B Gravity Between Two Stars 45 Billion Light Years Away?

[NODARman]

TL;DR

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Hi, mathematically in the F = GMm/r^2 equation r can be very close to infinity (or the size of the universe), but gravitational force always will be some number.
But how is that in the real world? Let's say we have a perfectly empty universe but only with two sun-like stars. If they are away from each other like 45 billion light years, then is their gravitational force still some number or 0?
I took a zero because I'm wondering if space has rubber-like physics. 2D rubber plane with x, y geometrical dimensions and z gravity. If its area is huge (like millions of kilometers ^2), then a small metal ball cannot deform this stretched elastic thing thousands of kilometers away, right? (or it can but it will approach zero?)
If you know what I mean...
Thanks.

 

[Ibix]

The reach of gravity is infinite in both Newtonian and relativistic gravity, so its effect is always non-zero.

Don't try to reason from a rubber sheet model - it is (at best) illustrative of some situations, but is not remotely rigorous. It would be like trying to deduce how a steam train works by looking at a child's wooden toy.

 

[PeroK]

TL;DR Summary: .

Hi, mathematically in the F = GMm/r^2 equation r can be very close to infinity (or the size of the universe), but gravitational force always will be some number.
But how is that in the real world?

The equation of Newtonian gravity you quote is a mathematical model. We can test it in the real universe to some extent. But, there are limits to the experimental data we can collect. For example, there are always multiple sources of gravity and we cannot empty the region of space around the Earth or a star to test the theory beyond what nature allows us to. All we can say is that the equation holds to the extent that we have been able to test it. Moreover, on the largest scales we have an expanding universe and we need GR to model that.

Note that there is no such thing as "close to infinity". We can only say that $r$ is large compared to other factors.

 

[phinds]

... can be very close to infinity

To expand slightly on what Perok pointed out (because this is a concept that throws people when they're new to it) here's the problem. If you take the largest number that has ever been written down, in whatever exotic form so that it huge beyond belief, the distance from that number to infinity is still ... infinity. So, you aren't any closer than you were at zero.

 

[Herman Trivilino]

If they are away from each other like 45 billion light years, then is their gravitational force still some number or 0?

Many things in physics and engineering can be negligibly small. Theory may predict a small nonzero value, but that doesn't mean it can be measured, because measuring instruments have limitations.

 

[haushofer]

In your model you should always consider for which values of your parameters the model is still valid. Physicists often do that implicitly. In quantum field theories this question becomes hugely important due to renormalization issues, but already in Newtonian physics this issue plays a role.