Black holes

[Bowman_David/tiger_mpc@tiger.gtc.georgetown.ky.us]

Item Subject: Black Holes
Chris Jones wrote:
> Can anybody explain to me why the calculation of the Schwarzschild radius:
> R(sch) = (2GM/c^2)
> is valid? I have trouble explaining why this should form the event horizon 
> ...

Paul Camp's response nicely answered many of the questions raised by Chris'
post concerning nonrotating uncharged black holes and what happens to photons
and other objects as they move radially toward and away from the hole.  There
is a point regarding Chris' question above which Paul (perhaps wisely) did not
see fit to address.  Unfortunately, my answer concerning the point will (as
usual) be long.

The formula for the Schwarzschild radius: R(sch) = 2GM/c^2 for a (nonrotating,
uncharged) black hole is not necessarily valid.  Its validity depends on what
one means by "radius".  This formula gives the "radius" of the event horizon
in, so called, Standard Coordinates.  In other coordinate systems the "radius"
for the event horizon will have other values.  The reason for this is that the
spacetime surrounding the mass of the black hole is curved (distorted) so that
space is radially "compressed" and time is "stretched" near the hole with the
amount of distortion increasing as one goes toward it and decreasing to zero
as one goes asymptotically far from it.  This effect is present around all
isolated masses; its just that it becomes VERY significant for black holes.

What I mean by "stretching" time is simply the fact that standard clocks tick
more slowly close to a gravitating mass than far from it.  What I mean by
compressing space is that if one draws a circle around an isolated mass and
accurately measures its circumference (with standard meter sticks) and then
accurately measures the distance across the circle (again with standard meter
sticks) one discovers that the measured circumference divided by pi is less
than the measured diameter.  In other words for a given boundary size
(circumference) there is more interior (diameter) than what would be the case
for a flat space.  Similarly for a sphere of volume V and area A in flat space
are related by: V = (4 pi/3)(A/(4 pi))^(3/2), but if the sphere surrounds an
isolated mass one finds that V > (4 pi/3)(A/(4 pi))^(3/2).  Thus there is more
space inside the sphere than one would have guessed by measuring its outside.
It is in this sense that space is "compressed".  (This effect for the Earth is
quite small in that a circle drawn around the Earth, near its surface, has
a circumference divided by 2 pi less than its radius by 1.5 mm.  This excess
radius jumps to a half a km for the Sun however.)

Because of the real curvature of spacetime produced by the mass of the black
hole no coordinate system (normally used to describe a flat spacetime) will
accurately reflect its true properties everywhere with out some distortions.
This is like how a Mercator projection of the Earth's surface distorts the
distances between points on the flat map.  There is no "scale of miles" for
the map which is accurate over the whole map and in all directions.  To make
accurate measurements on the map one needs a locally variable distance scale.
There are different projections of the curved surface of the Earth onto a
plane which may preserve some features at the expense of others (i.e., areas,
distances, angles, etc.).  The same is true of coordinate systems describing
the curved spacetime of a black hole.  The Standard Coordinate system
mentioned above inaccurately measures radial distances but gets circular arc
lengths at fixed "radius" correct.  Thus dr IS NOT the differential of proper
radial distance but r d(theta) IS the proper arc length distance differential 
at fixed r in Standard Coordinates.  It is the radial coordinate r of Standard
Coordinates which has the value of 2GM/c^2 at the event horizon.  If the event
horizon's proper area was measured and then it was divided by 4 pi and the
result was raised to the 1/2 power, then this result would be the value
2GM/c^2.  It does NOT represent the actual radius of the black hole, however,
(whose measurement is problematic in practical terms since anyone who entered
it with his meter stick to measure its radius would be destroyed at the
singularity at the center and could not come back out and report what value he
obtained).

Because of the distortion of radial distances in Standard Coordinates this
coordinate system is anisotropic.  Another coordinate system which IS
isotropic in its distance measurements is called (naturely) Isotropic
Coordinates.  In Isotropic Coordinates the "radius" at which the event horizon
occurs for the black hole is at: R = GM/(2c^2) rather than the formula above.
In Isotropic Coordinates the radial differential dr and the angular arc length
differential r d(theta) both do not directly measure proper distance, but they
both use the same isotropic "fudge factor" (i.e. metric coefficient).  Both
Standard Coordinates and Isotropic Coordinates are different analogs of
ordinary spherical coordinates used in flat space.  We can also define analogs
of flat-space Cartesian coordinates, called Harmonic Coordinates, and use them
to describe the spacetime around the black hole.  In Harmonic Coordinates the
event horizon occurs at the "radius" R = GM/c^2.  In Harmonic Coordinates the
radial r value is just the square root of the dot product of a spatial
3-vector with itself as measured from the spatial origin.

So we see that the value of the Schwarzschild radius of a black hole is
ambiguous and ill-defined without further qualifications describing the
measuring scheme used to calculate it.  The reason the value given (by Chris)
from Standard coordinates is most commonly used is that it has the physically
understandable meaning of being the radius of a flat-space sphere whose proper
surface area (and circumference) is the same as that of the event horizon of
the black hole.

David Bowman
Georgetown College
dbowman@gtc.georgetown.ky.us