"Tom Roberts" wrote in message
news:tOCJg.4299$ The Einstein field equation is symmetric in time, and
cannot determine


Remember that r is TIMELIKE there [interior region], and increasing r does
NOT mean "coming out". |

This is interesting. I'm trying to get a better feel for the interior
region in Schw. coordinates where r is timelike and t is spacelike. What
would be an example of values of the Schw. coordinates r and t that would
correspond to a point that is spatially located 'just inside' the horizon?
Or is that even a valid question? Thanks.

T. S.

Tom Roberts says...

Daryl McCullough wrote:

If you consider a test particle moving along a radial geodesic,
then the particle's "position" r as a function of proper time s
will satisfy a differential equation
(dr/ds)^2 - A/r + B = 0
where A and B are constants of the motion. (It's interesting
that this looks exactly like the differential equation for
nonrelativistic motion in Newtonian gravity.)
Without solving the equation explicitly, we can immediately
read off the qualitative behavior: [...]

In the interior region r2M, r is timelike and you did not handle that
properly.

Yes, in fact, I did.

In the white hole region +d/dr is future pointing, in the
black hole region -d/dr is future pointing.

I'm not sure in what sense you think I didn't "handle that
properly". In the interior of the black hole, the line element
looks like this: (with the convention that the signature is
-+++)

ds^2 = (2m/r - 1) dt^2 - 1/(2m/r - 1) dr^2 + r^2 dOmega^2

So dr is timelike and dt is spacelike. What difference does
that make? Every event has an associated value for r, and
given a geodesic parameterized by proper time s, we can ask
the value of r corresponding to proper time s. What is the
problem?

In any case, it is still true that there are geodesics
parameterized by proper time r(s) and t(s). It is still true that
dr/ds for a "radial" geodesic (meaning one such that
dtheta/ds = dphi/ds = 0) satisfies

(dr/ds)^2 - A/r + B = 0

for the appropriate constants A and B. It is still true that
there is a solution r(s) with the qualitative feature that
r(s) increases up until some maximum value for r, and then
r(s) decreases.

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. 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.

--
Daryl McCullough
Ithaca, NY

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.

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.

(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 :-)

* 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.

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.

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. 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.

--
Henning Makholm "Ambiguous cases are defined as those for which the
compiler being used finds a legitimate interpretation
which is different from that which the user had in mind."

In sci.physics.research Tom S. asked
I'm trying to get a better feel for the interior
region in Schw. coordinates where r is timelike and t is spacelike. What
would be an example of values of the Schw. coordinates r and t that would
correspond to a point that is spatially located 'just inside' the horizon?
Or is that even a valid question? Thanks.

r=1.5m, t=anything

However, the Schwarzschild time coordinate isn't very well suited
for this purpose. You would probably find Kruskal-Szekeres (u,v)
coordinates more instructive, ditto Eddington-Finkelstein (t,r).
These are all explained very clearly in Misner, Thorne, and Wheeler.

ciao,

--
-- "Jonathan Thornburg -- remove -animal to reply"
Max-Planck-Institut fuer Gravitationsphysik (Albert-Einstein-Institut),
Golm, Germany, "Old Europe" http://www.aei.mpg.de/~jthorn/home.html
"Washing one's hands of the conflict between the powerful and the
powerless means to side with the powerful, not to be neutral."
-- quote by Freire / poster by Oxfam

Edward Green wrote:
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? I have some questions
about this, but, elsewhere... *

Finite future according to proper time (which is what the observer
represented by such geodesic experiences). Infinite future only in the
faux sense of the Schwarzschild t-coordinate which - like all
coordinates - is just a labelling of events without own physical
meaning.

Those geodesics
that approach r=2m for large positive t join up with geodesics in the
_black_ hole after a finite amount of proper time. 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. The only mass implied by this part of
the solution is a mass M (or is that, just possibly, -M, as Daryl
suggests?) associated with the (white hole) singularity.

M0 for both.

Real-world black holes formed by gravitational collapse do not have a
white hole counterpart, because the Schwarzchild (and Kruskal)
solutions are _vacuum_ solutions, and if you go back in time to look
for the white hole you reach a time before the collapse where the
vacuum did not reach all the way to r=2m and therefore the vacuum
metric did not apply.

Aha! _That_ is the best reason I have seen suggested (in my admittedly
very scanty researchs) to justify neglecting the white hole
singularity. The boundary conditions are not met: GR lives.

Yes but OTOH non-vacuum solutions could possibly have white holes too.
Even the Schwarzschild vacuum with just a small test particle going
from the WH to the BH wouldn't qualitatively change things by much? Now
how small is mall? An electron? A planet? A billion galaxies?

* 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.

Most relevant proper times are finite and most relevant infinities
result either from light signal delay or using bizarre coordinates (and
Schwarzschild's t and r are bizarre).

--
Jan Bielawski

Scripsit "Tom S."

I'm trying to get a better feel for the interior region in
Schw. coordinates where r is timelike and t is spacelike. What
would be an example of values of the Schw. coordinates r and t that
would correspond to a point that is spatially located 'just inside'
the horizon? Or is that even a valid question?

