A gas of photons in thermal equilibrium

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Relativistic stars and compact objects · Relativistic stars and compact objects

The Geometry of Gravity, Ch Matter in curved spacetime: minimal coupling and relativistic hydrodynamics

Problem

A gas of photons in thermal equilibrium, black-body radiation, is a perfect fluid with

p=\rho/3

, and its energy density is fixed by its local temperature,

\rho=a_{\rm rad}T^4

with

a_{\rm rad}

a constant. The gas sits at rest in a static metric with

g_{tt}=-e^{2\mathcal{A}(x^i)}

, where it obeys the relativistic hydrostatic equilibrium equation

\partial_i p=-(\rho+p)\,\partial_i\mathcal{A}

. ((a)) Substitute

p=\rho/3

and show that

\rho\,e^{4\mathcal{A}}

is the same at every point of the gas. ((b)) Hence show that the local temperature satisfies

T\,e^{\mathcal{A}}=\text{constant}

. Is the gas hotter where

\mathcal{A}

is lower, deeper in the gravitational field, or where it is higher? ((c)) A tall sealed column of radiation stands in the Earth's weak field, where

\mathcal{A}\approx\Phi/c^2

and

\Phi

increases by

gh

over a height

h

. Find the fractional temperature difference

(T_{\rm bottom}-T_{\rm top})/T_{\rm top}

between the bottom and the top of a column

h=100\ \mathrm{m}

tall, with

g=9.81\ \mathrm{m\,s^{-2}}

and

c=3.00\times10^{8}\ \mathrm{m\,s^{-1}}

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