Consider a fluid with 4-velocity u(vector) and rest-mass density o as measured in the fluids rest frame.(a) From the physical meanings of u(vector), o, and

Consider a fluid with 4-velocity u(vector) and rest-mass density ρo as measured in the fluid’s rest frame.(a) From the physical meanings of u(vector), ρo, and the rest-mass-flux 4-vector S(vector)_rm, deduce Eqs. (2.62).

(b) Examine the components of S(vector)_rm in a reference frame where the fluid moves with ordinary velocity v. Show that

Explain the physical interpretation of these formulas in terms of Lorentz contraction.

(c) Show that the law of conservation of rest mass ∇(vector) · S(vector)_rm = 0 takes the form

where d/dτ is derivative with respect to proper time moving with the fluid.

(d) Consider a small 3-dimensional volume V of the fluid, whose walls move with the fluid (so if the fluid expands, V increases). Explain why the law of rest-mass conservation must take the form d(ρ_oV )/dτ = 0. Thereby deduce that

S = Poy, and Si= si y = Poyv, where y = 1/1-v. poyvi, where