@user1247: Two photons moving in parallel photons do not attract no matter how boosted--- this is a consequence of special relativity, but it is also confirmed in GR, and it is the basis of plane wave exact solutions which propagate in a fixed direction and do not collapse.

@Ron Maimon -- But the essence of my question (for which I'm itching for an answer) is why. I thought that in GR energy is a source of gravity. Why is the energy of a photon an exception?

@Ron Maimom -- According to accepted answers to another stackexchange question you aren't correct about this.

@user1247: Two nonparallel beams of light do attract, there is no contradiction. Only parallel light beams are nonattractive. The energy of the photon is attractive, but gravity is a tensor force, and a photon also has momentum along its direction, and the analog of magnetic forces in gravity make the photon beams stable. The same is true in electromagnetism--- two electrons highly boosted in the same direction have an electric repulsion exactly balanced by a magnetic attraction.

@Ron Maimon -- OK, just to make sure I understand: you are saying that if we have two photons moving anti-parallel (approaching each other from opposite directions), their deflection angle is invariant with respect to the photon energy. If the two photons are 500nm they will be deflected the same angle as if they are blue shifted to have the energy equivalent of a supermassive black hole. The invariance is because the deflection due to the gravitational field of the supermassive black hole is compensated by gravitomagnetic effect. Is this right? I'll accept your answer if you edit to add this.

@user1247: This is not right, because angle is not relativistically invariant. Also, boosting photons doesn't change their center of mass energy, because one is blueshifted and the other redshifted. If you boost them perpendicular to their motion, they become nearly parallel, and time dilation slows the collision. The momentum transfer squared is the relativistically invariant analog of angle. The compensation is for parallel beams. I don't know what you want me to add, because what you ask me to add is wrong.

OK, I will re-state my question. If we are in the CM frame watching two photons come in from opposite directions, does the angle of deflection due to the gravitational interaction depend on the CM energy?

@user1247: Yes, of course. The angle of deflection is equal to the center of mass energy divided impact parameter (the distance between the pencils), up to constant factors, for small angles.

@Ron Maimon -- If two photons are moving parallel and one has more energy than the black hole at the center of the milky way, then I understand from your comments that gravitomagnetic effects cancel the gravitational attraction. OK, but what about hawking radiation, and other effects one would expect in GR of an object with that much energy density?

@user1247: Hawking radiation time dilated with boost, so it only depends on the invariant rest mass of the object, in this case, zero. There is no Hawking radiation from a transplanckian photon, although there will be as soon as you scatter it off a stationary object.

@Ron Maimon: So the rate of hawking radiation is exactly compensated for by the gamma factor? I've never seen a formula for hawking radiation consistent with that. Also, if there is Hawking radiation between a transplanckian photon and a stationary object, then any stationary observer should see Hawking radiation from a transplankian photon? Or are you somehow distinguishing between a "hard scatter" in which the hawking radiation is highly off-shell vs elastic scattering?

So, thanks for taking the time to try to explain this to me. I'm not intentionally trying to poke holes in your argument, but I'm also frustrated because I'm trying to get my question answered, and it feels like you keep answering it obliquely rather than trying to help understand what I am asking and get to the heart of my confusion.

I think I figured out though why I didn't understand your last statement about time dilation -- you said "Hawking radiation time dilated with boost, so it only depends on the invariant rest mass of the object, in this case, zero" -- now I realize that the time dilation is infinite for a photon, right? That was a brain-freeze on my part.

So maybe I'm beginning to get what you are saying. The heart of the matter seems to be that the photon is special because of its unique reference frame, so that no matter what it's energy density, tricky things like gravitomagnetic effects and infinite time dilation cancel out what would otherwise be GR effects like gravitational attraction or hawking radiation.

I understand that all of this can be assumed from Lorentz-invariance, but the way in which the Lorentz-invariance is enforced turns out to be very interesting!