# Newton v. Einstein gravitation

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Newton gives a formula for calculating a gravitational force. Is there a similar formula for GR ?

Is the einsteinian force (always) greater than, or less than, the newtonian ?

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GR does not consider gravity as a force. Gravity is the outcome of the curvature of space-time caused due to mass.

Newton did not saw such a picture. He considered it as a purely attractive force.

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The 'model' is certainly different, but the predictions made by GR and Newtonian gravity are the same in the low mass/energy limit.

At higher mass/energies, such as close to a deep gravitational well, we start seeing deviations due to 'curvature' ( Mercury's perihelion and its procession for example )

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"We start seeing deviations...". My question concerns these deviations. I asked how the GR force is calculated, and if it is greater or less than the Newtonian formula predicts.

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You cannot directly calculate force in GR because gravity is not treated as a force.

And it isn't as simple as greater or less; the results are just different. In most cases, the differences are too small to be noticed. For example, in Newtonian gravity an object can be in a stable elliptical orbit around a star. The path will not change; the planet will keep going round the same ellipse. (Apart from external influences like other planets.) But if you work out the path using GR then the ellipse will change its position slightly (precess).

Newtonian gravity is good enough in most cases but (as noted above) we see a small anomaly for Mercury which can only be explained by GR.

There are various other things that the two theories give different results for, such as the amount of gravitational lens, "frame dragging, etc.

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One famous example: the gravitational bending of star light towards the Earth is twice the amount predicted based on Newton's light particle theory.

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I like to imagine two masses A and B, far enough from all other bodies that the gravitational attraction of the latter is negligible.

Newton gives us a formula for calculating the attractive force between A and B. Is the formula exact, even for very large / small masses, and for a very large / small distance between A and B ?

OR, does GR kick in and predict discrepancies even in this simplest of scenarios ?

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In GR there is a class of solutions the the Einstein field equation called the Newton limit. The lecture notes by Sean Carroll gives a good coverage

https://arxiv.org/pdf/gr-qc/9712019.pdf

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