Expanding Glome to Special Relativity

In Big Fat Coincidence and Problems that Don’t (Seem to) Exist, I laid out how the FUGE model works.  What I didn’t do, because I didn’t think about it at the time, was explain how one reaches an explanation of (Special) Relativity from an expanding glome.  So here goes …

The equation for a glome is:

Δx² + Δy² + Δx² + (cΔτ)² = r² = (cΔt)²

where x, y and z are spatial units, τ is a temporal unit and r is the radius, which is given by the change in time, t, times c, which is a constant required to mediate the units.

We can use this equation to consider a change in spatial location on the glome in the period Δt:

v² = Δx²/Δt² + Δy²/Δt² + Δz²/Δt²

so:

v².Δt² + (cΔτ)² = (cΔt)²

and then, rearranging:

c²Δτ² = c²Δt² - v².Δt²

Δτ² = Δt² - (v²/c²).Δt²

Δτ = √(1 - (v²/c²)).Δt

Which is the equation for temporal dilation where Δt is by convention expressed as t' and Δτ as t.  Note that “(a)fter compensating for varying signal delays due to the changing distance between an observer and a moving clock (i.e. Doppler effect), the observer will measure the moving clock as ticking slower than a clock that is at rest in the observer's own reference frame”.  If we are counting ticks, we are actually measuring a frequency (at a rate of one tick per second) and this is why the time dilation equation will usually appear as something like this:

t' = t / √(1 - (v²/c²))

To get length contraction, one simply multiplies through by c:

Δτ.c = √(1 - (v²/c²)).Δt.c

ΔL = √(1 - (v²/c²)).ΔL_o

Alternatively, given that the surface volume of the expanding glome is flat in the FUGE model, one could merely use the approach described in Galilean to Special in One Page.

As for mass-energy, the total energy of a mass is given by:

E_total = m.v_spacetime² = m.v² + m.c²Δτ² = mc² ≈ m_o.c² + ½m_o.v²

See On Time where I explain why m.v² + m.c²Δτ² ≈ m_o.c² + ½m_o.v².

Big Fat Coincidences and Problems that Don't (Seem to) Exist

If our universe were to be undergoing flat expansion (Flat Uniform Granular Expansion, or FUGE) then, I suggest, a lot of what I have recently identified as “Big Fat Coincidences” would not be coincidences at all but would rather be the natural consequences of the process of that FUGE.  In addition, in the past few posts, I have identified a couple of problems that, given FUGE, aren’t actually problems – namely the Flatness Problem and the Cosmological Constant Problem.  Please note carefully, I am not suggesting that I have solved the problem, I am merely saying that if FUGE is true, then these are not problems.

I am going to summarise how I envisage that FUGE works and try to hit as many big fat coincidences and problems that don’t (seem to) exist as I can on the way.  If I miss any, or it’s just too awkward to address them at the time as they come up, I’ll list them at the end.

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First and foremost, we know that the universe expands.  We can see evidence of that as we look at distant galaxies that are receding from us at a rate proportional to their distance from us.  This rate is what we all the Hubble parameter (H) or, more often, the Hubble constant (H₀) which merely is the value of the Hubble parameter today (meaning that it isn’t really a constant, since it changes with time).

Secondly, our measurements of the curvature of the universe imply that it is flat.  This flatness of the universe (which must be more extreme as we go back in time, per the Flatness Problem) tells us that the density of the universe is critical and that, per the Friedmann equations, is ρ_c = 3H²/8πG.

As can be seen above, curvature of the universe and the Hubble parameter are linked.  In the FUGE model, the universe is expanding and it remains flat throughout that expansion.

The final element of the FUGE model is its granularity.  This granularity is at the Planck scale, which is the scale “below which (or beyond which) the predictions of the Standard Model, quantum field theory and general relativity are no longer reconcilable”.  To be more precisely, in the FUGE model, the universe is granular at the Planck length and Planck time.

