# Lorentz transform for random v

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Both of the books I have only derive the LT for motion of the frames parallel to the x axis. OK, this keeps the maths simple and also covers motion parallel to any other axis. But nowhere have I seen the LT for a relative velocity v which has components in each of the 3 spacial dimensions.

I understand that the x values will only be effected by the x component of the velocity - and similarly for y and z. But what about t?

So I guess my question is what is the "t" transform in this case?

John

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Got it - thanks

John

Well I was not expecting that! So if there is only motion in one direction then all is simple but as soon as thats not true it gets a whole lot more complex. Need to study these a bit more to try and understand.

John

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You can only have motion in one direction at a time — velocity is a vector. If the vector has y and/or z components, then one option is to re-define your coordinate system. Some problems don't lend themselves to that, though, so you'd have to use the generalized transform in Atheist's link.

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I was recently reading some online lecture notes on classical mechanics that said basically the same thing. That a poor choice of coordinate system can make a solution virtually unrecognizable. It was an analysis of the motion of a pendulum bob in 2 dimensions. The original answer was a mess, but simply rotating in the x-y plane turned it into the standard equations for an elipse, circle or line, depending on the major and minor axis parameters.

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Yes thanks for that - as I work through more problems I am realizing that the choice of frame if critical.

John

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