Do photons obey the 1/r^2 gravity law?

Barry_G

Has such experiment been performed?

PeterDonis

I'm talking about an electromagnetic field, as described here:

http://en.wikipedia.org/wiki/Electromagnetic_field

As you can see from the section on the mathematical description of the EM field, it can be described by two 3-vectors, the electric field and the magnetic field. Each 3-vector has 3 independent components, for 6 components total.

Electrons are not electromagnetic fields, so no, this doesn't apply to them.

The EM field associated with a photon (more precisely, associated with a classical electromagnetic wave, which is the best classical approximation to a photon) is a particular type of EM field called a "null electromagnetic field", as described, for example, here:

http://en.wikipedia.org/wiki/Classif...agnetic_fields

As shown in that article, the two invariants that are used to classify EM fields are

[tex]
P = E^2 - B^2 \\
Q = \vec{E} \cdot \vec{B}
[/tex]

For a null electromagnetic field, P = Q = 0; this constrains the components of the electric and magnetic field vectors so that there are only two independent ones. You should be able to work that out from the equations above.

Um, one of each?

It depends on the energy of the photon.

It means the electric and magnetic fields satisfy the source-free Maxwell's Equations:

http://en.wikipedia.org/wiki/Maxwell's_equations

See the section on Vacuum equations, electromagnetic waves, and the speed of light. "Source-free" means there is no charge or current present.

PeterDonis

No, it will be twice as "influenced". The angular deflection of a light beam passing close to a massive object is given by:

[tex]\delta \phi = \frac{4 G M}{c^2 b}[/tex]

where b is the distance of closest approach. Since this is a function of 1/b, not 1/b^2, the bending is only doubled if b is halved. See here:

http://en.wikipedia.org/wiki/Kepler_...ght_by_gravity

Note that this formula is only valid for b very small compared to GM / c^2.

Barry_G

Thank you. I'd like to talk about that in more detail so I'll start a new thread in Classical Physics forum.

PAllen

No, and it's not likely feasible, in practice. However, there is no doubt about what GR predicts for this. So called 'box of light' examples are used in classic GR exercises and papers.

I think there are astrophysical observations that provide evidence that radiating EM radiation cause as body to lose gravitational mass. That's probably the closest you can come to observational support.

PeterDonis

Yes, that's the right place to post questions about general properties of EM fields.

DaleSpam

The reason I provided you the information about null dust and pp-waves is that those are the names you will find in the GR literature for the appropriate spacetimes that you are interested in. They won't use the word "photon" because the spacetimes are general for all massless radiation, not just EM radiation. Also, the authors in the literature generally know better than to mix quantum terminology with classical theories.

As far as the numbers go, unfortunately your question is too vague to answer with a concrete number. What photon energy are you considering, what is the geometry you are interested in, what measurement technique to you plan on using to generate the number of interest, etc.? Until you completely specify the problem then all that can be done is to provide you a link to the relevant concepts and solutions.

Barry_G

That's great. I think that answers the question, for me at least.

Thank you everyone, I'll leave it here and move to classical physics forum to discuss photon electromagnetic field components and Maxwell's Equations.

pervect

Not directly. Carlip argues in Kinetic energy and the equivalance principle that we do have good experimental reason to believe that electromagnetic binding energy contributes to gravity, and that this implies that electromangetic fields must gravitate. Classically (and GR is a classical theory), light is made up out of electromagnetic fields, so we know it's made up of something that's been observed to contribute to gravity (albeit indirectly).

Carlip also briefly discusses the "box of light". Carlip shows that in weak field gravity, the total system (box + light) must gravitate according to the total energy. There is a similar result for strong fields , but it requires that the metric be stationary (i.e. not a function of time). The argument is different in detail from Carlip's. While I'm not aware of any paper that specifically does the strong field calculation for a box of light, the calculations aren't hard to perform.

The non-technical summary of the strong field argument is that in some sense the interior of the box, the light, does "weigh" twice as much, but that the stress in the box walls compensates for this giving a negative contribution to the weight, due to the tension in the container walls.

As an aside, recall that tension and pressure are part of the stress-energy tensor - so here we see an example of stresses contributing to gravity.

The more technically accurate way removes the words "in some sense" by saying that it is the Komar mass of the interior of the box that doubles for the "box of light". This makes the argument more precise, at the cost of introducing a new term that seems to scare people away from understanding the point to be made. On the other hand, some "scariness" is perhaps warranted, at least if the fear induces some caution, for reasons which will be explained below.

As previously mentioned, even though the contents of the box weight twice as much, the stresses in the walls subtract from this "extra" mass, and you recover the value E/c^2 for the mass of contents + walls.

It's worth mentioning at this point, at the risk of confusion, that there are several definitions of "mass" in general relativity, and NONE of them is completely general (including the Komar mass). ALL of them require certain preconditions to be applied. Understanding the conditions where they are applicable may take some work, this is where the "scariness" factor comes in.

The "big three" sorts of mass in GR are Komar mass, ADM mass, and Bondi mass - you'll see a brief discussion of them in the wiki at http://en.wikipedia.org/w/index.php?...ldid=514908524

Barry_G

Perhaps I should be more specific. I think what PeterDonis said answers the question as "yes, photons obey 1/r^2 gravity law". Because, I think what that equation describes is analogous to gravity potential which is 1/r, so that if we somehow worked out the force or acceleration we would get 1/r^2 relation.





