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The electromagnetic energy tensor
Consider a continuous volume distribution of charged matter in the
presence of an electromagnetic field. Let there be particles
per unit proper volume (unit volume determined in the local rest frame),
each carrying a charge . Consider an inertial frame in which the
3velocity field of the particles is . The number density of the
particles in this frame is
.
The charge density
and the 3current due to the particles are and
, respectively.
Multiplying Eq. (1564)
by the proper number density of particles, , we obtain an expression

(1568) 
for the 4force acting on unit proper volume of the distribution
due to the ambient electromagnetic fields. Here, we have made use of
the definition
. It is easily demonstrated, using some of the
results obtained in the previous section, that

(1569) 
The above expression remains valid when there are many
charge species (e.g., electrons and ions) possessing different
number density and 3velocity fields. The 4vector is
usually called the Lorentz force density.
We know that Maxwell's equations reduce to
where is the electromagnetic field tensor, and
is its dual. As is easily verified, Eq. (1571)
can also be written in the form

(1572) 
Equations (1568) and (1570) can be combined to give

(1573) 
This expression can also be written

(1574) 
Now,

(1575) 
where use has been made of the antisymmetry of the electromagnetic
field tensor. It follows from Eq. (1572) that

(1576) 
Thus,

(1577) 
The above expression can also be written

(1578) 
where

(1579) 
is called the electromagnetic energy tensor. Note that
is a proper4tensor. It follows from Eqs. (1481), (1484), and (1516)
that
Equation (1578) can also be written

(1583) 
where is a symmetric tensor whose elements are
Consider the timelike component of Eq. (1583). It follows from Eq. (1569)
that

(1587) 
This equation can be rearranged to give

(1588) 
where and
, so that

(1589) 
and

(1590) 
The righthand side of Eq. (1588) represents the rate
per unit volume at which energy is
transferred from the
electromagnetic field to charged particles.
It is clear, therefore, that Eq. (1588) is an energy conservation
equation for the electromagnetic field (see Sect. 8.2). The proper3scalar can be identified
as the energy density of the electromagnetic field, whereas
the proper3vector
is the energy flux due to the electromagnetic field: i.e., the Poynting flux.
Consider the spacelike components of Eq. (1583). It is easily demonstrated
that these reduce to

(1591) 
where
and
, or

(1592) 
and

(1593) 
Equation (1591) is basically a momentum conservation equation for the
electromagnetic field (see Sect. 8.4). The righthand side represents the rate per unit volume
at which momentum is transferred from the electromagnetic
field to charged particles. The
symmetric proper3tensor
specifies the flux of electromagnetic
momentum parallel to the th axis crossing a surface
normal to the th axis. The proper3vector represents
the momentum density of the electromagnetic field. It is clear that
the energy conservation law (1588) and the momentum conservation law
(1591) can be combined together to give the relativistically invariant
energymomentum conservation law (1583).
Next: Accelerated charges
Up: Relativity and electromagnetism
Previous: The force on a
Richard Fitzpatrick
20060202