(1762) |

This immediately implies that all of the diagonal components of the tensor are zero. In fact, there are only three independent non-zero components of such a tensor. Could we, perhaps, use these components to represent the components of a pseudo-3-vector? Let us write

(1763) |

It is clear that transforms as a contravariant pseudo-3-vector. It is easily seen that

where , et cetera. In this manner, we can actually write the components of a pseudo-3-vector as the components of an antisymmetric proper-3-tensor. In particular, we can write the components of the magnetic field in terms of an antisymmetric proper magnetic field 3-tensor which we shall denote .

Let us now examine Equations (1762) and (1763) more carefully. Recall that and . It follows that we can write Equation (1762) in the form

Likewise, Equation (1763) can be written

(1766) |

Let us multiply this expression by , making use of the identity

We obtain

(1768) |

or

because .

Let us define a proper-4-tensor whose covariant components are given by

It is clear that this tensor is antisymmetric:

(1771) |

This implies that the tensor only possesses six independent non-zero components. Maybe it can be used to specify the components of and ?

Equations (1767) and (1772) yield

(1772) |

Likewise, Equations (1771) and (1772) imply that

(1773) |

Thus,

(1774) | ||

(1775) |

In other words, the completely space-like components of the tensor specify the components of the magnetic field, whereas the hybrid space and time-like components specify the components of the electric field. The covariant components of the tensor can be written

Not surprisingly, is usually called the

The contravariant components of the electromagnetic field tensor are given by

(1777) | ||

(1778) |

or

Let us now consider two of Maxwell's equations:

(1780) | ||

(1781) |

Recall that the 4-current is defined . The first of these equations can be written

because . The second of these equations takes the form

(1783) |

Making use of Equation (1769), the previous expression reduces to

Equations (1784) and (1786) can be combined to give

(1785) |

This equation is consistent with the equation of charge continuity, , because of the antisymmetry of the electromagnetic field tensor.