From: bbowen@megatest.com (Bruce Bowen)
Newsgroups: sci.physics
Subject: Black hole insights
Date: 22 Jan 93 01:28:25 GMT
Organization: Megatest Corporation

Most discussions on black holes I've seen here focus mainly on
kinematics of infalling bodies.  The following are various thought
questions/answers.  Unless explicitly stated otherwise, all of what
follows applies to an uncharged, non-rotating black hole.



1.  Proper distance to the event horizon.

From any point outside the event horizon, the proper length (ruler
distance) to the event horizon is finite and can be obtained by
integrating ds/dr of the Schwartzchild metric from the Schwartzchild
radius to your radial point, while keeping t, theta and phi constant.
The derivative though (ds/dr) diverges as one approaches the horizon,
so it is an improper integral.


2.  Time to fall into black hole.

The coordinate time for an infalling infinitesimal testpoint to reach
the horizon is infinite.  The proper time is finite.  The testpoint
falls through the horizon perpendicularly, independent of what angle
and speed it started, and locally at the speed of light.

For a massive object, what horizon are you talking about?  The horizon
of the original black hole, or the horizon of the black hole + object?


3.  Spacetime curvature.

Is finite as one approaches the event horizon.  This is reflected in
the tidal forces experienced by an object in free fall.  An object not
in free fall and/or moving at a different speed will experience
different tidal forces since lorentz contractions, and hence the
object's spatial extent, will be different.  An (infinitesimal) object
falling through the event horizon falls through at speed "c", so it is
lorentz contracted to zero thickness and doesn't experience tidal
forces resulting from #4 below.


4.  Local static gravitational force (Weight of a 1 gram object).

This diverges (goes to infinity) as one approaches the horizon.


5.  Suspended DC electrical transmission line.

Voltage, current and total energy delivered in the line are
proportional to the gravitational redshift.  The power delivered is
proportional to the redshift squared.


6.  Extractable potential energy.

If you lower a massive object into a black hole in a controlled manner
and extract its released potential energy as you do so, say, by
beaming it back up the above suspended DC transmission line, the total
energy you can get back is equivalent to the mass of the original
object. This could in theory be reconstituted into the original object.
Which leads one to wonder, "What actually fell into the hole."  Again,
this is only true in the limit of an infinitesimal object.


7.   Stable orbits.

Stable orbits are impossible below a point which is significantly
above the event horizon.  I forget the exact value (2R, 3R?) but it's
in MTW.  One way to think of this is that you have to curve very
little to go around the hole since space is already curving for you.
So you generate very little "centrifugal" force to counteract gravity.
All you have to do to circle the hole is go straight.  Think of a
cylinder with gravity pointing along the axis.  No matter how fast you
circle the cylinder, you generate no "centrifugal" force to oppose the
gravity (or equivalently, the "centrifugal" force has no component in
the cylindrical surface).  Stable orbits ARE possible on a cone,
provided the conic angle isn't to sharp (due to maximum speed of "c").
Space is flat far away from the hole, "conic" as one approaches the
hole, and in the limit, "cylindrical" at the horizon.  If you plot
only r and one circumferential coordinate, the actual surface is the
top (or bottom, does YOUR rubber sheet bulge up or down? :) ) half of
a parabaloid of revolution.

For orientation only: y = x^2 + 1, x = (0,inf), rotated about x axis.
Schwartzchild radial coordinate = "y". singularity at y=0. Proper
radial length = arc length of parabola.  Event horizon at y=1.


8.   Not only curved spacetime, but curved space!

In Euclidean space the surface area of a sphere is 4*pi*r^2 or 

r = 1/2 * sqrt(area/pi).

The volume of a sphere is 4/3 * pi*r^3.

r = (3/4 * volume/pi)^(1/3)

If you measure the volume and surface area of a sphere surounding a
massive object, and calculate r in both ways above, you get different
answers.  The volume equation gives you a bigger "r".  Gravity packs
more volume into a sphere!  The hump in the surface is real, not just
an anology.  It's the plot of what you get when you drop out time and
one spatial coordinate, or equivalently, a slice of spacetime with
time and one spatial coordinate held constant, then imbedded into 3
dimensions.
Newsgroups: sci.physics