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Eddington already proved that starlight passing our Sun is deflected more according to GR compared to the outcome of Newton. But does this rate of deflection also depend on the frequency the photons has?

A photon doesn't have a rest mass but his energy could perhaps be considered as equivalent to mass. So the higher the energy of the photon the larger the gravity works? Should the answer be the same according to Newtons'law and GR?

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It does not depend on the frequency of the photon. Formally, the deflection angle is a function of the mass of the body deflecting the light (the Sun in your example):

$$ \theta = \frac{4GM}{rc^2} $$

Intuitively, even in Newtonian gravity how much a particle is deflected does not depend on its mass, the reason being the fact that gravitational mass is the same as inertial mass, and so $m a = mGM/r^2$ is the same as $a = GM/r^2$

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