Earth's atmosphere near sea level is modelled as a static perfect fluid in the weak…
PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Relativistic stars and compact objects · Relativistic stars and compact objects
The Geometry of Gravity, Ch Matter in curved spacetime: minimal coupling and relativistic hydrodynamics
Problem
Earth's atmosphere near sea level is modelled as a static perfect fluid in the weak field of the Earth, where gtt=−e2A with A≈Φ/c2 and Φ(r)=−GM⊕/r (take GM⊕/c2=4.44 mm and R⊕=6.37×106 m). The relativistic hydrostatic equilibrium equation is drdp=−(ρ+p)drdA. Air near sea level has mass density ρm=1.2kgm−3 and pressure patm=1.0×105Pa, so its rest-frame energy density is ρ=ρmc2≈1.08×1017Jm−3.
((a)) Compute the dimensionless ratio patm/(ρmc2) and the ratio ∣Φ∣/c2=GM⊕/(R⊕c2) at the Earth's surface.
((b)) Take the formal limit p/ρ→0 and Φ/c2→0 in the equilibrium equation above, keeping the gradients finite, and show it reduces to the Newtonian hydrostatic equation dp/dr=−ρmdΦ/dr.
((c)) State which of the two ratios found in part (a) is smaller, and hence which relativistic correction, the pressure added to the inertia or A replacing Φ/c2, is the tighter approximation for Earth's atmosphere.