Atomic Clock Speed graph between two stationary masses

[MatthewKM]

TL;DR Summary

Effect of Equal and Opposing gravitational forces on a clock vs near zero gravity

A question about general relativity and clocks. Clocks run slower in lower gravity. Or said a different way, the oscillation frequency of a subatomic particle approaches maximum as gravitational effect approaches minimum.The further away from the centre of mass of a planetary body, the faster a clock runs (to a limit) My understanding of this is it has historically always been a reference to clocks orbiting large masses and who’s centrifugal inertia holds them at said distance. And of course faster moving clocks run slower so the instantaneous linear velocity of the orbiting clock must be calculated as a function of time such that the decreased clock speed that results can be accounted for.

So the question is: Two non rotating earth sized masses are held suspended in space by some imaginary force analogous to me holding two apples, one in each hand. These planetary size masses are say 500,000 km apart. Running between them is a cable of near infinite strength held taught. I attach an atomic clock to this cable at the surface of mass #1 The attachment has a little motor that takes the clock along the cable at a constant and slow non-relativistic speed from one planetary mass to the other.

What does the speed of the atomic clock do as it traverses the cable from one mass to the other? What does that graph look like? When the clock is equidistant does the clock run at the same speed as it would if it was floating in deep interstellar space far removed from the influence of any massive body.

 

[PeterDonis]

Clocks run slower in lower gravity.

More precisely, a clock at a lower gravitational potential will run slower as compared to a clock at a higher gravitational potential. (Note that this requires that gravitational potential is well-defined, which is only the case for a special class of spacetimes in GR.)

The further away from the centre of mass of a planetary body, the faster a clock runs (to a limit)
My understanding of this is it has historically always been a reference to clocks orbiting large masses and who’s centrifugal inertia holds them at said distance.

I don't know where you are getting that from, but it's wrong. The only clocks for which gravitational time dilation (i.e., the effect of gravitational potential described above) is the only significant effect are stationary clocks--clocks that are "hovering" at a constant altitude and do not rotate with or revolve around the central mass.

As you note, any clock that is moving relative to stationary clocks, for example because it is rotating with the central mass (such as a clock at rest on the surface of the rotating Earth) or revolving about it (such as a clock in Earth orbit) will have two effects on its "rate of time flow": the gravitational potential effect due to altitude, described above, and a slowing of its rate due to its speed relative to stationary clocks. Both must be taken into account to properly understand the behavior of these clocks.

Two non rotating earth sized masses are held suspended in space by some imaginary force

You can't just make up imaginary forces. Anything that produces a force has energy, and anything that has energy curves spacetime. So you have to include whatever is producing the force in your model; otherwise you will get wrong answers.

Since your model is not well-defined, it is impossible to answer your questions because a model that is not well-defined cannot be used to make predictions.

 

[PeroK]

TL;DR Summary: Effect of Equal and Opposing gravitational forces on a clock vs near zero gravity

A question about general relativity and clocks. Clocks run slower in lower gravity. Or said a different way, the oscillation frequency of a subatomic particle approaches maximum as gravitational effect approaches minimum.The further away from the centre of mass of a planetary body, the faster a clock runs (to a limit) My understanding of this is it has historically always been a reference to clocks orbiting large masses and who’s centrifugal inertia holds them at said distance. And of course faster moving clocks run slower so the instantaneous linear velocity of the orbiting clock must be calculated as a function of time such that the decreased clock speed that results can be accounted for.

So the question is: Two non rotating earth sized masses are held suspended in space by some imaginary force analogous to me holding two apples, one in each hand. These planetary size masses are say 500,000 km apart. Running between them is a cable of near infinite strength held taught. I attach an atomic clock to this cable at the surface of mass #1 The attachment has a little motor that takes the clock along the cable at a constant and slow non-relativistic speed from one planetary mass to the other.

What does the speed of the atomic clock do as it traverses the cable from one mass to the other? What does that graph look like? When the clock is equidistant does the clock run at the same speed as it would if it was floating in deep interstellar space far removed from the influence of any massive body.

This post appears to be the result of a steady diet of pop-science material. I suggest consuming from a genuine source on the theories of special and and general relativity. That may cure the intellectual dyspepsia.

 

[phinds]

TL;DR Summary: Effect of Equal and Opposing gravitational forces on a clock vs near zero gravity

Clocks run slower in lower gravity.

