I have intended to write something for quite a while about something
that I wrote in Mathematics to Address an Apparent Problem
with Imagining a Universe has been bugging me. That post was “a very brief explanation as to
how mass/energy enters to the (…) universe at a rate of one unit of mass/energy
per unit of time”.
The problem here is that the mathematics tells us that, to remain flat in the presence of expansion, a universe with a volume equal to the Hubble volume (where the Hubble “constant” is the inverse of the age of the universe) must accrue an additional unit of (Planck) mass/energy every two units of (Planck) time. If the universe got one unit of mass/energy per unit of time, then it would never have done more than collapse in on itself because it would have had a density twice that of a black hole.
In Mathematics to Address an Apparent Problem
with Imagining a Universe, I wrote:
We can get circles from a sphere
by sectioning it. The greatest circle we
can create has the same radius as the sphere itself. The sectioning effectively creates two
hemispheres. Note that I remain aware
that the surface area of the curved section of the hemisphere is not equal to the
surface area of the circle created by the section. By analogy, the universe could be notionally
sectioned by a spherical section, creating two halves, meaning two
(three-dimensional) Hubble volumes, meaning that the nice one “unit” of energy
for each “unit” of time during which the radius increases by one “unit” of
length is obtained for the universe as a whole.
Usch.
Recently, PBS Space Time published the video Could the Universe be Inside a Black Hole? which ended with a question
about what exists outside of the black hole. I slapped together a quick (and somewhat
imperfect) comment to the effect that outside our universe – if it were to be a
black hole – is not a where-question but rather a when-question. There were universes before ours and there
will be universes after ours, but they would be orthogonal to our universe.
I also riffed, in that comment, about the problem above that
has been bugging me, writing:
My working hypothesis at the
moment is that if you project four space (spacetime) onto three space (space),
there are two equal solutions. To
understand this imagine looking at a monochrome sphere - ignoring any shading -
and it would look like a circle. That
circle incorporates precisely half of the sphere with a complete other circle
on the other side that you can't see.
This still doesn’t get me to where I want to be, because
there is a suggestion of contiguous spacetime, with a surface volume that would
have a density that is equivalent to that required (for flatness) for half
that volume and I am back to having a similar problem – too much space now
rather than too much mass/energy.
However, it did get me thinking and – coincidentally – there
was something else that has been bugging me for a while. I hinted at it in The Messiness of Layered Spheres, the
notion I had in my head, was that for each instant (Planck time) that the
universe expanded, there would be a cascade of Planck volumes filling in the gaps
created by the expansion. But it’s messy
and the mechanism was utterly obscure.
Recently, although little less recently than the PBS video,
I also saw an episode of QI in which Alan Davies argues with Sandi Toksvig about
the shape that of the cells that bees make.
Hexagonal, obviously. But no. The cells are circular, they
just make hexagons because that’s the shape that arises when you pack circles (and
are minimising the use of resources such as wax). Those facts together got me thinking about things
a differently.
I provided some visualisations about the lifecycle of a universe back in Apparent Problems with Imagining a Universe, with the last image looking like this:
The idea is that time in the universe to the left is upwards, and time in the universe to the right is rightwards (thus orthogonal). Implied is that the spatial dimensions are also orthogonal with no intention to imply that any spatial dimension in the second universe would be aligned with the temporal dimension in the first.
The equations that support this notion relate to the
approach of the event horizon of black hole:
and
These become zero at the event horizon (where r = 2GM/c2)
and then, inside the event horizon, become the square root of a negative number. Now, when we are first introduced to the idea
of the square root, we are channelled into the thinking that, for example, the
square root of 4 = 2. However, it is
more accurate to think that the two solutions to the square root of 4 are +2
and -2 – because (+2)2 = 4 and (-2)2 = 4. That is, the absolute value of the square root
of 4 is 2, but it could be positive or negative.
I was thinking that what goes on, on the other side of the
event horizon, was a new universe shooting off in the i direction (that
is, orthogonally, so really off in the i, j, k, τ direction). However, it makes more sense mathematically
if there are two directions – the i, j, k, τ direction and the negative i,
j, k, τ direction*.
That would imply that our universe is one of a matched pair – linked in a way, without any way to communicate or interact – which are being “fed” at a rate of one unit of mass/energy per unit of mass time, equally split between the two of us, resulting in both universes being eternally flat.
--
Why were the bees relevant? Because I realised that I had been trying to solve the wrong question - How do bees make hexagons? They don't, they make circles.
It puts me in mind of one of my favourite jokes. How do you get down from an elephant? You don't, you get down from a duck.
--
* This is deliberately vague. I'm still thinking about
it. One issue that remains is that neither +τ nor -τ would
have precedence, which opens up the possibility that neither would have
precedence over any other potentially orthogonal temporal dimension. A
possible solution to that is that at the event horizon all time and space is
effectively compressed to zero - meaning that everything on the outside would
have been fed through a singularity simultaneously and the orientation of i,
j, k, τ and their opposites selected - and set -
instantaneously. As far as the universe(s) on the inside of the event
horizon is (are) concerned, the orientation of i, j, k, τ (and
their opposites) are and have always been what they are.
--
Update: 3 April 2022
I
just listened to the Skeptic's Guide to
the Universe episode from 2 April 2022 (audio) - which was live and recorded
some days earlier, somewhere in northeastern US. Bob Novella speaks of a
new theory in published in the Annals of Physics (reported
at Live Science), in which there is a mirror (or
anti-)universe running "backwards in time". The physicists involved have come to this conclusion a different way, because they were looking for CPT (charge, parity, time) symmetry as an explanation for dark matter.
There are some interesting (and parallel) ramifications of
the theory, including "a CPT-respecting universe naturally expands and
fills itself with particles, without the need for a long-theorized period of
rapid expansion known as inflation". Compare this with my model
(Universe (in) a Black Hole - published
7 June 2019) in which our universe is "fed" with one unit of Planck
mass/energy per two Planck units of time (see Is the Universe Getting More Massive?
(Flatness, not Fatness) – published 30 July 2019) and the expansion
follows the Hubble radius = Schwarzschild radius metric, where the Hubble
radius is the speed of light multiplied by the age of the universe - which
implies that there is no inflation (see My Universal (and Expanding) Struggles
– published 9 July 2019).
Fascinating stuff. Note that the model that these physicists are suggesting (at least according to Live Science) is one in which there is a pair of universes in which one consists (largely) of matter and the other (largely) of antimatter, which would explain the dearth of antimatter in our universe. I don't think that my model necessarily implies that - I think that both universes would have mass/energy of some sort, as necessary to obtain one Planck unit of mass/energy per unit of Planck time. It should be noted that antimatter has positive mass and that equal parts of matter and antimatter do not cancel each other out, when they collide each is annihilated but they release energy. You could say that their matter (and antimatter) nature is destroyed, but not their energy.
