A comet's dust tail

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Relativity & Gravitation · Kinematics, conservation laws and ideal flow

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

Problem

A comet's dust tail, close to the nucleus, is modelled as a stream of identical dust grains all sharing one common velocity (dust: no random spread in velocity, no mutual pressure). In the tail's own rest frame the grains have number density

n_0 = 5.0\times10^{12}\ \mathrm{m^{-3}}

and each grain has rest mass

m = 1.0\times10^{-15}\ \mathrm{kg}

. A spacecraft passing through sees the stream moving at

v = 0.80c

along its

x

-axis. Take

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

. ((a)) By counting the grains crossing a surface of constant

t

and one of constant

x

, obtain

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

for dust, with

\rho = n_0 m c^2

the rest-frame energy density. Explain the physical origin of each factor of the Lorentz factor

\gamma

that appears in

T^{00}

as measured by the spacecraft, and evaluate

T^{00}

and

T^{0x} = T^{x0}

numerically in the spacecraft's frame. ((b)) Far from the nucleus the grains have spread out and slowed until their relative speed satisfies

v \ll c

, and random motions of the grains give the stream a small thermal pressure

p \ll \rho

. Starting from the perfect-fluid tensor

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

, show that

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

reduces to the continuity equation and

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

to Euler's equation in this limit, stating the order of every term dropped. ((c)) State the trace

T^\mu{}_\mu

of the dust stream's stress-energy tensor, evaluate it numerically, and explain why it does not matter which frame it is computed in.

Browse the problem library