Mathematics to Address an Apparent Problem with Imagining a Universe

Imagine a Universe contains only narrative with no equations.  Before I posted that narrative, I posted a piece that explained that I understood that there are at least three apparent problems with the narrative, which I archived for posterity before overlaying it.  Just as with the narrative itself, I tried to minimise the use of equations – which was a little tricky with regard to the glome.  What follows is a very brief explanation as to how mass/energy enters to the inner universe at a rate of one unit of mass/energy per unit of time.

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Below is what I find most problematic to explain without recourse to equations (and even with them, a little):

·        To be nice and neat, it would be great if the inner universe receives one “unit” of energy for each “unit” of time during which the radius increases by one “unit” of length.  That does not initially seem to be the case though, it’s one Planck mass worth of energy for each two units of Planck time during which the universe expands by two Planck lengths, per Hubble volume (which is the sphere defined by the radius of the universe at that time, recalling that universe is a glome).  This gets a little confusing in four dimensions and I am not entirely convinced by people who say they can imagine what a four-dimensional object looks like, so let’s consider a sphere as an analogy.  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.

I described how, if the universe is spatially flat, the mass (or “mass-energy”) of the universe increases by one unit per two units of time in Mathematics for Imagining a Universe, under the rubric “Critical Density and Expansion”.  In that section I wrote (emphasis added):

Which means that, within a Hubble volume, mass increases at a rate of half a Planck mass per Planck time to maintain critical density.

The challenge is to understand how the universe might have a volume of two Hubble volumes, thus making the mass increase at a rate of one Planck mass per Planck time (to maintain critical density).

What I am describing above with the sphere is the perspective of a 2D character living on the surface of that sphere, let’s call him Fred.  Say that Fred occupies the x and y dimensions, while the sphere occupies the x, y and z dimensions.  The sphere can be described as x² + y² + z² = r², where r is the radius, but Fred cannot perceive the z dimension so as far as he is concerned the relevant equation is x² + y² = r², which is a circle, with himself at the centre – or at coordinates (0,0).  Due to his position and dimensional limitations, however, Fred can only see one half the sphere on which he lives.

Say that Fred is actually at (0,0,r) and that a fellow sphere dweller, Freda, is at (0,0,-r), also perceiving herself to be at (0,0).  Freda too will only perceive a circle, but that circle has no overlap with Fred’s despite also being described, by Freda, as x² + y² = r².  In three dimensions, the two circles are clearly different being (x² + y² = r², z=r) and (x² + y² = r², z=-r) and are descriptions of two separate halves of the sphere, the z-positive hemisphere and the z-negative hemisphere.

Note that any other (non-collocated) 2D observer in that universe will also perceive themselves to be in the centre of a circle that describes half of the sphere, but that hemisphere with overlap with both Fred’s and Freda’s.

Precisely the same logic applies with 3D observers, like ourselves, living in the surface volume of a glome described by w² + x² + y² + z² = r².  We cannot perceive the additional dimension (w), so we see ourselves as being in an apparent sphere (a Hubble volume), but we cannot access the other half (the -w hemiglome, if you like).  The division between positive and negative halves, however, seems irrefutable making the 3D perceivable volume of the universe twice that of the Hubble volume.

Given that the rate of increase in mass is half a Planck mass per Planck time per Hubble volume, then the rate of increase of mass into the universe as a whole is one Planck mass per Planck volume – if the universe is spatially flat.

And is the universe spatially flat?  It very much looks like it.

Apparent Problems with Imagining a Universe

The text below was initially posted where Imagine a Universe is now to be found (see Internet Archive version).  It has been edited slightly to bring it up to date and some retrospective rewording for clarity.

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This text (the content of the post "Imagine a Universe") was replaced shortly after it was posted, as I wanted to record it for posterity at archive.org.  The replacement content is the intended content, which is a narrative about a universe that undergoes expansion, until that expansion stops (for reasons that are unexplained at the time), then gravity takes over and the entirety of the universe eventually ends up in one ginormous black hole, which (effectively) shunts all the universe’s mass/energy into an orthogonal universe.  The incoming mass/energy (effectively) powers that inner universe’s expansion, until the mass/energy of the outer universe is entirely transferred into the inner universe, at which time the inner universe’s expansion stops.

The purpose of this “underlay” to that narrative, if you like, is to note that I am aware of at least five apparent problems with the story:

First, there is an implication of a meta-time.  As described elsewhere I have posited that the expansion of the universe is basically time as experienced in that universe.  However, if the expansion reverses, this is indicative of a sequence associated beyond expansion which in turn implies another sort of time, or a meta-time.  It is possible that this implication is a function more of how our brains work, immersed as they in actual time.  An alternative, at least as I see it, is block time which is linked to a form of hard determinism (or nomological determinism).  Maybe there’s a compromise position in between (that is a universe with no meta-time but without everything being effectively predetermined).

