Does the general-relativistic attraction between two optical cavities depend on the relative phase between the two oscillations?

Question

My understanding is that parallel beams of light do not attract but anti-parallel beams do attract (see, for example here). But if this is true, then consider a device in which a laser is bounced between two mirrors arranged along the Z axis. Such a device should attract another similarly oriented device *only* if the two laser trajectories are out of phase (if they are in phase, then the two beams are always parallel and never anti-parallel). It is interesting to consider that if the period of oscillation of the two devices is not the same, then there will be a beat frequency between 'parallel' and 'anti-parallel' beams, and the attraction between the two devices will oscillate. How is this behavior reconciled with the fact that in both cases the stress energy is the same for each system, and therefore the general-relativistic attraction should be independent of the relative phase?  

Answer 1

I like how this topic is treated in Tolman, Ehrenfest and Podolsky in PhysRev. To understand how gravity and null-matter works I always try to think about it as of pure gravitomagnetism. The parallel-inert and antiparallel-attracting behaviour of two rays of light is then analogous to magnetism between two wires (with the caveat that gravity can only be attractive). Another way to think about this is to ask: What else than a photon, a maximum-speed rest-mass-zero particle should exhibit pure dragging?

When you take a look in the article or think about it, you will realize that the stress-energy tensor is not equal in both the mirror-boxes you mention. Unless you are considering counter-moving streams of photons, the direction of the ray-pencil will always be reflected in the off-diagonal components (such as say $T^{tx}$ for rays moving in the $x$ direction). 

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I.e., yes, the attraction will depend on the relative phase between the pulses being reflected there and back again. However, consider that the situation cannot be most certainly thought of as stationary. The very variation of the attraction between the mirror boxes will be then associated with gravitational radiation and there might lurk something very unintuitive in the gravitational radiation from light rays (I don't know about any reference on this).