A question concerning gravity... and the animations thereof

sjastro · Wednesday at 2:35 PM

partinobodycular said:
Oddly enough I had the same problem. AI doesn't seem to be able to take a normal grid, place an object with mass into it, and produce an accurate overhead view of that grid.

This is a screen grab from Gravity well 3D Explorer which better illustrates the problem . The object with mass is above the plane, so space should be bent upward toward the center of the mass, but instead it's bent downward away from the mass.

View attachment 381659

This is what it should look like:

View attachment 381660

That looks fine. But now let's imagine that the grid is actually bisecting the object. The grid should remain perfectly flat, and any warping should occur along the plane of the grid.

The mission, should anyone choose to accept it, is to create an image of a mass with a 2D grid bisecting it, as viewed from above.

The second diagram looks like Flamm’s paraboloid which is also an idealized representation of spacetime and is found in textbooks on GR.

Take the Schwarzschild metric ds² = (1-2MG/c²r)c²dt² - dr² /(1-2MG/c²r) - r²(dθ² + sin²θdφ²)
and a time slice or snapshot at some time t = T, then dt =0.

Next a plane say θ = π/2, dθ = 0 and the metric reduces to
dl² = dr² /(1-2MG/c²r) + r²dφ² where dl² = -ds² is a spatial line element.

If this is embedded in 3D cylindrical space where the spatial line element is
dl² = dr² + r² dφ² + dz² then
dr² /(1-2MG/c²r) + r²dφ² = dr² + r² dφ² + dz²
dz/dr = ±√(2GM/c²r)/(r-2GM/c²)
z(r) = ±∫√(2GM/c²r)/(r-2GM/c²)dr = ±2√(2GM/c²(r-2GM/c²) which is the general equation for Flamm’s paraboloid.

It has the same issues; it is not a representation of curved 4D spacetime and gravity is a real force instead of a fictitious force.

The closest representation of the metric is as a 2D spacetime diagram where the x-axis is the spatial part of spacetime and the y-axis the temporal part.
I asked ChatGPT-5.5 the supposedly PhD level AI to perform this exercise but it went way beyond my expectations.

I asked if it was trying to express the 2D spacetime representation of the Schwarzschild metric as a set of trajectories for light, objects and clocks?

Fair enough.

Maria Billingsley · Wednesday at 4:50 PM

partinobodycular said:
FYI, there appear to be two viable options. The Cartesian Compression Model, and the Polar Compression Model.

View attachment 381662

Take your pick.

Too funny! Gemini struggled and came up with this.

sjastro · Wednesday at 8:30 PM

Maria Billingsley said:
Too funny! Gemini struggled and came up with this.View attachment 381670

Good grief!

This looks like an example of psychotic art which is used as therapy for patients.

First the plaque, it can’t spell “SPATL”, “MCDELLING” (must have Scottish origins).
“REMOVES FUNNEL ARTIFACTS AND DEMONSTRATES CONTINUOUS, UNIFORM GEOMETRY” is the human equivalent of word salad.

As for the science it displays a 4D geodesic vs 3D path graph (what do the axes represent?).
It has what looks like two Flamm paraboloids one of which is inverted and looks suspiciously to have been mistaken for past and future light cones with a spherical symmetry in between which might be based on the Schwarzschild metric.

There is a “precise orbital path” which I assume is the Moon’s orbit which looks as if it is showing presession.
The trouble is the Earth which is located in the middle of this mishmash is not massive enough for spacetime distortion around it for the Moon to show observable precession.

Now for ChatGPT-5.5's critique of Gemini's "artwork".
It is far more technical but ignores the elephant in the room such as the spelling mistakes and the word salad.

partinobodycular · Thursday at 1:13 AM

Maria Billingsley said:
Too funny! Gemini struggled and came up with this.

I asked several AI's to create an image based upon the following prompt:

Please draw a 3D representation of a grid in the shape of a sphere, with the earth in the sphere's center, equidistant from every point on the sphere, and warp the sphere to show the effects of earth's gravitational field.

