General relativity when can we approximate to Newtonian gravity?

Question

Lets consider this scenario in deep void of space where other curvatures of large objects are negligible in this case and we bring 2 objects lets say $A$ and $B$.

We give it a force slightly lower Force ($F$) than the gravitational force of attraction ($F_{g}$) of both objects in sense of Newtonian gravity in the opposite direction of A to B and vice versa. Now according to Newtonian gravity the force makes them move towards each other overpowering the force actually given to the objects and then collide (Image for reference)

But when we see in the sense of general theory of relativity we see that gravity is a curvature of space-time so when they move in opposite direction to each other and will never meet!

I have seen many people say at small instances GR can be approximated to Newtonian gravity but From this simple example it isn't the case , when can we approximate GR to Newtonian gravity?

You said, "we give it a force..." Did you maybe mean to say, we give it an impulse? "Force" implies a continuous pushing. "Impulse" is momentary. If you give a pair of mutually gravitating bodies an impulse that causes them to move away from each other at less than their mutual escape velocity, then they eventually will reverse direction and come back together again. Both Newton's mechanics and Einstein's mechanics predict it. — Solomon Slow, Apr 01 '23 at 15:15
Gravity is the curvature of space and time. Why do you think, “when they move in opposite direction to each other and will never meet!”?. — Ajay Shanmuga Sakthivasan, Apr 07 '23 at 09:26

score 9 · Accepted Answer · answered Apr 01 '23 at 15:34

Suppose we have an object in a gravitational field and the potential energy of the object is $\Phi$. Then the magnitude of the deviations from Newtonian physics due to general relativity are of order $\Phi/c^2$. So Newtonian physics will be a good approximation when:

$$ \frac{\Phi}{c^2} \ll 1 \tag{1} $$

This comes from the weak field limit of general relativity and the derivation is a little involved, but there is a way to see why this relationship exists. Imagine an object falling into a gravitational field from a large distance, then the change in the gravitational potential energy will be equal to the increase in its kinetic energy, so we get:

$$ m\Phi = \tfrac12 mv^2 $$

and we expect relativistic effects to become important as the velocity $v$ becomes comparable to the speed of light i.e. when:

$$ m\Phi \approx \tfrac12 mc^2 $$

and rearranging gives:

$$ \frac{\Phi}{c^2} \approx \tfrac12 \tag{2} $$

And you can see the similarity between this equation and our equation (1) above.

score 7 · Answer 2 · answered Apr 01 '23 at 20:28

But when we see in the sense of general theory of relativity we see that gravity is a curvature of space-time so when they move in opposite direction to each other and will never meet!

Why not?

As you said, they can come together until collide if the separating force is not great enough according to Newtonian gravity. And Newtonian law of gravity can be interpreted as the equations for a geodesic in a curved space-time of 4 dimensions. See for example this answer.

GR is a tiny correction for examples like that, where velocities and masses are not too big.

General relativity when can we approximate to Newtonian gravity?

2 Answers2