A white dwarf has mass and radius km

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Schwarzschild spacetime and the tests of general relativity · 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=0.70\,M_\odot

and radius

R_\star=8000

km. Static observers at the surface and at rest very far away compare the frequency of light emitted at the surface: it is related exactly by

\nu_\infty/\nu_{\rm em}=\sqrt{1-r_s/R_\star}

, with

r_s=2GM/c^2

and

z\equiv\nu_{\rm em}/\nu_\infty - 1

. ((a)) Starting from

z=(1-r_s/r)^{-1/2}-1

, expand to first order in the small quantity

r_s/r

and show it reduces to

z\approx r_s/(2r) = GM/(rc^2)

, the first-order redshift derived from the equivalence principle. ((b)) Using

GM_\odot/c^2=1.48

km, find

r_s

for this white dwarf, then evaluate both the exact redshift

z

and the first-order approximation at its surface, quoted to four significant figures, and state their fractional difference.

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