Do Photons Attract Each Other?

[ergospherical]

Then the equation of motion for a photon in the beam (moving parallel to the $x$ axis) is\begin{align*}
\ddot{y} = - \Gamma^y_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu} = -[\Gamma^y_{00} + \Gamma^y_{xx} + 2\Gamma^y_{0x}](\dot{x}^0)^2
\end{align*}and similar for $z$. In the linearised theory ($h_{\mu \nu} \equiv g_{\mu \nu} - \eta_{\mu \nu}$) we have $h_{00} = h_{xx} = - h_{0x}$ which after some algebra implies that the right hand side vanishes. This result should still hold in the non-linearised theory, but how can one prove this?

 

[anuttarasammyak]

This result should still hold in the non-linearised theory, but how can one prove this?

I do not think it hold in non linear orders because photons have energy momentum thus curve space time geometry.

 

[PAllen]

No, it does hold rigorously. There are a number of proofs in the literature. Here is the classic paper on this:

https://projecteuclid.org/journals/...vitational-field-of-light/cmp/1103841572.full

 

[PAllen]

Actually, I just realized I misunderstood something about the OP. The paper I referenced, and the result that is relatively well known, is that parallel light beams whose momentum is in the same direction do not attract each other. It is also known that for for beams with opposite momentum (anti parallel beams), there is attraction. The OP seems to be asking about perpendicular beams. I never thought about this case before, and I don’t know the answer beyond the linearized argument presented in the OP.

Here is a reference covering the antiparallel case:

https://arxiv.org/abs/gr-qc/9811052

 

[PeterDonis]

the equation of motion for a photon in the beam (moving parallel to the $x$ axis) is

It seems like there is some missing context here. What assumption is being made about the metric?

 

[anuttarasammyak]

It is also known that for for beams with opposite momentum (anti parallel beams), there is attraction.

For two light beams except parallel case we have COM system where two beams has opposite momentum. Then according to the above two light beams attract.

 

[ergospherical]

It seems like there is some missing context here.

This is another problem from Lightman’s book, the full statement is:

Show that in linearised theory there is no attractive gravitational force between two thin parallel beams of light.

which is fairly straightforward to answer once you figure out what the components of the SET are (because $\square \bar{h}_{ab} = -16\pi T_{ab}$ in linearised theory). The geometry I have in mind is nothing but two beams parallel to the $x$ axis but with different (fixed) $y$ and $z$ coordinates.

There’s an additional remark at the end saying that the result in fact also holds to higher orders - the quite surprisingly simple justification is in the first paper @PAllen linked to.

The question about the anti-parallel and perpendicular cases is interesting. I think I will come back to them on the weekend because I have too many other assignments to complete at the present!

 

[PAllen]

Oh, so you were interested in the parallel case. Your comment in OP about z direction confused me. Anyway, the analysis in the second paper I referenced establishes that any beam orientation except parallel will have net interaction between the beams. However, it only gives an actual solution for anti parallel and parallel cases.