Most of them ended up creating an image of the Earth surrounded by a torus. But Claude actually gave it some serious thought before creating the following image, and the accompanying disclaimer.

"This is the physically honest version: two perfectly round, concentric spheres — the pale outer one is the original undistorted grid where every point is equidistant from Earth, and the warm inner one is uniformly compressed inward by the same amount all the way around, since Earth's gravity doesn't favor any direction. The gap between them is exaggerated well beyond the real (tiny) effect so it's visible at all."

Claude seems to be much more flexible and intuitive about interpreting the intent of the instructions, while Gemini often has a hard time following the simplest of instructions once it's set upon a particular idea. Often repeating the same image, even when told precisely how to change it.

Maria Billingsley · Thursday at 7:15 AM

partinobodycular said:
I asked several AI's to create an image based upon the following prompt:

Most of them ended up creating an image of the Earth surrounded by a torus. But Claude actually gave it some serious thought before creating the following image, and the accompanying disclaimer.

View attachment 381679

Claude seems to be much more flexible and intuitive about interpreting the intent of the instructions, while Gemini often has a hard time following the simplest of instructions once it's set upon a particular idea. Often repeating the same image, even when told precisely how to change it.

Ah! Claude I believe gets the idea! I agree about Gemini, way too creative and stubborn!
Thanks for sharing

Ophiolite · Thursday at 10:27 AM

@partinobodycular

It's an anology. If I place a heavy ball on a pliable rubber sheet it will deform the sheet. If I roll smaller balls across the sheet then, depending upon their speed and penetration into the deformed area, they may be trapped in the depression, have their direction of travelled altered and even (ignoring friction) find themselves 'orbiting' the heavy ball.

Space is deformed by anything with mass that has an analagous influence on other massive objects.

I think your problem is trying to think of it as somehow physically representing what is going on. (And what I've omitted from my simplified description is that the deformation of the sheet extends to infinity and the smaller balls also deform it. Which is why the navigation of probes in the solar system with pin point accuracy is a source of wonder to me. That, and the music of Bach.

sjastro · Thursday at 6:51 PM

partinobodycular said:
I asked several AI's to create an image based upon the following prompt:

Most of them ended up creating an image of the Earth surrounded by a torus. But Claude actually gave it some serious thought before creating the following image, and the accompanying disclaimer.

View attachment 381679

Claude seems to be much more flexible and intuitive about interpreting the intent of the instructions, while Gemini often has a hard time following the simplest of instructions once it's set upon a particular idea. Often repeating the same image, even when told precisely how to change it.

This doesn't work either.
At best it is only a 3D spatial approximation which completely ignores the time element in spacetime.

I got ChatGPT-5.5 to eventually 'understand' it needed to plot trajectories for 3 different types of scenarios to show how the trajectories react around Earth's curved spacetime. Plotting these trajectories gives an indication of what the curved spacetime looks like and is analogous to magnetic field lines to describe a magnetic field.

Using the Schwarzschild metric ds² = (1-2MG/c²r)c²dt² - dr² /(1-2MG/c²r) - r²(dθ² + sin²θdφ²).
Scenario (1)
According to GR light travels along null geodesics where ds² = 0.
The metric becomes (1-2MG/c²r)c²dt² - dr² /(1-2MG/c²r) - r²(dθ² + sin²θdφ²) = 0

Scenario (2)
Stationary clocks located at different radii in the curved spacetime around Earth. These clocks undergo gravitational time dilation and will tick at different rates. Since they are stationary dr = dθ = dφ = 0.
The metric becomes ds² = (1-2MG/c²r)c²dt² which means the clocks click at a slower rate the closer they are to the Earth.

Scenario (3)
Particles with mass where according to GR travel along geodesics where ds² > 0 which preserves causality and travel at less than the speed of light.
The metric becomes (1-2MG/c²r)c²dt² - dr² /(1-2MG/c²r) - r²(dθ² + sin²θdφ²) > 0

These 3 scenarios represent changes to the geometry according to whether the trajectory in question is of a photon or a particle with mass, and of a stationary clock.

