John Baez or Lubos
Motl here. I am not like that AT ALL.

Steve Carlip

- Sabbir.

wrote:
We realize that we have been rather impolite to a number of people who,
as it turns out, know quite a bit more physics and mathematics than we
do. We politely ask their pardon.

I just realised that I cannot apologise to anyone on s.p.research as my
apology would be 'overly speculative'! :-)

LEJ Brouwer wrote:
I 'know' that the Schwarzschild
solution is wrong, and I also 'know' that my proposal must be either
correct, or if not completely correct at least on the right path.

The rest of us want to do physics, not whatever it is you are trying to
do. What God told you this? Why do you attempt to discuss such divine
revelations in a physics newsgroup?


I
can't tell you precisely how I know - it is just a very strong gut
feeling, and when I feel like this, I am usually right.

Here all you've shown is that you do not understand the MANY papers and
books that have been written about this. You merely re-hash old
objections long refuted, and old mistakes long corrected.


I actually admire you a great deal. You are like a walking
encyclopaedia on gravity, yet you do not appear to be at all
pretentious or arrogant about it.

Yes, Steve Carlip is all of that.


BTW, could you please explain what you mean when you say that my
infinite cone has an 'edge'?

I assume you mean your attempt to glue the two exterior regions of the
Kruskal manifold together. The "edge" occurs when one follows an
infalling timelike geodesic -- when it reaches r=2M all of a sudden it
is impossible to compute the geodesic, because the metric is not C^2
there. Steve implied there is a boundary there, but I believe this can
be done such that the manifold is continuous there, just not smooth.
This is not a viable physical model because the Einstein field equation
must be valid everywhere, and it cannot be valid on either a boundary or
a locus where the metric is not C^2.

One can glue the two regions together there topologically.
But in doing that one must clearly distort the Kruskal
plane (i.e. the U-V coordinate plane) -- that is OK because
that can be a diffeomorphism that carries the metric
along; but at best the metric can be only C^0: for the metric
to be C^n its first n derivatives must all be equal at the
join, and the symmetry of the two exterior regions means they
must vanish; for this metric the first derivative is nonzero.

Note that on physical grounds the metric must be C^2 for two
different reasons: to satisfy the EFE, and for geodesic paths
to be C^1 (a worldline must have a 4-velocity everywhere).

["C^n" means continuously differentiable n times.]

[Hmmm. The U-V plane suppresses the two angles; I am not 100%
certain that those suppressed dimensions do not prevent
the gluing I describe; I assume that it is OK. You also
implicitly assumed this is OK.]



The trouble with physic(ist)s is not that we are "not even wrong", but
rather, from your point of view, the trouble is that we don't accept
your "strong gut feeling" as evidence of anything except the fact that
you are not doing science. shrug


Tom Roberts