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I'm interested in deriving the relativistic relative velocity of two particles moving near the speed of light. It turns out to be (with ##c = 1##) $$\frac{V^{(1)} - V^{(2)}}{1 - V^{(1)}V^{(2)}}$$

How should I approach to this problem? Maybe I should not think of particles at all, but instead simply to consider the two reference frames as two set of coordinate values? So

Frame 1: ##(x^0,x^1,...)##; Frame 2: ##(y^0,y^1,...)##.

Then the velocities are ##\partial x^i / \partial t## and ##\partial y^i / \partial t'##.

##t \equiv x^0##, ##t' \equiv y^0##

Now, ##dy^\alpha = \Lambda^\alpha{}_{\beta}dx^\beta##. The solution for the ##\Lambda## are easy for the case when the displacement ##dx^i## are null in one of the frames, but this does not help in solving for the situation in question.

How should I approach to this problem? Maybe I should not think of particles at all, but instead simply to consider the two reference frames as two set of coordinate values? So

Frame 1: ##(x^0,x^1,...)##; Frame 2: ##(y^0,y^1,...)##.

Then the velocities are ##\partial x^i / \partial t## and ##\partial y^i / \partial t'##.

##t \equiv x^0##, ##t' \equiv y^0##

Now, ##dy^\alpha = \Lambda^\alpha{}_{\beta}dx^\beta##. The solution for the ##\Lambda## are easy for the case when the displacement ##dx^i## are null in one of the frames, but this does not help in solving for the situation in question.

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