In other words, if instead of two beams of light there were two beams of electrons or two streams of dust, where one is passing at double the distance from the planet than the other, then further away beam of electrons or stream of dust would too be "influenced" two times more less than the closer one, just like with two beams of light, but the force or acceleration between the planet and each electron or dust particle would be function of 1/r^2.

PeterDonis

The equation I gave describes the angular deflection of a light beam passing close to a massive object. I don't see any obvious analogy between that and gravitational potential.

The problem with that approach is that viewing gravity as a "force" that causes "acceleration" is an approximation that only works when all the objects involved are moving very slowly compared to the speed of light. Obviously that's not the case for a light beam passing close to a massive object.

A better way to phrase the question asked in the OP would be: does light respond to gravity? Or, is the path of a light ray affected by gravity? The answer to that is clearly "yes". But trying to salvage an interpretation of the bending of light as responding to a 1/r^2 force law may not work, because that force law is a non-relativistic approximation only, and light is relativistic.

I'm not sure this is true; the formula I gave is not exact, it's an approximation for the ultrarelativistic case where the object passing by is moving at or very close to the speed of light. The motion of a slower-moving object is more complicated.

If the electrons or streams of dust are moving slowly enough, a "force" interpretation would work.

pervect

Carlip's paper http://arxiv.org/abs/gr-qc/9909014 has an expression for the defliction of a particle moving at velocity v, and a caution about the "force" interpretation.

The defleciton angle is:
[tex]\theta = \frac{2GM}{bv^2}\left(1+\frac{v^2}{c^2}\right) [/tex]
where b is the impact parameter (it can be thought of as the distance of closests approach IIRC). and G,M, and v are the usual.

The caution is "Not to ignore the curvature of space" when calculating light deflections. This spatial curvature produces effects in GR that can not really be well described as a force - though thinking of it as a force proportional to velocity^2 comes at least very close to working. (I don't think the resulting "force" transforms properly even so, you wind up with coordinate dependencies this way.) Hopefully it's obvious why dependence on coordinates is bad in this context, and if it's not obvious, I'm afraid I don't have the heart at the moment for another long discussion of why it is bad.

Barry_G

I find analogy between the two in the word "influenced", which I interpret as 'feeling gravity potential' (at some distance from the planet). If there is some object in space that emits electrons that would travel near the speed of light then we could measure deflection of that electron beam similarly how we do it for light, I guess, and then we could compare it with that of light and I think we would get similar result, not in regards to the amount of deflection, but in regards to that the electron beam twice as far from the planet than the other electron beam would be two times less "influenced", just as it the case with two beams of light.


Which leads me to another question. If we were to measure "influence" or deflection of two beams of particles passing near some planet at the same distance away from it and with the same speed, but one beam is made of electrons and the other of particles with greater mass, say neutrons, would we be able to measure any difference?

The planet would have so much more mass compared to that of those particles that it would be kind of like "hammer and feather" thing, but then again, even a small difference in the change of angle when they pass next to the planet would grow larger with the distance, and so at the end we could actually measure even the smallest differences in mass of those particles in such beams. And if all this was true and possible, then I guess that would give us a way to measure photon gravity field (mass) too, or at least put it in some perspective compared to that of an electron.

PeterDonis

As you can see from the formula pervect posted, the result for an electron beam moving at v < c would be inversely proportional to b, yes. So I was too pessimistic when I said that result would be more complicated. I'm pretty sure pervect's formula still requires that b is much greater than GM / c^2.

The mass of the particles in the beam doesn't appear anywhere in the formula, so it wouldn't make a difference. Only the velocity of the particles in the beam matters. This is a manifestation of the fact that, in Newtonian language, all objects fall with the same acceleration in a gravitational field, regardless of their mass.

The "hammer and feather" thing doesn't depend on the mass of the planet being so much larger than the mass of the particles in the beam. It's always true for gravity, according to our best current theories, regardless of the mass of the gravitating object or the mass of the objects being deflected.

As I noted above, the mass of the particles in the beam doesn't appear in the formula, so you can't use bending of the beam by a massive object to measure anything about the mass of the particles in the beam.

Barry_G

Oops. You are right, force would be greater for particles with greater mass, but acceleration would be the same as it gets divided by proportionally greater mass. I got confused thinking about J. J. Thomson experiment and trying to make a parallel with measurement of electron mass in cathode ray tube. Which now makes me wonder how could that kind of thing measure any mass since the principle would be the same and so deflection of particles with different mass would be the same, having the same charge. Apparently I need to revisit that one.


On the other hand that perhaps makes for even more conclusive comparison regarding this topic. It seems it would mean that any beam of anything would be exactly deflected as much as any other beam of anything else, regardless of the strength of gravity field (mass) of the particles constituting any such beam. Which than means, I suppose, if everything else follows inverse square gravity law, and if beam of light bends exactly the same as a beam of anything else would, then photons too obey the same law. Does that follow?

PeterDonis

Yes, but that's only true of gravity. It isn't true for other forces.

No, it wouldn't, because Thomson's experiment was using the electromagnetic force, not gravity, to move electrons. So he was really measuring the ratio of the electron's charge to its mass; but since there were already independent measurements of the electron's charge, measuring the charge/mass ratio allowed him to calculate the electron's mass. If he had done the same type of experiment with, say, a proton, he would have measured a different charge/mass ratio and therefore a different mass.

Photons do obey "the same law"; it's the formula that pervect wrote down. But although the law does not depend on the mass of the particles in the beam, it does depend on their velocity. So photons, moving at c, will be deflected differently than particles moving slower than c, like electrons.

Barry_G

Uh, yes, I was too haste to write that. It all makes sense now.


That settles it then. Thank you for your patience.