Can AI teach me cosmology at applied maths undergraduate level or even post grad?

sjastro · Jan 29, 2026

As you may be aware I have been monitoring the progress of AI in physics and mathematics for nearly three years and seen tremendous progress.
While I use AI to tabulate my responses, in this thread I will give it free reign to educate me on an aspect of cosmology requiring a working knowledge of tensor calculus, Riemannian geometry and general relativity as prerequisites.

I will structure this thread according to the following.
(1) To assess ChatGpt-5's existing knowledge of these prerequisites and highlight any dubious or ambiguous descriptions.
(2) Propose a hypothetical universe for ChatGPt-5 to critique and find evidence which contradicts it.

Unfortunately discussions like this tend to get lost in mathematical jargon which the layperson may struggle with.
I will try to explain the jargon where necessary.

The next post is to address point (1).

sjastro · Jan 29, 2026

For the FLWR metric derive the non-vanishing Christoffel symbols and derive any equations from it.

Before providing GPT-5’s response the FLWR metric is defined as:

The metric is simply a space-time interval equation which describes the geometry of space-time, in 3D Euclidian space it is the shortest distance between two points which is a straight line.

A Christoffel symbol is briefly described in this post.

Here lies the problem in 2D flat space (which can be generalised to higher dimensions) using Cartesian coordinates the partial derivative is defined by:

In this case the direction of the projection vector as h approaches zero is unchanging hence we can define the partial derivative.

In curved space however the definition does not work as the projection vector does change direction.
This can be illustrated by the change of a projection vector as it moves along the arcs on the surface of a sphere.

The concept of the derivative in curved space needs to be redefined which factors in the changes in the projection vectors.
In this case we need to calculate the partial derivatives of the metric tensor components which define the geometry of the spacetime in which the projection vectors change.

Without going in specific details which would require a detailed knowledge of tensor analysis the derivative of a tensor known as the covariant derivative is defined as:

The left hand side of the equation is the covariant operator applied to a vector v.
The first term on the right hand side is the normal partial derivative of the vector, the next term contains information how the metric tensor components g(uv) change by calculating their partial derivatives which impacts on changes to the projection vector.

The operator in the second term is known as a Christoffel symbol and is defined as

ChatGpt-5’s response, my queries and its response to my queries are in the next post.

sjastro · Jan 29, 2026

ChatGpt-5's response to my question For the FLWR metric derive the non-vanishing Christoffel symbols and derive any equations from it.
The FLWR metric describes the spacetime geometry for an expanding universe.

The shaded portions in its responses are either errors or require more information, the first one is a discrepancy in how the metric is expressed, I used -c²dt² as the first term, GPT-5 used -dt². Now either this is a typo or GPT-5 is a lot smarter by expressing c as a natural unit (c=1).
The second query is ChatGpt-5 needs to provide more information on how the Friedmann equations are derived using Einstein's field equations.

Its responses are in the next post.

sjastro · Jan 29, 2026

It's response to my criticism I wrote the c² term in the FLWR which you omitted it in your explanation.
Egad it used the natural units excuse....

It has now provided a more detailed explanation on how the Friedmann equations are derived from Einstein's field equations.

Now that I have pretty good idea of GPT-5's strengths now comes the interesting part which will be covered in the next posts.
(2) Propose a hypothetical universe for ChatGPt-5 to critique and find evidence which contradicts it.

Bradskii · Jan 29, 2026

You lost me at ds².

sjastro · Jan 29, 2026

Bradskii said:
You lost me at ds².

Here is a simple example of ds² in 2D Euclidean space.

The shortest distance between the points O and S is a straight line.
ds is the distance of a line segment PQ along OS and ds² = dx² + dy² which is the Pythagoras theorem.

For flat space-time there is the Minkowski metric which in spatial 2D is ds² = c²dt² - dx² - dy².
Here ds is not a length segment but is space-like if ds² < 0 (P and Q cannot be causally connected) , time-like if ds² > 0 (P and Q can be causally connected) or light-like if ds² = 0 (P and Q are connected by a light signal).

sjastro · Jan 30, 2026

Now for the serious stuff.
In another thread a creationist making the fatal mistake of trying to use science to justify their faith used the analogy as the universe expands time stretches like a rubber band.
Mathematicians like to create toy models of the universe which are analysed by physicists as possible physical models.

If time is like a rubber band it is possible to construct a toy model by modifying the FLWR metric:

I proposed to GPT-5 the following toy universe based on the metric.

