# Curvature in space-time is shown as a "fabric"

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On 9/18/2021 at 2:23 AM, Markus Hanke said:

Correct - but with the caveat that the concept of ‘gravitational potential’ can only be meaningfully defined in certain highly symmetric spacetimes, such as Schwarzschild. It is not a generally applicable concept.

Resurrecting this idea, that I never ended up wrapping my head around...

Say we use a mountain as an analogy for an arbitrary spacetime. There's a clock at the top. Two travellers each transport a clock down the mountain along different routes, and meet at the bottom. There, they generally find their clocks have aged differently, but the two clocks, together at relative rest, are now ticking at the same rate. Their clocks tick at a different rate from the clock at the top, but of course the two clocks at the bottom share the same time dilation factor relative to the one at the top. Therefore there must be a scalar number that relates the two locations, that is independent of any differences in how spacetime is curved along the different paths between the two.

In this analogy, "height" works as that single scalar, and gravitational potential is meaningful. Is there another such scalar that relates two locations in an arbitrary spacetime, or is it just the "time dilation factor", and that factor isn't completely determined by any other scalar factor alone? Or, is it that gravitational potential loses meaning only where there aren't worldlines between the two locations?

Or to ask in a different way, if you have two particles in freefall with different velocities, that pass through some one event and later pass at some second event, is gravitational potential certainly meaningful along their respective freefall paths, and if so is it the same along both paths, no matter how asymmetric the spacetime is?

Edited by md65536
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5 hours ago, md65536 said:

Or to ask in a different way, if you have two particles in freefall with different velocities, that pass through some one event and later pass at some second event, is gravitational potential certainly meaningful along their respective freefall paths, and if so is it the same along both paths, no matter how asymmetric the spacetime is?

Basically, the concept is meaningful only if it is independent of the specific path taken between the two events. This requires the presence of certain symmetries - which not all spacetimes have.

For example, consider what happens if the spacetime is not stationary, such as in a binary star system. The work required to escape to infinity from any point within this system depends not only on where that point is, but also on when the escape happens, and what specific trajectory is taken. In other words, it depends on the path taken through spacetime, the metric of which now explicitly depends on both time and space coordinates. As such it isn’t possible to assign a single unique value that signifies gravitational potential to any point in that spacetime.

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18 hours ago, Markus Hanke said:

As such it isn’t possible to assign a single unique value that signifies gravitational potential to any point in that spacetime.

But there is a unique single value gravitational time dilation factor between two points (in your example at least). Elsewhere on the web I see that gravitational potential represents the metric only as an analogy. Then the resolution might be that "gravitational time dilation is determined by the metric," while it's not always applicable that "gravitational time dilation is determined by gravitational potential"?

Edited by md65536
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2 hours ago, md65536 said:

Then the resolution might be that "gravitational time dilation is determined by the metric," while it's not always applicable that "gravitational time dilation is determined by gravitational potential"?

Yes, exactly right.

Note that the metric determines all relevant gravitational phenomena, so there’s really no need at all to try and carry over gravitational potential from Newtonian physics.

2 hours ago, md65536 said:

But there is a unique single value gravitational time dilation factor between two points (in your example at least).

How so? Time dilation between two points would be a time-dependent function rather than a single value, since all of spacetime here is filled with gravitational radiation.

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14 hours ago, md65536 said:

But there is a unique single value gravitational time dilation factor between two points (in your example at least). Elsewhere on the web I see that gravitational potential represents the metric only as an analogy. Then the resolution might be that "gravitational time dilation is determined by the metric," while it's not always applicable that "gravitational time dilation is determined by gravitational potential"?

In your pursuit of gravitational potential are you interested in the application of 'the virial theorem' to cosmology ?

Markus' advice to move on is good when you consider relativity.

You have to look inside Einstein's tensor equation, which is not really one equation at all but sixteen coupled equations, minimum, to find the relevent part.
That is why I prefer matrix methods, which display them explicitly.

If you want them the derivation is about 10 pages and also involves derivatives (or jacobians) of the Einstein equation and its solutions.
This will give you something akin to the Newtonian gravitational potential (one of the gijk components) and the cross products will bring in time as Markus says. But this 'g' is a variable not a constant.

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On 12/2/2021 at 3:25 PM, Markus Hanke said:

How so? Time dilation between two points would be a time-dependent function rather than a single value, since all of spacetime here is filled with gravitational radiation.

I mean between two events. I'm reasoning that if two clocks are identical except for their histories (basically, same place, time, and speed), they must tick at the same rate, and have the same time dilation factor relative to some distant clock. Therefore the time dilation factor between two clocks can't depend on "how the clocks got there" (I'm generalizing), yet a calculation of something representing gravitational potential, could.

13 hours ago, studiot said:

In your pursuit of gravitational potential are you interested in the application of 'the virial theorem' to cosmology ?

[...] If you want them the derivation is about 10 pages and also involves derivatives (or jacobians) of the Einstein equation and its solutions.

I've never heard of the virial theorem. It looks not basic enough for me. I should look at the derivation but I'd probably give up after getting lost at the top of page 1.

