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I have been recently comparing Newton's and Einstein's laws of gravitation. Most of the authors in physics seem to claim that Newton's laws were incorrect. One of the important aspects that they consider about the laws is that they explain that the gravitational effect is instantaneous. My question is, is it something that should be obvious from the sets of equations that Newton propounded? How did physicists first come to realise that Newton's laws described the idea of instantaneous action?

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    $\begingroup$ As a minor remark, Newton himself knew that his laws very profoundly inherit the instantaneous action. It was something he couldn't really explain. If someone can find a reference regarding this, kindly comment. $\endgroup$ – Feynmans Out for Grumpy Cat Jun 16 '17 at 21:31
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    $\begingroup$ There just isn't any t in Newton's law of gravity. $\endgroup$ – Ben Crowell Jun 16 '17 at 22:08
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Look at the laws of motion for a particle at position $x(t)$ subject to the gravitational force of a particle of mass $M$ at position $X(t)$:

$$\ddot{x}(t) = -GM\frac{x(t)-X(t)}{\|x(t)-X(t)\|^3}$$

When $X(t)$ changes, $\ddot{x}(t)$ changes too: same time $t$, i.e. no delay. Such a delay would look like replacing $X(t)$ by

$$X_\text{retarded}(t) = X\left(t - \frac{\|x(t)-X(t)\|}{c_g}\right)$$

where $c_g$ would be a naive speed of gravity. Such a theory is completely ruled out.

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