No, I don't think it is a valid question. The interior region is not a
stationary spacetime, so thinking of "points" and spatial locations is
at best going to be confusing. If you want an *event* "just inside"
the horizon you can set r=2m-epsilon and t=whatever you like, but that
doesn't make you wiser, does it?

I can recommend the extremely nice and clear discussion of these
matters in chapter 12 of Rindler's relativity textbook.

--
Henning Makholm "Lad min høne være."

Tom S. wrote:
"Tom Roberts" wrote
Remember that r is TIMELIKE there [interior region], and increasing r
does NOT mean "coming out".

This is interesting. I'm trying to get a better feel for the interior
region in Schw. coordinates where r is timelike and t is spacelike.
What would be an example of values of the Schw. coordinates r and t that
would correspond to a point that is spatially located 'just inside' the
horizon?

Look at a Kruskal diagram, such as the one on p 834 of MTW. In this
diagram the future is upward, and light cones are 45 degrees, so a
future-pointing timelike trajectory is headed upward within that 45
degree cone (remember the angles are suppressed, so an "X" in the figure
is really a 3-d cone). Consider a vertical trajectory starting in
exterior region I (i.e. constant u, v increasing with proper time). Just
before it reaches the horizon at r=2M (u=v), it has r slightly greater
than 2M and t increasing unbounded to +infinity. Immediately after it
crosses the horizon it has r slightly less than 2M and t decreasing from
+infinity.

Several such trajectories are shown on the facing page of MTW.


Tom Roberts

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.

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, even though it is non-static and does not
happen to fit the form of the metric we derived after applying our
coordinate transformations.

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, and (b)does not fit the
form of

the metric that we have just derived. 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.

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. Mr ****wit of course starts dribbling
profusely with excitement at this point, exclaiming proudly that the
exterior solution we have correctly derived must be patched onto his
own dubious interior solution so that the particle can continue falling
inwards.

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

We also perform some calculations showing that the area of the horizon
is 16m^2, but that it is at distance zero from the central mass, which
must therefore be at (or, rather, just inside) the event horizon.
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. An observer in region I will
not see the particle travelling backwards in time in region II, but
rather an antiparticle travelling forwards in time, so that the whole
process looks like a particle-antiparticle annihilation event occuring
at the horizon asymptotically at time t = infinity. Of course the
particle in region II can in principle reach the event horizon again
and return to its original position in region I, resulting in the
possibility of closed timelike loops. We note however that this would
not imply any inconsistency in our picture, as this process would just
look like a pair creation and subsequent pair annihilation event at the
EH.

We infer that this is the correct picture of the spherically symmetric
solution outside of a pointlike mass. Of course we are too late as
bigmouth Mr ****wit has already convinced himself and his sheep-like
chums that he is right, and together they have made big profits on book
sales without even realising that they are in a state of perpetual and
irreversible mass self-delusion. To this day they take great pride in
explaining the virtues of their wonderful solution, singularities and
all, and in badmouthing all the 'crackpots' who reject their point of
view (as they like to call them for merely disagreeing with them as
they are so much smarter than the crackpots and so very certain that
they are right and everyone else is wrong). Of course the establishment
****wits and their Sheep-like followers will never stoop to admit they
have actually gotten it all wrong, and that that has been the reason
that they have failed to make any significant progress in their
understanding of GR in the last 90 odd years. No black holes, no white
holes, just assholes. Yes, you know who you are.

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

Tom Roberts says...

Daryl McCullough wrote:

If you consider a test particle moving along a radial geodesic,
then the particle's "position" r as a function of proper time s
will satisfy a differential equation
(dr/ds)^2 - A/r + B = 0
where A and B are constants of the motion. (It's interesting
that this looks exactly like the differential equation for
nonrelativistic motion in Newtonian gravity.)
Without solving the equation explicitly, we can immediately
read off the qualitative behavior: [...]

In the interior region r2M, r is timelike and you did not handle that
properly.

Yes, in fact, I did.

In the white hole region +d/dr is future pointing, in the
black hole region -d/dr is future pointing.

I'm not sure in what sense you think I didn't "handle that
properly". In the interior of the black hole, the line element
looks like this: (with the convention that the signature is
-+++)

ds^2 = (2m/r - 1) dt^2 - 1/(2m/r - 1) dr^2 + r^2 dOmega^2

So dr is timelike and dt is spacelike. What difference does
that make? Every event has an associated value for r, and
given a geodesic parameterized by proper time s, we can ask
the value of r corresponding to proper time s. What is the
problem?

In any case, it is still true that there are geodesics
parameterized by proper time r(s) and t(s). It is still true that
dr/ds for a "radial" geodesic (meaning one such that
dtheta/ds = dphi/ds = 0) satisfies

(dr/ds)^2 - A/r + B = 0

for the appropriate constants A and B. It is still true that
there is a solution r(s) with the qualitative feature that
r(s) increases up until some maximum value for r, and then
r(s) decreases.

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. 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.

--
Daryl McCullough
Ithaca, NY