Re: Speed of Light

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding the part of my initial post of yesterday on this thread where I
wrote:

> 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  ....

Although this formula *is* "manifestly less than c" it doesn't matter
because the formula is incorrect.  It seems I made a silly error in my
calculation.  The apparent speed calculated above, instead, should
obey this formula:

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

Although it is no longer so apparent, it is still the fact that this
speed *is* less than c.  It is just more work to prove it.

BTW, if instead of measuring the speed by taking r_B - r_A as the
apparent distance covered, we rather use the actual proper radial
distance between observers A & B we get a much more complicated formula
for the calculated speed of the photon going from A to B (where this new
speed is the actual proper distance covered by the photon divided by its
time of flight as observed in B's reference frame).  In this later case
the avg. speed of the photon as observed by B is:

c*(r_B*P - r_A*Q + r_s*(ln(r_B/r_A)/2 + ln((1+P)/(1+Q))))/(P*(r_B - r_A +

  + r_s*ln((r_B - r_s)/(r_A - r_s))))

where we have defined the quantities P == sqrt(1 - r_s/r_B) and
Q == sqrt(1 - r_s/r_A) to somewhat simplify the expression.  Although is
is not at all apparent, it is nevertheless possible to prove that this
speed *is* less than c and only approaches the limit of c itself when
r_A --> r_B (from below), or the limit when r_s --> 0, or both.

It is also amusing to calculate the speed of the photon as observed using
the clock of A rather than that of B.  IOW, we calculate the speed by
dividing the actual proper distance covered by the photon (in a frame in
which the hole is at rest) by the time of flight for the photon as
kept by A's clock (rather than by B's clock as before).  In this case the
expression for the speed, as seen by A, is nearly the same as above
*except* that one of the P's on the top line turns into a Q, i.e.:

c*(r_B*P - r_A*Q + r_s*(ln(r_B/r_A)/2 + ln((1+P)/(1+Q))))/(Q*(r_B - r_A +

  + r_s*ln((r_B - r_s)/(r_A - r_s))))

With this change in whose clock will be used to measure the time of
flight we find that in this new case the "speed" of the photon is now
*greater* than c.

David Bowman
David_Bowman@georgetwoncollege.edu