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How far does gravity's influence extend?

I've recently (re-)seen 'The Universe' season 1 where it's said that 'take these two dice, place them perfectly still in empty space 1 inch apart and within an hour they'll touch due to their gravitational attraction'.

I get the explanation and it makes sense.

Imagine a universe that has NOTHING AT ALL in it (except the properties of spacetime as we know it, static (therefore devoid of dark energy/matter) and the two die (dice)) - placed an inch apart, they'd eventually touch, a metre apart would they still eventually touch? a kilometre? A mile? A light year? xxx light years?

i.e. if there were no other gravitational influences, would the dice eventually touch if they were placed at opposite sides of the visible universe (given that the extended un-visible universe were also devoid of any mass)?

I believe yes as Newtonian physics states the attraction is the inverse square of distance - the attraction would be almost infinitely small, but surely there would be some and would take 'some time' to touch.

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In an expanding universe, the farther away an object is, the faster is it's recessional velocity relative to an other object at distance $r$:

$$v_{rec}=H(t) \cdot r$$

with $H(t)$ as the current Hubble parameter. On the other hand, for every given distance there is also an escape velocity

$$v_{esc}=\sqrt{\frac{2\cdot G\cdot M}{r}}$$

If you want your 2 bodies to stay at a fixed distance you have to solve for $r$ in

$$v_{rec}=v_{esc}$$

to cancel attraction and expansion, and you get

$$r=\sqrt[3]{\frac{2\cdot G\cdot M}{H(t)^2}}$$

where $M=m_1+m_2$, assuming of course that there are no other masses disturbing the interaction.

Since $H(t)$ decreases with time, but converges to a constant value in the future, technically such a distance where the 2 objects would keep their distance fixed does not yet exist, but it will in a few billion years when the ratio of dark energy to matter will be practically 100:0 and the rate of expansion almost constant.

If we lived in a non expanding universe there would be nothing to cancel gravity, so the bodies would always attract each other with $F \propto 1/r^2$.

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Well, in your very ideal system and from the classical point of view, yes the dices (considering they as an electromagnetic neutral system) will be in touch at some time independent of where are they at the beginning.

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  • $\begingroup$ It's a thought experiment (or perhaps a proposal) just as Einstein did in his 20's with the clocks. $\endgroup$ – Ken Alton Apr 11 '16 at 19:38

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