Electromagnetic force density and energy-momentum tensor in any continuous medium (Vol. 42, No. 5)

For more than a century, physicists have searched for a unique and general form for the force density that an electromagnetic field imposes on a medium. The existing expressions for this quantity, obtained, e.g., by Minkowski, Einstein and Laub, Abraham, and Helmholtz, are different, and, as such, give different predictions in particular physical situations. The theories of Abraham and Minkowski, for example, ignore the existence of electro- and magnetostriction. Moreover, real media with dispersion, dissipation, and nonlinearities have not been addressed much.

We present an unambiguous general equation for the electromagnetic force density f = -∇T- (dG/dt) expressed in terms of a new three-dimensional energy-momentum tensor T and momentum density G of the field. The tensor T can be written as T = T_M + I_V, where T_M is the Minkowski tensor, I the unit tensor, and V the density of the field-matter interaction potential that is responsible for electro- and magnetostriction. Remarkably, if the medium is not magnetic, the momentum density G is given by Abraham’s expression G = ExH/c². If the material obeys the Clausius-Mossotti law, the tensor T becomes the Helmholtz tensor that to our knowledge has not been contradicted in any experiment so far.

The general equation obtained for the force density can be applied to essentially any natural or designed material whether inhomogeneous, anisotropic, nonlinear, dispersive, or dissipative, and even to materials providing optical gain. We also calculate the rate of work done on a medium by an electromagnetic field, and using the result, obtain the four-dimensional energy-momentum tensor T₄ in spacetime. Interestingly, this tensor is physically very close to the almost forgotten tensor of Einstein and Laub.