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# Dual Electromagnetic Field Tensor

We have seen that it is possible to write the components of the electric and magnetic fields as the components of a proper-4-tensor. Is it also possible to write the components of these fields as the components of some pseudo-4-tensor? It is obvious that we cannot identify the components of the proper-3-vector with any of the components of a pseudo-tensor. However, we can represent the components of in terms of those of an antisymmetric pseudo-3-tensor by writing

 (1786)

It is easily demonstrated that

 (1787)

in a right-handed coordinate system.

Consider the dual electromagnetic field tensor, , which is defined

 (1788)

This tensor is clearly an antisymmetric pseudo-4-tensor. We have

 (1789)

plus

 (1790)

where use has been made of . The previous expression yields

 (1791)

It follows that

 (1792) (1793)

or

 (1794)

The previous expression is, again, slightly misleading, because stands for the component of the pseudo-3-tensor , and not for an element of the proper-3-vector . Of course, in this case, really does represent the first element of the pseudo-3-vector . Note that the elements of are obtained from those of by making the transformation and .

The covariant elements of the dual electromagnetic field tensor are given by

 (1795) (1796)

or

 (1797)

The elements of are obtained from those of by making the transformation and .

Let us now consider the two Maxwell equations

 (1798) (1799)

The first of these equations can be written

 (1800)

because . The second equation takes the form

 (1801)

or

 (1802)

Equations (1802) and (1804) can be combined to give

 (1803)

Thus, we conclude that Maxwell's equations for the electromagnetic fields are equivalent to the following pair of 4-tensor equations:

 (1804) (1805)

It is obvious from the form of these equations that the laws of electromagnetism are invariant under translations, rotations, special Lorentz transformations, parity inversions, or any combination of these transformations.

Next: Transformation of Fields Up: Relativity and Electromagnetism Previous: Electromagnetic Field Tensor
Richard Fitzpatrick 2014-06-27