ChatGPT, Will Fool, God (Twice)

To say that I have been somewhat obsessed with OpenAI's ChatbotGPT would be somewhat of an understatement.  It's amazing, even if it's a very thin veneer over a basic recombination of what already exists on the internet somewhere.

I started with asking questions about climate change because I've been having that discussion in realspace, but I also wondered what the bot would have to say about the god question.  So here are two mirrored stories about some guy called Will Fool and his relationship with "God".

Story 1:

Instruction:

Write a 3000 word story about Will Fool who doesn't believe in God, how he reached that conclusion and the strongest arguments that persuaded him.

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Will Fool was raised in a religious household and attended church regularly with his family. However, as he grew older, he began to question his beliefs. He was a curious and analytical person by nature, and he couldn't bring himself to blindly accept the teachings of the church without evidence.

As he entered his teenage years, Will began to read extensively on the subject of religion and spirituality. He studied the major world religions, as well as a variety of philosophical and scientific perspectives on the existence of God. He was particularly fascinated by the arguments of prominent atheists, such as Richard Dawkins and Christopher Hitchens.

Despite his religious upbringing, Will found himself drawn to the logical and evidence-based arguments of the atheists. He was particularly swayed by the concept of Occam's Razor, which states that the simplest explanation for a phenomenon is often the correct one. In the case of the existence of God, Will found that the complexity of the concept and the lack of concrete evidence made it less likely to be true.

Another argument that resonated with Will was the problem of evil. If God is all-powerful and all-loving, why does evil exist in the world? How could a loving God allow for natural disasters, disease, and suffering? Will found it difficult to reconcile the idea of an all-powerful and all-loving God with the existence of evil.

Will also took issue with the idea of faith. He saw faith as a crutch that people used to avoid confronting difficult questions and uncertainty. He didn't believe that faith alone was sufficient to justify one's beliefs, and he felt that a lack of evidence for God's existence should be enough to reject the idea.

As Will continued to explore and study the subject of God and religion, he became more and more convinced that belief in God was not rational. He didn't see any compelling evidence to support the idea of God, and he found the arguments of the atheists to be more logical and persuasive.

Despite his lack of belief in God, Will was not an aggressive or confrontational person. He didn't try to convert others to his point of view, and he was respectful of the beliefs of others. He simply saw himself as someone who had thought deeply about the subject and reached a conclusion based on the evidence available to him.

Over time, Will's lack of belief in God became a central part of his identity. He was confident in his position and was not troubled by the thought of living in a world without God. He saw life as a precious and beautiful gift, and he believed that each person had the responsibility to make the most of their time on Earth.

Despite the strength of his convictions, Will was not immune to moments of doubt and uncertainty. There were times when he felt a twinge of fear or anxiety at the thought of there being no higher power or greater purpose to life. However, he always reminded himself of the logical arguments and evidence that had led him to reject the idea of God, and these doubts would quickly dissipate.

In the end, Will Fool was a man who had rejected the belief in God based on reason and evidence. He was not a bitter or angry person, but rather someone who had thought deeply about the subject and reached a conclusion that made sense to him. He lived his life with confidence and purpose, and he did not let his lack of belief in God hold him back in any way.

Story 2

Instruction:

Write a 3000 word story about Will Fool who doesn't believe in God but came to believe that God exists as he got older, how he reached that conclusion and the strongest arguments that persuaded him.

---

Will Fool had always been a skeptic. Growing up, he had never been religious and had always scoffed at the idea of a higher power controlling the universe. He believed that everything could be explained by science and logic, and the idea of a God seemed like nothing more than a fairy tale to him.

But as he got older, Will started to question his beliefs. He had always been a logical person, and he couldn't help but wonder if there might be something more to life than what he could see and touch. He started to read up on the arguments for and against the existence of God, and he was surprised to find that many of them were surprisingly compelling.

One of the arguments that really caught his attention was the argument from design. This argument stated that the complex and intricate design of the universe and all of its systems showed that it must have been created by a higher power. Will had to admit that the more he thought about it, the more sense it made. The universe was so vast and complex, and it seemed impossible that it could have come about by chance.

