B Is the effect of gravity limited by distance?

[lifeonmercury]

Assume the universe is flat and neither expanding nor contracting whatsoever, and the only matter in it are two marbles separated from each other by a distance of 10 million light years. Would the marbles eventually begin pulling toward each other (after 10 million years)?

 

[Bandersnatch]

What do YOU think would happen, and why?

 

[lifeonmercury]

I would guess they would pull toward each other because the effect of gravity never truly tapers off to zero. Now will you please tell me what YOU think would happen?

 

[Bandersnatch]

they would pull toward each other because the effect of gravity never truly tapers off to zero

There you go, you've answered your own question. :)
Your intuition was good here, as it should be if one's ever seen the gravitational force equation: $F=GmM/R^2$ - no matter how large a distance you substitute for R, the resultant force never equals 0.

 

[Mechanic]

Interestingly enough, according to Chernin, et al. (https://arxiv.org/abs/0706.4068) the marbles will recede away from each other if they are beyond a certain distance apart. Per Chernin, gravity and antigravity balance each other out at what they refer to as the “zero gravity surface” and they refer to the distance to this surface as Rv. At distances less than Rv the marbles would accelerate towards one another. At distances greater then Rv they would accelerate away from each other. A smooth transition from the tendency to accelerate towards one another at relatively short distance to the tendency to accelerate away from each other over relatively large distances has been proposed.

 

[Bandersnatch]

the marbles will recede away from each other if they are beyond a certain distance apart

Not in the setup as stated in the OP, i.e. in a flat, static universe.

 

[Vanadium 50]

Mechanic, this is a B-level thread. And, as Bandersnatch points out, it's for a different setup.

 

[grauitate]

There you go, you've answered your own question. :)
Your intuition was good here, as it should be if one's ever seen the gravitational force equation: $F=GmM/R^2$ - no matter how large a distance you substitute for R, the resultant force never equals 0.

By the way, there is an answer of this interesting question from another aspect, too: in an universe filled with only two marbles of some kg, regardless in which distance they are located, the gravitational constant G (6.674e^-11 m³/(kg s²)) would not have this value but would be nearly exactly zero instead. Regard that the estimate mass (matter and energy) of the real universe is approx. 10⁵² kg and that the gravitational interaction of all this stuff still results in this small number of G.
So, the gravitational attraction of these two 1 kg-marbles would be theoretically F = Gm1m2/R² = G/9*10⁴⁵ = 7.5e^-57 N
and now assuming G as nearly zero as e.g. ≅10^-40 m³/kgs² F would reach 10^-100 N, a number which reliably is at least in the vicinity of 0 with some confidence. Or spoken the other way round: these tiny force of 10^-100 N would be much lower than any quantum fluctuations which might be present and within this bias of vibrating froth nothing (i.e. 0) could be detected.

 

[Bandersnatch]

By the way, there is an answer of this interesting question from another aspect, too: in an universe filled with only two marbles of some kg, regardless in which distance they are located, the gravitational constant G (6.674e^-11 m³/(kg s²)) would not have this value but would be nearly exactly zero instead. Regard that the estimate mass (matter and energy) of the real universe is approx. 10⁵² kg and that the gravitational interaction of all this stuff still results in this small number of G.

Can you provide a reliable source for the claim that the value of G is dependent on the energy content of the universe?

 

[grauitate]

Can you provide a reliable source for the claim that the value of G is dependent on the energy content of the universe?

All comes from the foundation of Einstein's General Relativity, which is fundamentally based on the phenomenon that heavy mass is exactly equal inertial mass and the Mach's statement that the inertia comes from the gravitational effect of the sum of all masses in the universe.
The inertial movement of the Foucault pendulum simply shows the rotation of the earth, because it does not follow the Earth's rotation but its polarisation plane stands still against the masses of the universe.
Or citing a joke from H. Pfister, M. King: Inertia and Gravitation, Lecture Notes in Physics 897, Springer Switzerland 2015, p. 137 (fine print), where E. Schucking is quoted as "When one of Mach's principle promotors, Dennis Sciama, slammed on the brakes of his car, propelling his girlfriend, seated next to him, towards the windshield, she was said to be heard moaning, `All those distant galaxies´."
Its just a logical and axiomatic reason that e.g. the gravitational acceleration around a heavy mass is a = GM/R² (still non-relativistic) where G is the proportional constant determined by the gravitational (= inertial) effects of all masses of the universe, arriving locally at every moment simultaneously from all cosmic distances (i.e. from all past times). Reading Einstein's papers of 1915...1922 and todays experimental verifications e.g. of the frame dragging effect confirms this reason.
So, for the first, I'd recommend this source (among uncountable others):
H. Pfister, M. King: Inertia and Gravitation, Lecture Notes in Physics 897, Springer Switzerland 2015, p133 ff, Chapter 4.3 "Realisation of Machian Ideas in Cosmology and in Nature"
Or answering with a counter-question: What other than the mass and inertia of the world shall determine the value of G? Just a number by accident?

