See Part 1 to understand what this is about.
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In this Letter, we treat background spacetimes with
where n is the lapse, which may be set constant by reparameterizing t, and a(t) is the scale factor.
I think they are
treating space as if it were three dimensional here (by eliminating one spatial
dimension). Possibly for ease of
calculation. (This is wrong. See Part 6.) A lapse is the “proper”
time separation between two events – where “proper” means “as per a clock that
travels between those two events”. Note
that “proper” in this sense means “own time” rather than “correct time”. Also note use of the term events (locations
in spacetime that refer to both spatial and temporal coordinates). An object that is “stationary” in
(coordinate) space for a fixed period of (coordinate) time, is travelling
between events in time and coordinate time equals proper time. Another object that travels in one direction
in coordinate space won’t arrive back at the second event. There needs to be a change in direction and
if that happens, then coordinate time for the “stationary” object won’t be the
same as proper time for the object that left and returned.
Comoving
3-space is assumed to be maximally symmetric, with metric γij(x)
and Ricci scalar 6κ.
Comoving distances
between objects in the universe factor out the expansion of the universe. The term comoving can also be thought of as
being (relatively) stationary in respect to “the Hubble flow” – such that the cosmic
microwave background appears isotropic (neither red- nor blue-shifted). Note that it is this implied frame in which
the stay-at-home twin in the “Twin
Paradox” is stationary and the travelling twin is
in motion.
3-space is just
space with three dimensions (so we are thinking only of the spatial component
of spacetime). Maximally symmetric just
means that you can’t get any more symmetric (so we’re talking about symmetry in
all axes). A search for the string “Ricci
scalar 6κ” produces precisely one result – this paper. There is, however, a paper that mentions “Ricci
scalar 5κ” and further clarifies that they are talking about an Einstein manifold,
M5. I presume that all
that is being said here is that there is an implication of a tighter curving of
the manifold (where κ is the fundamental curvature).
For
κ > 0, it is S3, with volume V = 2π2
κ -3/2.
For κ < 0, we assume a compact subspace of H3,
whose volume is 2π2
|κ| -3/2
times a topology-dependent constant. For ease of presentation, we generally
leave the constant implicit.
Where
κ = 0, the manifold is flat.
Note that “flat” is pretty simple when we think of a line or a plane but
gets a bit more complex as we increase in dimensions. This is probably non-standard, but I think of
this way. If you take a line and look at
it from one end, and it’s a point, then it’s flat (mathematically all lines are
flat, if they aren’t they are called curves).
This is, in effect, just rotation to reduce by one dimension. (Say your line is y = 3x + 4. Rotate to make the line parallel with the x-axis
and you have y = 4. When you make it just
a number line, then it’s a point at y = 4.)
You
can do the same thing with a plane, look at it from the side on, and it’s a
line (which is flat as per above). One
of the characteristics of a flat manifold (in 2d, a plane) is that two lines
that are parallel at any point, they don’t converge and they don’t diverge. On a
closed curved manifold, they do converge and they diverge on an open curved manifold.
I
have no idea why they have defined volumes that way. We can revisit it if it becomes pertinent.
--
Hopefully it’s
quite obvious that this series is more of an open pondering session rather than
any statement of fact about what the authors of Gravitational entropy and the flatness,
homogeneity and isotropy puzzles
intended to convey. If I have
misinterpreted them, then I’d be happy to hear about it.
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