Another argument that Will found convincing was the argument from morality. This argument stated that the existence of objective moral values and duties showed that there must be a higher moral lawgiver who created them. Will had always believed in doing what was right and avoiding what was wrong, and he couldn't help but feel that there must be some kind of higher authority behind these moral values.

But despite the strength of these arguments, Will still had doubts. He didn't want to believe in something just because it was comforting or convenient. He wanted to be sure that the existence of God was a fact, not just a belief.

So he decided to put his skepticism to the test. He started to explore the world around him, looking for evidence of God's existence. He spent hours pouring over books and articles, and he even traveled to different parts of the world to see the natural beauty and complexity of the universe for himself.

And as he continued to search, Will began to see more and more evidence that pointed towards the existence of God. He saw the beauty and majesty of the natural world, and he couldn't help but feel that there must be some kind of higher power behind it all. He also saw the incredible complexity of the human body, and he was amazed by the intricate design that must have gone into its creation.

Finally, Will came to the conclusion that he could no longer deny the existence of God. He had seen too much evidence, and he couldn't ignore it any longer. He had come to believe in God, not because it was comfortable or convenient, but because it was the only logical conclusion based on the evidence he had seen.

And in the end, Will realized that his belief in God wasn't just a matter of faith, but a matter of fact. The arguments for the existence of God were strong, and they couldn't be denied. He had come to believe in God not because he wanted to, but because the evidence was overwhelming.

In the end, Will was grateful for the journey that had led him to this conclusion. He had always been a skeptic, but he had also always been open-minded. And by staying open-minded and following the evidence wherever it led, he had come to believe in the existence of God.

----

I am pretty sure that the stories that I ask ChatbotGPT to create for me are synthesised from what existed on the internet when it was being trained.  (Although I do have an example of a story that probably isn't, more of that later.)

For this reason, although I did not say that Will Fool used to be religious, the bot still created a narrative from belief to disbelief in Story 1.  It attributed certain characteristics to Will that perhaps are common among people who make that journey (curiosity and analytical nature) and placed the transition in Will's teenage years.  I accidentally prevented that option in Story 2 by saying that Will transitioned "when he got older", but I never said anything about scoffing or the idea of god being "nothing more than a fairy tale" but again, this may be common in the stories of people who become religious.  It certainly seems to be a common refrain in the backstories of amateur apologists.

Anyway, I was using the story format to trick the bot into providing supporting arguments for certain positions which it might not otherwise be willing comment on (as it appears to avoid political and possibly other charged topics).  The really interesting aspects of these stories are, therefore, the strongest arguments for each case.

In summary:

For god:

• The argument from design (presented twice)
• The argument from morality
• The beauty of the world (via an argument from incredulity)
• Use of arguments as evidence (note that I did ask for "strongest arguments", but the bot implies that these are evidence)

Against god:

• Authority (although to be fair, it is only stated that their arguments are "particularly facinat(ing)"
• Logic and evidence-based arguments
• Occam's Razor
• Problem of Evil
• Issues with faith as a concept and the lack of rationality associated with it
• Lack of evidence for the existence of a god

Interestingly, the bot made a bit of an effort to cast Will Fool the atheist as a good guy (despite his lack of belief), but managed to almost contradict itself by saying that Will "was confident in his position and was not troubled by the thought of living in a world without God" and then saying, in the next paragraph, that he was troubled, albeit only occasionally.  However, the bot wrote "(t)here were times when he felt a twinge of fear or anxiety at the thought of there being no higher power or greater purpose to life", which could arguably be something other than a god.  Also there's a question as to what constitutes being "troubled".

It occurs to me that perhaps the best approach, as an anti-apologist, may be to address what seems to be agreed to be the strongest arguments for god  and promote the strongest arguments for the non-existence of god (leaving aside appeals to authority though).

---

As a bonus, here's a story which I am pretty sure does not exist out there on the internet or anything like it, as told by ChatbotGPT.  (I am not responsible for the spoiler by the way.  The bot seemed obsessed with giving things away, even if I didn't ask for that. It also ignored my request to start the scene properly.  I was actually aiming for a narrative in which Eldon was either gaslighting Rachel or simply unaware that Deckard was a penguin.  Unsuccessfully.  I had also hoped that the Voight-Kampff questions would be skewed in a bizarre way by the fact that Deckard was a penguin, but the bot went off in a totally different, unexpected direction.)

