A white dwarf has mass and radius

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Differential geometry of curved spacetime · Schwarzschild spacetime and the tests of general relativity

The Geometry of Gravity, Ch The Schwarzschild geometry and Birkhoff's theorem

Problem

A white dwarf has mass

M = 1.02\,M_\odot

and radius

R_\star = 8000\ \mathrm{km}

. Take

GM_\odot/c^2 = 1.48\ \mathrm{km}

. ((a)) In the general static, spherically symmetric ansatz

ds^2 = -e^{2\mathcal{A}(r)}c^2dt^2 + e^{2\mathcal{B}(r)}dr^2 + r^2d\Omega^2

, name the Killing-vector property that forces (i)

\mathcal{A}

and

\mathcal{B}

to be independent of

t

, and (ii) the absence of any cross-term

dt\,dr

. ((b)) Using the weak-field metric derived from linearised gravity, matched to the Schwarzschild metric at large

r

, evaluate the Schwarzschild radius

r_s

of this white dwarf in kilometres. ((c)) Independently estimate the fractional frequency shift

z

of a photon climbing from the white dwarf's surface to a distant observer, using the Newtonian-potential redshift relation

z \approx GM/(R_\star c^2)

from the equivalence-principle argument for gravitational redshift, and check that it equals

r_s/(2R_\star)

to the precision quoted.

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