Are photons affected differently by gravity?

[George Jones]

We need to abandon this line of thinking and look at the equations of motion as derived from the Euler-Lagrange equations. You can see that the Lagrangian for the massless photon (corresponding to the Maxwell equations, page 149, eq.1) is quite different from the Maxwell-Proca Lagrangian (by the presence of the term in "m"). Earlier in this thread, before the discussion veered into speculations relative to "massive" photons, I posted the equation of motion for the massless photon as derived from the Maxwell Lagrangian.

No, the Maxwell Lagragian gives the equation of motion (i.e., the field equation) for the electromagnetic field, not equation of motion for the zero rest mass particles in Schwarzschild spacetime.

I would like to challenge someone else, to write down the equations of motion as derived from the Maxwell-Proca Lagrangian. Only then, we can answer if the "massive" photons are "rainbowed" or not in a gravitational field. Unfortunately, since they do not exist, we cannot test the predictions of the Proca theory on this particular case.

The Maxwell-Proca Lagrangian gives the equation of motion (i.e., the field equation) for a massive spin-1 field, not the equation of motion for the zero rest mass particles in Schwarzschild spacetime.

Math gives us the precise answer to your question.
The trajectory of a non-charged test particle in a non-rotating gravitational field is given by:

d^2u/dphi^2+u=m/h^2+3mu^2

where h=angular momentum/unit of rest mass and m=GM/c^2 is related to the Schwarzschild radius

For ANY frequency photon, rest mass=0 so h=infinity

The equation becomes:

d^2u/dphi^2+u=3mu^2

These equations of motion for massive and zero rest mass particles are derived from Lagrangians constructed from the Schwarzschild metric, not from the Maxwell and Maxwell-Proca Lagrangians. The first equation is appropriate for massive spin-1 quanta (massive "photons"), while the second equation is appropriate for massless spin-1 quanta (massless photons).

The dispersion relation for massive photons written in the form

k = \sqrt{\omega^2 - m^2}

shows that spatial momentum (as measured in a particular frame) depends on frequency, and thus h depends on frequency.

 

[starthaus]

These equations of motion for massive and zero rest mass particles are derived from Lagrangians constructed from the Schwarzschild metric, not from the Maxwell and Maxwell-Proca Lagrangians.

Correct, so we can narrow the challenge to a simple question: find out the expression for "h" in the Proca formalism. Does "h" in post #38 depend on the speed of "massive" photons or not?

 

[George Jones]

Correct, so we can narrow the challenge to a simple question: find out the expression for "h" in the Proca formalism.

This can be done fairly easily, but I am quite drained right now. If no one does it sooner, I will post tomorrow.

Does "h" depend on the speed of "massive" photons or not?

Yes, it depends on spatial velocity (with respect to a particular frame).