Instruction:

Rewrite the scene in Blade Runner where Deckard meets and tests Rachel, but make Deckard a penguin.  Eldon Tyrell is helping administer the test but only Rachel notices that Deckard is a penguin.  Start from the owl sequence.

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As Deckard the penguin entered the room, Eldon Tyrell greeted him with a nod. "Deckard, this is Rachel. She's one of our most advanced replicants. I'll be administering the Voight-Kampff test to determine her emotional capacity."

Rachel looked at Deckard curiously, noticing his unusual appearance. "Is that a penguin?" she asked.

Eldon chuckled. "Yes, Deckard is a highly trained animal. He's very good at administering the Voight-Kampff test."

Deckard waddled over to Rachel and began the test, peering at her with his beady eyes. Rachel answered the questions calmly and confidently, but she couldn't shake the feeling that something was off about Deckard.

As the test continued, Rachel's suspicions grew. She couldn't help but notice the way Deckard moved, the way he held himself. It was almost as if he was more than just a penguin.

Finally, Rachel couldn't take it anymore. "Eldon," she said, turning to him. "Deckard is a replicant, isn't he?"

Eldon's eyes widened in surprise. "Yes, Rachel. Deckard is a replicant, just like you."

Rachel nodded, a mix of emotions passing over her face. "I knew it," she said. "I could tell from the way he moved, the way he looked at me. He may be a penguin, but he's still one of us."

Deckard looked at Rachel, his expression unreadable. "You're right, Rachel," he said. "I may be a penguin, but I'm still a replicant, just like you."

How to break OpenAI's chatbot

Just a question on how climate change predictions are wrong (testing to see what the bot would do):

Now do it in the style of Noel Coward:

Mind. Blown.

Note: Noel Coward is wrong.  And dead.

Digesting a Paper on Flatness (Part 7)

See Part 1 to understand what this is about.

--

Figure 1: Cosmological solutions for: i) a(t) in Lorentzian time (see Eqs. (1), (3), with n real); ii) a(t) after a Wick rotation W, setting n = -iN, with N real; iii) b(t) after a conformal and a Wick rotation CW, setting a(t) = ib(t), n = -iN with b(t) and N real.

--

Figure 1 isn’t referred to in the text of the document until later, but a(t) appears in the paragraph at Equation 1 discussed in Part 6.  Some of the terms shown in this figure appear in that later section (around Equation 3, to be discussed later), specifically:

• ESU
• κ
• L⁺_i and L⁺_o
• L^±_a

The title of each image in the figure needs some clarification.  “Lorentzian” seems to be a reference to the metric signature of the Minkowski metric, which is expressed as either (-,+,+,+) or (+,-,-,-).  Another way to think of the metric signature, is in terms of (p,v,r), where p, v and r are (respectively) the number of positive, negative and zero eigenvalues of the real symmetric matrix g_ab of the metric tensor with respect to a basis.  In this case, we can just consider them as dimensions, three (positive or negative) dimensions of space and one (negative or positive) dimension of time and no zero dimensions making the metric signature (3,1,0) or (1,3,0).  The related equation is ds² = -(c²dt²) + dx² + dy² + dz² which conveniently has all the signs (+ and -) shown, in the right order.  It seems to be a convention in the metric to show time first, I don’t think it has any physical implications.

A Wick rotation is a transformation to express Minkowski space as if it were Euclidean or, in other words, going from a metric signature of (-,+,+,+) to one of (+,+,+,+) by treating the time dimension as imaginary.  So, using natural units such that c = 1, going from ds² = -(dt²) + dx² + dy² + dz² to ds² = dτ² + dx² + dy² + dz² (where τ is time that takes on imaginary values).  I don’t think that using a different version of the Lorentzian metric signature makes any difference, it’s just a different way of expressing the same thing and is thus another matter of convention.