To expand, the radius of the universe must increase.  Note however that this is the radius of a 4D shape, specifically a glome.  The 3D universe does not have a radius in the same way that a sphere has a radius, but the volume of the 3D universe is linked to the radius of the glome (r) thus:

V_surface = 2π²r³

It is this radius that increases.  Clearly the universe, to expand, must expand at a rate.  In the FUGE model, the expansion is given by:

Δr/Δt=c

At the granular level, this is equivalent to the radius of the universe (as a glome) expanding by one Planck increment each Planck time.  I say one Planck increment because it could be said that the expansion is time, so the expansion rate is one Planck time per Planck time however given the interchangeability of space and time, this is equivalent to one Planck length per Planck time, which is the value of r that can be used to determine the surface volume of the universe – r = ct, where t is the age of the universe.

As the universe expands, it is filled with Plank atoms – where a Planck atom is the 4D equivalent of Planck volume, so l_pl³.t_pl.  The surface volume of the universe is the current layer of Planck atoms, which appear to us, at the macro level, as 3D space.

Note that as the surface volume of the glome expands, room will become available for more Planck atoms.  In other words, gaps will open up.  These gaps will open up everywhere with a random distribution at a rate proportional to ct.  The consequence of this is that, within the surface volume of the universal glome, for a sphere of r_LH = ct (that is a sphere defined by a radius equal to the light horizon, which is the distance that light could have travelled in the age of the universe, t, to reach an observer in the centre of that sphere), the rate of expansion would be c.  The recession of any object at distance D, as observed from the centre of that light horizon sphere, would be given by:

v = (c/r_LH).D = (1/t).D = H.D

This accounts for one big fat coincidence, namely that the value of the Hubble parameter today, H_o, is the inverse of the age of the universe.  In the FUGE model, the value of the Hubble parameter is always the inverse of the age of the universe.  It also accounts for the fact that the Hubble length (l_H = c/H) is the same as the light horizon.

Note that our observations are based on the light horizon in which we exist.  The volume of that light horizon is V_LH = 4πr³/3, where r=c.t.

As mentioned above, is the universe is flat, then ρ_c = 3H²/8πG.  Given that H = 1/t  and ρ_c = M/V and Δr/Δt = c and V_LH = 4πr³/3, consider the change in mass (ΔM) over a period of Δt:

ΔM/(4π(cΔt)³/3) = 3/8πGΔt²

ΔM/Δt = (c³/G)/2 = (M_pl / t_pl)/2

This is telling us that mass (and thus mass-energy) is increasing within the light horizon at a rate of one half Planck mass per Planck time.  This resolves the flatness problem since, if this is true, the universe will be maintained at precisely critical density forever – and will be flat forever.  Note that this applies to whichever light horizon we choose, from either here in our current spacetime location, or from any other event location.  This implies that the entire universe is increasing in mass at rate that is greater by a factor of 3π/2.

Note that this rate of mass increase is not time constrained, so it would apply even for very small values of t.  This means that there is no singularity problem.  For sure, the universe would have been significantly denser at the beginning, when Hubble parameter values would have been very high, but that density would not have been infinite.

In the FUGE model, the universe does have a size because, while it’s not bounded (in 3D), it’s not infinite.  The volume of the universe is given by 2π²(ct)³ = 4.39x10⁷⁹ m³.  It should be noted that the volume inside a sphere defined by a radius equal to the “comoving distance to the edge of the observable universe” is greater than this.

The radiation that is received by us today as cosmic microwave background radiation has travelled across an expanse of space that now has a comoving distance of 46 billion light years despite the fact that the radiation has travelled at precisely the speed of light for only 13.8 billion years.  Comprehending this difference is a little mind-bending but, in short, it is incorrect to think of a sphere painted with something like the image below receding from us at H_o.cd (where H_o is the current value of the Hubble parameter and cd is the comoving distance to the origin of the cosmic microwave background radiation (not to the “the edge” of the observable universe)).

Instead, think of an expanding sphere, on which there is a “flat” (unbounded but not infinite) 2D surface area and imagine that 2D information travels along the surface in a straight line (or rather a geodesic or a “straight arc”).  Imagine further that the sphere is expanding in such a way that information can’t quite circumnavigate the sphere:

The value r_p is what could be called the “pseudo-radius” in that is the apparent radius of the large circle that the observer (denoted by the star) perceives herself to be the centre of.  Note that there is overlap, in that the observer can spin around to face the other direction and “see” the same expanse again so, to express the area of the apparent surface, we must use only half of the pseudo-radius, r_p/2 – even if the observer would think she is looking at r_p.

