Avoiding a Contravention of the Extended Consistency Principles

In the previous post, I concluded that there is an apparent contravention of what I called the extended consistency principles and asked whether that contravention might be resolved?  I think it might.  To explain, I need to beg the indulgence of the reader, to just go with me in the scenario below (at least until I mention Sean Carroll).  There are some issues that might come up, a few of which I am already aware of and have resolved, but I can address them at a later stage.

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Let us say that there are two conjoined volumes, joined at a null point.  Initially, they are infinitesimally small (ie tending towards zero volume, perhaps even at zero volume) and empty.  Imagine that we can look very, very closely at these volumes at an angle such that they appear to be two areas, purely for illustrative purposes, as below.  (Alternatively, you could imagine a variant of the image illustrating the history of the universe show in the previous post, extending downwards as well as upwards. Just don’t include the exponential curvature.) 

 

Let us say then, that we allow a certain amount of mass-energy into these volumes over the smallest meaningful period of time.  Say that amount is one unit of Planck mass-energy over one unit of Planck time and let us share the mass-energy evenly between the two volumes. 

 

Let us also say that, with every unit of Planck time, the radius of each volume increases by one unit of Planck length.

The equations for Planck mass, Planck time and Planck length are as follows:

This means that after one Planck time, there are (notionally) two tiny volumes of V=4/3.π.l_p³ inside each of which is half a Planck mass (½m_p).  That means there’s a (notional) density of:

After a certain number of Planck times, let us say ꬱ=8.0619x10⁶⁰, the density will be:

Now, these values are:

Planck time – t_P = 5.39x10^-44 s

Gravitational constant – G =6.674x10^-11 m³/kg/s²

Pi – π = 3.14159 (-ish)

Plugging them in, we get 9.47x10^-27 kg/m³.  This is precisely the critical density calculated for of our universe – at this time.  It should be noted that there are other values given, such as by NASA (in 2013), who gave the critical density as 9.9 x 10^-30 g/cm³ but note that they also say that “WMAP determined that the universe is flat, from which it follows that the mean energy density in the universe is equal to the critical density”.  Most of the results that come up, when you search for <<most recent value of the density of the universe>>, are for the critical density, but given that the consensus is that the universe is flat, then the density and the critical density are the same anyway.

Note also that, expressed in energy density, this is 0.85133x10^-9 J/m³ (which is approximately the estimated value for vacuum energy, at 10^-9 J/m³).

The only question of course is: what is 8.0619x10⁶⁰ Planck times in more recognisable units?  It is, of course, 13.77 billion years – the age of our universe as calculated by WMAP.

I just want to summarise here for a moment and make some comments.

In the previous section, I indicated that it would appear that (given the inflationary model of cosmology) we are in a privileged era, because the Hubble parameter is roughly equivalent to the age of the universe.  There are some related facts, so these can’t be thought of as additive coincidences: the universe is (currently) flat, which means that the density is equal to the critical density and the density of the universe is equivalent to the density of a Schwarzschild black hole with a radius of the age of the universe multiplied by the speed of light (which is a consequence of the universe being flat).

However, if the expansion of the universe is currently accelerating (and went through exponential expansion early), then there would not have been any era in the past nor will there be any era in the future when the density would be critical, unless something very strange was happening with the quantity of mass-energy in the universe.  Note that, in the link to Sean Carroll’s article on why we are not inside a black hole, he notes (without support or clarification, unfortunately) “that a spatially flat universe remains spatially flat forever”.  This aligns with the extended consistency principles, but the inflationary cosmological model doesn’t – again unless something odd is happening with mass-energy.

Fortunately, something odd is happening with mass-energy, basically that energy changes when spacetime changes – which is to say that density of the mass-energy of the universe is maintained at the critical density or, as the universe increases in size, the amount of energy increases (at what rate is not explained by Carroll).

Note that this does not mean that the laws of conservation of energy and momentum are broken in such a way that you can get free energy.  What it does mean is that the notion of allowing 1 unit of Planck mass-energy into the universe per unit of Planck time is not a contradiction of existing laws of physics.  Existing laws of physics already imply that mass-energy is being added at rate necessary to ensure that the universe remains flat.

That said, the way that mass-energy would need to enter the universe in order to keep an inflationary/accelerated expansion universe flat (noting that “a spatially flat universe remains spatially flat forever”) would not be consistent.  What we would see is an initial period of exponentially accelerating increase in the mass-energy (well above one unit of Planck mass-energy per unit of Planck time), the rate would then slow down for a while (to less than one unit of Planck mass-energy per unit of Planck time) and then, later, resume a more leisure acceleration (reaching a rate today that is, coincidentally, one unit of Planck mass-energy per unit of Planck time) and, according to predictions, continue to accelerate (to eventually significantly exceed one unit of Planck mass-energy per unit of Planck time).

Perhaps, if the rate of mass-energy entering the universe is entirely governed by rate of expansion (whatever that may be at the time), then I guess that flatness could be maintained.  However, it would not be such that the amount of energy in the universe were proportional to the age of the universe, ever, except for right now.  And the rate at which mass-energy enters the universe would never equal one unit of Planck mass-energy per unit of Planck time, except for right now.  Which puts us back in the situation of being in a privileged era.