No, they do not. Clocks always tick away at one second per second. What IS true, as @PeterDonis pointed out is that COMPARED to clocks higher in a gravity well, lower positioned clocks APPEAR to run slower.

Possibly you are confusing time dilation with differential aging. I second @PeroK 's suggestion that you read some actual physics. Pop-sci presentation are entertainment, not education.

 

[MatthewKM]

I was trying to ask a question about the isolated effect of gravity on relative subatomic clock speed. That is all I am asking I picked a bad hypothetical I though an answer to that question might help me. Guilty as charged.

 

[MatthewKM]

This post appears to be the result of a steady diet of pop-science material. I suggest consuming from a genuine source on the theories of special and and general relativity. That may cure the intellectual dyspepsia.

Please temper your comments. I have not consumed any PopScience" whatever that is. (Books read in the last 6 months: Love and Math by Edward Frenkel, Helgoland and The Order of Time, by Rovelli, Something Deeply Hidden by Sean Carroll, Fundamentals by Frank Wilezek, The End of Everything by Katie Mack, and have listened to the 10+ hr series by Carroll on Physics of TIme and his 6 hr series on Higgs Boson plus a few other 10+ hr Wondrium courses on particle physics and such. Non of this is "PopScience" I suppose my "PopBrain" didn't assimilate deeply enough to resonate here. I forgive you guys as I am quite sure your "Up To Here" with hairbrianed theory but please do temper your comments Save the "PopScience Sarcasm for the third or fourth level rebut rather that the first attack.

 

[PeroK]

I have not consumed any PopScience" whatever that is. (Books read in the last 6 months: Love and Math by Edward Frenkel, Helgoland and The Order of Time, by Rovelli, Something Deeply Hidden by Sean Carroll, Fundamentals by Frank Wilezek, The End of Everything by Katie Mack, and have listened to the 10+ hr series by Carroll on Physics of TIme and his 6 hr series on Higgs Boson plus a few other 10+ hr Wondrium courses on particle physics and such. Non of this is "PopScience"

Those are all pop science. Examples of "real" physics textbooks are:

https://scholar.harvard.edu/david-morin/special-relativity

https://www.preposterousuniverse.com/grnotes/

It's a bit like the difference between listening to a piano recital and learning to play the instrument.

PS here's an example of a question from a physics student (just one I picked from today). Compare this with your question:

https://www.physicsforums.com/threads/on-the-magnetic-dipole-radiation-in-griffiths-book.1053390/

The student could spend 6 hours just on that one problem!

 

[MatthewKM]

OK

 

[Dale]

TL;DR Summary: Effect of Equal and Opposing gravitational forces on a clock vs near zero gravity

What does the speed of the atomic clock do as it traverses the cable from one mass to the other? What does that graph look like?

Assuming that you are in a spacetime that can be represented using a potential, then the potential you are describing would be shaped like a W (but curved). The clock speed would be the central portion of the W, or an upside down v (but curved).

TL;DR Summary: Effect of Equal and Opposing gravitational forces on a clock vs near zero gravity

When the clock is equidistant does the clock run at the same speed as it would if it was floating in deep interstellar space far removed from the influence of any massive body.

Typically no. Typically the middle peak of the W for a potential like you describe will be lower than the potential at infinity.

 

[PeterDonis]

Assuming that you are in a spacetime that can be represented using a potential

Which would not be the case for two planets if they were the only bodies present, since they would fall towards each other and the spacetime would not be stationary.

Two planets held apart by a rod that is under compressive stress could in principle form a stationary system, but, as I noted in post #2, you would have to include the stress-energy of the rod, which would have a significant effect on the spacetime geometry. I'm not sure what the shape of the potential would be for this kind of scenario. However:

Typically no. Typically the middle peak of the W for a potential like you describe will be lower than the potential at infinity.

I think this is a reasonable general expectation for any kind of stationary system where you have massive bodies with space in between them. The obvious simple case is two planets in circular orbits about their common center of mass. In GR this solution is not quite stationary because the system will emit gravitational waves, but the time scale for that is much, much longer than the time scale of the mutual orbits, so one can treat such a system as "quasi-stationary" on the orbital time scale and ask what the gravitational potential looks like. Your description will be true for this case.

