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## Volume elements in position and momentum space

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Hi there,

I think the fact that the 4-momentum is orthogonal to the mass hyperboloid follow from the fact that p^2=-m^2 and that it is constrained to lie on this as time passes.

For example (assuming a non-zero rest mass so I can use proper time τ) the 4-momentum travels on a path along the mass hyperboloid as time passes, so p = p(τ). The 4-vector dp/dτ is tangent to this path and thus the hyperboloid. Taking the proper time derivative of p^2=-m^2 gives p•dp/dτ=0, meaning that p is orthogonal to the hyperboloid.

Let me know what you think.

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That’s great thanks. I was trying for ages to figure out how to find a tangent to the hyperboloid in the Minkowski metric, completely forgetting that it’s just dp/dτ. I’ve added your comment to the original post.

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