Model Jupiter as a static

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Einstein's field equations · Einstein's field equations

The Geometry of Gravity, Ch The Einstein field equations and the Newtonian limit

Problem

Model Jupiter as a static, uniform ball of matter at rest whose pressure, although it holds the planet up, is negligible next to its energy density,

p \ll \rho

, so that in the source it acts as dust with

T^{\mu\nu} \approx \rho u^\mu u^\nu/c^2

. Take mass

M_J = 1.90\times10^{27}\ \mathrm{kg}

and mean radius

R_J = 6.99\times10^7\ \mathrm{m}

. ((a)) Using the trace-reversed field equations with

\Lambda=0

and the relation

R_{00}=\nabla^2\Phi/c^2

for the weak, static tidal field, show that a dust source with

T_{00}=\rho

and

T=-\rho

gives

\nabla^2\Phi = 4\pi G\rho_{\rm m}

, where

\rho_{\rm m}=\rho/c^2

. ((b)) Treating Jupiter as a uniform sphere of density

\rho_{\rm m}

, use the exterior field

g_J = GM_J/R_J^2

to estimate its surface gravity. ((c)) For exact dust (

p=0

\nabla_\mu T^{\mu\nu}=0

is equivalent to every dust particle following a geodesic. Explain why this is not a separate postulate here but is forced by the field equations. Then explain why Jupiter could not be exact dust at rest, and what its small pressure must do instead.

Browse the problem library