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Re: falling through the horizon of a black hole



At 02:19 AM 3/2/01 -0800, Michael Bowen wrote:

a) Can a chunk of ordinary matter (say, a brick or an
electron) initially located somewhere outside the event horizon (and
totally free to move under gravitational influence from an initial
velocity of zero) actually pass through to the horizon's interior, or
b) does such matter approach the horizon in some asymptotic manner?


The answer to (a) is yes in one frame, and the answer to (b) is yes in
another frame!

Let's first consider the brick's proper reference frame. In this case, I
like to use the bug-sucker analogy. I once saw an insect trap built along
the following principles, with a fan and a funnel.

. \ /
. \ /
. \ h /
. \ /
. \ /
. \ /
. fan
. | |
. | | porous filter bag
. |_|

There was also some sort of UV light to attract bugs to the area.

The air velocity will obviously be a function of position within the
funnel. A bug can fly into some parts of the funnel and then fly away,
perhaps with some difficulty. But there will, somewhere within the funnel,
be a "horizon" such that if the bug crosses the horizon, there can be no
return, because the air velocity exceeds the bug's max flight speed.

Now the interesting thing is that nothing special happens at the point of
no return! The bug cannot tell when the horizon is crossed, without doing
some subtle calculation. At some later time, the bug will be shredded by
the wind shear and/or the fan blades, but there is absolutely no reason why
this should happen at the horizon.

So it is with bricks falling into black holes. From the brick's point of
view, nothing special happens at the horizon. Some time later, the brick
will be shredded by tidal forces, and then crushed to infinite density, as
it approaches the singularity... but the singularity is far, far inside the
horizon. Observers riding on the brick can calculate their velocity
(perhaps by integrating accelerometers) and they report a quite finite
velocity as they cross the horizon.

===========

Now for a much different viewpoint: Let's look at this from the viewpoint
of an observer who is stationary at a safe distance from the black
hole. We see the brick approach the horizon asymptotically. We get
messages from the brick-riders. Every message reports a finite
velocity. We are worried because there gets to be a longer and longer time
between reports. Also the "carrier frequency" on which the reports are
transmitted is getting strongly redshifted. We will receive only a bounded
number of messages. The last message will take a divergently long time to
receive, as the carrier gets redshifted almost to DC.



I ask this because if the chunk must gain an infinite amount of
gravitational potential energy to pass outward through the horizon,

Be careful; the observers riding on the brick don't think they have
infinite energy.

Does the idea of "passing through" the event horizon even make sense?

It does to observers riding on the brick.

Or is the process, if it
occurs, more akin to quantum mechanical "tunneling"?

It is not even remotely related to tunneling.

I am assuming that the matter chunk in question is sufficiently small
(or of sufficiently rugged construction) that destruction due to tidal
forces while still outside the event horizon is not of concern, unless
of course it is pertinent to the explanation.

For a large-enough black hole, your ordinary Home-Depot type brick would
survive crossing the horizon. For homework, calculate
1a) the magnitude of the gravitational acceleration, and
1b) the magnitude of the tidal stress
...at a distance R from an object of mass M. Use the Newtonian formulas. Then
2) Using your tidal-stress formula, evaluate the special case of R = 2 G
M / c^2. Estimate how much mass it takes to create a hole large enough
that it can eat bricks without shredding them.