Fallacy of Relativistic Doppler Effect

Eric Gisse says...

On Mar 26, 2:42=A0pm, Koobee Wublee wrote:

What is the transverse Doppler effect under relativity? =A0According to
the energy transformation and also your derivation, it should predict
a blue shift while experiments time after time all have indicated
red. oops! oshrug

In what way are your arguments credible? It has already been
established that you were COMPLETELY WRONG when discussing your
strawman derivation of the relativistic Doppler effect. What are the
odds you are correct about the transverse Doppler effect?

Koobee's problem is that physics is hard, and Koobee is lazy.
Whenever he runs into something that he doesn't understand, he
gives up, and declares it to be nonsense.

But Koobee has noticed an ambiguity in the interpretation of
the phrase "the transverse Doppler effect". That's actually
to his credit. But rather than trying to *resolve* the ambiguity,
he's taking at as yet another argument that relativity is nonsense.
I could explain the situation to him, but Koobee is incapable of
following arguments that require thought, and dismisses them as
"fudging".

The question is: What is the formula for the transverse relativistic
Doppler effect? Let's add some background information to make this
question more precise:

Suppose we have two observers, A and B, traveling inertially. Let
F be the frame in which A is at rest, and let F' be the frame in
which B is at rest. Assume that, according to the coordinate system
of frame F, B is traveling in the +x direction at speed v.
Rather than assuming that the separation between A and B is
in the x-direction, we will assume that they are at different
y locations. For definiteness, we will assume that, as measured
in frame F, B is traveling in the x-direction along the line y=L,
and A is sitting at x=0, y=0.

Assume that A is transmitting a periodic electromagnetic
wave in the +y direction. Let e_1 be the
event at A corresponding to the start of a cycle, and let
e_2 be the event at A corresponding to the end of that
cycle. Let T be the time between e_1 and e_2, in frame F,
which is also the period of the wave.

Now, about this set up, we can ask two *different* questions
about what things look like in frame F':

(1) Let T'' be the period of the electromagnetic wave produced
by A, as measured by frame F'. What is the ratio
T''/T?

(2) Let e_3 be the event at which the signal from event e_1
reaches the line y=L. Let e_4 be the event at which the signal
from event e_2 reaches the line y=L. Let T' be the time between
e_3 and e_4, as measured in frame F'. What is the ratio
T'/T?

If B were traveling in the same direction as the electromagnetic
wave, straight away from A, then there would be no difference
between T' and T''. But in the transverse case, they are not
the same. This is not an inconsistency; the two quantities
T' and T'' have different definitions, and there is no
logical reason for them to be equal, and they are not equal
according to SR.

Solution to (1).

The simplest to derive is T''. The phase phi of the
electromagnetic wave is given in frame F by
phi = k y - w t,
where k = w/c and where w = 2pi/T.

Phase is an invariant. So when we switch to frame F',
we have:

phi' = phi = k y - w t

We want to re-express this in terms of F' coordinates,
so we use the inverse Lorentz transform:

y = y'
t = gamma (t' + v/c^2 x')

to get

phi' = k y' - gamma w t' - gamma vw/c^2 x'

We can write this in the form: phi' = k_x' x' + k_y' y' - w' t'
with the definitions:

k_x' = - gamma vw/c^2
k_y' = k
w' = gamma w

Since w' = 2pi/period, we defined T'' to be the period in F',
we have:

T'' = 2pi/w' = 2pi/(gamma w) = 1/gamma (2pi/w) = T/gamma

So T''/T = 1/gamma.

So T'' is less than T, by a factor of 1/gamma.

Solution to (2).

To derive T', we need to compute the coordinates of
the events e_1, e_2, e_3 and e_4 in both frames.
Once again, e_1 is the event at sender A at rest
in the F frame at the start of a cycle. e_2 is
the event at sender A at the end of the same cycle
(time T later, according to frame F). e_3 is the
event at which the light from e_1 crosses the line
y=L. e_4 is the event at which the light from e_2
crosses y=L.

e_1 and e_2 take place at A, which we can assume
is the origin of the F coordinate system. We may
as well assume that e_1 takes place at t=0.
So we have:

x_1 = 0
y_1 = 0
t_1 = 0

x_2 = 0
y_2 = 0
t_2 = T

Light propagating in the y-direction will reach
the line y=L after a time period of L/c. So we
have:

x_3 = 0
y_3 = L
t_3 = L/c

x_4 = 0
y_4 = L
t_4 = T + L/c

Letting delta-x be x_4 - x_3, delta-y be y_4 - y_3,
and delta-t be t_4 - t_3, we have:

delta-x = 0
delta-y = 0
delta-t = T

Now, transform to frame F' to get:

delta-x' = - gamma vT
delta-y' = 0
delta-t' = gamma T

delta-t' is just the T' introduced earlier. So we have:

T'/T = gamma

So T' is greater than T, by a factor of gamma.

Reconciliation of (1) and (2).

As we saw, T' and T'' are not the same: T' T, but T'' T.
But the two results are completely compatible. Let's look
at the change in phase between e_3 and e_4 in both frames.
In frame F, the phase is given by: phi = ky - wt, so the
change in phase between e_3 and e_4 is
delta-phi = k delta-y - w delta-t
= 0 - wT
= - 2pi
(because w = 2pi/T)

Now, look at the same change in phase from the point of
view of frame F':

delta-phi = k_x' delta-x' + k_y' delta-y' - w' delta-t'

We've already calculated
k_x' = - gamma vw/c^2
k_y' = k
w' = gamma w
delta-x' = - gamma vT
delta-t' = gamma T

So

delta-phi = (- gamma vw/c^2)(- gamma vT) + 0 - (gamma w) (gamma T)
= gamma^2 v^2/c^2 wT - gamma^2 wT
= -gamma^2 wT (1-v^2/c^2)
= - wT
(since gamma^2 = 1/(1-v^2/c^2))
= -2pi

So the two equations T' = gamma T and T'' = 1/gamma T are
perfectly consistent, once you realize that T' is *NOT*
the period of the electromagnetic wave in frame F'. Why
not? It's because e_3 and e_4 are not at the same location
in frame F'; they don't have the same value for x'. To
directly compute the period of a wave, you have to have
two events such that the first event is at the start of
one cycle (which is the case with e_3), and the second
event is at the end of that cycle (which is the case
with e_4), and the two events are at the *SAME* location
(which is not the case with e_3 and e_4). So the time
between e_3 and e_4 is NOT the period in frame F'.

--
Daryl McCullough
Ithaca, NY