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.

Digesting a Paper on Flatness (Part 5)

See Part 1 to understand what this is about.

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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 is wrong.  See Part 6.)  A lapse is the “proper” time separation between two events – where “proper” means “as per a clock that travels between those two events”.  Note that “proper” in this sense means “own time” rather than “correct time”.  Also note use of the term events (locations in spacetime that refer to both spatial and temporal coordinates).  An object that is “stationary” in (coordinate) space for a fixed period of (coordinate) time, is travelling between events in time and coordinate time equals proper time.  Another object that travels in one direction in coordinate space won’t arrive back at the second event.  There needs to be a change in direction and if that happens, then coordinate time for the “stationary” object won’t be the same as proper time for the object that left and returned.

Comoving 3-space is assumed to be maximally symmetric, with metric γ_ij(x) and Ricci scalar 6κ.

Comoving distances between objects in the universe factor out the expansion of the universe.  The term comoving can also be thought of as being (relatively) stationary in respect to “the Hubble flow” – such that the cosmic microwave background appears isotropic (neither red- nor blue-shifted).  Note that it is this implied frame in which the stay-at-home twin in the “Twin Paradox” is stationary and the travelling twin is in motion.

3-space is just space with three dimensions (so we are thinking only of the spatial component of spacetime).  Maximally symmetric just means that you can’t get any more symmetric (so we’re talking about symmetry in all axes).  A search for the string “Ricci scalar 6κ” produces precisely one result – this paper.  There is, however, a paper that mentions “Ricci scalar 5κ” and further clarifies that they are talking about an Einstein manifold, M₅.  I presume that all that is being said here is that there is an implication of a tighter curving of the manifold (where κ is the fundamental curvature).

For κ > 0, it is S³, with volume V = 2π² κ ^-3/2. For κ < 0, we assume a compact subspace of H³, whose volume is 2π² |κ| ^-3/2 times a topology-dependent constant. For ease of presentation, we generally leave the constant implicit.

Where κ = 0, the manifold is flat.  Note that “flat” is pretty simple when we think of a line or a plane but gets a bit more complex as we increase in dimensions.  This is probably non-standard, but I think of this way.  If you take a line and look at it from one end, and it’s a point, then it’s flat (mathematically all lines are flat, if they aren’t they are called curves).  This is, in effect, just rotation to reduce by one dimension.  (Say your line is y = 3x + 4.  Rotate to make the line parallel with the x-axis and you have y = 4.  When you make it just a number line, then it’s a point at y = 4.)

You can do the same thing with a plane, look at it from the side on, and it’s a line (which is flat as per above).  One of the characteristics of a flat manifold (in 2d, a plane) is that two lines that are parallel at any point, they don’t converge and they don’t diverge. On a closed curved manifold, they do converge and they diverge on an open curved manifold.

I have no idea why they have defined volumes that way.  We can revisit it if it becomes pertinent.

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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.

Digesting a Paper on Flatness (Part 4)

See Part 1 to understand what this is about.

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One of us has given a formal argument, based on Picard-Lefschetz theory, that a path integral for a quantum gravitational transition amplitude can never yield a positive semiclassical exponent [18, 19, 20]. However, for a statistical ensemble, a formal argument indicates precisely the opposite.

See Part 3 for the semiclassical exponent, iS/ħ. A transition amplitude, also known as a probability amplitude, is a weighting of the probability of a particular state (or eigenstate).  This is a complex number (with a real and an imaginary component), which reflects the wave function.

Consider, for example, the partition function Z(β) = Tr(e^-βH). The time reparameterization invariance of general relativity means that the Hamiltonian H vanishes on physical states [^{we only consider cosmologies in which space is compact}].

The Hamiltonian H is the “sum of the kinetic energies of all the particles, plus the potential energy of the particles associated with (a) system”.  Note that it does not include the inherent energy of the particles (ie that pertaining to the mass of the particles).  Basically, all that is being said here is that if we consider a system as having no momentum and no potential, then the total energy is given only by E=mc².  Note that the Wikipedia article on partition functions states that “the dimension of e^-βH is the number of energy eigenstates of the system” (there’s a slight difference in that they give the Hamiltonian H a hat, maybe to distinguish it from the Hubble parameter, H).

Thus, Z = e^S simply counts the number of states. If Z is approximated by a saddle, the semiclassical exponent must be positive.

The equation Z = e^S is a rewording of a generalisation of the Boltzman equation (S = k_B log W generalises to S = -k_B Σp_i ln p_i – which can be rearranged to [dropping the subscripts i and B for ease of representation] S = k ln p^-Σp and so e^S = p^-Σp+k).  For any non-trivial (that is small) number of states, it’s true that S must be positive (and the larger, the more states) – if real (see Part 3), unless Z = e^S has been normalised.

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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.