2+1 gravity (was This Week's Finds 232)

Greg Egan · science forum addict Joined: 01 May 2005 Posts: 75

What happens if you take the holonomy around a loop in an expanding 2+1
FRW dust universe? This is a measure of the total momentum contained
within the loop, expressed as an element of SO(2,1).

All the non-static FRW universes expand forever, at a constant rate. If
we pick coordinates on the unit hyperboloid/sphere we then need to
multiply them by a scale factor that's linear with time if we want to get
actual distances. However, the advantage of using the unscaled
coordinates is that they're co-moving with the particles, so if we give
our loop a certain radius in these coordinates, it will enclose the same
world lines whatever the value of t.

In what follows, the constant C refers to the conservation equation:

4 pi rho(t) R(t)^2 = C

where rho(t) is the density and R(t) is the scale factor of the
expansion, giving the radius of the hyperboloid/sphere that is each
spacelike slice. For a closed universe, C is the total amount of matter.

The scale factor is linear:

R(t) = sqrt(2C - k) t

where k=-1, 0 or 1 for a universe whose spacelike slices are
hyperboloids, planes, or spheres.

OK, what are the holonomies, for a loop of radius r in unscaled
coordinates?

Hyperbolic spacelike slices
===========================

The scalar sum of the enclosed rest mass is (C/2)(cosh(r)-1).

For 0 < r < (1/2) arccosh(1+1/C), the holonomy ranges from a rotation of
zero up to a rotation by 2pi, corresponding to a mass of 1/4 (in units
where G=c=1).

For greater values of r, the holonomy is a boost that increases without
bound for increasing r. This corresponds to tachyonic total momentum for
the enclosed system.

Flat spacelike slices
=====================

The scalar sum of the enclosed rest mass is (C/4)r^2.

For 0 < r < 1/sqrt(2C), the holonomy ranges from a rotation of zero up to
a rotation by 2pi, corresponding to a mass of 1/4. At this point the
scalar sum of the enclosed rest mass is 1/8, so intuitively we could say
that half the mass is coming from kinetic energy.

For greater values of r, the holonomy is a boost that increases without
bound.

Spherical spacelike slices
==========================

In this case C is the total rest mass in the universe.

Note that because the expansion rate is sqrt(2C-1), we must have C>=1/2,
and C=1/2 is a static universe. Also, 0<r<pi, because r=pi takes us all
the way to the antipode.

The scalar sum of the rest mass enclosed in our loop is (C/2)(1-cos(r)).

For C=1/2, the static case, the holonomy is always a rotation by 2pi
(1-cos(r)), corresponding to a mass of (1/4) (1-cos(r)), which is exactly
in agreement with the enclosed rest mass.

For C>1/2, the expanding case, the holonomy is a rotation when r is less
than (1/2) arccos(1-1/C), or greater than pi minus the same value.

In between these values of r the holonomy is a boost, reaching a peak at
r=pi/2, where it corresponds to a "tachyonic mass" of (1/4) sqrt(2C-1) i.

Google