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Inspired from the commentary on this question.

How far apart do two objects have to be for the gravitational force between them to be negligible? By negligible I mean, that it could never be measured rather than a technical limitation that results in a measurement/force of zero and/or the accuracy of the measurement is too imprecise to be of use.

Basically when is the force to small to ever be measured, now or in the future? Is there a limiting distance?

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Unfortunately, since you changed the question after people answered, you're not getting a key part of your (edited) question included in answers, which is: how do we measure gravitational forces between distant objects, and what are the hard limits to that? – Chelonian Aug 18 '15 at 15:04
@Chelonian I changed it about 30 seconds after the first answer, so these people have had a long time to edit. – Zach466920 Aug 18 '15 at 15:05
up vote 5 down vote accepted

In classical gravity, the answer is "never". In general relativity, the answer is "never".

Now what about a quantum theory of gravity? We don't know how it'll work, but it should reduce to general relativity in the classical limit (i.e. the limit of weak fields and large distances, which is exactly what your question is about). So the answer is still "never".

What about when the force gets less then the "Planck force" or whatever? "The Planck foo" is completely meaningless; it's just the foo that can be formed out of $\hbar$, $c$, and $G$. The significance is that because it has all of those constants in it, the Planck foo might be a quantity that comes up in quantum gravity.

It turns out the Planck force is over $10^{44}$ Newtons, so it might be an indication that something weird happens when forces get that high. Again, it says nothing about what happens when forces get low; that should just continue to work.

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So you're saying there is a measurable force no matter how far apart the objects are? By measurable, I mean actually measurable, not a number that is greater than zero. – Zach466920 Aug 16 '15 at 21:10
I believe there's no reason in principle you can't do the measurement. Will any of us actually ever be able to actually do the measurement though? Nope. – knzhou Aug 16 '15 at 21:16
Well, what do you mean by practical? You've never defined it. If you mean, "I can measure it with a meter stick and a stop watch in an afternoon", well, there are lots of well-accepted forces which cannot be practically measured. – WhatRoughBeast Aug 16 '15 at 21:21
@WhatRoughBeast by practical I meant physically possible to do using the resources in reality. For instance, I'd presume traveling to the singularity of a black hole and back impractical in addition to not theoretically possible. – Zach466920 Aug 16 '15 at 21:28

No distance is far enough.

Among other things, if you are extremely far away, then there is room for lots and lots of things to be far away from you and even if they individually have little effect we can find the net effect of all of them. So we know the effect of each one is not zero.

So we can prove the effect of A on B is not zero even when they are far far apart by having many As and seeing the total effect on B.

When you are far away you are being affected by the distant past though. Since that is what affects you. But you are still affected.

Here is an example. For any star of any mass you can orbit it at any distance (provided you aren't too close to the event horizon). If you are far away the period of your orbit is just longer. Its like how the year is a certain duration here on earth but a planet farther out would just take longer to go around a circle.

You can go around the circle at a huge radius and the Planck length isn't a factor. The Planck length is about quantum mechanics and gravity but in the regime where quantum gravity reduces to the kinds of experience we see every day you get the weak field limit of classical GR and you can orbit at any large enough radius.

So gravity affects you even when you are really really far away and quantum mechanics doesn't change that. Since no one knows why you think otherwise or even why you might that's all I can tell you.

So you're saying there is a measurable force no matter how far apart the objects are? By measurable, I mean actually measurable, not a number that is greater than zero.

Yes. If you are a distance $R$ away and have a mass $M$ then you feel of force of $F=GMm/R^2$ when you are in the classical GR weak field limit (which exists at large lengths). And so you accelerate at $Gm/R^2$ which from $v^2/R$ for circular motion means $v=\sqrt{Gm/R}.$ So we can relate your period and total distance travelled which are all things that can be actually measured out therein the region where the classical GR weak field limit holds. We have $v=D/T$ which can be measured ($D$ and $T$ can be measured) then we can note that it will be equal to $\sqrt{Gm/R}.$ All that happens when $R$ is very very large is that $v=\sqrt{Gm/R}$ becomes very small which means $D$ will be small compared to $T.$

Note that if your speed gets large compared to $c$ you will need to use SR dynamics instead of Newtonian. But it is still classical GR weak field limit so the Planck length will not be relevant.

Now if you want to argue that the cosmological constant or dark energy is s limit then you have to argue if is constant (versus dynamic) and whether that is a technical issue that we can get around by placing matter in the space between the two objects.

And the whole light cone issue is still there, but if you have expansion you might only have access to a finite portion of the sending objects history. Is that your concern? I feel like I'm guessing here, which makes me thing the question might be too broad.

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I changed the question – Zach466920 Aug 16 '15 at 20:59
@Zach466920 It doesn't look very different to me (maybe the changes are unclear and/or vague). But I took that as a request for more details so I added an example. You asked "what distance" and the answer is "none." There isn't much more to say. – Timaeus Aug 16 '15 at 21:12
Your last sentence is hard to parse...So you're saying there is a measurable force no matter how far apart the objects are? By measurable, I mean actually measurable, not a number that is greater than zero. – Zach466920 Aug 16 '15 at 21:14
@Zach466920 Yes we can measure it. – Timaeus Aug 16 '15 at 21:28
much better, one more question. If we were to measure the force of two objects that are far apart such that the uncertainty is within say %1 then would taking the measurement upset the systems position by the uncertainty principle? For instance might observing the force from earth to say some far way galaxy upset the system in a macroscopic way? – Zach466920 Aug 16 '15 at 21:43

The question is ill-posed.

At the classical level, the force (gravitational or otherwise) between objects never becomes zero. It goes to zero as the distance goes to infinity, but it never really becomes zero.

At the quantum level, we don't have a theory of gravity, but already the concept of "distance" doesn't make precise sense anymore, since quantum states are very rarely position eigenstates (in fact, almost never, since the position eigenstates tend to lie outside the Hilbert space of states), so their "distance" is not well-defined anymore. Also forces don't work by being real number, but by providing certain types of interactions.

So, either way, there is no distance at which the force (gravitational or otherwise) would become zero.

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I changed the question – Zach466920 Aug 16 '15 at 20:59
@Zach466920: It is considered very bad style to change a question so that it invalidates (two!) existing answers. Also, the Planck scale is not a scale that limits measurements – ACuriousMind Aug 16 '15 at 21:01
I'll ask this to clarify: So you're saying there is a measurable force no matter how far apart the objects are? By measurable, I mean actually measurable, not a number that is greater than zero. – Zach466920 Aug 16 '15 at 21:15
@Zach466920: Well, at some point we won't currently have the instruments to measure it, but there is no theoretical limit to how precise measurements can be. – ACuriousMind Aug 16 '15 at 21:19
Ok, but so we're clear, Wikipedia states that the weak and strong force do have a range, which is why I question the "infinite" range of the other two forces. – Zach466920 Aug 16 '15 at 21:35

If the two objects are electrons then the gravitational force is always negligible.

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I can see why this was downvoted for not being an actual answer, however it does bring up the crucial point that "negligible" must be defined wrt something. – fqq Aug 16 '15 at 21:52

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