- #1

Greg Egan

>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 realize 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.