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I have this expression for the gravitational force between 2 masses when the gr term is added:

$$F = - \frac{GMm}{r^2} + \frac{4G^2mM(M+m)}{r^4c^2}$$

which I got from the internet since I haven't bee able to find a similar expression in any book. But i'm not sure if it's right or needs fixing.

Is the second term in the force correct?

6548873432486
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    The question I've linked gives the exact expression. If I take the equation from that answer and expand the square root then I find the first term is $G^2M^2m/c^2r^3$ which is close to your expression is you've made a mistake and put $r^4$ instead of $r^3$. As velut luna says dimensional analysis suggests the power of $r$ is wrong. – John Rennie Apr 29 '17 at 05:59
  • I see, so there shouldn't be a factor of 4 there, and $1/r^3$ instead of $1/r^4$ – 6548873432486 Apr 29 '17 at 06:00
  • If you look at the answer I linked, the Newtonian force is multiplied by $\left(1 - 2GM/c^2r\right)^{-1/2}$. Doing a binomial expansion as far as the first term gives $\left(1 + GM/c^2r\right)$. Multiply this by $GMm/r^2$ and you get the result I gave. – John Rennie Apr 29 '17 at 06:09

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I only know a little bit of GR. But by checking dimensions, it should be wrong.

$$\left[\frac{GM}{r^2c^2}\right]=\left[\frac{GM^2}{r}\right]\left[\frac{1}{Mc^2}\right]\left[\frac{1}{r}\right]=\left[\frac{1}{r}\right]$$

which is not dimensionless.

velut luna
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