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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 3-velocity field of the particles is . The number density of the particles in this frame is . The charge density and the 3-current due to the particles are and , respectively. Multiplying Eq. (1564) by the proper number density of particles, , we obtain an expression
 (1568)

for the 4-force 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 3-velocity fields. The 4-vector is usually called the Lorentz force density.

We know that Maxwell's equations reduce to

 (1570) (1571)

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 proper-4-tensor. It follows from Eqs. (1481), (1484), and (1516) that
 (1580) (1581) (1582)

Equation (1578) can also be written

 (1583)

where is a symmetric tensor whose elements are
 (1584) (1585) (1586)

Consider the time-like 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 right-hand 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 proper-3-scalar can be identified as the energy density of the electromagnetic field, whereas the proper-3-vector is the energy flux due to the electromagnetic field: i.e., the Poynting flux.

Consider the space-like 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 right-hand side represents the rate per unit volume at which momentum is transferred from the electromagnetic field to charged particles. The symmetric proper-3-tensor specifies the flux of electromagnetic momentum parallel to the th axis crossing a surface normal to the th axis. The proper-3-vector 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 energy-momentum conservation law (1583).

Next: Accelerated charges Up: Relativity and electromagnetism Previous: The force on a
Richard Fitzpatrick 2006-02-02