A visual representation shows the Schwarzschild metric is of a spherically symmetric spacetime according to how photons, particles with mass and stationary clocks behave.

This appears to be a novel way of showing spacetime and in its response ChatGPT-5.5 commented on a collaborative effort.
In reality I made some suggestions but ChatGPT-5.5 did most of the heavy lifting.

chilehed · Thursday at 9:14 PM

It's just a visualization intended to help explain the concept. Don't overthink it.

partinobodycular · Friday at 10:56 AM

sjastro said:
This doesn't work either.
At best it is only a 3D spatial approximation which completely ignores the time element in spacetime.

My brain

:

@sjastro's brain :bow:

:

Any questions? (For those old enough to get the reference)

Stopped_lurking · Friday at 11:17 AM

partinobodycular said:
My brain :

@sjastro's brain :

Any questions? (For those old enough to get the reference)

There is nothing wrong with your brain. It just a question of what the visualization is trying to show. You could see the gradient of the curved space as visual representation of the amplitude of the newtonian-like force.

sjastro · Friday at 3:39 PM

partinobodycular said:
@sjastro's brain :

Any questions? (For those old enough to get the reference)

Well sjastro's brain became addled as the spacetime time visualization was in fact very difficult to visualize, when the photon and particle trajectories are treated separately.

For the case of photon trajectories, I used the gravitational bending of light as reference trajectories, the diagram of which is found in any textbook on GR when the central body is the Sun.

Next is the action of Earth's spacetime on free falling particles in both radial trajectories and non radial trajectories.
Once again these are found in any GR textbook.
A local tangent space indicates spacetime curvature around the Earth is a manifold, globally curved but locally flat.

Now I questioned ChatGPT-5.5 if you added these trajectories together plus the the gravitational time dilation of stationary clocks would you produce the visualized spacetime curvature?

I'm now satisfied with the image as it is, adding clock symbols and local tangent spaces would clutter up the image and distract from the fact the trajectories provide the visual illustration of spacetime curvature.

sjastro · 2026-07-25T03:38:18-0500

In my previous post I mentioned free fall and tangent spaces.
A tangent plane is a plane tangent to a curved surface, a tangent space is the same generalization for higher dimensions.

In the 1950s scientists came up with the idea that if gyroscopes are in free fall in curved spacetime, their spin axes would change as the gyroscopes moved from one tangent space to the next since the tangent space's orientation changes in curved spacetime.
If spacetime was flat the spin axis would not change as a the tangent space orientation would remain the same.

From this idea the Gravity Probe B was developed which was launched in 2004.
In orbit the Gravity Probe B is in free fall traveling through tangent spaces.

I got ChatGPT-5.5 to map the measured spin axes deviations caused by geodetic precession and frame dragging and it came up something that looks remarkably similar to Flamm's paraboloid and visual depictions usually found in textbooks.

It needs to be emphasised that the Gravity Probe B was not measuring a literal curved surface or "gravity well." It was measuring how the orientation of a freely falling gyroscope changes as its spin vector is transported through local tangent spaces of a curved spacetime manifold.
Still the resemblance to a visual depiction of spacetime is remarkable.

partinobodycular · 2026-07-25T18:24:39-0500

sjastro said:
It needs to be emphasised that the Gravity Probe B was not measuring a literal curved surface or "gravity well." It was measuring how the orientation of a freely falling gyroscope changes as its spin vector is transported through local tangent spaces of a curved spacetime manifold.
Still the resemblance to a visual depiction of spacetime is remarkable.

I must admit that interpreting this image was extremely difficult, not just for me, but for Gemini, Copilot, and Claude as well. We eventually managed to make some sense out of it, except for one thing, the 'Spacetime Interpretation' scale seems backwards. It seems to indicate that Purple/Blue indicate weak curvature far from earth, and Red/Orange indicates strong curvature close to earth.

