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.