Frequency dependence of gravitational acceleration

[Dylan H]

Imagine a massive object emitting photons of various frequencies. Because the object is massive, it will exert gravitational acceleration on those photons. Because the energy of a photon is proportional to its frequency, it seems that higher-frequency photons will experience a higher magnitude of gravitational acceleration. Some questions:

1. Is it true that higher-frequency photons experience greater gravitational acceleration?
2. ...and does this mean that there is a "gravitational dispersion" effect even in vacuum?
   
3. From a classical viewpoint, such acceleration could potentially cause photons to move slower than c. How do relativistic accounts reconcile acceleration of light with the constancy of the speed of light?

 

[Matterwave]

Light travels at c, locally, always. This is true even in GR. As such, it can not "accelerate" in the way that we are familiar with the term. What does happen to light is that it gets red shifted as it climbs out of a gravitational well (its frequency decreases).

1) The gravitational red shift formula can be seen here: http://en.wikipedia.org/wiki/Gravitational_redshift as you can see, there is no dependence on the frequency of light. Each frequency is proportionately red shifted, where the proportion is dependent only the properties of the gravitating matter.

2) All light travels at c, so there's no "dispersion" in the speed of light if that's what you mean.

3) No they can't since there is no such "acceleration", only red shift. What can happen though is that over a larger patch of space, a light ray may seem delayed in making a trip in the presence of a gravitational body, than if that gravitating body weren't there. This is called the Shapiro delay. One can think of this effect in two different ways, as either an effect of gravitational time dilation on the non-local speed of light, or as a stretching of space near the gravitating body (so that the light ray had to travel a longer distance than anticipated). However, if you were near the light ray, you would always measure it move at c, there is no changing that, that is a basic postulate of SR and GR.

 

[Dylan H]

Thanks.
I've been reading about gravitational redshift and Shapiro delay, and have some clarifying questions.

1. Are photons affected by gravity? (To me, that seems to be how gravitational lensing works.)
2. If photons are affected by gravity, then photons have gravitational mass. Is this gravitational mass proportional to the frequency of the photon? (If not, where does it come from?) (Because the photon has zero rest mass, it seems like its mass-energy consists of its quantum of energy which is proportional to its frequency.)
3. If photons have gravitational mass proportional to their frequency and light axiomatically always travels at speed c then does gravity always accelerate photons in such a way that they change only in direction but not in magnitude (i.e. a kind of "centripetal" acceleration)?
4. If 1-3 are all true, then why wouldn't spacetime curvature deflect light in a frequency-dependent way?
5. Would a photon falling into a massive object appear to change in frequency? If yes, then can you say that there is no frequency-dependent acceleration of light because the frequency-dependent changes in frequency exactly counterbalance the would-be frequency-dependent acceleration?
   
6. Gravitational redshift seems to be a result of time dilation on account of the curvature of spacetime, which suggests that it's simply an observer-dependent phenomenon. Can you also interpret it as energy lost as a result of climbing out of the gravitational well (the energy in frequency form being converted into gravitational potential energy) or is that a separate phenomenon?

 

[Nugatory]

Yes, light is affected by gravity as you say in #1. We see this in gravitational lensing, gravitational red shift, Shapiro delay, and the like.

However, #2 does not follow from #1. In General Relativity "having gravitational mass" is not a requirement for being acted on by gravity. In fact, GR has no concept of gravitational mass; the closest we can come is to say that the energy of the light contributes to the stress-energy tensor. Thus, #3 and #4 are not valid.

#5: Yes, gravitational redshift works both ways; infalling light is blue-shifted. However, not only is there no frequency-dependent acceleration, there is no acceleration at all - it's moving at a constant speed, $c$, all the way down.

#6: All measurements of the frequency of anything, not just light, are observer-dependent. We're measuring the time between two different events (successive peaks in the cycle) and that's clearly observer-dependent.

(You may have noticed that I haven't used the word "photon" anywhere. That's because in relativity there are no photons; light is an ordinary classical electromagnetic wave. You can get away with saying "photon" when you mean "flash of light" or "light pulse", but it is a bad habit that will cause you much grief when you get into quantum mechanics. Photons just aren't what you think they are).

 

[Matterwave]

It sounds to me like you want to use a gravitational lens as a prism of sorts? The angular deflection of light by a central spherical mass is independent of the frequency, there is no frequency dependent index of refraction like there is for glass.

See here:
http://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity#Bending_of_light_by_gravity

 

[Dylan H]

you want to use a gravitational lens as a prism of sorts?

Yes; I like that description. It seems that, as evidenced by various equations, this sort of phenomenon doesn't happen. I'm still trying to understand why the energy of the photon is irrelevant to the gravitational force. I would have expected that just as objects of increasing mass will deflect a test mass more strongly, massless objects of increasing energy will deflect a test mass more strongly. This seems to follow from a mass-energy equivalence idea.

 

[Nugatory]

why the energy of the photon is irrelevant to the gravitational force.

The energy of the light is irrelevant to the gravitational force because in GR there is no such thing as "gravitational force". An object in a gravitational field changes its speed and/or direction of travel in three-dimensional space, but not because some force is acting to push it off the straight-line path that you'd expect from inertia in the absence of forces. It is following a straight line ("geodesic", in the lingo) in curved four-dimensional space-time precisely because there is no force to push it off that straight line.

Check out member A.T.'s (excellent, truly outstanding, impossible to overstate the pedagogical value) video showing how GR and curvature explain the falling apple... and note that the exact same explanation works for a flash of light.

Edit - this video:
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[Nugatory]

I would have expected that just as objects of increasing mass will deflect a test mass more strongly, massless objects of increasing energy will deflect a test mass more strongly.

They do - the energy of all particles, whether massless or not (and here I'm going to repeat my warning that photons aren't what you think they are, and that by not describing light as electromagnetic radiation you are setting a trap for yourself) contributes to the stress-energy tensor which in turn affects the curvature of spacetime and hence the deflection of a test mass.

But that's not what you've been asking about. You're not asking about the deflection of test particles caused by the energy of the light, you're asking about the deflection of the light itself. That is independent of the energy of the light for pretty much the same reason that all dropped objects fall at the same speed, regardless of their mass.

 

[Dylan H]

Perfect, thanks.

 

[PeterDonis]

I would have expected that just as objects of increasing mass will deflect a test mass more strongly, massless objects of increasing energy will deflect a test mass more strongly.

As Nugatory pointed out, this question (how light acts as a "source" of gravity) is a different question from the one in your OP (how does light move in a gravitational field produced by something else as a source, viewing the light itself as a "test object"). The answer to this different question is that the source of gravity is the stress-energy tensor, and yes, light has a stress-energy tensor. But if you want to know more about that, you should start a separate thread, so this one can stay focused on the question in your OP.

 

[A.T.]

That is independent of the energy of the light for pretty much the same reason that all dropped objects fall at the same speed, regardless of their mass.

Yes, one can deduce that independence from the Equivalence Principle:

Far away from any mass, two light rays of different frequency but same source & direction will follow the same straight paths. In the local free falling frame near a large mass light behaves in the same way as above, and if the light rays don't diverge in one frame, then they don't diverge in any frame.

 

[Matterwave]

Yes, one can deduce that independence from the Equivalence Principle:

Far away from any mass, two light rays of different frequency but same source & direction will follow the same straight paths. In the local free falling frame near a large mass light behaves in the same way as above, and if the light rays don't diverge in one frame, then they don't diverge in any frame.

This is a good way to think about this. :)