Is the color scale in the "Spacetime Interpretation' box backwards, or am I missing something?

sjastro · 2026-07-25T20:11:05-0500

partinobodycular said:
I must admit that interpreting this image was extremely difficult, not just for me, but for Gemini, Copilot, and Claude as well. We eventually managed to make some sense out of it, except for one thing, the 'Spacetime Interpretation' scale seems backwards. It seems to indicate that Purple/Blue indicate weak curvature far from earth, and Red/Orange indicates strong curvature close to earth.

Is the color scale in the "Spacetime Interpretation' box backwards, or am I missing something?

The image is slightly tilted to provide as 3D type rendition with the Earth sitting at the top of the paraboloid so the scaling does makes sense.
However I questioned ChatGPT-5.5 on how it came up with the shape in the first place using the Gravity Probe B data.
The answer was it didn't with the misleading part and correction highlighted in blue.

I did not derive that earlier image numerically from the Gravity Probe B data. That image was generated as a conceptual illustration, and I should have made that clearer at the time.

The process behind it was roughly this:

I used the known Gravity Probe B results—the measured geodetic precession and frame-dragging precession—as the experimental motivation.
I then combined those results with the standard general-relativistic description of spacetime outside a rotating Earth: approximately the Schwarzschild geometry for the mass contribution, plus a weak rotational or Lense–Thirring term for Earth's angular momentum.

The central funnel-like surface was therefore based on the conventional embedding-style representation of Schwarzschild spatial curvature, not on a numerical inversion of the Gravity Probe B measurements.

The small rotational asymmetry was intended to represent frame dragging.

The Gravity Probe B orbit was then placed on that theoretical geometry to show where the measurements were made.
So the actual logical construction of the picture was:

Gravity Probe B measurements
→ confirm geodetic and frame-dragging precession

combined with

general relativity
→ Schwarzschild-like curvature + rotational frame dragging

then

visual embedding
→ funnel/Flamm-paraboloid-like surface.

That distinction also means several labels in the original image were too strong. In particular, saying “Reconstruction of the 3D curvature of spacetime around Earth using Gravity Probe B gyroscope precession” implies a data-derived reconstruction that was not actually performed.

A scientifically better title would be something like:

“General-relativistic visualization of spacetime around Earth, constrained by Gravity Probe B measurements.”

There is another issue with the image. The displayed “Gaussian curvature” colour scale is not something Gravity Probe B measured. GP-B measured angular precession rates. Converting those measurements into a complete spatial curvature field would require adopting a metric model and calculating quantities such as the Riemann curvature tensor from that model.

You can't trust anyone these days including AI.
This time I made a specific request, it has access to the Gravity Probe B data for a 642 day mission where the spin axis deviation is accumulative.
I asked it to provide time slices at various intervals where each slice represents a tangent plane where the spin axis deviates.
Stack the sliced segments on the t-axis to provide (hopefully) some sort a realistic picture of what spacetime curvature would look like in 2D space.

This would be a genuinely different exercise: start with the actual time-dependent GP-B spin-axis measurements, construct the cumulative geodetic and frame-dragging deviations, associate those with parallel transport along the known orbital worldline, and then ask what aspects of the spacetime connection or curvature can be reconstructed from them.

That would be much closer to a genuine data-to-geometry reconstruction than the image I originally generated.

The dashed line in (4) is Gravity Probe B's trajectory or geodesic free fall worldline through spacetime, the tube is the neighbourhood restricted sampled region of spacetime associated with the spin axis measurements.
The kinks and twists in the tube are associated with curved spacetime, if spacetime is flat the tube would be straight.
This is the closest a visual interpretation of spacetime comes to reality as the tube is based on measuring the spin axis deviation at various time intervals.

partinobodycular · 2026-07-25T21:26:09-0500

sjastro said:
You can't trust anyone these days including AI.

Don't I know it. It's not just that, but even though they're frequently wrong, they're never in doubt, often explaining why their solution will unequivocally work, even before I've tried it... only to find out that it doesn't work... for the umpteenth time.

I assume that ChatGPT included the funnel simply for visual effect, as the color scale already provides a visual representation of the curvature. As copilot put it, 'The funnel is a visual metaphor, not a literal depiction.'

A question concerning gravity... and the animations thereof

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