Since this is only part of the whole energy story, there is no problem when it becomes superluminal. The overly rapid appearance of electromagnetic energy at one point and its simultaneous disappearance at another point merely indicates an exchange of energy between the electric field and the medium.
We need not be dazzled by a magician who invites the audience to look only at the field energy while energy transfers into and out of the ‘unwatched’ domain of the medium. Extra field energy seems to appear ‘prematurely’ downstream only if there is already non-zero field energy downstream to stimulate a transfer of energy from the medium. The actual transport of energy is strictly bounded by; superluminal propagation of a sharp signal front is impossible. In accordance with Poynting’s theorem, the total energy density stored in an electromagnetic field and in a medium is given.
As used increases, the energy in the medium increases. Conversely, as use decreases, the medium surrenders energy to the electromagnetic field. While it is possible for users to become negative, the combination used +u (−∞) (i.e. the net energy in the medium) can never go negative since material cannot surrender more energy than it possesses to begin with. Poynting’s theorem has the form of a continuity equation which when integrated spatially over a small volume V yields.
Note that this ratio of the Pointing flux to the energy density has units of velocity. When the total energy density u is used in computing, the energy transport velocity has a fictitious nature; it is not the actual velocity of the total energy (since the part is stationary), but rather the effective velocity necessary to achieve the same energy transport that the electromagnetic flux alone delivers. If we reduce the denominator to the subset of the energy that can move, namely the field, the Cauchy-Schwartz inequality (i.e. α 2 +β 2 ≥ 2αβ) ensures an energy transport velocity vE remains strictly bounded by the speed of light in vacuum c. The total energy density u is at least as great as the field energy density field. Hence, this strict luminosity is maintained.
The expression describes the net energy density transferred to a point in the medium after all action has finished (i.e. at t = ∞). It involves the power spectrum of the pulse. We can modify this formula in an intuitive way so that it describes the transfer of energy density to the medium at any time during the pulse. Since the medium is unable to anticipate the spectrum of the entire pulse before experiencing it, the material responds to the pulse according to the history of the field up to each instant. In particular, the material has to be prepared for the possibility of an abrupt cessation of the pulse at any moment, in which case all exchange of energy with the medium immediately ceases.
In this extreme scenario, there is no possibility for the medium to recover from previously incorrect attenuation or amplification, so it must have gotten it right already. If the pulse were in fact to abruptly terminate at a given instant, it would not be necessary to integrate the inverse Fourier transform beyond the termination time t after which all contributions are zero.
Last word
Causality requires that the medium be indifferent to whether a pulse actually terminates if that possibility lies in the future. Therefore, can apply for any time t (not just for t = ∞) if the spectrum is evaluated just for that portion of the field previously experienced by the medium (up to time t). The following is then an exact representation for the energy density transferred to the medium.
