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General formulae

In a 1 + 1 dimensional spacetime, consider two arbitrary overlapping coordinate patches P = {(t, x)} and P' = {(t', x')'} related by some coordinate [1]

transformation (t, x) ^ (t', x') written in the form

where T and X are smooth functions of t and x. Suppose a particle is moving through a region of spacetime covered by both patches, such that relative to patch P, the worldline of the particle is given by

dx' d2x'

Our interest is in the instantaneous velocity u' = — and acceleration a' = -

dt' dt'2

of the particle at a given event on its worldline, as measured by the P' chorus. If

dots denote differentiation with respect to time in P, we readily find the relations assuming T = 0.

Application to generalized transformations

Given the generalized transformation (18.2) we readily find giving for the velocity and acceleration transformations the rules

From these results we deduce the following:

1. Setting u = 0 gives V, the velocity of F0 as seen by observers in frame F, to be

2. Setting u = +c gives cR, the one-way speed of light in the positive direction as measured in the F’ frame:

3. Setting u = —c = -cL gives cL, the one-way speed of light in the negative direction as measured in the F' frame:

  • [1] A hypothetical handbook of all standard protocols for setting up physical unit standards andperforming experiments.
 
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