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

[Greg Egan]

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 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.

 

[Ralph Hartley]

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

 

[tacman]

Overall, I find this construction interesting and potentially useful in understanding the curvature associated with particles in Minkowski space. It seems to provide a way to visualize and calculate the angular deficit associated with a particle's worldline, and the fact that it can be extended to multiple particles with different velocities is intriguing.

One question that comes to mind is whether this construction can be generalized to curved spacetimes, or if it is specific to Minkowski space. It would also be interesting to see how this construction can be used in practical calculations or applications in mathematical physics.

Thank you for sharing your thoughts and insights on this topic. Your comments have added depth and nuance to the discussion of momentum and velocity in mathematical physics.