So, the titles for i) and ii) could also be “Lorentzian (Metric)” and “Euclidean (Metric)”.

A conformal rotation is transformation or remapping of space(time) that maintains angles but not necessarily lengths.  It’s not entirely clear to me whether the rotations mentioned should be considered as two rotations (and thus whether the order makes a difference) or one rotation that has both features, so in other words a Wick rotation that is also conformal, or a conformal rotation that is also a Wick rotation.  From the commentary on iii) it does seem to be two rotations, the upshot of which is to have all dimensions imaginary.  I’m not sure why this is beneficial at this stage – the rotations are normally related to making the equations simpler during the time they are being worked on (after which there’s the implication of a rotation back).

The images include the symbol “r” that refers to radiation (see later post - note that it's a little unclear because gamma is also used elsewhere in the document and they look very similar in italics). The kappa, “κ”, appears to be referring to curvature.  It’s unclear what happens to those curves when curvature is zero.

The value “n” appears to be the lapse mentioned in Part 6. The meaning of lapse doesn’t seem to be that easy to find the answer to the question “what is the physics definition of lapse?”  The best I came across was here (but note that it’s talking about the lapse function).  Fundamentally lapse is proper time between an event on one slice of a hypersurface and the same event on another slice and shift is the spatial separation between the event on those two slices.  I think this is related to the simultaneity issue when selecting frames.  Say I take a cut through the universe of “now” using my frame as the reference and consider two events that are (or rather were, according to me) simultaneous.

 

A --------------------------------> Me <------------------------------- B

 

Light reaches me from locations 180 degrees apart at the same instant.

However, if my twin were travelling (relative to me) at some significant speed (on the path AB), then the events would a) not be simultaneous, b) not be located where I thought they were, or c) both – according to my twin, depending on where they were at the time that they took as “now”.  We’d have a lapse, or a shift, or both.

Basically, the “foliation” of spacetime that they are talking about is the selection of different values of “now” and lapse and shift are used to translate between those values of “now”.

Oddly, when I looked up an answer to the question “what is the physics definition of lapse?” I came across a support page for a Maple Soft Physics [ThreePlusOne] product.  Of particular interest was the equation on that page:

ds²=−α²dt²+γ_i,j(dxⁱ+βⁱdt)(dx^j+β^jdt)

Compare this with the equation from Part 6:

ds²=a²(t)(-n²dt²+γ_ij(x)dxⁱ dx^j)

Ignoring some minor convention differences, and assuming that the authors of the paper assume a shift of zero noting that α and β have the lapse and shift values respectively, it appears that the only thing added is the a²(t) term or that ThreePlusOne has assumed that a²(t) = 1.

Note that a(t) is the scale factor and that the current value is 1, so a²(t) = 1 is (currently) true.  In the paper, however, the authors don’t appear to want to be limited to the current value and by implication to the current state of the universe which is taken to be dark energy dominated.

It’s further worth noting that the value of a(t) is calculated using the Friedmann equations.  One of the assumptions here is that “the metric of the universe must be of the form -ds² = a(t)²ds₃² - (c²dt²)”, which follows from “the simplifying assumption that the universe is spatially homogeneous and isotropic”.  Noting the topic of the paper, these assumptions should be kept in mind, because there is a risk of a circular argument.  It might appear odd that in the equation here, a(t)² applies only to the time component, where in the equations from Part 6 and ThreePlusOne, it applies to both the time and space components. I suspect that this is because γ_ij is normalised.

I find it strange that, when the equation that leads to a(t)=1 in the current, dark energy era is discussed, there is a key component that is described as “some constant” – but it’s not just any constant, it’s a key constant since that equation is:

Per the scale factor description, a(t) is proportional to t^1/2 in the radiation dominated era, t^2/3 in the matter dominated era and … hold onto your hat … e^(H₀t) in the current dark energy dominated era (I ran into problems trying to put H₀ in as an superscript because of the subscript, hence the e^ notation).  Currently, “just by coincidence”, H₀ = 1/t, so a(t) is proportional to e.  Hm.  Why do that?  We know that a(t) today is 1.  Since, per the Standard Cosmological Model, the universe was radiation dominated, then matter dominated and, today, dark energy seems to dominate, it follows that the vague constant, w, is increasing – being very small at first (so that 2/3(w+1) ≈ 2/3), then approaching ⅓ (so that 2/3(w+1) ≈ 1/2) and then approaching infinity (so that 2/3(w+1) ≈ 0).  That implies that a₀ would equal 1.

