Can Different Wavelengths Affect the Gravity Between Light Beams?

[calinvass]

In the experiment using pencils of light by Tolman, is has been shown that two parallel light beams do not attract but anti parallel beams do.
What happens if we use different wavelengths for the light beams. Will the higher frequency beam be less deflected ?

 

[mfb]

Energy density is the relevant parameter (technically energy and momentum density, but for light they have a fixed relation). Wavelength alone doesn't tell you what will happen.

 

[PAllen]

Just a nit pick - Tolman's work, and other similar work on this phenomenon are all theoretical. The deflections are many orders of magnitude too small to observe.

 

[calinvass]

Energy density is the relevant parameter (technically energy and momentum density, but for light they have a fixed relation). Wavelength alone doesn't tell you what will happen.

But isn't energy density proportional to wavelength for a stream of photons ?

 

[Ibix]

Energy per photon depends on wavelength, yes. The energy of a beam depends on the number of photons, too. And remember that GR is a classical theory. It can deal with a light beam as a source of gravity, but not a quantum particle. That's why @mfb is talking about the energy density (of a classical beam).

 

[mfb]

But isn't energy density proportional to wavelength for a stream of photons ?

And the gravitational force on Earth is proportional to the mass of Earth. That doesn't mean lifting a truck is as easy as lifting a grain of sand, even though the mass of the Earth is the same in both cases. The other mass matters as well, and only their product is relevant for the force.

Wavelength alone doesn't tell you anything, You need the energy density, the product of photon density and energy per photon.

 

[pervect]

In the experiment using pencils of light by Tolman, is has been shown that two parallel light beams do not attract but anti parallel beams do.
What happens if we use different wavelengths for the light beams. Will the higher frequency beam be less deflected ?

For the parallel beams, there isn't any deflection, so it won't matter. For the anti-parallel beams, one would need to work it out in more detail. It's probably coordinate dependent, if we imagine two almost-parallel light beams of the same frequency and energy density in a rest frame, by going to a moving frame we can make the energy densities and frequencies unequal.

 

[calinvass]

And the gravitational force on Earth is proportional to the mass of Earth. That doesn't mean lifting a truck is as easy as lifting a grain of sand, even though the mass of the Earth is the same in both cases. The other mass matters as well, and only their product is relevant for the force.

Wavelength alone doesn't tell you anything, You need the energy density, the product of photon density and energy per photon.

Ok, but, when I said a stream of photons I meant a constant homogeneous stream, because that is what a laser approximately does.
The energy per photon is hν.

The problem is, I understand, a photon can only be described as a quantum object, and GR doesn't deal with them.

 

[mfb]

That is not the "problem". Even with classical particles you would need the density. The energy per particle is not sufficient to determine what happens.

 

[calinvass]

Thanks, yes, the density of particles over a volume of space makes sense. This I suppose corresponds to the intensity of the beam.

 

[PeterDonis]

t's probably coordinate dependent

The numerical values of energy and momentum density are. But the fact that anti-parallel light beams attract is not.

if we imagine two almost-parallel light beams of the same frequency and energy density in a rest frame, by going to a moving frame we can make the energy densities and frequencies unequal

Yes, but that also makes the momentum densities and pressures unequal in the same way, and the action of the beams as a source of gravity depends on all of those things. The actual source of gravity is the stress-energy tensor of the light beam, whose components include all of those things and co-vary appropriately when you change frames.

 

[calinvass]

I'm not sure of this, but an observer sees these light beams curving space around them , therefore, why doesn't see them moving towards each other. Usually intuition doesn't work very well in relativity. Is it possible that as the beams curve spacetime around them, they also move further and the spacetime curvature manifests like gravitational waves always behind the light beams. However, if we use continuous beam (not short pulses) this explanation doesn't work anymore.

 

[PAllen]

I'm not sure of this, but an observer sees these light beams curving space around them , therefore, why doesn't see them moving towards each other. Usually intuition doesn't work very well in relativity. Is it possible that as the beams curve spacetime around them, they also move further and the spacetime curvature manifests like gravitational waves always behind the light beams. However, if we use continuous beam (not short pulses) this explanation doesn't work anymore.

