lecture notes on statistical mechanics - Scott Pratt - Michigan State ...

4.5 Relativistic HydrodynamicsSome of the most important applications of hydrodynamics involve relativistic motion. These includeastronomical applications as well as some laboratory applications such as relativistic heavyion collisions. In some cases, such as black holes, general relativity comes into play as well. Hydrodynamicscan incorporate special relativity in a most elegant fashion by considering the stressenergy tensor,T αβ = (P + ϵ)u α u β − P g αβ , (4.34)where ϵ is the energy density in the frame of the fluid ϵ = U/V , and u α is the relativistic fourvelocity. The equations of motion for hydrodynamics and for entropy conservation are expressed as∂ α T αβ = 0. (4.35)This is effectively four separate conservation equations, one for each value of β. To see that theserepresent the equations of motion for hydrodynamics, one can view the system in a frame wherethe velocities are small, and if they reproduce the non-relativistic equations in that frame, they willbe correct in all frames since they have the correct Lorentz form. To see this, we write the 4 × 4stress-energy tensor for the limit where u = (1, v x , v y , v z ),T αβ =⎛⎜⎝ϵ (P + ϵ)v x (P + ϵ)v y (P + ϵ)v z(P + ϵ)v x −P 0 0(P + ϵ)v y 0 −P 0(P + ϵ)v z 0 0 −P⎞⎟⎠ (4.36)By inspection one can see that the ∂ α T α0 = 0 leads to the equation for entropy conservation, Eq.(4.29). The other three equations, ∂ α T αi = 0, become,∂v∂t = − 1 ∇P. (4.37)P + ϵIn the non-relativistic limit P + ϵ ∼ ρ m , and these equations reproduce the non-relativisitic expressionfor acceleration. Finally, after writing the equation for current conservation relativistically,∂ α (ρu α ) = 0, all the non-relativistic equations are satisfied, and if they are satisfied in one frame,they must be satisfied in all frames given that they have a manifestly consistent Lorentz structure.One might ask about whether the energy density includes the vacuum energy. In that case thereis also a vacuum energy. By “vacuum”, one refers to the energy density at zero temperature andparticle density. In that case, the pressure isP = − dE vacdV = −d(ϵ vacV )dVwhere ϵ vac and P vac denote the vacuum energy density and pressure.relativistic hydrodynamics is insensitive to the vacuum energy density.4.6 Hydrodynamics and Sound= −ϵ vac , (4.38)Thus, the P + ϵ term inTo see how a sound wave is a hydrodynamic wave, we linearize the equations of motion and theequation of continuity, keeping terms only of first order in the velocity ⃗v, and first order in the64