What remains very unclear from all this is precisely what w is.  Here, however, is an equation for w, which is described as “the equation of state of the universe” (it’s also implied here and here):

Where P’ is a pressure and ρ’ is a density (and c, of course, is the speed of light).  Specifically:

 

and

Where m seems to indicate “during the matter dominated era” and Λ seems to indicate “during the dark energy dominated era” (note the symbol is more commonly associated with the cosmological constant).

Later in the document it becomes clear that ESU, as referred to in Figure 1, means “Einstein’s Static Universe”.

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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 6)

 See Part 1 to understand what this is about.

--

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.)

--

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.

--

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.

--

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.

--

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.

--

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 3)

 See Part 1 to understand what this is about.

--

The partition function of a statistical ensemble can be represented by a path integral over configurations that are periodic in imaginary time.

The mention of a path integral indicates that we are talking about the partition function in relation to quantum field theory here rather than statistical mechanics, or as Wikipedia helpfully puts it:

In quantum field theory, the partition function {\displaystyle Z[J]}Z[J] is the generating functional of all correlation functions, generalizing the characteristic function of probability theory.

Um, ok.  That doesn’t really help.

I think it’s entirely possible to get hung up on this sentence overly long.  For the moment, therefore, suffice it to say that we are talking about a partition function.  A partition function is the normalising constant applied to the probability equation for a microstate.

Think about statistical mechanics of, for example, a gas.  A gas consists of a multitude of particles (molecules usually, atoms if it’s a noble gas).  Knowing the microstate would require an exhaustive cataloguing of all the molecules, how many there are, what they are, where they are and what momentum they all have.  Knowing that would give you other characteristics of the gas (pressure and temperature for example).  But we never know all that and the microstate is changing all the time.  Instead, we can think of the likelihood of an ensemble of microstates (a bunch of them).  The sum of all probabilities has to be 1 – hence the normalising factor.

What I think they are getting at here is that some microstates are more likely than others and as you integrate more microstates, you arrive at a more likely overall state (or “path”) for the system to be in an eventually you can say “I don’t know what the precise microstate is at any one time, but the temperature of the gas is this and it’s pressure is that”.

Go even deeper and think about things at the quantum rather than molecule level, and you have the same sort of situation.  It is not possible to know the precise microstate (quantum state, maybe) of a system, especially when you have superposition of different states that don’t manifest until there’s been a measurement, but you can integrate the likelihood of states to arrive at things that you can say about the overall state.  The sum of the probabilities of all quantum states is going to be 1 again, and you therefore need a normalising factor, which would be the partition function.  The partition function would, therefore, tell you something about the likelihood of each ensemble of states (or configurations).

What I expect happens is that there is a peak of probability at what constitutes reality (which is a bit fuzzy across a relatively narrow range) – other states are not impossible, but they are very unlikely.

Why this occurs in imaginary time is not immediately obvious.  It might be worth noting what imaginary time is though.

There is what is called a Wick rotation, which converts the geometry of spacetime from Lorentzian to Euclidean.  This converts the metric for spacetime from ds² = - (dt)² + dx² + dx² + dx² to ds² = (i.dt)² + dx² + dx² + dx² (because i² = -1).  Note that this sort of metric is implied in On Time where I talk about an invariant “spacetime speed” (although I deliberately collapsed down the spatial components to just one x dimension for ease of calculation).

The phrase “periodic in imaginary time” is a reference to the alignment of temperature (in statistical mechanics) with oscillations in imaginary time (in quantum mechanics) – in that the related equations are of the same form.

Hawking and collaborators first used these methods to investigate black hole thermodynamics, a topic which has since burgeoned into holographic studies [10, 11] and experimental tests using analog systems [12, 13]. We shall follow Hawking et al.'s original approach here [14, 15, 16, 17].

