The Newtonian-limit derivation of matches (from the trace-reversed equation for dust)…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Newtonian gravitation as a field theory · Einstein's field equations

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

Problem

The Newtonian-limit derivation of

\kappa

matches

\nabla^2\Phi = \tfrac12\kappa c^2\rho

(from the trace-reversed

00

equation for dust) to Poisson's equation

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

, giving

\kappa=8\pi G/c^4

. Suppose instead someone forgets the trace-reversal factor of

\tfrac12

and matches

\nabla^2\Phi = \kappa c^2\rho

directly to

4\pi G\rho_{\rm m}

. ((a)) Find the resulting (wrong) constant

\kappa_{\rm wrong}

in SI units. ((b)) Re-derive Poisson's equation using

\kappa_{\rm wrong}

in place of the correct

\kappa

, and use it to predict the surface gravity of Mars, given

M_{\rm Mars}=6.42\times10^{23}\,\mathrm{kg}

and

R_{\rm Mars}=3.390\times10^6\,\mathrm{m}

. ((c)) Compare with the measured value

g_{\rm Mars}=3.71\,\mathrm{m\,s^{-2}}