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.

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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.

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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.

Is QED Wrong, or Just Wikipedia?

There seems to be an error either with quantum electrodynamics (QED) and stochastic electrodynamics (SED) or Wikipedia.

 This is the section in question (archived):

 

Note the critical density of the universe is in the order of 10^-26 kg/m³ which means that the critical energy density of the universe (given that E=mc²) is in the order of 10^-9 J/m³.

Note also for a universe with a radius of one Planck length, at an age of once Planck time and a corresponding Hubble parameter value of one inverse Planck time, the critical density would be in the order of 10⁹⁶ kg/m³ which corresponds with a critical energy density in the order of 10¹¹³ J/m³.

A vacuum energy of 10¹¹³ J/m³ beyond the spacetime origin of our universe is ridiculous.  As an example, the Earth has an average density of 5515 kg/m³, which is equivalent to an energy density of 5x10²⁰ J/m³ – meaning that if the vacuum had a density of 10¹¹³ J/m³, it would swamp us.  We’d not even be a rounding error.

Whether QED/SED is wrong, or Wikipedia contains a misinterpretation of the writings of Peter Milonni and/or de la Pena and Cetto, I don’t actually know.  But anyone suggesting such a huge magnitude of vacuum energy density should really go back and check their figures.

I am certainly not going to stay awake at night worrying about the cosmological constant problem (or whether I need to worry about it being a slam dunk for Fine Tuners).

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.