General Relativity: An Introduction for Physicists · Problem 8.4

The gravitational field equations — Hobson, Efstathiou & Lasenby

Physics · charged-dust-energy-momentum-tensor

Problem

Consider a cloud of charged dust particles. Show that the equation of motion of such a fluid is

\rho u^\nu \nabla_\nu u^\mu = \sigma F^\mu{}_{\nu} u^\nu,

where

\rho

is the proper matter density of the fluid,

\sigma

is its proper charge density and

u^\mu

is the fluid 4-velocity. Define an energy–momentum tensor

T^{\mu\nu}_{\rm d} = \rho u^\mu u^\nu

, where

\rho

is the proper density of the fluid. Hence show that

\nabla_\mu T^{\mu\nu}_{\rm d} = F^\mu{}_{\nu} j^\nu,

\nabla_\mu T^{\mu\nu}_{\rm em} = -F^\mu{}_{\nu} j^\nu,

where

j^\mu = \sigma u^\mu

is the 4-current density. Thus write down the energy–momentum tensor for charged dust,

T^{\mu\nu} = T^{\mu\nu}_{\rm d} + T^{\mu\nu}_{\rm em}