The Sun is treated as a static

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

The Geometry of Gravity, Ch Linearised gravity: weak fields, gravitomagnetism and the wave equation

Problem

The Sun is treated as a static, non-rotating sphere of mass

M_\odot

and radius

R_\odot = 6.96\times10^5

km, with

GM_\odot/c^2 = 1.48

km. Its interior and exterior are ordinary matter, for which the pressure satisfies

p \ll \rho_{\rm m} c^2

(an ideal fluid stress-energy tensor), and it is not rotating, so

T_{0i}=0

. ((a)) Starting from the linearised, Lorenz-gauge field equation

\Box\bar h_{\mu\nu} = -(16\pi G/c^4)T_{\mu\nu}

, use the stress-energy tensor for slow, low-pressure matter to identify which component of

\bar h_{\mu\nu}

is sourced at leading order, and write the static equation it obeys outside and inside the Sun. ((b)) Solve this static equation for

\bar h_{00}

outside the Sun in terms of

M_\odot

and the isotropic radial distance

\tilde r

, using the same Poisson-equation method that gives the Newtonian limit of the field equations. Undo the trace reversal to obtain

h_{00}

and

h_{ij}

. ((c)) Show that the

g_{00}

you obtain reproduces the gravitational time dilation relation

\Delta\tau/\Delta t \approx 1 + \Phi/c^2

established from redshift measurements, with

\Phi = -GM_\odot/\tilde r

. ((d)) Evaluate

h_{00}

and each diagonal component of

h_{ij}

at the Sun's surface (

\tilde r \approx R_\odot

), quoting both to two significant figures.