Weight of a Moving Body in SR

Problem: Consider a body, with rest mass m₀, moving in a uniform gravitational field aligned in the z-direction. Restrict the motion to constant velocity in the z = 0 plane. Define the magnitude of the weight as the magnitude of the force of support which balances the gravitational force in order to restrain the motion to the z = 0 plane. Determine the weight of the moving body.

Solution #1 - Heuristic approach: Consider a gas of N particles in a box. For simplicity let the gas be 2-dimensional by restricting the motion to the particles to sliding on the botton of the box. Let the box be a rest on a weight scale. Let each particle have the same rest mass m₀, and the same energy E₀. Then all particles will move with the same speed v. The total energy of the box will be then be

[Eq. 1a] E_box = S E_i = N E₀

Since the box is at rest the weight of the box must

[Eq. 2a] W_box = (E_box /c²)g.

The weight of the box, neglecting the weight of the boxes walls (assume they are massless, this will cause no difficulty since we can subtract this weight), is then the sum of the forces on the floor of the box due to the weight w_i of each particle. Let this weight of each particle be w₀. Thus for N particles, the total weight W of the box is then

[Eq. 3a] W_box = Sw_i = N w₀

The energy of each particle, moving with speed v is

[Eq. 4a] E₀ = g m₀ c²

where g = [1 - (v_x² +  v_y²)/c²]^-1/2 = [1 - b²]^-1/2

Equating Eq.1a withe Eq.3a and inserting Eq.4a we get

[Eq.5a] W_box = N w₀ = (E_box /c²)g = ( N E₀ /c²)g = ( N [g m₀ c²] /c²)g = N g m₀ g = N w₀

And hence we get our result

[Eq. 6a] w₀ = g m₀ g = m g = m₀g/[1 - b²>]^1/2

where m = g m₀ is the so-called relativistic mass of the particle.