As said above, the surface volume of a glome of radius ct is given by 2π²(ct)³.  This can be equated to a sphere with a radius of half the pseudo-radius, or:

    (4π/3).(r_p/2)³ = 2π²(ct)³

    r_p³ = 12π.(ct)³

    r_p = ³√(12π).ct

When ct=13.8Gly, this gives a value of r_p = 46.3Gly, which is (give or take a little) equal to the comoving distance to the “edge” of the observable universe.  Another big fat coincidence.

As discussed above, mass-energy enters the universe at a rate of one half Planck mass per Planck time within the light horizon (where the light horizon can be taken from any location within the surface volume of the universe.  Given that M = ((c³/G)/2).t, and V = (4π/3).(ct) ³, this gives us a mass-energy density of:

      E/V = M.c²/V = ((c⁵/G)/2).t / ((4π/3).(ct) ³) = 3c²/(8πG.t ²)

At the current age of the universe at 13.8 Gy or 4.35x10¹⁷s, that gives us a mass-energy density of 8.48x10^-10 J/m³.  Noting that baryonic and dark matter make up 32% of the mass-energy of the universe, that means 68% is dark energy.  Specifically, that is 68% of 8.48x10^-10 J/m³ which is 5.76x10^-10 J/m³.  According the WMAP Survey the “positive energy density (is) about 6 × 10^-10 joules per cubic meter”.  Another big fat coincidence.

Above I arrived at ΔM/Δt = (c³/G)/2 = (M_pl / t_pl)/2.  It can be seen that his represents the lower bound of the Heisenberg Uncertainty Principle, because:

      (ΔE/Δt) = (ΔM/Δt).c² = (c⁵/G)/2 = (ℏ/2)/t_pl²

Noting that the Planck time is the smallest division of time (equated with the lower limit of Δt), we can multiply through by t_pl² to find:

       ΔE.t_pl = ℏ/2

Given that Δt ≥ t_pl, then we have ΔE.Δt ≥ ℏ/2, which is the Energy-Time variant of the Heisenberg Uncertainty Principle equation.  Another big fat coincidence.

Finally, there are some aspects to the fine-tuned universe argument that are addressed by the FUGE model.  I only want to go into two here, as an example; the value of Ω and the value of Λ as argued by Martin Rees.

Omega (Ω) is the density parameter and, he argued, its value is very close to 1.  If the value of Ω deviated from 1 by any significant margin then gravity would be either too strong or too weak – too strong and the universe would collapse, too weak and stars would not have been able to form.  In the FUGE model, the value of Ω in not “close to 1”, it is precisely 1 as argued above in regard to flatness.

Lambda (Λ) is the cosmological constant and, Rees argued, its value is very, very small.  Making some assumptions, including that the dark energy density is constant, he arrived at 10^-122 as its approximate value which as a very small, but not zero value is curious.  In the FUGE model, however, one of the assumptions made by Rees does not hold.  Dark energy density is not a constant.  Overall mass-density is inversely proportional to time, so density of dark energy is must be decreasing albeit at a slower rate than the density of baryonic and dark matter.

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I’ve not progressed much further than this.  I don’t have anything to say, at this time, about how mass-energy coalesces into baryonic or dark matter.  I have some inkling about how the probability of a Planck atom appearing at any location is inversely proportional to the concentration of mass-energy in that location and the increased likeliness of space to appear where there is already space leads to localised curvature which manifests as gravity.  For the moment, these can remain as projects for the future.

Expansion on Uncertainty

In Is the Universe Getting More Massive?  (Flatness, not Fatness), I concluded that (if the universe is and has always been flat then) the mass of the universe is increasing at a rate of M / r = c²/2G.  I should have put deltas in there, ie:

ΔM / Δr = c²/2G

I went on to write:

This implies, to me, that if the universe is and has always been flat, then the mass of universe is increasing by one unit of Planck mass every two units of Planck time.  (Note that I reached the same conclusion in Is the Universe (in) a Black Hole? but I expressed it in terms of energy.)

