Ralph Hartley wrote: >There is no problem defining the group valued momentum for a particle, >as long as you specify both a loop, a base point and a coordinate frame >at the base point. You also need to be careful defining the "ordinary" >velocity. A reference frame is not enough. You need a reference frame at >a some chosen point, *and* a path from that point to the particle. By specifying a path and a loop, do you mean homotopy classes of these things? >An *arbitrary* spacelike surface can have curvature anywhere, not just >at the particles. Good point! Your comment made me think of the hyperbolic plane, and the opportunity it gives for the sums of angles in triangles to be less than 2pi. I'm not sure if the following construction is the kind of thing you had in mind ... Consider the unit timelike hyperboloid in Minkowski space, -t^2+x^2+y^2=-1. Suppose we have a particle moving with speed v e_x in some reference frame, so its world line lies in the xt plane, and punctures the hyperboloid at: u = (1, v, 0) / sqrt(1-v^2) Now consider the two planes with unit spacelike normals n1 and n2: n1 = (v, 1, v) n2 = (v, 1, -v) n1.n2 = 1 - 2v^2 (Lorentzian dot product) Note that n1.u = n2.u = 0, so the intersection of these two planes is the world line of our particle. These two planes intersect the unit hyperboloid along the curves: y = +/- (v^2 - (1 - v^2) x^2) / (2vx) t = (v^2 + (1 + v^2) x^2) / (2vx) which can be confirmed by checking that (t,x,y) with the above substitutions is always a unit timelike vector, and is orthogonal to n1 or n2 respectively. Both curves are asymptotic to the yt plane, with y->+/-infinity as x->0. So these two hyperbolas meet in a cusp on our particle's worldline, at u, and then spread out as they approach the yt plane. Their projection into the xy plane will be something like this: y ^ |. |. | . | . | . |______.____ x | . | . | . |. |. The Lorentz transformation that rotates around the particle's worldline and carries n2 into -n1 will carry the bottom curve into the top curve, counterclockwise around this diagram. So if we identify the bottom curve with the top one this way, we will have an angular deficit of pi+arccos(1-2v^2) associated with this particle (choosing the branch 0 < arccos < pi). We can turn this pair of curves into a pair of surfaces meeting in a cusp along the world line by linearly rescaling everything by a factor lambda over some range of positive values. The same rotation around the world line will take one surface into the other, and although the angular deficit will be greater than pi the excised wedge will never encroach into the region x<=0. (We have to stick to +ve lambda, so we can't extend things back into the indefinite past, though it might be possible to get around that with some further tricks.) We can then use a mirror-reversed version of the same construction to add a second particle. We don't have to use the same value for v, and we could displace the origin by some vector if we liked. And of course it's not compulsory to make either rotation exactly pi+arccos(1-2v^2), that's just an upper bound. I realise that the boundary in the past is a bit messy, but I don't have the energy to try to fix that up just yet.