How do you define the centre in a system of two photons?

**Tough question!** [

] I'll have to ponder this for a while. Let's not even think about the effect of the expansion of space; just confine the discussion to short distances and times where space does not expand appreciably.

For openers, let's consider only inertial reference frames whose relative motion is restricted to the x direction, which is the direction of relative motion of the two photons, which are moving in opposite directions parallel to the x-axis. (Later we may try to generalize to other reference frames.) Let's define the origins of all these reference frames as the point in space-time where the two photons pass closest to one another, at x = 0, t = 0. In all such reference frames, the center of the two-photon system is the origin, but the origins of different reference frames only coincide at the instant when the photons pass one another.

I guess you have to start with a reference frame in which both photons have equal energy and equal and opposite momenta. In that reference frame, the energy of each photon is E, and the momentum of each photon is p = E/c. Since the momenta are equal and opposite, the momentum of the system is zero.

Next, consider a reference frame moving in the +x direction at .866 c relative to the first reference frame. Gamma = 2; so in this reference frame, the photon moving in the +x direction has energy E'

_{1}= E/2, and the other photon has energy E'

_{2} = 2E. Do I have that correct? My brain is about to trip a circuit breaker, here.

**Someone, please let me know if I got the photons' energy right before I proceed to dig myself into a deeper hole. **My postulate is that, at low velocities, the two photon system has inertial mass M = dp/dv = E/c

^{2}. At higher velocities, I suspect that formula turns into a hyperbolic function. Rats! I hate hyperbolic functions! [xx(]

**Thinking ahead: **In any one reference frame, the center of the two-photon system is fixed, so a two-photon system can't have a non-zero velocity in an inertial reference frame. However, when you change to another reference frame, you move that center. (This gets into a gray area between SR and GR. I'm not sure I'll be able to handle the math.) A gradual change from one reference frame to another by small increments, dv, will gradually move the center of the system. So the rate of change of reference frame's velocity relative to the first reference frame (dv/dt) imparts motion to the system's center. I don't know yet whether the systems center has acceleration or uniform velocity while the reference frame's velocity is changing at a constant rate. If the momentum of the system changes at the rate dp/dv, that is the inertial mass of the system.