 

[PeterDonis]

All comes from the foundation of Einstein's General Relativity

No, it doesn't. In GR, G is a constant and can't change. There are alternate theories of gravity in which the equivalent of G can vary, but all of them are either ruled out by experiment or confined by experiment to a range of parameter values which makes them indistinguishable from GR and G effectively constant. (Brans-Dicke theory is an example of the latter.)

The inertial movement of the Foucault pendulum simply shows the rotation of the earth, because it does not follow the Earth's rotation but its polarisation plane stands still against the masses of the universe.

This is not correct. The pendulum's plane is affected by the Earth's rotation; the effect is just very small. Gravity Probe B tested for the same effect (frame dragging) and found it.

todays experimental verifications e.g. of the frame dragging effect confirms this reason.

It confirms it in the sense that it shows that the distribution of stress-energy in the universe affects the spacetime geometry (in this case the Earth's rotation affects the curvature of spacetime around the Earth), but we already knew that.

It does not say anything about whether G is constant, because frame dragging occurs in both GR, where G is constant, and alternate theories of gravity where the equivalent of G varies.

What other than the mass and inertia of the world shall determine the value of G?

In GR, nothing does; it's a constant that can't be predicted by the theory and has to be given a value from experimental measurements.

 

[nikkkom]

the Mach's statement that the inertia comes from the gravitational effect of the sum of all masses in the universe.

Mach principle is just an idea, it's not an established fact. It is neither confirmed not disproved by current experimental data.
For one, General Relativity is not a machian theory.

 

[PeterDonis]

General Relativity is not a machian theory.

I think this depends on your definition of "Machian". Cuifolini & Wheeler wrote a whole textbook, Gravitation and Inertia, arguing that GR is a Machian theory, using their definition of "Machian". Others have said it isn't, using their (different) definition of "Machian". So one really needs to explicitly say, mathematically, what they mean by "Machian" or "not Machian" in order to make a claim like this.

 

[john george pa]

would not quantum effects come into play also ?

 

[Nugatory]

would not quantum effects come into play also ?

Not in the problem posed here. They're well and thoroughly negligible when you're dealing with marble-sized masses in free-fall in otherwise empty space.

 

[ProfChuck]

would not quantum effects come into play also ?

Yes, quantum effects must be included to complete the model. There are two (that come to mind); photon pressure and uncertainty when the space time curvature is on the order of the Planck length.

 

[PeterDonis]

There are two (that come to mind); photon pressure and uncertainty when the space time curvature is on the order of the Planck length.

Both of which are irrelevant for the problem posed in this thread.

 

[nikkkom]

I think this depends on your definition of "Machian". Cuifolini & Wheeler wrote a whole textbook, Gravitation and Inertia, arguing that GR is a Machian theory, using their definition of "Machian".

My definition of "machian" is that the math of the theory in question should include effects of distant masses/objects on the local conditions. IIRC math of GR doesn't. Also, there are valid global GR metrics where entire Universe rotates, which shouldn't be possible if Mach principle is true.

https://en.wikipedia.org/wiki/Gödel_metric
https://en.wikipedia.org/wiki/Van_Stockum_dust

 

[ProfChuck]

Both of which are irrelevant for the problem posed in this thread.

I'm not so sure about that. In this wildly hypothetical scenario any effect whose value can be calculated should be included in the model. It may be that the problem is insufficiently constrained by the question. If each marble radiates photons, which it would do unless it was at zero Kelvin, they will exert photon pressure on each other. Also if space-time is quantized then gravitation becomes uncertain at a calculable distance.

 

[PeterDonis]

If each marble radiates photons, which it would do unless it was at zero Kelvin

If the marbles are alone in the universe (no CMBR, no other radiation), then they would be at zero Kelvin, since there is no radiation around for them to establish a thermal equilibrium at finite temperature. Of course this is highly idealized, but so is the scenario as a whole.

if space-time is quantized

Which is a speculative hypothesis that has nothing to do with this scenario, which, as I understand it, assumes classical gravity.

 

[PeterDonis]

then gravitation becomes uncertain at a calculable distance.

This is also not the case, since even on the assumption that spacetime is quantized, nobody knows how to calculate this distance; all we have is handwaving assumptions that it should be something around the general order of the Planck length.

 

[ProfChuck]

If the marbles are alone in the universe (no CMBR, no other radiation), then they would be at zero Kelvin, since there is no radiation around for them to establish a thermal equilibrium at finite temperature. Of course this is highly idealized, but so is the scenario as a whole.
Which is a speculative hypothesis that has nothing to do with this scenario, which, as I understand it, assumes classical gravity.

