Electromagnetic Energy Tensor

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

The previous 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

We know that Maxwell's equations reduce to

where is the electromagnetic field tensor, and is its dual. As is easily verified, Equation (1868) can also be written in the form

Equations (1865) and (1867) can be combined to give

(1868) |

This expression can also be written

(1869) |

Now,

(1870) |

where use has been made of the antisymmetry of the electromagnetic field tensor. It follows from Equation (1869) that

(1871) |

Thus,

(1872) |

The previous expression can also be written

where

(1874) |

is called the

(1875) | ||

(1876) | ||

(1877) |

Equation (1875) can also be written

where is a symmetric tensor whose elements are

(1879) | ||

(1880) | ||

(1881) |

Consider the time-like component of Equation (1880). It follows from Equation (1866) that

(1882) |

This equation can be rearranged to give

where and , so that

(1884) |

and

(1885) |

The right-hand side of Equation (1885) represents the rate per unit volume at which energy is transferred from the electromagnetic field to charged particles. It is clear, therefore, that Equation (1885) is an energy conservation equation for the electromagnetic field. (See Section 1.9.) 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: that is, the

Consider the space-like components of Equation (1880). It is easily demonstrated that these reduce to

where and , or

(1887) |

and

(1888) |

Equation (1888) is basically a