 

[PeterDonis]

I picked a bad hypothetical

I describe a better one in the latter part of post #10.

 

[FactChecker]

If you are just interested in an example, you might be interested in the adjustments done for GPS satellites. Their clocks are adjusted for the effects of both their velocity and the gravitational potential.
See this GPS result, and this accidental "experiment".

 

[PeterDonis]

this accidental "experiment".

Note that, as is too often the case with pop science articles, the article misstates the effect. It implies that altitude was the only relevant variable, but of course it wasn't: the orbital speeds of the satellites also varied, and that also contributes to the observed effect.

 

[Vanadium 50]

Two planets held apart by a rod that is under compressive stress could in principle form a stationary system, but, as I noted in post #2, you would have to include the stress-energy of the rod, which would have a significant effect on the spacetime geometry

Dumbell!

No, not you. I am describing the setup.

If I increase the size of the entire system by a factor $a$, the force between planets goes as $a^3/a^2$ or $a$ and the pressure goes as $1/a$. Energy density goes as the square of the pressure, and energy goes as the volume, so the energy goes as $a$. This compares to the mass, whch goes as $a^3$. So build the system large enough and your constraining rod doesn't matter.

 

[PeterDonis]

build the system large enough and your constraining rod doesn't matter

You have shown that the stress and kinetic energy density in the rod is negligible compared to its rest energy density for a large enough $a$. We can make a similar assumption about the planets themselves. But that's not the same as showing that the rod's rest energy density is negligible compared to the rest energy density of the planets. I don't think it can be.

 

[FactChecker]

Note that, as is too often the case with pop science articles, the article misstates the effect. It implies that altitude was the only relevant variable, but of course it wasn't: the orbital speeds of the satellites also varied, and that also contributes to the observed effect.

Good point. It has links to the journal article, which I assume has all the details correct. Unfortunately, it requires a journal subscription.

 

[Vanadium 50]

The require condition is that the rod's energy density (from compression) has the same scale dependence as the spheres,, Otherwise, I could reach any arbitrary balance between the two by making the system big enough or small enough.

The reason this works is that the rod's strength is ultimately set by α but the gravitational pull is ultimately set by G. These have different dimensions: so thee always exist a size beyond whch the connecting rod can be ignored.

 

[PeterDonis]

The require condition is that the rod's energy density (from compression) has the same scale dependence as the spheres

I'm not talking about the rod's (extra) energy density due to compression. I'm talking about its rest energy density. I don't see where anything you've said shows that the rest energy density of the rod can always be made negligible compared to the rest energy density of the spheres by making the system large enough. All I see being shown is that the other parts of the SET, i.e., kinetic energy, "internal energy" due to compression, stresses, for both the rod and the spheres, can be made negligible compared to the rest energy density.

 

[Vanadium 50]

The rod's rest energy density is just its mass, right? A piece of piano wire doesn't get crushed separating two cannonballs, so we have an existence proof.

 

[PeterDonis]

The rod's rest energy density is just its mass, right?

It's its mass density. The components of the stress-energy tensor are densities.

A piece of piano wire doesn't get crushed separating two cannonballs

Sure, and the spacetime curvature produced by all three is negligible compared to that produced by, say, the Earth. But that doesn't mean the spacetime curvature produced by the wire is negligible as compared to the spacetime curvature produced by the cannonballs. The mass densities of all three are comparable, so their Einstein tensor components will be comparable as well.

 

[mfb]

Two planets held apart by a rod that is under compressive stress could in principle form a stationary system, but, as I noted in post #2, you would have to include the stress-energy of the rod, which would have a significant effect on the spacetime geometry.

That rod can have a negligible mass and other contributions will be negligible, too. Just place the planets at a sufficient distance.
It's the equivalent to a massless pulley in classical mechanics problems. Yes, they don't exist, but they are a good approximation.

 

[PeterDonis]

That rod can have a negligible mass and other contributions will be negligible, too. Just place the planets at a sufficient distance.

Once more: the relevant quantity is the mass density of the rod, since that is the significant component of the stress-energy tensor. So far I have not seen an argument for the mass density of the rod being negligible compared to that of the spheres. Mass density is not the same thing as mass.

 

[PeterDonis]

Thread closed for moderation.

 

[PeterDonis]

Thread reopened.