  The Abraham--Minkowski controversy refers to a long-standing inability to
adequately address certain issues involving the conservation of the momentum of
an electromagnetic field in a linear dielectric medium. We apply the usual
assumption of a material subsystem that couples to the electromagnetic
subsystem such that the total energy and total momentum are conserved. We then
construct the total energy--momentum tensor from the total energy density and
the total momentum density. Applying conservation principles to the total
energy--momentum tensor, we construct the tensor energy--momentum continuity
equation. We show that one of the components of the tensor energy--momentum
continuity equation, the energy continuity equation, is manifestly false. We
conclude that the Abraham--Minkowski controversy is unresolvable because the
extant principles of conservation are inconsistent in a simple linear
dielectric medium.
