SR WITHOUT THE LIGHT POSTULATE?

With primed values specifying an event in one inertial frame,
and unprimed values specifying the same event in another inertial
frame, a general transformation giving primed values in terms
of the unprimed has the form x'(x,y,z,t), y'(x,y,z,t),z'(x,y,z,t),
t'(x,y,z,t). The transformation for x', for example will then have
an equation of the general form x'=f(x,y,z,t). To make this simpler,
let us suppose the event has coordinates (x',0,0,t'), and (x,0,0,t).
Then, the equation for x' will have the simpler form x'=f(x,t). We
know that the actual Lorentz transform equation for x' has the
form x'=g(x-vt). Some people, like *J.H.Field, and others, have
succeeded in deriving this transformation from the general form
x'=f(x,t), using only the most general postulates, such as
reciprocity, and single-valuedness, without reference to any light
postulate, and before g or v are identified as anything other than
general constants, independent of the variables.

I believe it can be shown immediately, however, that, where
the actual physics is supposed to be involved, a light postulate
is already indicated, as follows:

If the variable t represents time, then v is a velocity. Now,
differentiating the equation x'=g(x-vt) wrt t, we have
dx'/dt = g(dx/dt-v). If dx/dt=0, we get dx'/dt = -v, meaning
x' is moving in the opposite direction to the velocity (it would
turn out, in addition, that this would give g an infinite value).
If dx/dt = v, then x' = g(vt-vt) = 0. If dx/dt = p > v, then x' has
positive values, and this is the usual form of the transformation
equation: x'= g(pt - vt). However, here, x = pt and, since t is
a variable that indicates time, which constantly increases, and
v is a relative velocity, p must also be a relative velocity > v.
In this case, even the most generally derived equation already
indicates that x' is a coordinate that is actually being measured
by a probe moving at velocity p, and x signifies the existence
of such a probe, which was not mentioned in the general
postulates in deriving the equation from the general form, x'=f(x,t).
If, in the equation x'= g(pt - vt), we divide across by p, we get
x'/p = g(t - (v/p)t), which has the dimensions of time. If x'/p is to
refer to a transformation of the time in the primed frame, this can
be the case only if the probe velocity in the primed frame is also p,
as it is in the unprimed frame. Otherwise x'/p does not represent
anything that is actually happening in the primed frame. This
requirement, in effect, specifies the need for some 'probe velocity
postulate' and, if p=c, and the probe is light, we will have the light
postulate of SR, and the usual equation x'=g(ct - vt), where ct = x.
All this follows simply from the specification that the variable t
represents time.

It can also be argued from this that the transformation
equations are not as general as normally supposed. If x' = 0,
for example, in the equation for x', we have 0 = ct - vt, or v = c.
If the equation is written with ct as x, as x'=g(x - vt), we get
x = vt, when x' = 0, which makes it appear that the equation
can give a coordinate of the origin of the moving frame, x = vt.
This, however, is not so, for this form of the equation is valid
for only one value of v, that is, where v = c. To use this equation
for other values of v is a false use of the equation. The maths,
indeed, does give the correct result, but such a result really
indicates that t, in the general equation, does not represent
time, but is merely a general variable, and x does not have
to represent a probe moving at velocity c. It follows, therefore,
that, if the transformation equations are thus wrongly used,
they become pure mathematics, and are disconnected from
the underlying physics, in which t is a constantly advancing time.

Einstein's original derivation may not be the most general, but
it is the true derivation in the case where the variable t represents
time. In the context of Einstein's derivation, in which c is the
velocity of a probe, a rod AB, pointing along the x direction in
both frames, is moving at velocity v in the unprimed frame.
A probe, or light pulse, is emitted at A, creating event A,
transmitted along the rod to B, creating event B, and reflected
back to A, creating a second event A. The transformation
equations, thus derived, refer to events of the type B, and
cannot be applied to events of the type A, without a wrong use
of the equations, as argued above. Replacing ct with x in the
equation x'= g(ct - vt), is valid only where x > vt, because the
equation specifies the existence of a probe, with c > v, and
x = vt is valid only in the one case where v approaches c, and
not where c approaches v, which can never be the case.

It follows, therefore, that the general use of the transformation
equations, in which they are applied to ALL events, is invalid,
is purely mathematical, and disconnects them from the actual
underlying physics. It cannot be surprising if this false use of
the equations leads also to false interpretations of the actual
physics to which they are applied.

Alen

* "A New Kinematical Derivation of the Lorentz Transformation
and the Particle Description of Light." J.H. Field
arXiv:physics/0410262 v1 27 Oct 2004

.

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