Imagine a Universe contains mere narrative with only oblique reference to equations. That does not mean that the equations don’t exist.
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Time
(and Space)
The idea I
was playing with here was that time slows down as you approach the Schwarzschild radius of a black hole (it actually slows down as
you approach any mass, just not as much):
to/tf = √(1-2GM/rc2)
Where to
is proper time between two events for an observer at a distance of r
from the centre of a mass M, and tf is co-ordinate
time for an observer not under the influence of that mass (strictly speaking
not under the influence of any mass). G
is the gravitational constant and c is the speed of light. Buried in this equation is the Schwarzschild
radius rS = 2GM/c2, the event horizon of a
non-rotating black hole.
Objects can’t
get past this radius, but the narrative suggests that mass/energy (in the form
of energy) might – however, when r < rS, we have a ratio
of to/tf that is the square root of a negative
number and as r approaches 0, we have a ratio of to/tf
that approaches infinity times the square root of a negative number.
While the square
root of a negative number is formally called an imaginary number, this type of number is used in a number of
fields to denote orthogonality (for example in the use of quaternions) – the notion that two things (electric and
magnetic fields, for example) are perpendicular to each other. Time, as another example, is orthogonal to
space. With the four linked dimensions
of spacetime, each is orthogonal to the others.
When, in the narrative, I talk of an orthogonal universe, that universe (notionally)
has time that is orthogonal to all the dimensions of this universe and also to
the three dimensions space in that universe, each of which in turn is
orthogonal to all the dimensions of space in this universe.
There could
be no interaction between orthogonal universes, and to each the eternity of the
other would be a mere moment – with one at the end, and the other at the beginning. This permits us to hypothetically link
together a chain of effectively eternal universes.
The same
logic applies to space due to length contraction as one approaches a mass
although it’s not quite as simple, since the length contraction is only in one
dimension (parallel to the separation with the centre of the mass). However, masses that approach the Schwarzschild
radius are ripped apart and the resultant energy is smeared across the surface
of the black hole – acting as the “reservoir” for the inner universe.
Critical
Density and Expansion
In cosmology,
“critical density” is the density of
the universe at which it neither expands forever nor collapses. Such a universe is described as “spatially flat”. Critical density is given by:
ρc = 3H2/8πG
where H
is the Hubble parameter (note that H0
is also referred to as the Hubble constant, but that is just the value of the Hubble
parameter today) and G is the gravitational constant again. The Hubble parameter is the rate at which
distant objects are receding, given as a ratio between that rate and their
distance – usually given in km/s/MPc (kilometres per second per megaparsec).
There is
also Hubble time, currently calculated from
the measured value of the Hubble parameter
– either 67.4 or 74.0 km/sec/Mpc, with errors of about 1-2 km/sec/Mpc (or about
2%) – to be either 13.2 or 14.6 billion years, with error margins of about 0.3
to 0.4 billion years, the mid-range of which is 13.9 ± 0.3 billion years – which entirely covers
the range in which the age of the universe lies 13.77 ± 0.059 billion years).
The volume
of a “Hubble sphere”, where rH is the Hubble length (c/H, where c
is the speed of light), is
VH = 4πrH3/3 = 4πc3/3H3
The
universe appears, very much so, to be spatially flat. I make the simple assumption that that appearance
is reflective of reality.
If the
universe is spatially flat, then it always has been and always will be (according
to Sean Carroll, who wrote “a spatially
flat universe remains spatially flat forever, so this isn’t telling us anything
about the universe now; it always has been true, and will remain always true”). Consider then a “Hubble mass”, MH,
which is the mass inside a Hubble sphere given that the density in that sphere
is critical, so
ρc = MH/VH = 3H3MH/4πc3 = 3H2/8πG
So
MH = c3/2HG
Recalling
that rH = c/H, and rearranging,
rH = 2GMH/c2
Which is
the equation for the Schwarzschild radius of a Schwarzschild black hole of mass MH. Note however that there is a direct relationship
between the radius of a Hubble volume in a spatially flat
universe and the mass contained within that radius. This means that as the radius increases, so
too does the mass, or
ΔrH = 2GΔMH/c2
ΔMH/ΔrH = c2/2G
Another
assumption made in Imagine a Universe is that there is
a limitation of the rate of expansion to “quantum of length per quantum of time”. Using Planck units as our notional stand-ins,
this would mean one Planck length (LP) of additional radius
per Planck time (tP), noting that LP/tP
= c, therefore ΔrH/Δt = c, and
thus
ΔMH/Δt = c3/2G = ½c3/G
ΔMH/Δt = ½mP/tP
Which means
that, within a Hubble volume, mass increases at a rate of half a Planck mass
per Planck time to maintain critical density.
(More will be said about this in a later post.) Note that this does not necessarily
mean that the Planck time is the smallest increment of time, merely that even if
there are smaller increments, the ratio of mass added per increment will be
equivalent to half a Planck mass per Planck time (per Hubble volume).
Vacuum Energy and Critical Density
If, as discussed
above, mass (or as I prefer, mass-energy) is being added to the universe at the
rate of half a Planck mass (equivalent to half a Planck energy) per Planck time
(per Hubble volume), then we could work out how much mass-energy would exist in
the universe at this time. At 13.77 billion
years old, the universe has experienced 8.06x1060 Planck times and
so the Hubble volume that we live in would contain 4.03x1060 Planck
masses, which is 8.77x1052kg.
The currently estimated mass of the observable universe is “at
least 1x1053kg”, which is in the ballpark (more on this in a later post).
Note that
the energy equivalent of 8.77x1052kg is 7.88x1069J, in a
Hubble volume of 4πrH3/3, where rH is 8.06x1060 Planck lengths,
so 9.26x1078m3.
That makes the energy density 8.51x10-10J/m3, or
~10-9J/m3. Which
is the value of vacuum energy. Converting back into terms of mass, we get 9.47x10-27kg/m3,
or ~10-26kg/m3.
Which is our universe’s critical density (at this time). Note the comment here with regard to average density (“including contribution
from energy”).
Most of the
mass-energy in the universe is sitting there in the vacuum, with only a small proportion
manifesting as baryonic matter.
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The upshot
of all this is that if the universe is spatially flat, then the
introduction of mass-energy itself would drive expansion of the universe. It makes sense that this expansion would not
be instantaneous – and instantaneous expansion is not what we observe. What we observe instead is an expansion at a
fraction of the speed of light which is proportional to the fraction of a
Hubble length that the distant object (usually a galaxy) is from us. Which is also to say that we live in the centre
of a Hubble volume that is expanding at the speed of light (noting that this is
not a special place for us, every point in the universe is the centre of its
own Hubble volume, some of which overlap without ours). This in turn means that the universe is
expanding at a rate of one Planck length per Planck time (and, as calculated above, increasing in mass-energy at a rate of half a Planck mass per Planck time (per Hubble volume)).
Conversely,
if the universe is spatially flat, and the universe is expanding
(which we observe), then mass-energy must be being added to the universe.
It’s possible to imagine that mass-energy
would be pulled into existence (in our universe) if 1) something else were
driving the expansion of the universe while 2) some mechanism was constraining
the universe to be spatially flat, noting that this raises the question of where
the mass-energy comes from. But this
seems to be a less parsimonious hypothesis.
The far
more parsimonious hypothesis is that mass-energy is being added to a spatially
flat universe (which is observed) which explains the expansion (that we
observe) at the rate at which we observe it, resulting in a density that we observe
and a vacuum energy that we observe. We
are then left with only one question – where does that mass-energy come from?
This was
the question (to one level) that Imagine a Universe was hinting at.