The original question is much like a thought experiment. Thought experiments are useful not because of the questions they answer but because of the questions they pose. The question opens up many new ones. Does a universe that has only two particles in it have or even need three dimensions? What happens to the inverse square law in a universe with other than three dimensions? What is meant by a separation distance measured in light years in such a universe. Does it matter what the "marbles" are made of? and so on.

 

[PeterDonis]

The question opens up many new ones.

I don't think the OP intended to open up any of those questions, and they're not answerable anyway because we would need some theoretical framework in which to answer them and we don't have one. In the framework I was assuming the OP wanted to use (either GR or Newtonian gravity, they're basically equivalent for this problem), the questions you are asking can't even be posed. That's not to say they aren't interesting questions, just that they're out of scope for this discussion (and probably this forum, since, as I say, you would need a theoretical framework in which to pose them and there isn't one).

 

[ProfChuck]

I don't think the OP intended to open up any of those questions, and they're not answerable anyway because we would need some theoretical framework in which to answer them and we don't have one. In the framework I was assuming the OP wanted to use (either GR or Newtonian gravity, they're basically equivalent for this problem), the questions you are asking can't even be posed. That's not to say they aren't interesting questions, just that they're out of scope for this discussion (and probably this forum, since, as I say, you would need a theoretical framework in which to pose them and there isn't one).

Touche'

 

[newjerseyrunner]

There is way too much discussion about unknown variables in this thread. You know the old joke where you ask a bunch of people a question and they all get slightly different answers? It's usually about chickens and the punchline is that the physicists calculations have an exact value with the clause that all chickens are assumed to be perfect spheres in a vacuum. I always assume chickens are spherical unless otherwise stated (marbles are spherical, have positive mass, zero temperature...) Everything that's not relevant is idealized. Did you know that Dirac's formulation of QED only works if there is one electron in the whole universe?

If the question specifically mentions gravity and a flat universe, QM should never enter the thread. Gravity and QM don't play nicely and there is no reason to make assumptions about how they might. In Newtonian physics, the force can never go to zero without an infinite distance between them.

In GR, if you start with empty space that is perfectly flat and has no pressure, then adding anything to it will cause curvature which will not be zero. Sort of. The force between them will remain exactly zero until they have existed long enough for gravity (traveling at the speed of light) to reach from one to the other. QM fluctuations are useless to theorize here because we know that in our universe, empty space has a positive pressure. Since this scenario specifies a specific type of spacetime, we know it's not our universe and features of our universe such as vacuum energy should not be considered.

 

[ProfChuck]

This is an example of how powerful a thought experiment or question can be. A seemingly simple question has stimulated a flood of thoughtful responses. This is exactly what I always tried to stimulate when I was teaching. Wonderful.

 

[PeterDonis]

In GR, if you start with empty space that is perfectly flat and has no pressure, then adding anything to it

...violates conservation laws.

The force between them will remain exactly zero until they have existed long enough for gravity (traveling at the speed of light) to reach from one to the other.

Not if we obey conservation laws. Stress-energy can't just appear (or disappear); it is locally conserved. Which means that "gravity" can't just appear (or disappear). It is actually rather complicated to formulate a scenario, obeying the laws of GR, in which you can test "how fast gravity propagates", precisely because the obvious ways of doing it, like the one you are trying, violate the laws of GR and hence have no consistent formulation.

 

[GODISMYSHADOW]

I would guess they would pull toward each other because the effect of gravity never truly tapers off to zero. Now will you please tell me what YOU think would happen?

Assume the universe is flat and neither expanding nor contracting whatsoever, and the only matter in it are two marbles separated from each other by a distance of 10 million light years. Would the marbles eventually begin pulling toward each other (after 10 million years)?

If the marbles were orbiting each other at the correct rate of rotation to counteract gravity the distance between them would remain fixed at 10 million light years. However, there's nothing to use as reference to determine if the marbles are orbiting each other, there being no background stars. The marbles could approach each other or not approach each other, and there's no way to predict what they will do.

 

[PeterDonis]

there's nothing to use as reference to determine if the marbles are orbiting each other, there being no background stars.

You don't need one; the distance between the marbles can be established by bouncing light signals between them. So if they are orbiting each other, that distance will either remain constant or be periodic in time.

The marbles could approach each other or not approach each other, and there's no way to predict what they will do.

This is not correct. Given a set of initial conditions, specified in terms of the masses of the marbles, the distance between the marbles (determined as above), and its rate of change with respect to proper time for each marble, GR gives a perfectly definite prediction for how the system will evolve.

 

[rootone]

Two objects with mass can have an angular momentum around their common center of mass/gravity.
A background field is not necessary.