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6 hours ago, md65536 said:

I mean between two events. I'm reasoning that if two clocks are identical except for their histories (basically, same place, time, and speed), they must tick at the same rate, and have the same time dilation factor relative to some distant clock. Therefore the time dilation factor between two clocks can't depend on "how the clocks got there" (I'm generalizing), yet a calculation of something representing gravitational potential, could.

I've never heard of the virial theorem. It looks not basic enough for me. I should look at the derivation but I'd probably give up after getting lost at the top of page 1.

You should not have any trouble with this brief description.

The virial theorem was originally introduced in the mid 1800s in thermodynamics and describes the distribution of the total energy of a system of interacting particles between their energy of motion (KE) and their energy of configuration  (PE) but is applicable to any such system with these characteristics. Hence itcould be applicable to galaxies, universes etc.

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20 hours ago, md65536 said:

I mean between two events. I'm reasoning that if two clocks are identical except for their histories (basically, same place, time, and speed), they must tick at the same rate, and have the same time dilation factor relative to some distant clock.

I’m afraid I don’t follow you on this (maybe my fault). In this spacetime all the relevant components of the metric are explicitly time-dependent, so if the histories of these clocks differ, then they aren’t guaranteed to remain synchronised. Also, the dilation factor relative to the distant clock will be time and coordinate dependent.

But maybe I misunderstood your thoughts.

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Gravity is a conservative force, which means the path between two differing heights expends the equivalent work, independent of the path taken.
If we consider a light signal, or light clock if you will, that should mean that it expends the same amount of energy climbing out of a gravitational well irrespective of path taken, but depending only on the relative heights in the gravity well.
IOW, the red shift, or time dilation depends solely on the relative heights, not on path taken.

Or have I totally misunderstood your idea/analogy ...

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The work is path independent, but the accumulated phase of the clock depends on the frequency and the duration of the trip. The dilation only tells you what happens to the frequency.

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23 hours ago, MigL said:

it expends the same amount of energy climbing out of a gravitational well irrespective of path taken, but depending only on the relative heights in the gravity well.

Yes, but gravitational potential is a Newtonian thing, and that applies (always) in a Newtonian analysis.

I was going to use gravity assist / planet flyby as an example of how an object can climb out of a gravity well using less energy, in a changing gravitational field. However, in Newtonian physics, the planet's gravitational force acting on the object equals the force of the object acting on the planet, and the object still uses the same energy to climb, it just gets it from the planet. So it's not the case that a dynamic system alone breaks gravitational potential's path independence.

GR is a different system, and I don't understand where and how the analogy to gravitational potential fails. But for example, I read that if you separate two masses, the GR analog to gravitational potential depends on if the masses are spinning, but the Newtonian gravitational potential doesn't.

Edited by md65536
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21 hours ago, md65536 said:

But for example, I read that if you separate two masses, the GR analog to gravitational potential depends on if the masses are spinning, but the Newtonian gravitational potential doesn't.

That’s because the geometry of spacetime around a spinning mass is very different from that around one without angular momentum. In the former case, you need to perform more work to escape to infinity if you choose a path that runs counter to the direction of rotation, due to frame-dragging; in other words, there is no path-independence here.

On 12/5/2021 at 9:03 AM, MigL said:

Gravity is a conservative force

In GR this is true only in some highly symmetric special cases, but not in general.  In Newtonian gravity it is always true, of course.

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• 2 months later...

I personally think you need to look at the analogy in terms of a schematic view; the sort of thing a lecturer might use to demonstrate a 1st order 'solution' for an elastic surface curved by matter.

Since, in my further opinion the videos from NZ successfully demonstrate several aspects of the Newtonian model, I congratulate the faculty for the eye-opener: spacetime is an elastic medium, distorted by the presence of massive objects.

Since you can represent a lot of Newtonian orbital dynamics (modulo the friction), the elastic sheet is just the thing to let local gravity demonstrate an elastic force in action. Notice in the videos the initial vertical oscillations die out, the sheet is then in equilibrium so mg = kx for the background. The experiments in the videos are using Newtonian mechanics in a clever way, to show there's a good heuristic in there, somewhere.

The bending of light rays from distant objects around the deformations, shows that the heuristic of a two dimensional sheet of distorted spacetime works for rays of light in two dimensions. We know this is true independently of coordinates, so it's just one sheet but there's a lot more of them.

The UOW videos are serious science, except with a constant g everywhere where you only need one sheet, so why not use one with the right k, and stretch it as uniformly as practical to give it the same tension everywhere?

I personally would like to see measurements of sound profiles, when striking the sheet with and without a massive object in the middle: what does the tonal response say about the tension, i.e. how well tuned is it?

I just realised another problem with the analogy, the physical model is a catenary 'minimal surface' in a gravitational field; initially it looks like it's loosely tensioned, the problem is fixed by assuming it's actually flat, the bending in the sheet is because it has mass, you need to factor this out of the model to get a spacelike sheet.

Edited by SuperSlim
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One, maybe last, observation.

Because of the one-dimensional gravity field, determined by the local g, the rubber sheet model demonstrates curvature in an orbital plane. But an observer of curved geodesics for light rays can posit a 2d slice anywhere; the choice is arbitrary but determined by the direction of gravity in the experiments.

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