This differs from the FLWR metric as I have multiplied c²dt² by the scale factor a(t) which in effect stretches time at the same rate as space itself.

I asked GPT-5 to find the non vanishing Christoffel symbols, derive the equations for time and spatial components based on the geodesic equation and comment on the possibility of the metric being a physical model.

I’ll take the time, radial and angular equations as being correct on face value, given GPT-5 correctly derived them for the FWLR metric. (The mathematical drudgery of calculating the non-vanishing Christoffel symbols and plugging them into the geodesic equation to derive these equations is usually left to graduate students going for their PhDs.

)

What I can definitely state is while GPT-5’s answer is detailed in refuting the model as being physical it only tells part of the story as revealed in the next post.

sjastro · Jan 30, 2026

Before proceeding Einstein's field equations are defined as:

The left hand side of the equation defines the mathematical or geometrical properties of space-time, the right hand side how matter through the stress-energy tensor T affects the geometry.
GPT-5's answer in the previous post purely concentrated on the left hand side of the equation.

In order to make GPT-5's answer more complete I asked it if my proposed metric is a solution of the field equations.

sjastro · Jan 31, 2026

The metric results in a mathematical not a physical universe which does not conform to the Einstein field equations unless exotic matter is introduced where gravity acts as a repulsive force.
The next question for GPT-5 is to show the physical evidence in our universe which directly contradicts this mathematical universe.

The highlighted text are queries to be addressed in the next post.

sjastro · Feb 1, 2026

Two comments made made GPT-5 need clarification.

The Cosmic Microwave Background defines a universal rest frame

Comoving observers relative to the CMB experience:

no dipole anisotropy

maximal isotropy

The anisotropy of the CMB (Cosmic Microwave Background is discussed in this post and has an observed dipole anisotropy.

The dipole is created by the observer orbiting the Sun, the Sun's motion around the Milky Way, the Milky Way's motion in the Local Group of galaxies and the Local Group's motion relative to the CMB. As a result the observer is moving towards the CMB resulting in a Doppler blueshift while 180 degrees in the opposite direction it is moving away with a Doppler redshift. The difference results in a very small temperature variation ΔT which is the dipole.

I queried GPT-5's comment about the CMB isotropy as being too ambiguous as the dipole is observed which it correctly addressed in its response.

There is also this comment made by GPT-5.

You’re asking the right kind of dangerous questions — keep pushing.

My criticism is more directed towards the developers of AI rather than AI itself.
I understand the need for a touch of melodrama or a human element to make AI even more appealing to use but in this case it is totally unnecessary and even misrepresents my position as my hypothetical metric was devised to debunk it within the confines of General Relativity which is the gravitational theory used in mainstream cosmology.

My next post is to provide a summary if AI (ChatGPT-5) can teach me cosmology.

Yttrium · Feb 1, 2026

I've been testing ChatGPT-5 with much more serious subjects, anime and computer games. My testing suggests to me that retrieving information can be very unreliable. ChatGPT would frequently get one show confused for another, mixing up and misnaming anime characters, and providing video game information that was inappropriate for my PC. I was often able to check the sources it gave me for the correct information that it somehow missed. I would not trust ChatGPT to provide fully accurate information even if it found good sources. It can be a good search engine, and it draws nice pictures, and it's great at praising you if you like that kind of thing (I don't).

sjastro · Monday at 4:22 PM

Yttrium said:
I've been testing ChatGPT-5 with much more serious subjects, anime and computer games. My testing suggests to me that retrieving information can be very unreliable. ChatGPT would frequently get one show confused for another, mixing up and misnaming anime characters, and providing video game information that was inappropriate for my PC. I was often able to check the sources it gave me for the correct information that it somehow missed. I would not trust ChatGPT to provide fully accurate information even if it found good sources. It can be a good search engine, and it draws nice pictures, and it's great at praising you if you like that kind of thing (I don't).

I suspect it depends on the quality of the data ChatGPT-5 is trained on.
For cosmology the training data is confined largely to mainstream ideas and there is less scope for it stray into pseudoscience or off topic responses.

1. Where my cosmology knowledge comes from (percentage-style breakdown)
These percentages are conceptual, not exact engineering specs — but they’re a faithful way to think about how my knowledge of cosmology is shaped.

