Tom Roberts wrote:
Basically GR is nonlinear, and strange things can happen in a strong
field situation, such as inside the horizon in the Schw. manifold. Your
naive expectations and desires need not be valid in regions far removed
from your personal experience. shrug

Funnily enough Tom, you were right all along. I was like a little child
insisting that the sky was blue, and thus was confounded by clouds. But
you are like the sun, which has melted those clouds away, so that the
sky appears blue once again.

- Sabbir.

"Tom Roberts" wrote in message
om...
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


Thanks, Jonathan, Henning, and Tom.

I don't think I've managed to ask my question very well. This is
undoubtedly due to my confusion. I am trying to wrap my feeble mind
around the notion of whether or not it makes any sense to ask, ''Where
in *space* is the horizon from the point of view of the interior Schw.
coordinates.''

In the *exterior* region, the horizon seems to be located at a certain
*place* (namely r = 2m, where r is a space-like variable) and it sits at
that place for all eternity (all t).

In the *interior* region, the horizon seems to be located at a certain
instant of *time* (namely r = 2m, where r is now time-like) and it
exists at that instant 'everywhere' in the interior (i.e., for all t,
where t is spacelike). [This is where my thinking is probably most off
base and where I need the most help.]

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?

Outside the horizon, it seems that no one minds if you say that r =
2.001m is spatially closer to the horizon than r = 2.002m. But, inside
the horizon, I'm wondering if 'spatial closeness' to the horizon has any
meaning.

Thanks for any enlightenment.

Tom S

Is this NG a correct place for science fiction?

Henry Haapalainen

"Tom S." kirjoitti
viestissä:qPidnf4B7sr9NWHZnZ2dnUVZ_vmdnZ2d@insight bb.com...
"Tom Roberts" wrote in message
om...
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


Thanks, Jonathan, Henning, and Tom.

I don't think I've managed to ask my question very well. This is
undoubtedly due to my confusion. I am trying to wrap my feeble mind
around the notion of whether or not it makes any sense to ask, ''Where
in *space* is the horizon from the point of view of the interior Schw.
coordinates.''

In the *exterior* region, the horizon seems to be located at a certain
*place* (namely r = 2m, where r is a space-like variable) and it sits at
that place for all eternity (all t).

In the *interior* region, the horizon seems to be located at a certain
instant of *time* (namely r = 2m, where r is now time-like) and it
exists at that instant 'everywhere' in the interior (i.e., for all t,
where t is spacelike). [This is where my thinking is probably most off
base and where I need the most help.]

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?

Outside the horizon, it seems that no one minds if you say that r =
2.001m is spatially closer to the horizon than r = 2.002m. But, inside
the horizon, I'm wondering if 'spatial closeness' to the horizon has any
meaning.

Thanks for any enlightenment.

Tom S

JanPB wrote:
I think the manifold can be glued smoothly (one can write down a smooth
atlas on the quotient manifold) - it's the metric that has a "crease"
at the glued horizon. I posted an outline of an argument few minutes
ago. The problem is essentially with the Kruskal-Szekeres function
r(T,X) (the one given by the usual implicit equation) - it has a
nonzero slope at the horizon which would "crease" the metric there when
glued.

I agree with your analysis, and am no longer happy with the
implications of my original proposal. There was another possibility
which I had been trying rather to avoid (with hindsight maybe I
shouldn't have), in that the reflection of the infalling particle could
occur not at the event horizon, but rather at the central singularity -
this would also give rise to the same predictions, and I guess will not
upset the cart as the standard Schwarzschild interior solution remains
intact.

Has the possibility that a particle hitting the singularity is
reflected off of it both in space and time (i.e. into the white hole
interior solution) been suggested before that you know of? Are there
any good books or (esp. survey) articles discussing what might happen
(classically) at the Schwarzschild singularity to an infalling
particle?

Thanks,

Sabbir.

"Henry Haapalainen" wrote in message
news | Is this NG a correct place for science fiction?

Yes, of course.
Some people even claim acceleration isn't real.

Androcles

LEJ Brouwer wrote:

Has the possibility that a particle hitting the singularity is
reflected off of it both in space and time (i.e. into the white hole
interior solution) been suggested before that you know of?

