Red-shifted gamma rays and pair production

Factual Questions
1

The recent thread on how does a gamma ray create a positron-electron pair got me thinking about something I haven’t figured out at first glance.

Let’s say that the amount of energy it takes to produce an electron-positron pair is E. If a gamma ray has just a little more energy than E, then it should be able to produce a pair.

The question is, what will be observed in the following case?

[ul][li]A steady supply of gammas is emitted by the sun and passes by the earth, each with energy E + epsilon (epsilon = a small number.)[/li][li]An observer (Alice) on Earth watches the area the photons are traveling through and observes pair production of some photons, and also measures the energy of photons that do not produce pairs.[/li][li]An observer (Bob) on a spaceship flys by the earth at a high percentage of the speed of light, and observes the same area of space at time T. Because he is flying away from the sun, he measures photons of energy E - epsilon, insufficient energy to create pairs.[/li][/ul]

Bob must see that the gammas are producing pairs, because if Alice can see a pair so can Bob. The question is why? In his frame of reference the gamma rays do not have sufficient energy.

My off the cuff guess is that a photon cannot create a pair if it is in a true vacuum, because an arbitrary reference frame could be selected in which the photon has insufficient energy. In this scenario it is the photon interacting with a medium or other photons which produces the pair. So, perhaps Bob notices other particles in the vicinity and does the math and figures that the photon interacted with one of them. In his frame, the other particles are traveling towards the photon such that the center of mass energy is E + epsilon.

Any ideas?

2

From what I understand, your guess is correct. Both Alice and Bob will see gamma rays hitting the Earth and producing electron-positron pairs. Neither will see pair production happening on Bob’s spaceship, or in vacuum.

Also, if pair production happened in vacuum, you can always find a reference frame in which momentum is not conserved. If you select a reference frame where the resulting electron and positron has exactly opposite velocity (i.e. their center of mass is at zero velocity), then the total momentum of the pair is zero, but the original gamma ray cannot have had zero momentum in this frame. Therefore, the gamma ray photon must hit another particle to undergo pair production.

3

I believe that your question was answered in the original thread: this reaction actually requires two photons moving in opposite directions. If you move in the direction of one photon, thereby decreasing its apparent energy, you will be moving in the opposite direction of the other photon, increasing its apparent energy. Being “at rest” results in the least total apparent energy.