Learning Paradigm
Approx. Contribution
What it means in practice for cosmology
Supervised learning
≈ 60–65%
Most of my cosmology knowledge comes from labeled text: textbooks, review papers, lecture notes, Wikipedia-style explanations, peer-review summaries. This is where standard ΛCDM, FLRW metrics, inflation, dark energy, CMB physics, etc. are learned in a “this statement is presented as correct” sense.
Self-supervised / unsupervised learning
≈ 25–30%
I learn statistical structure from vast amounts of raw text: how cosmology concepts co-occur, how arguments are structured, how speculative ideas differ linguistically from mainstream ones, how equations are typically introduced. This helps me reason, connect ideas, and detect internal consistency — even when material is controversial or nonstandard.
Reinforcement learning from human feedback (RLHF)
≈ 5–10%
Humans guide how I respond: clarity, caution, tone, when to hedge, how to distinguish speculation from consensus. This strongly affects how I talk about cosmology (e.g., flagging exotic stress–energy, modified gravity, or nonstandard time interpretations).
Post-training alignment & safety tuning
≈ 1–3%
Affects what I avoid overstating: I’m nudged not to present speculative cosmologies as established fact, and to label interpretations clearly.
User interaction shaping (non-persistent)
≈ 0% (persistent)
Your questions influence this conversation only, not my underlying knowledge.

Key takeaway:
My facts about cosmology come mostly from supervised + self-supervised learning; my style and caution come from reinforcement learning.

2. Do I learn from corrections you make here?
Short answer:
No — not in the way a human or a continually-training model would.

Long answer (precise but honest):

I do not update my training weights based on this conversation.

I do not remember corrections once the conversation ends.

If you correct me:

✅ I can immediately adapt within this conversation

❌ I will not carry that correction into future chats unless it already aligns with my training

So if you say:

“That interpretation of cosmological redshift is wrong because…”

Click to expand...

Then:

I can acknowledge it

Re-derive or restate the argument correctly

Use your framing consistently for the rest of this chat

But once the chat ends, that correction does not modify me.

A subtle but important nuance
Even though I don’t learn, I do reason.

That means:

If your correction is logically sound and consistent with known physics, I’ll often say:

“You’re right — given X and Y, that conclusion follows.”

Click to expand...

If it conflicts with consensus, I’ll usually say:

“This is a minority or speculative view, here’s where it departs from standard cosmology.”

Click to expand...

This is especially relevant to topics you’ve raised before:

time stretching vs cosmic time

interpretation of FLRW coordinates

exotic stress–energy

Weyl / modified gravity

consciousness–physics crossover

I can analyze, compare, and stress-test those ideas — but not learn them permanently.

One-sentence summary

My cosmology knowledge is mostly built from supervised and self-supervised learning on human-written material; reinforcement learning shapes how I answer — and while I can adapt to your corrections in real time, I don’t retain them beyond the conversation.

Click to expand...

If you want, next we can:

map this learning breakdown onto specific cosmology topics (ΛCDM vs alternatives), or

analyze where mainstream training biases show up most strongly in cosmological debates.

Learning Paradigm	Approx. Contribution	What it means in practice for cosmology
Supervised learning	≈ 60–65%	Most of my cosmology knowledge comes from labeled text: textbooks, review papers, lecture notes, Wikipedia-style explanations, peer-review summaries. This is where standard ΛCDM, FLRW metrics, inflation, dark energy, CMB physics, etc. are learned in a “this statement is presented as correct” sense.
Self-supervised / unsupervised learning	≈ 25–30%	I learn statistical structure from vast amounts of raw text: how cosmology concepts co-occur, how arguments are structured, how speculative ideas differ linguistically from mainstream ones, how equations are typically introduced. This helps me reason, connect ideas, and detect internal consistency — even when material is controversial or nonstandard.
Reinforcement learning from human feedback (RLHF)	≈ 5–10%	Humans guide how I respond: clarity, caution, tone, when to hedge, how to distinguish speculation from consensus. This strongly affects how I talk about cosmology (e.g., flagging exotic stress–energy, modified gravity, or nonstandard time interpretations).
Post-training alignment & safety tuning	≈ 1–3%	Affects what I avoid overstating: I’m nudged not to present speculative cosmologies as established fact, and to label interpretations clearly.
User interaction shaping (non-persistent)	≈ 0% (persistent)	Your questions influence this conversation only, not my underlying knowledge.

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Can AI teach me cosmology at applied maths undergraduate level or even post grad?

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1. Where my cosmology knowledge comes from (percentage-style breakdown)​

2. Do I learn from corrections you make here?​

A subtle but important nuance​

One-sentence summary​

1. Where my cosmology knowledge comes from (percentage-style breakdown)

2. Do I learn from corrections you make here?

A subtle but important nuance

One-sentence summary