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' d^{2}x'

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 F_{0} as seen by observers in frame F, to be

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

3. Setting u = —c = -c_{L} gives c_{L}, the one-way speed of lightin the negative direction as measured in the F' frame:

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