The solar wind near Earth's orbit is modelled as a stream of protons flowing outward…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Relativistic dynamics and fields in flat spacetime · Relativistic dynamics and fields in flat spacetime

The Geometry of Gravity, Ch The stress-energy tensor and relativistic continua

Problem

The solar wind near Earth's orbit is modelled as a stream of protons flowing outward with one common bulk velocity. In the wind's own rest frame the proton number density is

n_0=1.0\times10^7\ \mathrm{m^{-3}}

, the proton rest mass is

m_{\rm p}=1.67\times10^{-27}\ \mathrm{kg}

, and the stream carries a small thermal pressure

p=2.0\times10^{-11}\ \mathrm{Pa}

. Relative to Earth the wind moves at

v=4.0\times10^5\ \mathrm{m\,s^{-1}}

. Take

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

. ((a)) Treating the protons first as pressureless dust, obtain

T^{\mu\nu}=\rho u^\mu u^\nu/c^2

by counting particles crossing a surface, with

\rho=n_0m_{\rm p}c^2

the rest-frame energy density. Evaluate

\rho

numerically. ((b)) Restore the pressure and use the perfect-fluid tensor

T^{\mu\nu}=(\rho+p)u^\mu u^\nu/c^2+p\eta^{\mu\nu}

, with

x

along the flow. Evaluate, in Earth's frame, the wind's

x

-momentum density

T^{x0}/c

and its momentum flux

T^{xx}

. A sail facing the wind absorbs every proton that reaches it: what pressure does the wind exert on the sail? ((c)) Evaluate the two ratios that control the terms dropped in the Newtonian limit of

\partial_\mu T^{\mu\nu}=0

p/\rho

and

v^2/c^2

, and say which of the two approximations (small pressure or small speed) is better satisfied. ((d)) Give the trace

T^\mu{}_\mu

of the full perfect-fluid tensor and evaluate it numerically.

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