In the linked article I wrote (where ꬱ is the age of the universe):

Of interest is the fact that, with the assumption that the universe is a black hole and that it is expanding at the speed of light, we can recall the equation for the Schwarzschild radius and get this result a little more easily:

r_s = ꬱ.c = 2GM/c²  =>  M = ꬱ.c³/2G

This last equation is for mass of the universe now but the implication is that for a given period of time Δt, ΔM = Δt.c³/2G, or

ΔM / Δt = c³/2G

From which we can conclude that Δr / Δt = c, but all this is saying is that the universe is expanding at c, which we already know.

We can go further though, using this relationship, noting that l_pl / t_pl = c and thinking of incremental changes (increments of Planck length and Planck time):

ΔM / Δr = c²/2G

=> ΔM.c . Δr = c³/2G . Δr² (multiplying through by c. Δr²)

=> ΔM.c . Δr = c³/2G . ℏG/c³ (noting that Δr² = l_pl²)

=> Δp . Δx = ℏ/2

This is the lower limit of Heisenberg’s Uncertainty principle (Δp . Δx ≥ ℏ/2).  Alternatively:

ΔM / Δr = c²/2G

=> ΔM.c² . Δr/c = c³/2G . Δr²

=> ΔE . Δt = ℏ/2

Which is the lower limit of an alternate expression of Heisenberg’s Uncertainty principle (ΔE . Δt ≥ ℏ/2). 

How then should this be interpreted?  The way I understand it is that if we consider the tiniest meaningful increment of time, by which I mean one Planck time, then we are being told by the Heisenberg Uncertainty Principle that the minimum change in energy must be half a Planck energy.  Now this might be the wrong way around, since the expansion and the flatness of the universe point to the lower limit of the Heisenberg Uncertainty Principle, so it could be that this principle is merely pointing to an emergent feature of “flat expansion”.  Or it could just be another big fat coincidence.

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Interestingly, if you find your way to the vacuum energy page at Wikipedia, you will find that there is an “unsolved problem in physics” note:

Why does the zero-point energy of the vacuum not cause a large cosmological constant? What cancels it out?

The cosmological constant is the energy density of space and in Is the Universe Getting More Massive?  (Flatness, not Fatness) I concluded that that the density of mass-energy of the universe which is not baryonic or dark matter is about 6x10^-10 J/m³, which is precisely what is measured.  In my model, this “unsolved problem in physics” is not a problem.

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I should point out that when I was looking for more information on “Planck atoms”, a term that I think was used in The Story of Loop Quantum Gravity - From the Big Bounce to Black Holes (as mentioned in Another Teeny Tiny Struggle), I chanced upon some documents by José Garrigues-Baixauli.  It was when I was perusing those that it struck me that my one Planck energy per two Planck time result was reminiscent of the Heisenberg Uncertainly Principle equation.  I am not in a position to agree with everything that José has written there, but I do notice some parallels in that he has arrived at a couple of similar ideas from a different direction.

This image is particularly evocative considering the contents of Spherical Layers, the image that followed and the rather opaque follow up in The Messiness of Layered Spheres (I promise that it made sense to me even before the clarifying edit that I have just performed, but I was inside my head at the time of writing so I had an advantage).

Vacuum Energy, Dark Energy and the Units of the Planck Parameter

So, I’ve been asking some questions and getting answers which indicate that the questions are somehow annoying (but which don’t actually address the questions asked).  In the process, the topic of vacuum energy came up, which is something that I had not even considered.

I sat at my desk for a while pondering how I would work out the amount of energy entering the universe at a given time and then get the average amount per cubic metre.  Then I intended to compare that value to the value given for vacuum energy, which I naïvely thought I’d just look up (it’s never that simple).

But as I sat there pondering, I thought: I already have a value that I could work with.  I concluded in Is the Universe Getting More Massive?  (Flatness, not Fatness) that mass-energy enters the universe at a rate of one Planck mass per Planck time.  I worked out that the density of the universe, if flat, after 13.8 billion years of this process would be the critical density at that time, which is approximately 10^-26 kg/m³.  Now we know that E=mc² (it’s really E_o=m_oc², since we need to consider rest mass but I’m sure we can get past that).  Given that I already say mass-energy, I don’t have any problem expressing a mass in terms of its energy equivalent and in this case that is approximately 9x10^-10 J/m³.  According to current estimates, 32% of the universe is either baryonic matter or dark matter, so … if the rest is just dark energy burbling away in “empty space”, then that would 68% of 9x10^-10 J/m³, or about 6x10^-10 J/m³.

