As a non physicist I can understand how Newtonian mechanics falls short in cases of high velocity etc. and is properly generalized by the special theory of relativity.

What is not clear to me is how universal gravitation falls short: under what conditions does it fail to make accurate predictions? What are the cases where the inverse square law actually fails to give accurate results and the general relativity is needed? I do understand that Einstein's theory provides the mechanism, whereas Newton's did not.

Assuming a hypothetical system involving a black hole and a planet like the earth revolving around it (far beyond the horizon, of course) would universal gravitation fail to accurately predict the behavior of the system?

## Best Answer

Generally speaking, and provided you don't stray too close to black holes, you can imagine GR as making small modifications to Newton's law. For example Newton tells that the acceleration of a body falling towards a planet of mass M is:

$$ a = \frac{GM}{r^2} $$

i.e. the famous inverse square law. If you calculate the acceleration for a Schwarzschild metric (which describes spherically symmetric bodies) you get:

$$ a = \frac{GM}{r^2}\frac{1}{\sqrt{1-\frac{2GM}{c^2r}}} $$

(This was calculated in this answer; do feel free to upvote as I think this is a nice calculation, though there are some subtleties to it that you need to read that answer to understand.)

So GR modifies the inverse square law by the factor:

$$ \frac{1}{\sqrt{1-\frac{2GM}{c^2r}}} $$

The modification is small as large distances i.e. when the distance $r$ is much greater than $c^2/2GM$. To give some idea of scale, at the Earth's distance from the Sun the factor is about 1.00000001 so it's almost negligable. Even right at the Sun's surface the factor is still only 1.000002. You need to get close to a black hole for the factor to increase much, and indeed right at the event horizon it goes to infinity!

These corrections to Newton's law may seem small, but they are responsible for phenomena like the precession of Mercury. Although the precession is a well known test of GR, people tend to forget just how small is is. It would take over 4,000 years for the precession to rotate Mercury's orbit by the angle subtended by the Moon from Earth (half a degree).