Digesting a Paper on Flatness - Introduction (Part 1)

I was pointed to the paper Gravitational entropy and the flatness, homogeneity and isotropy puzzles by the secondary author, Latham Boyle, in response to my question as to whether his team had got any further with considering flatness in respect to an earlier paper The Big Bang, CPT, and neutrino dark matter (ignore the date in the vertical text to the left of the pdf and consider instead the date of the first version, 23 Mar 2018).

This paper is a tough, maybe even totally impenetrable read for a non-professional, interested amateur like myself.  Therefore, if I have any hope of working through it, I will need a structured method for extracting meaning from the text.  This, along with following posts, is that structured method.

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Introduction:

Pictures of the earth from space reveal it to be remarkably round and smooth, with a curvature radius much larger than the distances we explore in our everyday lives. Hence, on those scales, it is often a good approximation to treat the earth’s surface as flat. The explanation of this flatness involves thermodynamics. First, gravity pulls matter inwards, so its potential energy is minimised in a spherical configuration. Second, relative motions of the earth’s constituent atoms generate friction: as a mountain is pulled downwards or a rock falls, gravitational potential energy is converted into heat. Even if the earth is regarded as a closed, conservative system, there are vastly more ways of distributing its internal energy among its ∼ 10⁵⁰ atoms as heat than there are of creating more complicated, less spherical geometries. Taken together, gravity, friction and the earth’s many atoms explain why it is locally flat [1].

This seems fair enough, massive bodies in space tend towards flatness (although not necessarily perfect sphericality – due to the effects of rotation).

In this Letter, we provide a similar, entropic explanation for the observed flatness, homogeneity and isotropy of the cosmos. Our argument rests on a new calculation of gravitational entropy, along the lines advocated by Hawking and others in the context of black hole thermodynamics [2 3 4].

An entropic explanation is one that rests on the notion that in a closed system entropy tends to increase – per the second law of thermodynamics.  That is that “potential energy is minimised” as per the first paragraph.  Gravitational entropy, therefore, is a tendency to the minimum gravitational potential energy state.

Flatness, in cosmological terms, means the absence of curvature – which isn’t particularly useful.  Per COSMOS at Swinburne University, a flat universe in one that is perfectly balanced between eventual collapse and eternal expansion – and this is a question of density.

Homogeneity, in cosmological terms, is smoothness at a large scale.  While it is clear the universe has lumps in it at our scale (the universe, galaxies, solar systems, planets, individual people, molecules and so on), at a grand scale, the universe is surprisingly smooth.  Together with isotropy, this constitutes the cosmological principle – which is the notion that we humans are not privileged observers.  No matter where one is in the universe, things look pretty much the same (at a grand scale).

Isotropy is invariance to orientation.  Again, at large scales, it doesn’t matter what direction you look in and observe the universe, things look the same in every direction – as if we were in the centre of smooth, monochrome bubble.

Note that the combined flatness, homogeneity and isotropy of the universe is not an assumption about it, it’s an observation, and therefore something to be explained.

I don’t believe that it is necessary to delve into the references here as they are just acknowledgements by the authors that the work that follows has been presaged by earlier physicists.  The links are provided in case anyone else wants to look at that preceding work.

The calculation is made possible by our new approach to the boundary conditions for cosmology, implementing CPT symmetry and analyticity at the bang, quantum mechanically, to solve many puzzles [5 6 7 8].

The term “boundary conditions for cosmology” appears to be a reference to the balancing act that the universe is engaged in – its flatness for example – that have been referred to by others as “fine tuning”.  More broadly however, boundary conditions are the constraints which permit solutions to equations – in this case those equations would be those that describe the universe.

CPT symmetry is the symmetry associated with charge (positive-negative), parity (see the following) and time (future-past).  Parity is related to chirality or handed-ness but more broadly related to a change in sign (and hence direction) – but it should be understood as different to a rotation.  A parity transformation is akin to creating a mirror image rather than turning something around.  Note that a consequence of the Standard Model is the implication that parity is not a symmetry of our universe – which is why there is an implication of a second, paired universe in which antimatter predominates.

The term “analyticity” is again about descriptive equations which, in this case, are such that the related “Taylor series (for each) about x₀ converges to the function in some neighbourhood for every x₀ in its domain”.  What I take from this is the universe is contiguous and complete, meaning that it doesn’t have locations in which there are sudden breaks or discontinuities.  Another way to say that is that the universe is “smooth” (in addition to being flat).  Yet another way, perhaps less formal and precise, and at the danger of being simplistic, is to consider what it takes to analyse something with confidence – basically you would not want to have elements of the relevant data missing or uninterpretable.

The term “the bang” appears to be a stylistic affectation on the part of the authors – most of us call it “the big bang”.  Realistically, given the current size of the universe (and the amount of energy in it now compared to then), the big bang was not really that big a bang.

