Conservation of momentum versus general relativity in an E2 steel cage smackdown or a silly physics question I've had for a long time.

So I'm imagining that this is a painfully easy (read: stupid) question, but I'm the sort of person who approaches these sorts of things with much thought but little rigor, and this is a nagging question that I haven't been able to get my head around. E2 has some great physics people and I'm sure you'll have me sorted in no time.

Leon Lederman explains in his excellent book The God Particle that particle accelerators have two basic strategies for slamming particles together: firing high velocity particles (like, say, protons) at a stationary target, or crossing two beams of particles accelerated in opposite directions and hoping the smack into each other in the middle. This second approach produces far few events, for obvious reasons (these particles being both amazingly small, and moving at terribly high speeds) but the energies of the collisions are much higher because (this is his explanation) the momentums of the particles cancel out, leaving all of the energy from the collision free to smash the hell out of each other, whereas in the former case, conservation of momentum dictates that much of the energy of the collision has to go into accellerating parts of the stationary particle.

And that explanation makes sense to me, in an intuitive way. Run a car into a brick wall, and a lot of stuff goes flying along the original vector of the car's momentum. Run two cars into one another head on, and you get two significantly smaller cars. But then I start trying to incorporate notions of relativity into my head, and things bog down fast.

General relativity as I understand it tells us that any inertial frame of reference is valid. Which as I understand it is basically saying that there is no such thing as a universally still place, or no universal (0,0,0) coordinate from which to base measurements of position, velocity, etc. Which sounds quite reasonable to me. But it brings me to my question:

How do the particles "know" their momentum? How do they know if particle 1 was moving and particle 2 was sationary, or if each was moving at half the total velocity in opposite directions (towards each other) or what? Doesn't general relativity say that the outcome of all of those scenarios has to be the same because there is no fixed point of reference for those judgements to be made against?

This is a losing-sleep cycle of thought for me. I'm counting on one or several of you to save me before it's too late... I will clarify my question if needs be, but I must stop writing now before I totally lose it and start playing "particle accelerator" with my computer parts in search of an experimental answer...

Purvis: I don't have the book here, so I can't be sure, but the impression I've had since I first read the book (and I've read it several times) is that he was not making the argument that the difference is simply due to a higher collision speed (and higher kinetic energy). It's a possibility I'm forced to consider until I can find my copy of the book, though...
Actually, its just good old fashioned Galileo-Newtonian relativity that says all inertial frames of reference are equally valid. No Einstein needs to be invoked in this paradox.
In the way of an answer, this is what I can come up with: Particle accelerators operate by taking advantage of the fact (explained by Special Relativity) that mass is convertible into energy and vice versa. So the kinetic energy lost when a formerly rapidly moving particle stops dead in its tracks due to a collision, is free to be used to form the mass of new particles created. What matters is the total kinetic energy of the system, and if in a stationary (laboratory) frame of reference I have two accelerated particles moving toward each other at high speed, that's more total velocity and therefore more total kinetic energy in the system than one accelerated particle moving toward a stationary one at high speed in the lab frame, no matter what frame of reference I switch to to measure the velocities.

The gist is that you are going to get out what you put in, energy wise, and if you put in enough energy to accelerate two beams of particles to near the speed of light you are going to get more energy, and therefore more mass and more new particles out than if you just accelerate one beam of particles to near the speed of light. Even different inertial reference frames moving at different constant velocities will recognize that acceleration has taken place, and therefore that energy has been added to a system. I don't see why momentum has to be invoked, but who am I to argue with Leon Lederman.

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