Question

A sphere moves with a relativistic velocity v through a gas whose unit volume contains n slowly moving particles, each of mass m. Find the pressure p exerted by the gas on a spherical surface element perpendicular to the velocity of the sphere, provided that the particles scatter elastically. Show that the pressure is the same both in the reference frame fixed to the sphere and in the reference frame fixed to the gas.

Answer 1

In the frame fixed to the sphere :- The momentum transferred to the eastically scattered particle is
`(2mv)/(sqrt(1-v^2/c^2))`
The density of the moving element is, from `1.369`, `n(1)/(sqrt(1-v^2/c^2))`
and the momentum transferred per unit time per unit area is
`p=` the pressure`=(2mv)/(sqrt(1-v^2/c^2))n(1)/(sqrt(1-v^2/c^2))*v=(2mnv^2)/(1-v^2/c^2)`
In the frame fixed to the gas :- When the sphere hits a stationary particle, the latter recoils with a velocity
`=(v+v)/(1+v^2/c^2)=(2v)/(1+v^2/c^2)`
The momentum transferred is `((m*2v)/(1+v^2//c^2))/(sqrt(1-(4v^2//c^2)/((1-v^2//c^2)^2)))=(2mv)/(1-v^2/c^2)`
and the pressure is `(2mv)/(1-v^2/c^2)*n*v=(2mnv^2)/(1-v^2/c^2)`