Hi all,

In recent threads I was trying to show velocities
do not add as Einstein's addition formula states,
but strictly as euclidean vectors,

w = u + v,

and the correct Doppler formula for that sum is

f' = Exp( - w/c) f.

So, this Doppler formula is telling us there is a relative
anisotropy of light for moving bodies, and we avoid
time dilation and length contraction.

Let's see this interesting issue. Rapidity r is defined as
a hyperbolic angle, r = arctanh(v/c). Under SR, c is
regarded as universal constant, so the beta is always
in the range 0 v/c 1. This means, we can write

r = arctanh(v/c) = ln( sqrt( ( 1 + v/c)/(1 - v/c) ) )
so,
Exp( - r) = sqrt( ( 1 - v/c)/(1 + v/c) ),

which is the relativistic Doppler factor. I was wondering why
the relativistic Doppler factor was implicitly hidden into the
rapidity, but that's only a feature of the function arctanh. SR
can predict a relativistic Doppler effect, for speed w, as

f'' = sqrt( ( 1 - w/c)/(1 + w/c) ) f

IOW, under S we can add rapidities as being euclidean,
and express the relativistic Doppler in function of the sum
of rapidities,

r_w = r_u + r_v ,
f'' = Exp( - r_w/c) f

These two equations form a system that is isomorphic to the
system I've provided

w = u + v,
f' = Exp( - w/c) f

Now, we can clearly guess where is the trick SR uses to hide
the relative anisotropy of light. The sum rapidity r_w can be
rescale to yield w, as w = c_w r_w, for real c_w. So we can
write

f'' = Exp( - w/c_w) f, but now c_w is not neccesarily equal
to the constant c.

Likewise, we can rescale to u = c_u r_u and v = c_v r_v,
for reals c_u and c_v. We get the system of equations

w/c_w = u/c_u + v/c_v,
f'' = Exp( - w/c_w) f.

We can show this system of equations is physically meaningless,
because there are two different physical phenomena involved in
it. Firstly, there is the real phenomenon of the relative motion of
bodies, regardless they emit or reflect off light. Secondly, there
is
the real phenomenon of emission of light, regardless the bodies
are at rest or in relative motion. In this rationale, it is assumed
three
inertial bodies move along the same straight line. Now, we can
split both phenomena into different equations, as

w = u + v, [1]

c_w' = (c_u' c_v')/c, [2]
with
c_w' = c Exp(-w/c), [3]
c_u' = c Exp(-u/c), [4]
c_v' = c Exp(-v/c) [5]

So, the former equation [1] describes the relative motion
of bodies, regardless the propagation of light, and the latter
ones [2][3][4][5] describe anisotropic propagations of light.
This means the relative motion of bodies is independent of
the light they may emit, but the propagation of light they emit
is dependent of their relative motion. Under SR, we find the
opposite view, propagation of light is independent on the
relative motion, but relative motion is dependent on propagation
of light. So, are both views equivalent for describing the same
relativistic effects? The short answer is NO, they aren't.

summarizing, under SR we have the system of equations

w/c_w = u/c_u + v/c_v,
f'' = Exp( - w/c_w) f,

and in my model, we get the simpler system

w = u + v,
f' = Exp( - w/c) f

Both systems only match (f'' = f') for the case

c_w = c_u = c_v = c,

it is saying they only match when rapidities equals
their respective betas,

w/c = r_w , u/c = r_u, v/c = r_v

So, where is the trick used by SR? Under SR, we add
velocities of inertial bodies through the addition of their
respective rapidities. But, a rapidity has embeded the
anisotropic propagation of light. For example, w/c_w, is
a rapidity for the beta w/c, it is saying arctanh(w/c) = w/c_w.
Once that addition of rapidities is performed, SR replaces
the constant c in the Doppler formula by the anisotropic
velocity c_w to compound another rapidity. In conclusion,
the relative anisotropy that SR is hidden is

c' = w / arctanh(w/c)

This means SR is equivalent to Galilean relativity equipped
with that rule for the relative anisotropy of light. The limit of
that c' when w tends to c or to -c is, of course, c' = 0, which
can be interpreted as an infinite time dilation. In my model,
the relative anisotropy has a different form, it is

c' = c Exp( -w/c)

The limit of c', in this model, when w -oo is c' = 0, and
c' = oo for w - -oo. In this model, the relative anisotropy
can yields values that may be interpreted as time dilations
or even as time contractions, depending on the sign of w.
If that speed is an approaching speed (w0), then c' is
superluminal, and c' is subluminal if it is a recessional
speed (w 0).

Regards