Quote:

> Black holes arise in GR in situations where classical physics would

> result in escape velocities higher than the speed of light. To an

> escape velocity of c corresponds a difference in gravitational

> potential of 0.5 c^2.

That's not the right way to think of it at all. The defining feature of a

black hole is an event horizon which acts as a "one way gate": objects

(and light) can pass through it in only one direction. The simplest way

of thinking about this (in the simplest case, a nonrotating black hole

with mass m) is to visualize the light cones shearing over with decreasing

radius until at radius r = 2m, one side is vertical, and for 0 < r < 2m,

even radially -outgoing- light must fall in. Specifically:

ds^2 = -dt^2 + (dr + sqrt(2m/r) dt)^2 + r^2 (du^2 + sin(u)^2 dv^2)

^^^^^^^^^^^^^^^^^^^^^^^^^^

= -(1-2m/r) dt^2 - 2 sqrt(2m/r) dt dr

+ dr^2 + r^2 (du^2 + sin(u)^2 dv^2)

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Here, one can take the ONB of Lemaitre observers (freely infalling

observers who "start from rest at r = infty") as

e_t = d/dt - sqrt(2m/r) d/dr

e_r = d/dr

e_u = 1/r d/du

e_v = csc(u)/r d/dv

In the tr plane, you can draw the light cones by first drawing the basis

vectors for say -5m < t < 5m and 0 < r < 5m. Note the the vector e_r is

horizontal, pointing in the direction of increasing r, but the e_t vector

has constant height 1 but is sheared over toward r = 0. You can then draw

the light cones using these vectors the same way you would use the vectors

d/dt, d/dr to draw light cones in Minkowski spacetime. I can't illustrate

that in ASCII but you should figure out how to do this. If you have

trouble, looking at Geroch, General Relativity from A to B might help.

Quote:

> The gravitational potential decreases further if one approaches the

> center of a star. If a spherical object has a constant density, then

> the gravitational potential at the center is 1.5 times the one of the

> surface.

You are no doubt referring to the Schwarzschild incompressible fluid

solution, which can be matched along surfaces r = r0, r0 suffiently

greater than 2m, to the Schwarzschild vacuum just discussed. However, the

Painleve coordinates given above are suitable for matching to a Friedmann

dust with E^3 hyperslices (Oppenheimer-Snyder collapse); you should use

the original "static" coordinates of Schwarzschild to match to a fluid.

However, "Gravitational potential" has no meaning in gtr. You probably

mean the acceleration of fluid particles, which causes their world lines

to be nongeodesics. This acceleration does indeed to decrease to zero at

the center of the fluid drop, but the pressure increases to a maximum, and

the curvatures R^t_(wtw) etc. are negative and decrease to a mininum (a

maximum in modulus) at r = 0. The gravitational time dilation relative to

the gravitational time dilation at the surface also increases to a maximum

at r = 0. But the ratio of the time dilation at the center to the time

dilation at the surface (both relative to clocks at r = infty) is not 3:2.

Quote:

> In the case of real stars the factor is much higher than 1.5 because

> of the much higher density near the center.

> At least a classical consideration leads to the conclusion that a

> visible star does somehow contain a black hole if the escape velocity

> is higher than c at its center.

The Schwarzschild model is oversimplified, but ordinary stars do not

contain black holes at their center. (Think about it: would hydrostatic

equilibrium be possible if there were was a black hole at the center of a

fluid drop?)

Quote:

> This problem results from a more general question: are black holes

> absolute or are they relative (i.e. observer-dependent)?

I think the best short answer is that according to gtr, black holes are

most definitely not observer dependent in the sense you mean. The event

horizon is a real phenomenom and there is no question about whether any

given event is inside or outside it, in any spacetime containing a black

hole.

(But see my post on the Vaidya solution, on my relativity pages, for a

cautionary note about the ontological nature of event horizons as compared

with apparent horizons--- you might think you are outside the horizon when

actually it has unexpectedly expanded out beyond your location, even

though you can still hover motionless above the hole by firing your rocket

engines. You only find out what has happened when some time later (by

your clocks) a collapsing spherical shell of radiation (moving at the

speed of light, so you can have no warning that it is coming) falls past

your location, after which you find you can no longer keep yourself from

falling into the hole, i.e., the new apparent horizon is -outside- your

present location, and you are doomed. See Hawking and Ellis for a related

discussion, and see also Frolov and Novikov; the full citations are in my

reading list.)

This is not to deny that the physical phenomena experienced by various

observers during a close encounter with a black hole depend very much on

the details of their motion relative to the hole. For example an

"ultrarelativistic observer" experiences a near encounter with a black

hole as something like an "impulsive" gravitational plane wave. But this

does not contradict what I just said about every event either being inside

or outside (or on) the event horizon.

Chris Hillman

Home Page: http://www.math.washington.edu/~hillman/personal.html