On 3/23/24 9:23 AM, gharnagel wrote:
[...]
I think that playing with the Lorentz transformation (LT) and the
relativistic velocity composition equation (RVCE) are futile, because
these are relationships among DIFFERENT COORDINATE SYSTEMS. But nature
clearly uses no coordinates whatsoever, so coordinate relationships are
irrelevant at the fundamental level [#]. That's why all modern physical
theories are expressed in terms of tensors, which are naturally
independent of coordinates [@]. For the LT and RVCE the relevant tensor
is the 4-velocity of an object -- the LT and RVCE are formulated
specifically so when an object's 4-velocity is projected onto the
different inertial coordinate systems, the tensor itself remains
unchanged (aka invariant).
[#] Not to mention the artificial requirement of inertial
coordinates.
[@] There are other mathematical methods to ensure coordinate
independence. Tensors are the simplest, due to their
requirement of multi-linearity. Nonlinear approaches are MUCH
more complicated, and to date no need for them has been
demonstrated.
Specifically, for a given inertial coordinate system (x^i) and an object
with 4-velocity U, the components of U are:
U^i = dx^i/dtau. {i=0,1,2,3}
where tau is the proper time of the object, and d is
partial derivative.
We also have:
U = U^i d/dx^i d is still partial derivative
Note that dx^1, dx^2, and dx^3 need standard rulers for their
definition, and dx^0 needs standard clocks, all of which must be at rest
in the inertial frame. So it is impossible to define these relationships
for coordinates moving faster than c relative to any inertial frame. IOW
a tachyon cannot have a rest frame.
Essential exercise for the reader:
Given two inertial frames A and B, calculate the components
of a given tachyon's 4-velocity relative to each; check
whether the LT and RVCE hold for the tachyon. The essential
question is: what does the norm of its 4-velocity mean?
As A and B are inertial frames, the metric components for
each are diag(-1,1,1,1).
If you can't/won't do this exercise, you'll never really understand
tachyons.
Tom Roberts
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