Imagine a
universe in which the inhabitants, those sufficiently intelligent and motivated,
and technologically advanced, can look out into space and observe that distant
galaxies appear to be moving away at a speed directly proportional to their
distance. This expansion of the universe,
as the inhabitants can determine, is sufficient to overcome the effect of gravity
that, they calculate, would otherwise eventually draw all the galaxies together.
Then
imagine that, after a certain amount of time has passed, at least a dozen or so
billion years, that expansion stops. And
gravity takes over.
If
perfectly balanced, the galaxies would remain where they are, because they would
experience attraction from all directions, which cancel out, but the balance is
not perfect. So, slowly but inevitably,
the galaxies coalesce, collapsing into black holes the like of which the
universe, increasingly lifeless, has never seen.
Eventually,
approaching effective eternity, all mass-energy in the universe is contained
within a single black hole which has captured everything that universe contained.
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Imagine a
notional particle as it approaches the outer reaches of this final black hole. Due to relativity, time dilation effects mean
that the time experienced by that particle (relative to a notional observer
outside of the influence of the black hole) approaches the square root of zero,
before flipping into square root of a very small negative number, or a time
that is orthogonal to the time in the universe.
Between the outer reaches of the black hole and its centre is an eternity
of orthogonal time. Length too extends inside
the black hole, orthogonally to infinity.
Imagine
that, inside this black hole then … is another universe.
From the
standpoint of the inner universe, the outer universe is entirely in the past, the
outer universe’s entire eternity occurring during the inner universe’s very
first instant. All the mass-energy of the
outer universe was placed into “a channel” and began appearing in this universe
– from that very first instant.
However,
there are physical limits on the rate at which mass-energy can enter a
universe, given by the critical density (hence the notion of “a channel”, of
restricted dimensions). This means that there
is a tension which leads the inner universe to expand as mass-energy makes its
way in. In the first instant, the first
quantum of time, the absolute minimum space fills with the absolute maximum
mass-energy for that volume. From then
on, half that amount of energy appears each quantum of time, as space expands such
that the radius increases one quantum of length per quantum of time – thus maintaining
the critical density.
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In the same
way as, for the inner universe, the outer universe’s eternity has already occurred,
from the notional perspective of mass-energy entering the inner universe from
the outer universe, the entire expanse is as it were a single quantum of
volume. Mass-energy (in the form of
energy) is spread between all “grains” of inner universe space.
Initially,
this means that the inner universe is incredibly dense and hot, but that density
decreases as the universe expands despite the constant feed of mass-energy
because the volume increases with cube of the radius (where the radius is
proportional to the linear rate of increase in mass-energy).
Eventually
the density and heat decrease to the point at which photons manifest. Then particles. And eventually stars. All the time mass-energy continues to feed
into the universe, driving its expansion.
By the time
that intelligent beings are considering the expansion of the inner universe,
say a dozen billion years or so after the beginning, the rate at which energy
is entering that universe (half the mass of the smallest possible theoretical black
hole per quantum of time) verges on infinitesimal. However, the accumulation of this energy over
a dozen billion or so years does manifest when the energy of empty space (vacuum
energy) is considered.
Because space
is largely empty, with galaxies and even stars and planets being massive exceptions,
the energy of space at any one time is roughly equivalent to the total amount
of energy that has entered the inner universe divided by the total volume.
And so, the
inner universe continues to expand, at a rate of one quantum of space per quantum
of time, which is also the maximum speed at which anything can move in such a
universe. Intelligent observers with
sufficient technological advancement notice this upper limit on speed (via the
fact that photons, in vacuo, are limited to it) and come to understand relativity,
gravity and the nature of black holes.
They develop
sophisticated equations that explain how their universe expands (although
perhaps not why), what the critical density of a universe is and how gravity affects
time and space, including what happens to time and space as objects approach a
black hole. They develop notions of the granularity
of space and time, which they come to understand as intrinsically interconnected. Some notice that the equations for a black
hole of a certain radius and the critical density of a universe with the same
radius are linked.
But one
day, the mass-energy that was effectively queued up by the final black hole in
the outer universe runs out. The total mass-energy
in the inner universe is now equal to the total mass-energy that was in the
outer universe.
And the inner
universe stops expanding.
