# How do I extend the Lorentz transformation metric to dimensions>4?

How do I extend the general Lorentz transformation matrix (not just a boost along an axis, but in directions where the dx1/dt, dx2/dt, dx3/dt, components are all not zero. For eg. as on the Wikipedia page) to dimensions greater than 4?

Thanks

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The general boost formula given on wikipedia will hold in any dimensions. e.g. in n+1 dimensional Minkowski space you can simply take r to be n dimensional spatial vector and v to be n dimensional velocity vector. – user10001 Jul 22 '12 at 4:49
He means the transformation formula using cross products. If you use dot-products, it works in any dimension. – Ron Maimon Jul 22 '12 at 9:56
That is not an easy task. What you want is a representation of the Lorentz group in R^n (with the Minkowski metric) for which you need to work with the Lie algebra of that group, get their n-dimensional real representations, and exponentiate it. I have notes on the derivation for the boost Lorentz matrices in dimension 4 (rotations of axis are just the action of the rotation group SO(n)) but they are not easily generalized to dim>4, requiring involved and deep knowledge of Lie groups and algebras. I do not know whether there are easier derivations. – Javier Álvarez Jul 23 '12 at 19:05