Digesting a Paper on Flatness (Part 6)

 See Part 1 to understand what this is about.

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In the previous part, I initially wrote this:

“In this Letter, we treat background spacetimes with

where n is the lapse, which may be set constant by reparameterizing t, and a(t) is the scale factor.

I think they are treating space as if it were three dimensional here (by eliminating one spatial dimension).  Possibly for ease of calculation.” 

This was wrong.  I was thinking about i and j being something different (two dimensions of imaginary numbers).

I subsequently realised that this is a reference to a matrix, or more specifically a tensor, of two dimensions, that can be used to describe a manifold of many dimensions.

It’s a little disturbing – for one not steeped in the subject – to note that i (which is not the imaginary number notation i), together with j, is an index from 1 to n (which is not the lapse n).  In other words, it could be that both i and j are indices from 1 to n=3, meaning that γ_ij could thus describe a three dimensional manifold.  It’s not necessarily as simple as that but, for example, the metric tensor (or metric) for Minkowski spacetime can be described as:

This makes most sense when thinking about a spacetime interval, which is given by ds² = c²dt² - dx² - dy² - dz².

Anyway, having revisited the equation above, I can see that it refers to (inside the parentheses) a time component and some unspecified number of space components (note that the related metric tensor has a different metric signature, so the time component is negative and the space components are positive).  They are all multiplied by the time dependent scale factor a²(t).

It's further worth noting that γ_ij has a specific meaning, or at least an implication.  Given that we are thinking about curvature here, it’s probably worth noting that the arc length parameterisation of a circle can be expressed as:

I am wondering, therefore, whether γ_ij could be referring to an n-sphere (where i and j are indices from 1 to n) – or at least manifold that is based on that.  It’d not surprise me if γ_ij were given as below:

Note I got to this a strange way.

I was thinking of a sphere, basically as the form directly above and thinking about the “scale factor” (although I was thinking of it in terms of a cosmological constant and I am not saying that this is correct).  I imagined that the sphere, with all positive values, represents a closed space.  Think of the surface of the sphere having two sides – an inside and an outside.  Then I thought about that same sphere with negative values.  The effect would be to flip the sides the other way, so that the “inside” of the sphere would include everything that otherwise would be outside of it.  That corresponds with the notion of positive and negative curvature.  In between the two you have zero values, and this corresponds with zero curvature – or flatness.

Flatness in Minkowski space, therefore (if my intuition is correct, which it may well not be) would mean that ds² = c²dt² - dx² - dy² - dz² = 0.  (And this would explain why it doesn’t matter if you write the equation as ds² = c²dt² - dx² - dy² - dz² or, with a different metric signature, ds² = dx² + dy² + dy² - c²dt².  I note that these equations are used in the development of Lorentz transformations. If time is considered to be inherently imaginary, with an inbuilt i, the equations converge on one form, ds² = dx² + dy² + dy² + c²dt² = 0 – because the squaring of the inherent i results in a negative value.  Alternatively, ds² = dx² + dy² + dy² + c²(i.dt)² = 0.  Note that this is called a “Wick rotation”, which the paper mentions in the comment on an image that one can easily skip past but probably shouldn’t.  See Part 7.)

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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.