Two anti parallel beams do move towards each other, while parallel beams do not. You don't notice this only because the amount of deflection is too small.

 

[calinvass]

Two anti parallel beams do move towards each other, while parallel beams do not. You don't notice this only because the amount of deflection is too small.

Yes, I know. Is there any contradiction to what I said?
If the curved spacetime is always behind the pulses of light, the beams don't get attracted. But this explanation doesn't work when we use continuous beams.

When antiparallel, the spacetime curvature clearly affects both pulses but after they pass by.

 

[mfb]

Light beams moving in opposite directions attract each other, a bit like massive objects moving in opposite directions would do. If we increase the intensity of light beam A, the deflection of light beam B increases. If we increase the intensity of light beam B, the deflection of light beam A increases.

 

[PeterDonis]

Usually intuition doesn't work very well in relativity.

Yes. That means you should not be trying to use intuition to analyze this problem, as you are doing. You need to actually look at the math.

 

[PAllen]

Yes, I know. Is there any contradiction to what I said?
If the curved spacetime is always behind the pulses of light, the beams don't get attracted. But this explanation doesn't work when we use continuous beams.

When antiparallel, the spacetime curvature clearly affects both pulses but after they pass by.

Yes, it directly contradicts what you seem to be saying. You say the beams don't get attracted. I say they do, and (given ridiculous precision), you would measure this. Light in oppositely moving light pulses would also mutually deflect each other (again, by an amount too small to be observed with current technology).

 

[calinvass]

Yes. That means you should not be trying to use intuition to analyze this problem, as you are doing. You need to actually look at the math.

Sometimes I do that but usually, as Feynman says, I guess it, then I compute the consequences. Unfortunately for this case, it is harder to guess.

 

[calinvass]

Yes, it directly contradicts what you seem to be saying. You say the beams don't get attracted. I say they do, and (given ridiculous precision), you would measure this. Light in oppositely moving light pulses would also mutually deflect each other (again, by an amount too small to be observed with current technology).

But on post #13 you said parallel beams do not attract.
Ps
Oh, I understand now, you said they don't but in fact it is only that we can't measure this.

I still don't understand what is the solution given by GR.

 

[PAllen]

But on post #13 you said parallel beams do not attract.
Ps
Oh, I understand now, you said they don't but in fact it is only that we can't measure this.

I still don't understand what is the solution given by GR.

Maybe I am misunderstanding you.

Anti-parallel beams attract and deflect. Parallel beams do not. Anti-parallel pulses defect each other. For parallel pulses I have never actually studied or performed a calculation. My guess is they don't deflect, but I would not have much confidence in this guess without further analysis. Even with some experience, intuition is unreliable in GR.

As for your attempts at intuitive explanations in terms of curvature, I don't think these are useful.

 

[pervect]

The numerical values of energy and momentum density are. But the fact that anti-parallel light beams attract is not.

Agreed

Yes, but that also makes the momentum densities and pressures unequal in the same way, and the action of the beams as a source of gravity depends on all of those things. The actual source of gravity is the stress-energy tensor of the light beam, whose components include all of those things and co-vary appropriately when you change frames.

The fact that the anti-parallel beams attract each other is coordinate independent, but I can't think of any coordinate independent way to describe "how much deflection" the light beams undergo.

The example I was trying to sketch out is a simple case where we have two identical (in some particular frame) light beams. In the center of momentum frame, which is the easiest to analyze, each beam will be deflected equally.

Then we move to a different reference frame, that's moving in a direction parallel to the beams, so that one beam gets red-shifted and the other beam gets blue shifted. Then in the new frame the energy densities won't be equal any more, and we'd expect one beam to deflect more than the other. (I think we agree on this?).

One challenge here is how to present this in a way that's understandable at a "B" level, if possible. Though before we can do that, we do have to know what the right answer is, and restricting ourselves to B-level tools isn't conducive to making sure we have the right answer.

