A steel wire lies along the axis under a tension of

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

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

Problem

A steel wire lies along the

x

axis under a tension of

\sigma=1.0\times10^9\ \mathrm{Pa}

, close to the breaking stress of a strong steel. In the wire's rest frame its stress-energy tensor is

T^{\mu\nu}=\mathrm{diag}(\rho,-\sigma,0,0)

: energy density

\rho=\rho_{\rm m}c^2

with

\rho_{\rm m}=7.9\times10^3\ \mathrm{kg\,m^{-3}}

, a tension (a negative stress) along the wire, and no stress across it. This is not a perfect fluid, so the perfect-fluid forms of the energy conditions do not apply directly. Take

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

. ((a)) An observer moves along the wire at speed

v

. Boost

T^{\mu\nu}

and find the energy density

T'^{00}

this observer measures, in terms of

\rho

\sigma

and

\beta=v/c

. Say also what an observer moving across the wire, along

y

, measures. ((b)) Using the weak energy condition in its general form, that every observer measures a non-negative energy density, find the condition on

\sigma/\rho

for the wire to satisfy it for every observer. Evaluate

\sigma/\rho

for the steel wire. ((c)) For a hypothetical material with the same

\rho

, find the tension, as a multiple of

\rho

, at which an observer moving along it at

0.6c

would measure

T'^{00}=0

, and say whether such a material satisfies the weak energy condition.

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