See Part 1 to understand what this is
about.
--
Figure 1: Cosmological solutions for: i) a(t) in Lorentzian time (see Eqs. (1), (3), with n real); ii) a(t) after a Wick rotation W, setting n = -iN, with N real; iii) b(t) after a conformal and a Wick rotation CW, setting a(t) = ib(t), n = -iN with b(t) and N real.
--
Figure 1 isn’t
referred to in the text of the document until later, but a(t) appears in
the paragraph at Equation 1 discussed in Part 6.
Some of the terms shown in this figure appear in that later section
(around Equation 3, to be discussed later), specifically:
- ESU
- κ
- L+i
and L+o
- L±a
The title of each
image in the figure needs some clarification.
“Lorentzian” seems to be a reference to the metric signature of the
Minkowski metric, which is expressed as either (-,+,+,+) or (+,-,-,-). Another way to think of the metric signature, is in terms of (p,v,r),
where p, v and r are (respectively) the number of positive,
negative and zero eigenvalues of
the real symmetric matrix gab of the metric
tensor with respect to a basis. In this case, we can
just consider them as dimensions, three (positive or negative) dimensions of
space and one (negative or positive) dimension of time and no zero dimensions
making the metric signature (3,1,0) or (1,3,0).
The related equation is ds2 = -(c2dt2)
+ dx2 + dy2 + dz2 which conveniently has
all the signs (+ and -) shown, in the right order. It seems to be a convention in the metric to
show time first, I don’t think it has any physical implications.
A Wick rotation is
a transformation to express Minkowski space as if it were Euclidean or, in
other words, going from a metric signature of (-,+,+,+) to one of (+,+,+,+) by
treating the time dimension as imaginary.
So, using natural units such that c = 1, going from ds2
= -(dt2) + dx2 + dy2 + dz2 to ds2
= dτ2 + dx2 + dy2
+ dz2 (where τ is time that takes on imaginary
values). I don’t think that using a
different version of the Lorentzian metric signature makes any difference, it’s
just a different way of expressing the same thing and is thus another matter of
convention.
So, the titles for
i) and ii) could also be “Lorentzian (Metric)” and “Euclidean (Metric)”.
A conformal
rotation is transformation or remapping of space(time) that maintains angles
but not necessarily lengths. It’s not
entirely clear to me whether the rotations mentioned should be considered as two rotations (and
thus whether the order makes a difference) or one rotation that has both
features, so in other words a Wick rotation that is also conformal, or a
conformal rotation that is also a Wick rotation. From the commentary on iii) it does seem to
be two rotations, the upshot of which is to have all dimensions imaginary. I’m not sure why this is beneficial at this
stage – the rotations are normally related to making the equations simpler
during the time they are being worked on (after which there’s the implication
of a rotation back).
The images include the symbol “r” that refers to radiation (see later post - note that it's a little unclear because gamma is also used elsewhere in the document and they look very similar in italics). The kappa, “κ”,
appears to be referring to curvature.
It’s unclear what happens to those curves when curvature is zero.
The value “n”
appears to be the lapse mentioned in Part 6. The
meaning of lapse doesn’t seem to be that easy to find the answer to the
question “what is the physics definition of lapse?” The best I came across was here (but note that it’s talking about the lapse
function). Fundamentally
lapse is proper time between an event on one slice of a hypersurface and the
same event on another slice and shift is the spatial separation between the
event on those two slices. I think this
is related to the simultaneity issue when selecting frames. Say I take a cut through the universe of
“now” using my frame as the reference and consider two events that are (or
rather were, according to me) simultaneous.
A
--------------------------------> Me <------------------------------- B
Light reaches me
from locations 180 degrees apart at the same instant.
However, if my twin
were travelling (relative to me) at some significant speed (on the path AB),
then the events would a) not be simultaneous, b) not be located where I thought
they were, or c) both – according to my twin, depending on where they were at
the time that they took as “now”. We’d
have a lapse, or a shift, or both.
Basically, the
“foliation” of spacetime that they are talking about is the selection of
different values of “now” and lapse and shift are used to translate between
those values of “now”.
Oddly, when I looked up an answer to the question “what is the physics definition of lapse?” I
came across a support page for a Maple
Soft Physics [ThreePlusOne] product. Of particular interest
was the equation on that page:
ds2=−α2dt2+γi,j(dxi+βidt)(dxj+βjdt)
Compare this with
the equation from Part 6:
ds2=a2(t)(-n2dt2+γij(x)dxi
dxj)
Ignoring some minor
convention differences, and assuming that the authors of the paper assume a
shift of zero noting that α and β have the lapse and shift values
respectively, it appears that the only thing added is the a2(t)
term or that ThreePlusOne has assumed that a2(t) = 1.
Note that a(t)
is the scale
factor and that the
current value is 1, so a2(t) = 1 is (currently) true. In the paper, however, the authors don’t
appear to want to be limited to the current value and by implication to the
current state of the universe which is taken to be dark
energy dominated.
It’s further worth
noting that the value of a(t) is calculated using the Friedmann
equations. One of the assumptions here is that “the
metric of the universe must be of the form -ds2 = a(t)2ds32
- (c2dt2)”, which follows from “the simplifying
assumption that the universe is spatially homogeneous and isotropic”.
Noting the topic of the paper, these assumptions should be kept in mind,
because there is a risk of a circular argument.
It might appear odd that in the equation here, a(t)2
applies only to the time component, where in the equations from Part 6 and
ThreePlusOne, it applies to both the time and space components. I suspect that
this is because γij is normalised.
I find it strange
that, when the equation that leads to a(t)=1 in the current, dark energy
era is discussed, there is a key component that is described as “some constant”
– but it’s not just any constant, it’s a key constant since that equation is:
Per the scale factor description, a(t) is proportional to t1/2 in the radiation dominated era, t2/3 in the matter dominated era and … hold onto your hat … e^(H0t) in the current dark energy dominated era (I ran into problems trying to put H0 in as an superscript because of the subscript, hence the e^ notation). Currently, “just by coincidence”, H0 = 1/t, so a(t) is proportional to e. Hm. Why do that? We know that a(t) today is 1. Since, per the Standard Cosmological Model, the universe was radiation dominated, then matter dominated and, today, dark energy seems to dominate, it follows that the vague constant, w, is increasing – being very small at first (so that 2/3(w+1) ≈ 2/3), then approaching ⅓ (so that 2/3(w+1) ≈ 1/2) and then approaching infinity (so that 2/3(w+1) ≈ 0). That implies that a0 would equal 1.
What remains very unclear from all this is precisely
what w is. Here,
however, is an equation for w, which is described as “the equation of
state of the universe” (it’s also implied here
and here):
Where P’ is a pressure and ρ’
is a density (and c, of course, is the speed of light). Specifically:
Where m seems to indicate “during the
matter dominated era” and Λ
seems to indicate “during the dark energy dominated era” (note the symbol is more
commonly associated with the cosmological constant).
Later in the document it becomes clear that
ESU, as referred to in Figure 1, means “Einstein’s Static Universe”.
---
---
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.
