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.

---

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.

--

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.

---

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.

Mathematics for Imagining a Universe

Imagine a Universe contains mere narrative with only oblique reference to equations.  That does not mean that the equations don’t exist.

---

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):

 

t_o/t_f = √(1-2GM/rc²)

 

Where t_o is proper time between two events for an observer at a distance of r from the centre of a mass M, and t_f 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 r_S = 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 < r_S, we have a ratio of t_o/t_f that is the square root of a negative number and as r approaches 0, we have a ratio of t_o/t_f 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:

 

ρ_c = 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 r_H is the Hubble length (c/H, where c is the speed of light), is

 

V_H = 4πr_H³/3 = 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”, M_H, which is the mass inside a Hubble sphere given that the density in that sphere is critical, so

 

ρ_c = M_H/V_H = 3H³M_H/4πc³ = 3H²/8πG

 

So

 

M_H = c³/2HG

 

Recalling that r_H = c/H, and rearranging,

 

r_H = 2GM_H/c²

 

Which is the equation for the Schwarzschild radius of a Schwarzschild black hole of mass M_H.  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

 

Δr_H = 2GΔM_H/c²

ΔM_H/Δr_H = 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 (L_P) of additional radius per Planck time (t_P), noting that L_P/t_P = c, therefore Δr_H/Δt = c, and thus

 

ΔM_H/Δt = c³/2G = ½c³/G

ΔM_H/Δt = ½m_P/t_P

 

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 4πr_H³/3, where r_H 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.

---

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.