The only B-level explanation that comes to mind is to appeal to length contraction, pointing out that lengths in the direction of the motion change with the frame, making the angles vary. But I don't think it's a totally complete or satisfactory explanation - I'm waffling if it's good to pursue this B-level sort of explanation more, or to risk loosing the audience by attempting a better explanation.

 

[PeterDonis]

I can't think of any coordinate independent way to describe "how much deflection" the light beams undergo.

I can't either off the top of my head, and I don't have time to consult references right now. IIRC this problem is treated somewhere in MTW.

 

[votingmachine]

Two anti parallel beams do move towards each other, while parallel beams do not. You don't notice this only because the amount of deflection is too small.

I am puzzled by why the direction matters. I may be envisioning the proposed light beams incorrectly:

Say I have a laser aimed to strike a small gap away from a laser source a light year away. And that laser is aimed to a strike the same small gap away from that first laser. And they are parallel paths, when run separately. Now if they are run together, the theory says the gap gets slightly smaller.

... From "gravitational" attraction based on the energy of the photons.

Now alter that set up, and put the lasers both at one end. Why is the gap between the end point spot NOT (note, the word "not" was added in a later edit) slightly smaller when they are run at the same time? What is the difference between parallel and anti-parallel?

I could see the situation if you were hypothesizing a single photon. Then the other photon is always outside of the cone of possible interactions, while the anti-parallel photons are passing thru a cone of the past of the other photon. But a beam? That implies both directions exist within the cone of the past of the other beam.

Sorry for the poor wording. The light cone idea of the past seems to be part of this, and I don't see why it is.

Another thing occurs to me. In the set-up I describe, if you turned on both lasers simultaneously, the beans would travel for a half a light year each before entering space that had the anti-parallel beam. So the gap would start be smaller when the light appears a year later, and then the gap should move inward over the next half year. Right or wrong?

 

[mfb]

General relativity uses the energy-momentum tensor as source of gravitational interactions. The energy of a light beam does not depend on the direction, but momentum does.
This does not have a non-relativistic analog.

 

[PAllen]

I am puzzled by why the direction matters. I may be envisioning the proposed light beams incorrectly:

Say I have a laser aimed to strike a small gap away from a laser source a light year away. And that laser is aimed to a strike the same small gap away from that first laser. And they are parallel paths, when run separately. Now if they are run together, the theory says the gap gets slightly smaller.

... From "gravitational" attraction based on the energy of the photons.

Now alter that set up, and put the lasers both at one end. Why is the gap between the end point spot NOT (note, the word "not" was added in a later edit) slightly smaller when they are run at the same time? What is the difference between parallel and anti-parallel?

I could see the situation if you were hypothesizing a single photon. Then the other photon is always outside of the cone of possible interactions, while the anti-parallel photons are passing thru a cone of the past of the other photon. But a beam? That implies both directions exist within the cone of the past of the other beam.

Sorry for the poor wording. The light cone idea of the past seems to be part of this, and I don't see why it is.

Another thing occurs to me. In the set-up I describe, if you turned on both lasers simultaneously, the beans would travel for a half a light year each before entering space that had the anti-parallel beam. So the gap would start be smaller when the light appears a year later, and then the gap should move inward over the next half year. Right or wrong?

This whole effect is not intuitive and was not expected; Tolman's original work, followed by many further analyses by others was a surprise consequence of GR. I am not going to answer your speculations in detail, because a correct understanding of this phenomenon must be based on general relativity, not intuitive speculations based on photons (which are not elements of any GR treatment; within GR as a classical theory, light must be treated as an EM field, or, in some cases, may be approximated as 'null dust').

A vague verbal description of what GR actually says about this phenomenon is that for parallel beams, gravito-magnetic effects exactly cancel attraction. This does not happen for anti-parallel beams. An example paper describing this analysis is: https://arxiv.org/abs/gr-qc/9811052

A pedagogic argument of mine provides intuition on this, while not being rigorous. The lack of rigor is that the argument is primarily SR based, rather than using the full machinery of GR. Consider two parallel beams of light. There exist frames in which the energy density of both beams can be made arbitrarily small, as close to zero as desired. These are as valid for analysis as any other. For zero energy density, one does not expect gravitational interaction. On the other hand, for antiparallel beams, all frames have one or both beams having significant energy density.

