Forgive this naive question, but I just can't seem to think of a way
that any kind of surface or manifold can give a path into an area but
not out of the area unless you append some artificial constraint like
isotropy or non-conservative fields, which to my knowledge do not
apply to time-space geometry. A one-way path on a continuous surface
seems like a topological impossibility. Can anyone give me a lower
dimensional geometric analogy that I can think about? Is it a
topological phenomena that only can happen in higher dimensions? All
the scenarios I run in my head either lead to a black hole that
pinches itself off from our universe or a mass whose escape velocity
asymptotically approaches but never quite reaches the speed of light.
I have a feeling that I am misunderstanding something about the
relationship between the light cone and curved space.

Thanks

Michael

Dear ben.lintschinger:

wrote in message
oups.com...
Forgive this naive question, but I just can't seem to think
of a way that any kind of surface or manifold can give a
path into an area but not out of the area unless you
append some artificial constraint like isotropy or non-
conservative fields, which to my knowledge do not
apply to time-space geometry. A one-way path on a
continuous surface seems like a topological impossibility.
Can anyone give me a lower dimensional geometric
analogy that I can think about?

Sure, swim up a waterfall like Niagara Falls, and keep even your
"propulsion fluid" from exceeding the speed of sound in water.
Use signalling limited to "water waves" for generating your
metric.

David A. Smith

On 2007-05-25, wrote:
Forgive this naive question, but I just can't seem to think of a way
that any kind of surface or manifold can give a path into an area but
not out of the area unless you append some artificial constraint like
isotropy or non-conservative fields, which to my knowledge do not
apply to time-space geometry. A one-way path on a continuous surface
seems like a topological impossibility.

Hawking proved that any spacetime which is causally stable,
deterministic, expanding with gravity always being attractive,
must be singular. Another way of saying this is that the spacetime
is geodesically incomplete; there are timelike vectors, not defined
for all R.


Can anyone give me a lower
dimensional geometric analogy that I can think about? Is it a

How about topological defects in crystals.

topological phenomena that only can happen in higher dimensions? All
the scenarios I run in my head either lead to a black hole that
pinches itself off from our universe or a mass whose escape velocity
asymptotically approaches but never quite reaches the speed of light.
I have a feeling that I am misunderstanding something about the
relationship between the light cone and curved space.

Possibly. Inside the black hole, the radial direction inside the horizon
is the time coordinate, so the _future_ lies at R=0. That is why all future
pointing timelike paths which cross the horizon, must end at the singularity.
Time ends at the sigularity which is spacelike and occupies all of space.