Why does the bending of light not depend on the wavelenght or energy of the photon?

Question

(First paragraph deleted, according to comments see below, because erroneously assuming that gravitation is described by Newton's three basic laws of motion.)

Can it be derived from Newtonian laws that the angle of the bending of a photon's path moving close to earth does not depend on its energy and does not vary according to the wavelength of the photon? Even if all objects underlie one and the same acceleration in speed (termed g) they gain different momentum and energy when gravity exerts (red-shift, blue-shift). How can it be excluded that this gain of momentum and energy does not translate into differences in angles of inflections (as with Raleigh scattering which is a phenomenon of electro-magnetic, not gravitational field)?

In relativistic physics, there is the principle of equivalence of energy and mass. A photon that has no rest mass but has energy should be accelerated and bent by gravitational force according to its energy thus wavelenght.

However, the formula on the angle of bending and on the shift of frequency show that both angle of curvature of path and change of wavelength/frequency (redshift/blueshift) do not depend on the energy/wavelength/frequency/relativistic mass of the photon.

Why do mass and energy "cancel out"? The Newtonian deflection angle is: θ=2GM/rc2.

Related:

"Does 'special relativity + newtonian gravity' predict gravitational bending of light?

Answer 1

Basics of Newtonian gravitational force

• Bodies are attracted by gravitational source. If the body initially does not move towards the center of force, its trajectory will be curved.
• A higher initial velocity causes a lower curvature.
• The component of motion perpendicular to the center of force remains constant (in airless space).

Basics of the constancy of the speed of light

• All photons move with the same velocity. Thus, they all follow gravitogeodesic paths.

Basics of Einstein's understanding of gravitation

• Gravity exerts an attraction on other bodies, but is not a force. On the falling body no acceleration acts (we are always weightless at the free fall in the airless space!, we feel no force).
• The gravitational potential is not only composed of the surrounding masses but is self-amplifying, the larger the surrounding masses are.
• With c, the gravitational potential in the space can be described unambiguously.
• With increasing gravitational potential the speed of light slows down (from an external observer far away from the point of changing gravitational potential).

To your question about the redshift

• The geodesic of bodies changes seriously with the changing value of the mass of the central body.
• The geodesic also changes seriously, if the body is in relation to the other gravitational source in the same order of magnitude.
• The geodesic changes - not so seriously - also still because of the self-amplifying gravitational effect. For example, our earth orbit would be more curved in the area of an umpteen times heavier sun than to be expected with the Newtonian formula.
• What has this to do with the redshift of photons? Emitted photons of heavier masses (etalon: our earth) are redshifted.

To your question about the geodesic of photons of different frequency

• The gravitational potential between two bodies is self-amplifying (thus not simply additive).
• If another body is added, the speed of light decreases (as always, of course only for an external observer from a space area with invariable gravitational potential).
• If one agrees that also the presence of energy increases a gravitational potential, thus photons of different energy content change the gravitational potential differently and thus also the local c. But this effect is purely academic and will play a role at most for gamma bursts etc..