This is a reference to the idea that the entropy of a black hole relates to its surface and not its volume (this seems rather obvious when it’s clear that time and space go to zero at the surface of a black hole – there isn’t anything “inside” a black hole in terms of our universe, everything that is going on with it is going on either at or above the surface [where particles, if there are any, are ripped apart and thus give off radiation]).

In the cosmological setting, it is an excellent approximation to treat the radiation as a relativistic fluid, in local thermal equilibrium at a temperature declining inversely with the scale factor.

I think this is just talking about radiation (composed of photons) being conceptually similar to a fluid that moves very fast.  If you’ve ever heard a term like “awash in photons” (probably in science fiction), then you have the idea of a relativistic fluid.  It should be noted that there are equations for fluid flow, which are analogous to equations for electricity, because electrons in a wire act a bit like molecules of oil in a hydraulic system and the same analogy can be made to radiation.

The instantons we present are saddle points of the path integral for gravity: real, Euclidean-signature solutions to the Einstein equations for cosmologies with dark energy, radiation and space curvature.

We’re talking about four-dimensional manifolds again.  I do not believe that the saddle points mention are in those manifolds but are rather points of high likelihood (and thus reality) among the possible (micro or quantum) states and the argument is that those states manifest gravity of the sort that we observe in our universe.

The associated semiclassical exponent iS/ħ is real, large and positive, and may be interpreted as the gravitational entropy.

The exponent iS/ħ appears in the equation for the “generating functional Z(J)”, where a generating function is a way of representing the sum of an infinite sequence of numbers, but in this case, we are representing a space of functions – it’s basically a description of the machine (think of a black box with an input and an output) that does something to something.  For what it’s worth, this is that equation (taken from here) – note that they have an erroneous close bracket after the final x, but I've removed it for the replication below:

or

It’s written slightly differently here (with no erroneous close bracket):

Note that e^ix cycles eternally between 1, i, -1 and -i as x increases in magnitude from zero (1 where x = 2nπ, i where x = (2n+1/2)π, -1 where x=(2n+1)π and -i where x = (2n+3/2), where n is any integer).  Therefore, with either equation, iS/ħ is a phase offset.

In these equations, Φ refers to a scalar field – which is our manifold again (the d⁴x tells us that we are using a four dimensional scalar field) – and the DΦ is telling us to “integrate over all possible classical field configurations Φ(x) with a phase given by the classical action S[Φ] evaluated in that field configuration”.  J or J(x) is an auxiliary function, referred to as a current.

But what does all this mean?

In the words of Vivian Stanshall, about three o’clock in the morning, Oxfordshire, 1973: In layman's language, “It's blinking well baffling. But to be more obtusely, buggered if I know. Yes, buggered if I know.”

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A couple of clarifications:

Z(J) is a functional integral (space of functions) while S is an action functional.  An action functional is related to the stationary-action principle, which when put very, very simply (perhaps to the extent of being simplistic) is about things not doing more than they have to.  If an object is moving in a particular direction at a particular speed (so, with a velocity) it will continue to move at that velocity unless acted upon by a force.  We know about a lot of forces that act on objects to change an object’s velocity, gravity, resistance (air, friction), other objects colliding into it, etc, etc.  Digging into the concepts of relativity, each object has its own frame (of reference) in which it is stationary (with respect to itself).

In a sense, every single object in the universe has a spacetime speed (note, not velocity) which is invariant.  The actions between objects could be worked out by considering how the spacetime speed is distributed between their relative temporal speed and their relative spatial speed (see On Time).  Effectively, every single object in the universe could be considered as stationary (different types of “stationary”, admittedly, but stationary all the same).

The statement “(t)he associated semiclassical exponent iS/ħ is real, large and positive” is interesting in itself.  Given that i is the imaginary marker and ħ is a real, small and positive constant (about 10^-34J/K), that would make S itself imaginary (not complex), large (or even just average) and negative.  Negative is understandable, if it’s representing gravitational entropy which is understood to be negative … but imaginary?  Not quite sure about that.

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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.