At the Wikipedia article on vacuum energy, the first value given for the vacuum energy of free space is 10^-9 J/m³.  This is the value estimated “using the upper limit of the cosmological constant”, and Sean Carroll is cited as the source (via C-SPAN’s Cosmology at Yearly Kos Science Panelbroadcast, Part 1).  The same value is quoted by John Baez, and he goes on to write:

One can know something is very close to zero without knowing whether it is positive, negative or zero. For a long time that's how it was with the cosmological constant. But, recent measurements by the Wilkinson Microwave Anisotropy Probe and many other experiments seem to be converging on a positive cosmological constant, equal to roughly 7 × 10^-27 kilograms per cubic meter. This corresponds to a positive energy density of about 6 × 10^-10 joules per cubic meter.

Interesting, huh?  Another big fat coincidence.

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In a parallel discussion in which I was accused of saying that there’s a speed (distance/time) associated with the expansion of the universe despite having carefully written, in reference to a hypothetical universe:

The universe expands such that the radius increases by 1 Planck increment every 1 Planck time (possibly with smaller increments depending on at what point the granularity kicks in).

There is a lack of clarity with respect to that statement but I am not saying that the universe expands at any specific rate, I am just saying 1) the universe expands and 2) due to that expansion the radius increases at a rate that looks like it could be a speed.  In reality, I think the universe expands at a rate of 1 Planck time per Planck time, and that’s not a rate at all, it’s dimensionless.  Note that I am not currently thinking of the universe as a simple sphere, but even if the universe were a glome, the surface volume of that universe would still expand in direct proportion to the radius of the glome.  Anyway …

I pointed out to my interlocutor that the Hubble parameter (today) is cited as ~70 because it’s expressed in km/s/Mpc, I assume because these are convenient figures in cosmology.  However, if you express this figure in Hubble lengths (where HL = c/H = 13.8 billion light years) and meters, rather than megaparsecs and kilometres, you get a value of 300,000,000 m/s/HL.  And, to more significant figures than is strictly necessary, this is the speed of light.  So, the expansion of the universe is associated with a very important speed, a speed when expressed in Planck units is 1.  But the expansion itself is not a speed, by its dimensions it’s more of a frequency – once every Planck time.

And the question that arises when thinking of the expansion of the universe as being a frequency is … a frequency of what?  It implies, strongly to me at least, that something is happening to the universe every unit of Planck time.  And for me, the answer is obvious, it’s expanding by an increment (be that a unit of Planck time, or a unit of Planck time multiplied by c, or a Planck length, or however you prefer to think of it).

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Finally, when looking up the amount of dark energy in the universe, I found the NASA webpage on the issue.  On that page is the following text (for the purposes of transparency I should advise that it is followed immediately by a section of text that I am still a little dubious about although I plan to give it some more thought and I should highlight that, even though from NASA, they are only talking speculatively):

One explanation for dark energy is that it is a property of space. Albert Einstein was the first person to realize that empty space is not nothing. Space has amazing properties, many of which are just beginning to be understood. The first property that Einstein discovered is that it is possible for more space to come into existence. Then one version of Einstein's gravity theory, the version that contains a cosmological constant, makes a second prediction: "empty space" can possess its own energy. Because this energy is a property of space itself, it would not be diluted as space expands. As more space comes into existence, more of this energy-of-space would appear.

I recall reading that, in terms of the FLRW metric, dark energy increases but I can’t find it again.  However, the Wikipedia article on dark energy quite clearly indicates that dark energy increases:

when the volume of the universe doubles, the density of dark matter is halved, but the density of dark energy is nearly unchanged (it is exactly constant in the case of a cosmological constant)

This is entirely consistent with my model – at least now that I have got a better handle on how dark energy might fit in (ie all the energy that is entering the universe today is in the form of dark energy).

Oh, and by the way, I do understand that I am implying that dark energy and vacuum energy might be the same thing.  It’s clearly not outside the realm of possibility though, since actual scientists in the field have made similar claims.