Extended Consistency Principles

There are (at least) three related principles that, in the following posts, I will refer to as “the consistency principles”:

• the Copernican principle,
• the cosmological principle, and
• the principle of relativity.

 

The Copernican principle is the notion that we “are not privileged observers of the universe” and that, therefore, “observations from the Earth are representative of observations from the average position in the universe”.  In other words, we are not the centre of the solar system, nor the centre of the galaxy, nor the centre of the universe.  And we’re not special (although, for some of us, our mothers still love us).

 

The cosmological principle is more technical than the Copernican principle, but basically says the same thing, that “the spatial distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale”.  In other words, what we, as humans, can view from our non-privileged location in the universe … is not special either.  It’s pretty much the same wherever we look.

 

The principle of relativity is the notion that “the equations describing the laws of physics have the same form in all admissible frames of reference”.  Wikipedia says it’s a “requirement” rather than a notion, but this is a requirement in the sense that for science to advance, for any of our testing and collated observations to make sense, the universe cannot be capricious – with different laws (of mathematics and physics and hence all subsequent scientific laws) in different times or locations (or frames).  It’s called a principle of relativity because it’s a necessary postulate for developing relativity, but it really applies to science as a whole.  Unfortunately, there no succinct principle that states that, but we could call it the “the fundamental laws of nature apply throughout the universe” principle.

 

In a sense, the cosmological principle is a consequence of the “the fundamental laws of nature apply throughout the universe” principle since, if we had no expectation that the laws of nature off in the direction of Polaris would be the same as in our solar system or in the direction of, for example, Sigma Octantis or Rasalhague, then we would have no expectation that space (at a sufficiently large scale) would be the same in those directions.

 

There is an extension of the cosmological principle, called “the perfect cosmological principle” – which (as applied) is anything but, since its application infers a steady-state universe.  However, the notion that a principle applies both spatially and temporally could be applied to the Copernican principle without such problem (and it could be modified, one could say “perfected”, to apply to the cosmological principle without implying a steady-state universe).

 

To clarify, the “perfect” cosmological principle is “an extension of the cosmological principle” that “states that the universe is homogeneous and isotropic in space and time”, and that “the universe looks the same everywhere (on the large scale), the same as it always has and always will” (or that “the observable universe is practically the same at any time and any place”) which therefore “underpins Steady State theory”.  To reword the Wikipedia entry slightly, this misguided principle states that the observable universe apparently never changes (at a sufficiently large scale).

 

There is, however, a possible alternative extension to the cosmological principle – that, at all times during its development, the universe is spatially homogenous and isotropic at sufficiently large scales.  This allows the universe to both change (ie develop) and look different at different times (so long as it remains homogenous and isotropic at appropriate scales).  In a sense, it’s not even an extension to the cosmological principle but rather a mere clarification.  The cosmic microwave background (CMB) is predicted to be isotropic by the ΛCDM model and is shown to be isotropic (“to roughly one part in 100,000”).  The implication here is that the universe has been (homogenous and) isotropic since at least the surface of last scattering (when the CMB was generated).  There’s no reason to expect that the universe was not homogenous and isotropic prior to that.

 

An extension to the cosmological principle implies an extension to the Copernican principle, namely that not only are we “not privileged observers of the universe” but also, we do not inhabit a “privileged era” of the universe.  Note that I don’t mean a “privileged era” here as meaning an era that supports life of our type (that is that stars have progressed sufficiently to create the necessary constituents of our bodies, and those constituents have not been destroyed or scattered too thinly).  What I mean is that we should not observe a universe that is in a special condition, beyond that which is necessary to permit our existence as observers.

 

I was toying with calling the extended version of the cosmological principle the “properly perfect cosmological principle”, but I eventually settled on the more obvious “extended cosmological principle” (with the understanding that the “perfect” cosmological principle would become the “overextended cosmological principle”).  Similarly, the notion that we should not be in a privileged era would become the “extended Copernican principle”.

 

There could also be an “extended principle of relativity”, positing that the laws of nature have always and will always be the same everywhere.  I understand that there could be resistance to this notion as the laws of physics are understood to emerge from the state of the universe – but maybe the “extended principle of relativity” could be thought of as applying since the “phase transition” broke the symmetry of a single primordial unified force into a number of distinct forces.  There were no and could be no observers until well after that event, and thus no science being conducted, so it would be an acceptable limitation of the principle’s applicability.

 

Alternatively, we could accept that certain dimensionless constants might change, but the underlying mathematics of the universe would be consistent throughout.  There is a suite of constants that, when expressed in terms of Planck units, all resolve to unity – but that resolution to unity cannot happen with dimensionless constants.  It’s possible that those constants could vary over time.  Similarly, there are different solutions to quantum mechanics, and perhaps some of those solutions dominate in different eras (such as prior to the phase transition mentioned above).

 

In combination, the extended cosmological principle, the extended Copernican principle and the extended principle of relativity would be the “extended consistency principles”.

 

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The question that one must immediately ask, of course, is whether there is any apparent contravention of any of these extended principles.

 

I would say that there is.

 

If standard cosmology is to be believed we live in a privileged era.  Consider this representation of our cosmological history:

 

 

The top-most rim of the bowl represents the universe as it is today, after 13.7 billion years.  (Ignore the placement of apparent galaxies, they are misrepresented in the illustration in an effort to distinguish between the CMB and the universe of today.)

 

The rest of the bowl shows the Big Bang (unfortunately implying that the Big Bang is the inflationary period), followed by the inflationary epoch (a period of approximately 10^-32 seconds, just prior to which the vacuum had a much higher density than now), the photon epoch, the surface of last scattering (which is effectively the CMB), the cosmological dark age (prior to the ignition of the first stars) and, more recently, since about 5 billion years ago, and accelerated expansion – forming the lip of the bowl.