Second, which has two parts (neither of which is really a problem, more of an explanation):

•    The inner universe is a glome which has a “surface volume” which is greater than the volume of a sphere of the same radius.  The idea of critical density that is mentioned is related to the radius of a sphere, specifically the sphere that defined by the distance that light could travel in the time that the universe has been in existence (and, relatedly, since it started expanding).  On the “surface volume” of glome, however, the radius is a little vexed, much the same as it is when considering drawing a (relatively large) circle on the surface of a sphere.  Is the radius that of the flat circle created by sectioning the sphere, or the arc length between the point on the sphere directly above the centre of that section and its outer rim?  That doesn’t really create a circle anyway, and its area is both greater than the circle created by the section and less than a circle as defined by a radius equal to the arc length.  Would two dimensional beings on the surface of such a sphere notice?  I don’t think so, since they would only be sensitive to two dimensions, which would notionally be aligned with a plane passing through the locus of the sphere. 

•    To be nice and neat, it would be great if the inner universe receives one “unit” of energy for each “unit” of time during which the radius increases by one “unit” of length.  That does not initially seem to be the case though, it’s one half Planck mass worth of energy for each unit of Planck time during which the universe expands by two Planck lengths, per Hubble volume (which is the sphere defined by the radius of the universe at that time, recalling that the universe is a glome).  This gets a little confusing in four dimensions and I am not entirely convinced by people who say they can imagine what a four-dimensional object looks like, so let’s consider a sphere as an analogy.  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 there is a nice whole single “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.

Third is the orthogonality.  What I am really talking about here is a orthogonal rotation of spacetime, such that while events in the outer universe could be described by the quaternion P = xi + yj + zk+ t, events in the inner universe would be described by a related quaternion P’ = x’l + y’m + z’n + t’(o).  Where the dimensions represented by i through to (possibly) o are all orthogonal to each of the others.  I say “possibly” about the dimension represented by o, which is linked to time, because of issue described below.

Fourth, and this is probably most obvious, at least to some: in my conception of this whole thing, time is equivalent to expansion.  Once expansion stops then, we have a problem.  Also, I do have a notion of gravity being linked to the expansion, since gravity is what happens where the expansion is resisted by a concentration of mass/energy, rather than being an attractive force per se.  Which means that once expansion stops … either something really bad happens, or nothing else happens.  Unless, that is, the process begins again but in a sort of reverse.  Just how that would play out is not something on which I intend to speculate other than to suggest that, if this is the mechanism, then conceptually the “reservoir” in which mass-energy resided during the inner universe’s expansion, or a similar if not the same reservoir, would begin filling with mass-energy that leaves the inner universe and, in the process, powers a contraction as the critical density is maintained.

The “really bad” is similar to something that I’ve not really considered to be a realistic option, until now – vacuum decay.  The effect, as I have envisioned it, would be to negate the structure of the universe, including not only time (via expansion stopping), but also space, possibly twisting the whole universe in an orthogonal direction and thus seeding a new, inner-inner universe that goes through the same process (with the expansion again being representative of time and thus not requiring the dimension o mentioned above).  I don’t, however, see this event as something that would initially happen in one part of the universe before spreading out at the speed of light, like some sort of cosmic cancer.  It would happen instantaneously across the entire universe as the mass/energy feed ends.  (For fans of the Marvel Universe, imagine a much more substantial snap of the fingers of an uncaring god.)

Fifth, it’s not obvious in the narrative because I’ve deliberately avoided – as much as possible – reference to equations, but the expansion of the universe is related to the critical density of a universe and this is mathematically linked to the mass and radius of a Schwarzschild black hole.  A black hole that contains the final mass of the inner universe will have the final radius of the inner universe.  We would not see, therefore, the entire universe being scrunched down into a tiny black hole.  Instead, as the black hole absorbed all the mass-energy of the universe, it would expand to fill the universe.  At that time though, relative to the outer universe, the entirety of that universe would be smeared over the surface of the universal black hole.

An image might make this slightly more comprehensible.  Say this is a standard universe:

There are some models which expect the universe to stop expanding, then reverse and trigger another universe in the opposite (notionally temporal) direction, setting up an oscillation (or bounce):

There there is a continuously cyclic universe (conformal cyclic cosmology), which is a bit like this:

The universe which is implied in “Imagine a Universe”, would look a bit like this, similar to the oscillating (or bounce) universe, but with the next universe rotating off into an orthogonal direction:

However, what I (currently) have in mind is more like this:

The universe expands out to its maximum, then a “really bad” thing happens (maybe like vacuum decay, maybe the entirety of the universe being absorbed into one final black hole) and the universe becomes, to the next universe in the sequence, a big bang-like event.  The universe that follows in the sequence but is not shown is also orthogonal, so if it helps you can imagine it emerging from the screen, but keep in mind that this is merely a representation of the notion involved.

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In any case, the description of both universes in “Imagine a Universe” is wrong – but it was intentionally, knowingly wrong, while potentially pointing to an underlying truth.  A sort of lie-to-children.