Digesting a Paper on Flatness (Part 2)

See Part 1 to understand what this is about.

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I skipped “gravitational instantons” in Part 1 because they seem to warrant their own post.  They were raised in the following context:

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.

Clearly there’s more in there than just gravitational instantons.

An instanton is described as “a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory”.  A gravitational instanton is an extension of the concept (or notion): “a four-dimensional complete Riemannian manifold satisfying the vacuum Einstein equations”.

So, while an instanton is sometimes referred to as a pseudoparticle, a gravitational instanton is more like a 4-dimensional topological space.  For a manifold, think of a surface, that’s a 2-dimensional manifold.  It can be flat (as in a plane) or curved (as in the surface of a sphere).  So spacetime, which is four dimensional, could be described by a gravitational instanton (or perhaps a range of gravitational instantons).

The term “macroscopic parameter” is not specific to this topic, but could be referring to temperature, pressure and entropy – as these are features of many particles in aggregate, as opposed, for example, to a quantum level parameter such as spin or charge.  This would make sense as the sentence finishes with a mention of gravitational entropy.

What is “gravitational entropy”?  As I understand it, there is a delicate ballet between entropy in general and the contribution to total entropy of a system as provided by gravity.  Imagine a free gas in empty space – it will tend to expand, and thus entropy is increased.  However, we know that that doesn’t happen at macro scales, because gas (and dust etc) will clump into galaxies, stars, planets, asteroids and comets (and everything in between).  So the idea is that gravity itself is effectively an entropy increaser such that a clump of gas in a star represents more entropy than the same gas just floating around dispersed. How much entropy there is in the universe (or a system) – due to gravity – is the value that the authors of the paper appear to be trying to calculate.

It's worth returning to the term “instanton” here for a moment.  Another description of an instanton is a saddle point, where there are two equally valid directions that something can go (in reference to quantum tunnelling between equally valid states).  There’s an implication of a saddle point with the notion of gravitational entropy – there are two potential, diametrically opposed ends to the universe that are posited.  Either it clumps back up, with all the mass/energy crushed down to a singularity (the big crunch) or it expands forever becoming increasingly disparate (the big rip).  As a saddle point, the universe could be balanced between those two options – if it is flat, which it appears to be.  But note: those two options both represent increasing entropy, so the question might be – is gravitational entropy sufficiently large as to drive the big crunch or sufficiently small as to allow the big rip?  Or just right so that neither happens?  It seems to be that if the universe is flat (remembering that the definition is such that the universe is balanced on a knife edge between those two options), then that would point towards the third option.

In reading about negative entropy (see if there was a possibility that the total entropy is balanced out by twinned universes – there’s isn’t), I noted that where there is local negative entropy it relates to an energy increase.  For a salient example, to lift a mass in a gravitation field, you need to add energy to the system (effectively giving the mass potential energy).  If mass-energy is entering the universe – what effect does this have on entropy?  Is the expansion more than enough to cover it such that the second law of thermodynamics is not violated?  Or would the fact that the universe would not be a “closed system” (due to energy coming in) be sufficient?

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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.