I haven't heard of it (which isn't saying much).

Are there
any good books or (esp. survey) articles discussing what might happen
(classically) at the Schwarzschild singularity to an infalling
particle?

From what I've seen people have no answers to this because this is
where reality stops in GR: singularity in the literal sense does not
exist, there are _no events_ there. You can of course compactify the
manifold but you still cannot extend the metric there. So you'd have to
come up with an extended sort of dynamics which does not require
"normal" spacetime with a metric. Sounds mighty implausible. And you
still would have to cross regions of curvature increasing without
bound. BTW, what about closed timelike geodesics?

--
Jan Bielawski

JanPB wrote:
LEJ Brouwer wrote:

Has the possibility that a particle hitting the singularity is
reflected off of it both in space and time (i.e. into the white hole
interior solution) been suggested before that you know of?

I haven't heard of it (which isn't saying much).

Are there
any good books or (esp. survey) articles discussing what might happen
(classically) at the Schwarzschild singularity to an infalling
particle?

From what I've seen people have no answers to this because this is
where reality stops in GR: singularity in the literal sense does not
exist, there are _no events_ there. You can of course compactify the
manifold but you still cannot extend the metric there. So you'd have to
come up with an extended sort of dynamics which does not require
"normal" spacetime with a metric. Sounds mighty implausible. And you
still would have to cross regions of curvature increasing without
bound. BTW, what about closed timelike geodesics?

--
Jan Bielawski

Well, it seems like closed timelike geodesics can exist consistently
(my guess is that they have to exist for quantum mechanical
interference to work properly, though I will need to sit down and think
about that more deeply when I get to that) - the changes in time
direction only occur at the singularities of the black holes according
to this proposal, which would look like pair creation events in the
infinite past and pair annihilation events in the infinite future (not
that anyone would be around to see them happen).

- Sabbir.

"Sorcerer" kirjoitti
. co.uk...

"Henry Haapalainen" wrote in message
news | Is this NG a correct place for science fiction?

Yes, of course.
Some people even claim acceleration isn't real.

Androcles

Who are those people? HH

"Henry Haapalainen" wrote in message
.. .
|
| "Sorcerer" kirjoitti
| . co.uk...
|
| "Henry Haapalainen" wrote in message
| news | | Is this NG a correct place for science fiction?
|
| Yes, of course.
| Some people even claim acceleration isn't real.
|
| Androcles
|
| Who are those people? HH

Pay me US$1000 and I'll tell you, ****head.
Androcles

LEJ Brouwer wrote:

JanPB wrote:
I think the manifold can be glued smoothly (one can write down a smooth
atlas on the quotient manifold) - it's the metric that has a "crease"
at the glued horizon. I posted an outline of an argument few minutes
ago. The problem is essentially with the Kruskal-Szekeres function
r(T,X) (the one given by the usual implicit equation) - it has a
nonzero slope at the horizon which would "crease" the metric there when
glued.

I agree with your analysis, and am no longer happy with the
implications of my original proposal. There was another possibility
which I had been trying rather to avoid (with hindsight maybe I
shouldn't have), in that the reflection of the infalling particle could
occur not at the event horizon, but rather at the central singularity -
this would also give rise to the same predictions, and I guess will not
upset the cart as the standard Schwarzschild interior solution remains
intact.

Has the possibility that a particle hitting the singularity is
reflected off of it both in space and time (i.e. into the white hole
interior solution) been suggested before that you know of? Are there
any good books or (esp. survey) articles discussing what might happen
(classically) at the Schwarzschild singularity to an infalling
particle?

I'm not sure that it makes sense to talk about the behavior of a classical
particle at the singularity, where tidal forces are infinite. But what
you are talking about now might be related to a quotient of the Kruskal
extension described by Gibbons in "The Elliptic Interpretation Of Black
Holes And Quantum Mechanics," Nucl. Phys. B271 (1986) 497. (See also
Chamblin and Gibbons, gr-qc/9607079.) In this geometry, the two exterior
regions are identified, as are the two interior regions. You lose time
orientability, though -- there's no global definition of a past and future
direction.

There's another "single exterior" identification one can make as well,
the RP^3 geon. The best introduction I know of off hand is Louko,
gr-qc/9906031.

Steve Carlip