The puzzles referred to appear to be:

• apparent CPT symmetry breaking, 
• dark matter, 
• the arrow of time, 
• and vacuum energy and the Weyl anomaly (both have links to symmetry breaking)

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We outline the connection to Penrose’s classical Weyl curvature hypothesis [9] in the conclusion.

This will be addressed in the conclusion (as by the time I work on that I will almost have certainly forgotten that I covered it here) but … the Weyl curvature hypothesis “is rooted in a concept of the entropy content of gravitational fields” and assumes a very low initial entropy of the cosmological gravitational field.  Note that Weyl curvature is related to the curvature of spacetime.

It might be worth noting that Penrose has a conception of the universe in which is cyclic (incorporating the Weyl curvature hypothesis) in which there are separate aeons or instances or (timelike) sectors of the universe.  Effectively, the “future timelike infinity” is the big bang of the next – as our future eternity is functionally equivalent to the zero point (or singularity) of the next universe from which it emerges.  This has some aspects of similarity to my model, since an event horizon, although it appears to be a surface from the outside is effectively a point from which all the mass/energy that passes through it would emerge into another universe.  The main difference we have is that I don’t think that it would be possible to reach the “future timelike infinity” because by that point the universe would have infinite mass/energy … and that probably comes with some attendant problems.  What makes more sense to me is that when the pressure on the universe to expand evaporates (because the total mass/energy from the “previous” universe is expended), this leads to an effective collapse, a rescaling and thus the initiation of a new universe (or rather a new pair of universes).

Note that the event horizon concept is such that mass/energy that enters our universe does so across its entirety, making the universe inherently homogenous – negating the need for inflation as an explanation for homogeneity of the cosmic microwave background.  (Penrose’s conformal cyclic cosmology also implies no inflation.)

Using these new boundary conditions, we showed that cosmologies with radiation, dark energy and curvature are periodic in imaginary proper time, with a Hawking temperature given by that of the corresponding de Sitter spacetime [7]. This is a strong hint that the solutions should be interpreted thermodynamically.

The term “periodic in imaginary proper time” makes sense to me, but I might be misunderstanding it since it aligns so closely with my model (so I might be motivated to misinterpret).  Imaginary time is time that is perpendicular to real time.  Remember there are “real numbers” and “imaginary numbers”, where an imaginary number is i with a value attached to it, so 2i is an imaginary number of magnitude 2 – there are a few examples in physics and engineering where imaginary numbers have very real usefulness, so don’t think of them as “made up” – the thing to remember is that an imaginary number represents orthogonality to the real number plane.

The term “proper time” has some meaning in relativity, but the use here is a bit perplexing.  In a sense, time is already imaginary, being orthogonal to the spatial dimensions (look at complex Minkowski spacetime for example) – but proper just implies the time a clock following a timelike world line would record, or “dilated time”.

I’m tempted to think that this is a reference to “actual” imaginary time, and that what is being referred to is a sequence of time sectors which are all imaginary in relation to each other (that is, orthogonal to each other).  But I accept that this might well be wrong.  It may become clear as we get deeper into the paper.

A Hawking temperature is a reference to the amount of Hawking radiation from a black hole, the smaller a black hole is, the higher its Hawking temperature – meaning that micro black holes dissipate quickly, and supermassive black holes don’t.  The reference to a Hawking temperature does seem to imply the involvement of a black hole.

Here, in an appropriate time slicing we find the solutions describe new gravitational instantons, one for each value of the macroscopic parameters, allowing us to calculate the gravitational entropy. For a given, positive dark energy density, with radiation included the gravitational entropy can be arbitrarily large. As the total entropy is raised, the most likely universe becomes progressively flatter. Furthermore, inhomogeneous or anisotropic perturbations are suppressed. Hence, to the extent that the total entropy exceeds the de Sitter value, the most probable universe is not only flat, but also homogeneous and isotropic on large scales.

I am going to skip over “gravitational instantons” – not because they are easy but because they are hard and appear important.  The “macroscopic parameters” should become clear as we get into the body of document.

The remainder of the paragraph basically just indicates that as the universe ages, it should – due to increasing entropy (and thus disorder) – become increasingly homogenous and isotropic, implying that we should not be surprised to find ourselves in such a universe.  What the de Sitter value of entropy actually is (other than its notation, S_Λ) … is unclear.  It may become clearer as we get deeper into the paper.  Some casting around various paper indicates that it might be a reference to a bound below which de Sitter spacetime is not stable, but as far as I can see the fame of de Sitter entropy is quite limited.  It’s not clear what the ramifications on the early universe (of low entropy) would be.

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With that we are a whole two paragraphs into the paper – or about half a page.  With the exception of “gravitational instantons”, which I will look into next.  And these were the easy paragraphs!

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Hopefully it’s quite obvious that this series is more of an open pondering session rather than any statement of fact about what the authors of Gravitational entropy and the flatness, homogeneity and isotropy puzzles intended to convey.  If I have misinterpreted them, then I’d be happy to hear about it.