 

[Traruh Synred]

Boost along thw parrallel beams. As the boost inncreases the energy-momentum density drops to zero. As any motion of the beams toward each other has to exist in all such frames and a it approaches zero as boost increases, it has to be zero in all frames.

 

[jartsa]

I am puzzled by why the direction matters.

Relative things can never be causes of absolute things. When kinetic energy is relative it will not cause two things to collide. I said when, because there seems to exist relative and absolute kinetic energy.

If we store 10 micro-grams of energy in a flywheel hanging on a wire that can only support 9 micro-grams, then the flywheel will collide with the ground. So rotational kinetic energy seems to be absolute.

Every atom of a flywheel is attracted by the rotational kinetic energy stored in the flywheel, right?

We can make flywheels out of light too, I mean the moving part of the flywheel can be light. So then the energy of the flywheel attracts every photon of the flywheel, right?

 

[votingmachine]

This whole effect is not intuitive and was not expected; Tolman's original work, followed by many further analyses by others was a surprise consequence of GR. I am not going to answer your speculations in detail, because a correct understanding of this phenomenon must be based on general relativity, not intuitive speculations based on photons (which are not elements of any GR treatment; within GR as a classical theory, light must be treated as an EM field, or, in some cases, may be approximated as 'null dust').

A vague verbal description of what GR actually says about this phenomenon is that for parallel beams, gravito-magnetic effects exactly cancel attraction. This does not happen for anti-parallel beams. An example paper describing this analysis is: https://arxiv.org/abs/gr-qc/9811052

A pedagogic argument of mine provides intuition on this, while not being rigorous. The lack of rigor is that the argument is primarily SR based, rather than using the full machinery of GR. Consider two parallel beams of light. There exist frames in which the energy density of both beams can be made arbitrarily small, as close to zero as desired. These are as valid for analysis as any other. For zero energy density, one does not expect gravitational interaction. On the other hand, for antiparallel beams, all frames have one or both beams having significant energy density.

I see it now. I was thinking of the problem of two photons traveling parallel. They would never affect each other because the other does not exist within the allowed light cone. That was traveling down a misleading path, but one which seemed contradicted by the use of "beam". So I was thinking how to carefully ensure that photons traveled within that cone.

The direction part is still a bit puzzling. Would the geometry be equivalent to attraction along the 45-degree angle to the direction of travel, or directly towards the anti-parallel beam?

 

[PAllen]

Relative things can never be causes of absolute things. When kinetic energy is relative it will not cause two things to collide. I said when, because there seems to exist relative and absolute kinetic energy.

If we store 10 micro-grams of energy in a flywheel hanging on a wire that can only support 9 micro-grams, then the flywheel will collide with the ground. So rotational kinetic energy seems to be absolute.

Every atom of a flywheel is attracted by the rotational kinetic energy stored in the flywheel, right?

We can make flywheels out of light too, I mean the moving part of the flywheel can be light. So then the energy of the flywheel attracts every photon of the flywheel, right?

The distinction you are looking for is invariant mass of system. Rotation is not directly part of it. It is computed in any frame by summing 4 momenta, and then doing E² - P² (using c=1 convention). Note that in the COM frame, since the sum of momenta is zero by definition, the invariant mass includes all kinetic energy as measured in the COM frame. Computations in other frames will produce the same number for invariant mass.

 

[PAllen]

Boost along thw parrallel beams. As the boost inncreases the energy-momentum density drops to zero. As any motion of the beams toward each other has to exist in all such frames and a it approaches zero as boost increases, it has to be zero in all frames.

As I noted in my post #25, this is a heuristic argument. It helps make a surprising result less surprising, but that is all. It is not a derivation of anything in GR. To see this, consider applying it to the anti parallel case. Hope that in the COM frame, the energy density of the beams will determine the attraction, similar to dust beams. Find that the error in this approach is a factor of 4 (!) Also note that the COM frame for parallel beams does not exist. Without other justification you cannot treat as valid an argument that pretends this exists via a limit. That other justification must be based on GR.
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