There is another issue that I touched on only very lightly.  I pointed out that the critical density, right now, expressed in terms of energy density, is 0.85133x10^-9 J/m³.  In what has been referred to as “the worst theoretical prediction in the history of physics”, quantum field theory calculates a value for the vacuum energy that is 120 orders of magnitude higher than measured (which is in the order of 10^-9 J/m³, and thus in the order of critical density but expressed in terms of energy rather than mass).

Purely out of curiosity at the time, a while ago, I pondered what the energy density would be after the very first unit of Planck time and used my method to get the answer 5.5331x10¹¹² J/m³, which is (approximately) 6.4994x10¹²¹ times higher than currently measured (or within spitting distance of 120 orders of magnitude).  It may come as no surprise to some that this multiplier is the square of 8.0619x10⁶⁰ which the more keen-eyed will remember is the value I assigned to ꬱ above and is the number of units of Planck time in 13.77 billion years.  So again, we look like we might be in a privileged era (since some argue that the vacuum density is invariant and this coincidence would therefore only occur right now).

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We would not, however, be living in a privileged era – if we were to be living in a two volume universe with smooth consistent expansion, at one unit of Planck length per unit of Planck time, with one unit of Planck energy entering per unit of Planck time (divided equally between the volumes).  If that were the case, then we get the results that we observe without needing to contravene the extended consistency principles.  No matter where or when an observer found herself observing the universe, the universe would be at critical density (which would the density of a Schwarzschild black hole with a radius equal to the age of the universe at the time observed multiplied by the speed of light), a Hubble parameter value that would be the inverse of the age of the universe, the rate at which mass-energy entering the universe would always be one unit of Planck mass-energy per unit of Planck time and, I fully expect, the vacuum energy would be in the ball-park of the critical density (the most significant difference being the units in which they are expressed).

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The very keen eyed will notice that I use the symbol ꬱ rather than the more easily found æ.  Initially, I took this out of the equation generated in Word, because the rendering looks pretty awful but I have since resurrected it for consistency (which I had lost having inadvertently reverted to using the easy option in later articles).

I prefer ꬱ because it hints at age with the a, along with a reflection of it extending into the other direction, as an echo of the notion of two equal but opposite universe halves, extending in different directions from the same null point (without necessarily being perfect reflections of each other). 

What is the Planck Constant?

Recently, the Planck constant was mentioned in the news due to an agreement to change the definition of the kilogram.  Rather than returning to the reference kilogram mass, the kilogram is now to be determined based on the value of the Planck constant, which is now a defined value in the same way as the speed of light is a defined value.

The question that is a little difficult to find the answer to in all the new reports is “what precisely is the Planck constant?”  Of course you could go and ask Google, but when you’re reading an article, you sort of want all your answers in one place.

I did see one article which tried to address the question, with the following:

The Planck constant is the amount of energy released in light when electrons in atoms jump around from one energy level to another, explained physicist Tim Bedding of Sydney University.

Well, yes, sort of.  I am pretty sure that this is a journalist error rather than a physicist error.

The Planck constant (h) can be used to work out the energy of a photon, when we know its wavelength or its frequency: E=h.f=h/λ.  But that’s not all.  The Planck constant is often used as part of a sort of exchange rate, allowing you to convert from everyday units like seconds, metres and kilograms into quantum level units: Planck units like Planck time, Planck length and Planck mass.

A particularly useful thing about using natural units, like Planck units, is that fundamental constants resolve down to unity, that is, they equal 1.  For example, using Planck units:

•         Speed of light, c = 1
•         Gravitational constant, G=1
•         Coulomb constant, k_e=1
•         Boltzmann constant, k_B=1

As it is, the Planck constant itself is not a fundamental constant that resolves to unity, but the reduced Planck constant is and does – meaning that the value of the Planck constant in Planck units is 2π, so the reduced Planck constant is ħ=h/2π.  There are two other key constants, the permittivity and permeability of free space, or the electric constant and the magnetic constant, ε₀ and μ₀, which don’t quite resolve down to 1 either, but these resolve down to ε₀=1/4π and μ₀=4π, so you could have a raised permittivity and a reduced permeability of ε₀-bar=1 and μ₀-bar=1.

In an earlier article, I wrote about how it is possible to resolve even the fine-structure constant and the gravitational coupling constant (which are both dimensionless) to 1.

None of this would be possible without the Planck constant, so the added role of defining the kilogram should not be too much for it to bear.

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I thought I might add here that for many purposes in physics what matters more is angular frequency rather than just frequency per se.  That is to say its not just how many whatevers per unit of time, but how many rotations per unit of time.  A full rotation is 2π radians so the relationship between vanilla frequency (f) and angular frequency (ω) is ω=2π.f which means that there is another equation for the energy of a photon, E=(h/2π).ω=ħ.ω.  There is an argument that the unreduced Planck constant is only used for historical reasons and that the reduced Planck constant is that one that we should be using primarily.

I'll go into this a bit more in a later article.