where p is the four-momentum, defined as p α α = ( p 0 , p 1 , p 2 , p 3 ) = ( γ γ m c , p x , p y , p z ) , τ the proper time of the particle, F the contravariant electromagnetic tensor F α α β β = ( 0 − − E x / c − − E y / c − − E z / c E x / c 0 − − B z B y E y / c B z 0 − − B x E z / c − − B y B x 0 ) and U is the covariant 4-velocity of the particle, defined as: U β β = ( U 0 , U 1 , U 2 , U 3 ) = γ γ ( c , − − v x , − − v y , − − v z ) , in which γ γ ( v ) = 1 1 − − v 2 c 2 = 1 1 − − v x 2 + v y 2 + v z 2 c 2 is the Lorentz factor.