I've worked out the FRW dust solutions in 2+1 gravity, that is the(adsbygoogle = window.adsbygoogle || []).push({});

homogeneous, isotropic solutions where the universe contains a continuum

of matter that approximates the presence of pressure-free particles at

the same density everywhere.

As in 3+1 gravity there can be negative, zero, or positive curvature for

the spacelike slices (i.e. hyperboloids, planes or spheres), but the time

evolution is extremely simple: either the solution is static (which is

only possible for spherical spatial geometry or zero density), or it is

expanding at a constant rate.

The metric is:

ds^2 = -dt^2 + R(t)^2 dw^2

where dw^2 is the metric on the unit hyperboloid, plane, or unit sphere.

If we put k=-1,0,1 for these alternatives, the relationship between R(t)

and the density of matter is:

R'(t)^2 + k = 8 pi rho R(t)^2

while the lack of pressure gives:

R''(t) = 0

For spherical geometry, the radius of the sphere is R(t), so the total

amount of matter present is M = 4 pi rho R(t)^2, and:

R'(t)^2 + 1 = 2 M

This implies that a static solution is possible iff:

M = 1/2 (in units where G=c=1).

In these units, the angular deficit associated with a mass of M is 8piM,

so this static solution has a total angular deficit of 4pi, in agreement

with the finite particle polyhedron solutions Ralph Hartley has described.

The only other *homogeneous, isotropic* static solution is the vacuum

solution. If we put rho=0 in the planar FRW solution, we get

R(t)=constant, and that's obviously Minkowski space, but if we put rho=0

in the hyperboloid solution we get R(t)=t, and the solution is the

interior of the light cone in Minkowski space, i.e. a portion of the same

vacuum solution.

For the non-static solutions, we note that

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

by conservation of mass; for the spherical geometry we have C=M, the

total mass in the universe, but for the other geometries where there's an

infinite amount of matter in the universe C will still be constant. So

we have:

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

rho(t) = C / [4pi (2C-k) t^2 ]

It'd be nice to be able to relate this to various discrete particle

solutions. I previously posted a solution with a ring of n particles

moving away from the origin with equal speed v in a symmetrical manner,

and noted that if we look at what's happening on the unit hyperboloid in

Minkowski space, the angular deficit associated with each particle

becomes much less at hyperbolic infinity, i.e. if we cut out a wedge by

arranging two planes to intersect along each particle's world line, the

sum of the angles between these planes can be much more than 2pi because

they take much smaller "bites" out of the light cone than the angles they

subtend around the world lines.

Specifically, if the angle of the "bite" is B, it turns out that for

small deficit angles A around the world line we have:

B = A sqrt( (1-v) / (1+v) )

Now, imagine a cloud of particles with world lines starting from the

origin of Minkowski space with all possible velocities, spreading out at

all angles. By cutting out wedges around their world lines, we ought to

be able to get one of the expanding hyperboloid FRW solutions, where each

spacelike slice at time t is isometric to the unit hyperboloid multiplied

by sqrt(1+2C) t, and the world lines of all the particles are normal to

every spacelike slice.

In hyperbolic geometry the area and circumference of a circle increase

with distance faster than in the Euclidean plane, but as we move out

through our cloud of particles in Minkowski space with wedges cut out,

we're *losing* area and circumference. So all the hyperboloids in the

original Minkowski space get flattened out. I guess the trick is to find

a way to make sure that their geometry is still that of a hyperboloid,

but a slightly larger, hence flatter, one ... by a factor of sqrt(1+2C).

But I haven't been able to figure out yet how to get the particles

distributed in the original Minkowski spacetime in such a way that

everything works out nicely.

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

Can you offer guidance or do you also need help?

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