It is well-known that Newtonian's law of gravitation(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\Phi(\vec{r}) = -\frac{Gm}{|\vec{r} - \vec{a}|}[/tex]

for the gravitational potential ##\Phi## generated by a mass ##m## located at ##\vec{a}## is in several ways inconsistent with special relativity.

What one could do is to modify the definition of ##\Phi## in several ways

[tex]m \to E/c^2[/tex]

[tex]d(\vec{r},\vec{a}) = |\vec{r} - \vec{a}| \to d_\text{SRT}(\vec{r},\vec{a})[/tex]

...

In the 2nd modification one would have to introduce a definition of "distance" respecting Lorentz covariance. So for a gravitating mass at ##\vec{a}(t)## and a test mass at ##\vec{r}(t)## the relative velocity of the two masses should be calculated w.r.t. the relativistic velocity addition formula.

Another reasonable approach is to introduce a "mass-" or "energy-density" ##\rho## and a retarded potential

[tex]\Phi(\vec{r},t) \sim \int d^3r^\prime \frac{\rho(\vec{r}^\prime,t_r)}{|\vec{r}-\vec{r}^\prime|}[/tex]

[tex]t_r = t - \frac{|\vec{r}-\vec{r}^\prime|}{c}[/tex]

I am sure that Einstein worked on these ideas after having published his famous paper in 1905.

My question is whether there are articles where these ideas are discussed and where the various reasons why the fail are shown.

Does the construction not make sense theoretically? (for electrodynamics it does) Or does it fail phenomenologically? (e.g. for deflection of light, perihelion precession)

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# Special relativity and gravitation

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