Koobee Wublee says...

On Mar 28, 3:53 am, Daryl McCullough wrote:

p_x' = gamma (p_x - Ev/c^2)
p_y' = p_y
p_z' = p_z
E' = gamma (E - p_x v)

Is the transverse Doppler equation not the following?

** E' = gamma (E - p_x v)

As I have explained to you multiple times, there
are two different questions:
(1) If an electromagnetic wave has frequency f in
one reference frame, then what is its frequency
f' in another reference frame?
(2) If two observers are in relative motion, and the
one observer sends signals at a characteristic rate
of f signals per second, as measured on his own clock,
then what is the rate f' that the other observer will
receive those signals.

These two questions are related, but are not the same.
They don't have the same answer. Now, we can show that
*IF* the line separating the two observers is parallel
to the relative velocity between the two observers,
*THEN* the two ratios will be the same. If the line
separating the two observers is *NOT* parallel to
the relative velocity between the two observers, the
two ratios will *NOT* be the same.

If you ask two different questions, in general, you
get two different answers. That's to be expected.

--
Daryl McCullough
Ithaca, NY