# 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' d ^{2}x'*

Our interest is in the instantaneous velocity *u'* = â 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

*Fâ*frame:

**3. Setting ***u **= â**c **= -**c*_{L}** gives ***c*_{L},** the ***one-way speed of light* **in the negative direction as measured in the ***F**'*** frame:**

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