Gravity & Photon Trajectory: Is Wavelength a Factor?

[chimera27]

This seems like an intuitive question, but I've seen some rather contradictory answers on it and am not sure what to think. What I'm curious about is if two photons, A and B, with wavelengths 900nm and 200nm respectively, both start from the same point on the same trajectory passing near a massive object, will photon A be refracted at a larger angle than B? I would think yes, but I'm not sure, as I've seen both remarks that say the velocity (in this case 'c') alone is factored, and remarks that it would indeed affect it. Both answers raise interesting questions, and I'm not sure which better describes the situation. Any thoughts?

 

[PeterDonis]

will photon A be refracted at a larger angle than B?

You should not use the word "refracted" in this connection, because it invites the (incorrect) comparison with the refraction of light in a material medium that you are being tempted to make. Unfortunately there isn't a fancy word for the bending of light by gravity; it's usually just called "the bending of light by gravity".

There are at least two ways to understand why the answer to your question is "no"--i.e., why photons of different energies are bent the same by gravity. The first is by analogy with objects with nonzero rest mass; all such objects have their trajectories "bent" the same by gravity. If you shoot two rocks past the Earth, one more massive than the other, with the same initial positions and velocities, they will both follow identical trajectories. By analogy, the same should be true for two photons of different energies which are shot with the same initial positions and velocities (the velocities have to be the same in this case because they're both photons).

The second way is to recognize that, in GR, gravity is not a force; it's spacetime geometry. The "bending" of the paths of objects is a property of the geometry of spacetime, not a property of the objects. The geometry of spacetime is the same for all objects, photons included.

 

[bcrowell]

The second way is to recognize that, in GR, gravity is not a force; it's spacetime geometry. The "bending" of the paths of objects is a property of the geometry of spacetime, not a property of the objects. The geometry of spacetime is the same for all objects, photons included.

To amplify on this for the OP, the track of a photon through spacetime is a geodesic, which in a curved space is the equivalent of a straight line. A geodesic is as straight as possible. (The equivalent on a sphere would be an arc of a great circle.) Two photons with different energies and otherwise identical initial conditions will both go "straight," i.e., both will follow the same geodesic.

 

[pervect]

Basically, the answer is no - the light will travel the same path (called a null geodesic) regardless of energy or frequency. Energy and frequency are related by quantum theory by the relation $E = h \nu$, E being energy and $\nu$ being frequency. Technically, "photons" are outside the scope of GR, but if one replace "photon" with "weak light pulse" one gets the answer above, that the deflection is independent of frequency. So gravitational lenses don't show any chromatic aberration, for instance.

The fine-print exception to the above is that if you had gravitationally significant amounts of energy in the light (atypical, and probably not what you had in mind when you asked about "photons"), the approximations used to get the above result would be invalid, and the path would depend on the total energy in the light pulse (though it wouldn't depend directly on the frequency even so, but it might depend on the total energy, which in quantum terms would be the number of photons multiplied by the energy per photon. But see the remarks about GR being a classical theory.)

 

[chimera27]

You should not use the word "refracted" in this connection, because it invites the (incorrect) comparison with the refraction of light in a material medium that you are being tempted to make. Unfortunately there isn't a fancy word for the bending of light by gravity; it's usually just called "the bending of light by gravity".

There are at least two ways to understand why the answer to your question is "no"--i.e., why photons of different energies are bent the same by gravity. The first is by analogy with objects with nonzero rest mass; all such objects have their trajectories "bent" the same by gravity. If you shoot two rocks past the Earth, one more massive than the other, with the same initial positions and velocities, they will both follow identical trajectories. By analogy, the same should be true for two photons of different energies which are shot with the same initial positions and velocities (the velocities have to be the same in this case because they're both photons).

The second way is to recognize that, in GR, gravity is not a force; it's spacetime geometry. The "bending" of the paths of objects is a property of the geometry of spacetime, not a property of the objects. The geometry of spacetime is the same for all objects, photons included.

Thanks, that's what I thought. just wanted to confirm since I had heard a few contrary remarks when researching it. Also, my use of 'refracted' was simply keeping in terms of the gravitational 'lensing' analogy, and was purely a semantical error, not a genuine misunderstanding xD

 

[jtbell]

IIRC this has also been tested experimentally by comparing the deflection of radio and/or light signals with different frequencies by the sun. No differences (dispersion) has been found.