Near the centre of the Sun the mass density is and the pressure is

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

Near the centre of the Sun the mass density is

\rho_{m,\odot}=1.5\times10^{5}\ \mathrm{kg\,m^{-3}}

and the pressure is

p_\odot=2.4\times10^{16}\ \mathrm{Pa}

. In the core of a typical neutron star the mass density is

\rho_{m,\mathrm{NS}}=6\times10^{17}\ \mathrm{kg\,m^{-3}}

and the pressure is

p_{\mathrm{NS}}=6.5\times10^{33}\ \mathrm{Pa}

. Treat both as perfect fluids at rest, with

\rho=\rho_m c^2

the proper energy density appearing in

T^{\mu\nu}

, and take

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

. ((a)) Find the dimensionless ratio

p/\rho

at the Sun's centre and in the neutron-star core. ((b)) The inertia of a fluid element in the relativistic Euler equation is

\rho+p

, and the local curvature source found for

R_{00}

\rho+3p

. State, as a fractional excess over

\rho

alone, the size of each correction for the Sun's centre and for the neutron-star core. ((c)) By roughly what factor is the neutron-star core's pressure correction larger than the Sun's?

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