Re: This Week's Finds in Mathematical Physics (Week 232)

Greg Egan wrote:
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?

Yes.

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 ...

I didn't really have any construction in mind. With one exception, I am
not convinced that it *is* possible to have a (connected) manifold with
total deficits > 2Pi. I had just noticed that my proof that there aren't
had a hole in it.

The one exception is a big bang with a total deficit of *exactly* 4Pi.

Consider a polyhedron inscribed in a sphere of radius 1, centered at the
origin. Let the surface of the polyhedron inherit the metric from R^3
(which will be flat except at the vertexes).

For any point p other than the origin, let p_1 be the intersection of
the polyhedron with the ray from the origin through p. Let t(p) = |p|/|p_1|.

The metric (on R^3-O) ds^2 = -dt(p)^2 + dp_1^2 is flat except at the
rays from the origin through the vertexes, and any timelike surface has
total deficit 4Pi.

Any loop divides the surface into two parts, either of which can be
viewed as "inside". The holonomy around the loop can only have one
value, and should be the sum of the deficits of the points it encloses.
This only works if holonomy is modulo 2Pi.

This solution is static, but there should be dynamic variants as well.
They shouldn't be too complicated in principle, but I would have to
abandon pencil and paper and start programing to figure them out (which
I don't have time to do).

I would also have to do a great deal more reading than I have so far.

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.

I'm afraid you lost me about there.

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.)

It is possible that you have a piece of the solution I outlined above.

I can't see how to extend this solution to infinite -ve time, but why not
make a virtue out of necessity and consider a family of Big Bang style
solutions which start from a singularity?

I'm not sure if what you describe is the same as mine.

If we pick an origin in Minkowski spacetime, we can take the topological
interior of the forward light cone of the origin and declare that this
set, minus some wedges excluded along particle world lines, will be our
entire solution.

It looks like your construction has a boundary, other than the origin.
If so, then it can't be the same as my big bang, but it could be a piece
of it (e.g. with a hole cut out).

Ralph Hartley

.

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