A static spherical star of mass has a weak external field

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · The equivalence principle and gravity as geometry · The equivalence principle and gravity as geometry

Relativity and Gravitation, Ch Gravity as curved spacetime: redshift and local inertial frames

Problem

A static spherical star of mass

M

has a weak external field,

GM/rc^2 \ll 1

everywhere of interest. Outside it the gravitational potential is

\Phi(r) = -GM/r

, zero at infinity, and the local field strength is

g(r) = GM/r^2

. Light is emitted by a source at rest at radius

r_1

and received by an observer at rest at radius

r_2 > r_1

. Work throughout to first order in

GM/rc^2

. ((a)) Starting from the local result

\Delta\nu/\nu = -g\,\Delta r/c^2

for a short radial climb

\Delta r

, stack short climbs into an integral and show that

\frac{\nu_{\rm rec}}{\nu_{\rm em}} = 1 - \frac{GM}{c^2}\left(\frac{1}{r_1} - \frac{1}{r_2}\right).

((b)) Evaluate the redshift

z

for light leaving the Sun's photosphere and received at the Earth's orbit. Take

M_\odot = 1.989\times10^{30}\ \mathrm{kg}

R_\odot = 6.960\times10^{8}\ \mathrm{m}

r_2 = 1.496\times10^{11}\ \mathrm{m}

G = 6.674\times10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}

and

c = 2.998\times10^{8}\ \mathrm{m\,s^{-1}}

; ignore the Earth's own field and its orbital motion. ((c)) Obtain the same formula a second way, from energy conservation: a mass

m

is lowered slowly from

r_2

r_1

with the work captured by a generator, is then converted entirely into a light pulse which is beamed up to

r_2

and there converted entirely back into a mass at rest. Show that the cycle can create no energy only if the pulse's energy obeys the ratio derived in (a), and hence, since a pulse's energy is proportional to its frequency, that the frequency ratio is the same.

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