Speed of Light

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding David Abineri's questions:

> A high school student asked about the speed of light escaping from just
> beyond the event horizon of a black hole.  Is the speed slower than  c?

Yes *and* no.  It depends on just how you propose to measure it.  Like
*any* other light ray anywhere else in the universe, if a photon passes
by (or through?) any observer that photon will *locally* be observed to
travel exactly *at* speed c as it goes by.  OTOH, if one makes a timing
measurement of the time of flight over a large distance then the speed
depends on whose time scale is used and on just how the distance marked
off is to be measured.

For a particular way of doing the measurement you could say that the
speed is slower than c for that particular measurement scheme.  For
instance suppose that the escaping light ray is directed radially
outward from a point A near (but outside) the event horizon to your
location B which is much farther out.  Suppose both points A and B are
fixed w.r.t. the hole.  Also suppose that you choose to label the
'radius' r from the center by an *indirect* method whereby an observer at
A follows a circle (fixed on the symmetry point of the hole) around the
hole at the same radius.  After completing the circular trip the measured
circumference is divided by 2*[pi] and the result is declared to the the
"radius" r_A.  Suppose that you do the same thing at B and declare r_B to
be the circumference of a symmetric circle ("centered") about the hole
which passes through the point B divided by 2*[pi].  Suppose you
synchronize your clock (via prior established signaling methods) with
observer A. (BTW, A's clock will run slower than your clock so you will
have to account for this when synchronizing them.)  And suppose A
releases the light photon toward you at time t_1 on *your* clock.
Suppose that at time t_2 on your clock it arrives to you at point B.  If
you calculate the quotient:

(r_B - r_A)/(t_2 - t_1)

you will realize that this is *less than c*.  In fact, the actual value
of this quotient will be:

c*sqrt(1 - r_s/r_B) - r_s*ln((r_B - r_s)/(r_A - r_s))/(t_2 - t_1)

which is manifestly less than c because it is of the form: c*D - E
where D is is positive constant less than 1 and E is another
positive constant.  In the above equation the value of r_s is the
Schwarzschild radius: r_s = 2*G*M/c^2 where M is the mass of the hole.

It should be noted that the value of r_B - r_A is *not* the actual
proper distance between points A & B.  The actual proper distance is
*greater* than this difference.  The reason why the difference between
the outer circumference and the inner circumference of an annulus
divided by 2*[pi] is not the radial thickness of the annulus is because
of the geometric distortion of the space around the hole caused by its
curvature (which was induced by the hole's mass).

BTW, the specific example above is only supposed to apply to an uncharged
and nonrotating black hole.  Otherwise things are much more complicated.

> How should we talk about that situation?

Only *very* carefully.

David Bowman
David_Bowman@georgetwoncollege.edu