Relativistic Motion

The Lagrangian of a relativistic particle in a static external potential U(r) reads p L = −mc c2 − v2 − U(r) , (1) where m is the particle mass, c is the velocity of light (a global world con- stant), v = r˙ is the velocity of the particle, and v = |v|. The Lagrange equation of motion d ∂L ∂L = (2) dt ∂v ∂r then yields d v ∂U m p = F(r) , F(r) = − . (3) dt 1 − v2/c2 ∂r p In the non-relativistic limit, v ¿ c, we have 1 − v2/c2 ≈ 1 and the equation of motion reproduces the Newton’s equation. To trace the rela- tivistic regime v ∼ c, and especially to describe the ultra-relativistic limit of (c − v)/c ¿ 1, it is convenient to employ the Hamiltonian formalism. To this end we ﬁrst need to ﬁnd the energy (which is interesting on its own). By the general formula ∂L E = v · − L (4) ∂v we get mc2 E = p + U(r) . (5) 1 − v2/c2 Note that in the non-relativistic limit we have mv2 E ≈ mc2 + + U(r)(v ¿ c) . (6) 2 The second term is the non-relativistic kinetic energy. The ﬁrst term is the famous relativistic rest energy. Since it is just a constant, there is no way to directly observe it in the non-relativistic physics. Now we introduce the momentum as the variable canonically conjugated to the coordinate: ∂L ∂L p = ≡ . (7) ∂r˙ ∂v

1 The explicit relation between p and v then is

mv p/m p = p ⇔ v = p . (8) 1 − v2/c2 1 + (p/mc)2

It is not a surprise that in the non-relativistic limit v ¿ c we just reproduce the known expression p = mv. We also see that in terms of momentum the condition of non-relativistic regime is p ¿ mc. Correspondingly, the condition of the ultra-relativistic regime is p À mc, in which case the relation between v and p yields p v = c (p À mc) . (9) p No matter how large is p, the velocity in the ultra-relativistic limit saturates to (but never reaches) c. To get the Hamiltonian, we need to express energy in terms of the mo- mentum and coordinate. This way we get q H(r, p) = mc2 1 + (p/mc)2 + U(r) . (10)

Given the Hamiltonian, we immediately get the equations of motion:

∂H p/m r˙ = = p , (11) ∂p 1 + (p/mc)2

∂H ∂U p˙ = − = − = F(r) . (12) ∂r ∂r The ﬁrst of the two equations is nothing but the relation between the ve- locity and the momentum that we have established previously. The second equation that relates the derivative of the momentum with the force turns out to be exactly the same as in the non-relativistic mechanics. This, in particular, means that if the force is coordinate-independent, then the mo- mentum is a linear function of time which can grow arbitrarily large (in sharp contrast to the velocity).

2