*disclaimer* I am not suggesting any crazy theories here, I am merely examining some of the more extreme situations allowed in relativity! ok so first off, lets allow particles to travel in timelike trajectories backwards through time relative to us. Although this does have some strange causal effects, relativity remains perfectly consistent as it is symmetric under time reversal. If we assume these particles have positive rest mass, then in our reference frame they would appear to have negative mass. I stress the word "appear" because the invariant mass remains positive. However, dt/dτ = -γ so you end up with negative signs every place "relativistic mass" appears. Now, my question is if there is any conservation law (or any other mechanism to prevent this) broken by the spontaneous "creation" of two particles of equal mass, where one is traveling back in time and one is traveling forward in time. The "creation" point is the space-time point where the particle's worldlines intersect. Before this point space was empty, and afterwards we see two particles moving away from each other, one with positive energy and the other with negative energy. It is pretty trivial to show that both energy and momentum should be conserved in this process, and similarly in the time-reversed process (annihilation). However, there is one thing I find very bothersome about this process, and I can't shake the feeling that it shouldn't be allowed on the grounds of conservation laws. If you draw the spacetime diagram of the process, you will see one particle going backwards in time from infinity, and then spontaneously start moving forwards in time back to infinity. How is it possible that momentum is conserved in a process where there is a clear change in direction of the particle! The only explanation I can think of is that special relativity is actually degenerate in the sense that for any 4-momentum and initial conditions there are 2 possible worldlines that are "solutions" (the normal one and the time reversal of it). So what we're actually seeing here is a "spontaneous" switch between two intersecting worldlines with identical 4-momenta.