 

The problem is that, today, the Hubble parameter, which is a measure of the expansion of the universe according to Hubble’s law, is currently equivalent to the inverse age of the universe.

 

This would never have been the case is any past era of the universe and, if the rate at which the universe is expanding is accelerating, will never be the case in any future era.  Which makes right now, just when we are around, able to observe the universe, a privileged era.  There is no reason (that I know of) to think that the value of the Hubble parameter makes it possible for us to inhabit the universe.

 

So, there is an apparent contravention of the extended consistency principles.  Can this be resolved?  I think it can, but I will put that in a separate post.

Constants that Resolve to Unity

Note that all the non-grey items in the table below resolve to unity or are unity by definition, if one uses the Planck units.

The two grey items are included because they are linked to a constant that isn't formally recognised, as far as I know, which I have dubbed the "charge to structure ratio" for ease of reference.

Note that I also made up the terms “reduced permeability” and “raised permittivity” as I have not noticed any similar usage anywhere.  Perhaps the terms are already used, or something similar and, if so, again I’d appreciate knowing.

Normalising permeability and permittivity to unity is certainly not a new concept.  I use the term “resolve” because I am not really doing anything to these values above, other than expressing them in Planck units.  I consider the step of dividing or multiplying by 4π to be a “normalisation”.  This normalisation is applied to my selection of the Planck charge, such that q_Pl²=4πε₀ħc.  Note also that the term 4πε₀ appears in the expression for Coulomb's constant, k_e=1/4πε₀.  In other words, Coulomb's constant is merely the inverse of raised permittivity.  For that reason I have placed them together and generally consider them to be the same constant.

Wikipedia's page on Planck units, at time of writing, indicates that the selection of Planck charge as either q_Pl=√(4πε₀ħc) or q_Pl=√(ε₀ħc) is at the author's choice.  I disagree for two reasons.

First, Planck charge can be written in terms of the Boltzmann constant, noting again that k_e=1/4πε₀:

q_Pl=√(ħc/k_e)

Compare this with the equation for Planck mass:

m_Pl=√(ħc/G)

Second, it can be seen clearly that when q_Pl=√(4πε₀ħc) is selected, the expression for the fine structure constant becomes α=e²/q_Pl², indicating that the value is merely a representation of the ratio between the elementary charge and the Planck charge.  (See also Why I Like Planck, Constants that Resolve to Unity and Coupling Constants.)

 

Note that the precise value of the reduced permeability in SI units at 1.00x10^-7 is merely an artefact associated with the definition of the ampere (which flows through to the coulomb).  That definition included a 2, which is why the reduced permeability involves 4π, while the reduced Planck constant involves 2π.  If the ampere were “the constant current which, if maintained in two straight parallel conductors of infinite length of negligible circular cross section and placed one metre apart in a vacuum, would produce between these conductors a force equal to 1 × 10⁻⁷ newton per metre of length” (rather than the actual 2 × 10⁻⁷), then the normalisation adjustments would all involve 2π.

A Simple Question

 Here’s a simple question, although getting to the question is quite so simple.

 

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There is a radius associated with a non-rotating, spherical black hole, called the Schwarzschild radius, which can be shown to be the radius at which light, travelling in a vacuum, cannot escape.  This is given by r_S=2GM_S/c², where G is the gravitational constant, M_S is the mass of the (Schwarzschild) black hole and c is the speed of light.  There is, therefore, a radius to mass ratio such that M_S = r_S.c²/2G. 

 

Such a black hole, by definition is spherical, and the volume of it can be given by V_S = 4/3.πr_S³.  This gives us the opportunity to calculate the density of a Schwarzschild black hole – given by

 

ρ_S = M_S/V_S = 3.c²/(8π.G.r_S²)

 

Interestingly, the density of a Schwarzschild black hole, therefore, is inversely proportional to the square of its radius (and also to the square of its mass, because the mass and radius are directly proportional).  The larger and more massive a black hole is, the less dense it is. Note that the gravitational time dilation equation is:

 

 

Note that a critical value of r, at which this equation becomes t_dilated = 0, is r=2GM/c².

 

You could wonder, then, what the density of an enormous black hole would be.  Let’s say one with a radius of 13.77 billion light years (the distance that light could have travelled in the time since the beginning of the universe, assuming flat space and something with uniform rectilinear motion that it was travelling relative to).  13.77 billion light years is equal to 13.03x10²⁶ m.  So, we have:

 

r_S = 1.303x10²⁶ m

c = 299792458 m/s

G = 6.6743x10^-11 m³/kg/s²

π = 3.14159 (-ish)

 

Plugging these in, we get a density of 9.47x10^-27 kg/m³ – or 9.47x10^-24 g/m³ – or 9.47x10^-30 g/cm³.  This is precisely the critical density of the universe (that should be of little surprise, since ρ_c = 3H²/(8π.G), where H is the Hubble parameter, which happens to be the inverse of the age of the universe, so in our equations above H = 1/(age of the universe) = c/r_S).

 

So: my simple question is this, why is it that the density of our universe is pretty much precisely (within the bounds of experimental uncertainty) the density of a Schwarzschild black hole that is precisely the size that a universe would be after 13.77 billion years, if it expanded at the speed of light?

 

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A slightly more complicated question is as follows.

 

Noting that, for any volume with a radius r_S, the density of that volume is critical when ρ_S is given by ρ_S = 3.c²/(8π.G.r_S²) – and the “criticality” thus only gets worse when the radius is greater, how can it be that the (observable) universe is 46.508 billion light years in radius and, according to WMAP measurements, 9.9x10^-27 kg/m³ in density, when a Schwarzschild black hole of that radius would have a density of 8.3x10^-28 kg/m³ (or about 1/12 of what we have)?