POYNTING'S THEOREM AND LUMINAL TOTAL ENERGY PHYSICAL REVIEW E 64 046610 nuity in the truncated signal(56). In this sense, we may say that the medium responds dynamically to"virtualfrequen- time cies, i.e., to frequencies that would be produced if the signal were suddenly turned off. It is as if the causal medium must be prepared for this possibility and responds accordingly The instantaneous spectra contribute in Eq.(54) to the total energy density of the medium-field system through summation over all frequency contributions. Of course, the pe c imaginary parts of the permittivity and permeability are also present in the integrand giving the energy density stored in the medium. The energy reactively stored in the dissipative medium is greatest, then, when the instantaneous spectrum produces the most overlap with the medium resonances L=0 which resonances are given by peaks in the imaginary parts of (the eigenvalues of) the permittivity E and permeability u 3D-space Depending on the detuning of the incident radiation from these resonances (i.e, depending on the asymptotic [-00 value taken on by its instantaneous spectrum) this time of FIG. 1. The space-time"cone"of a spherical region of space greatest energy storage can be before or after the peak of the that is initially free of energy. Three-dimensional space is repre propagating components of the pulse(which are given solely sented by the horizontal dimensions and time proceeds vertically by the fields e and H)have arrived at a specific position x This"temporal"' disparity of energy storage in the medium Given some final time apex, we prescribe an initial time (and subsequent retrieval from the medium)caused by the Ii(t <laper)at which u vanishes inside an x ball of radius mediums response to virtual frequencies then leads to spa- c(lapexr-ti) centered at position xaper tial redistribution of the field energy, giving rise to a(poten tially anomalous) global energy transport mechanism x∈b(x t;) It can be shown, though, that when this spatial redistribu- Here the notation is defined by tion of energy makes the pulse appear to move superlumi nally the redistribution does not constitute a signal in the B(xo, ro):=xx-xoll srl (58) direction of energy transport. Rather the redistribution is due to a change in the form of the energy-a change from me- [Note that in Fig. 1 the coordinates of the cone's apex are dium to field energy, for example. Thus no matter how fast (xaper faper Given this initial state, we can now show that the pulse may appear to move in a global sense(e.g, in the the energy density u, and thus the fields, vanish in the cone sense of center of mass), the associated signal velocities are depicted in Fig. 1, i.e., in the forward light cone defined by lways luminal [3, 9]. In this sense, the anomalous speeds apparently produced by these spatial redistributions are com- mapex, taper)={(x,)川x- kaper‖ pletely analogous to the phenomena in which two detectors can be made to"click" simultaneously regardless of their ≤c(apex-1,1=t≤1apex},(59) separation simply by irradiating them simultaneously with he same source. The clicking of the two detectors in this thereby establishing luminal front velocity example does not, of course, constitute superluminal cor To this end consider the energy in the various x balls munication between those detectors, rather it merely constI- Eut) denote these energies and note they are defined by tutes simultaneous luminal communication between the source and the detectors n(x,1)d2x,t≤t≤tapx A dynamical energy density implies a maximum front speed In this section we show by looking at energy fow that the support of fields satisfying the Maxwell equations (1) and Note also that since u is positive definite, Eu(t) is always (2), with constitutive relations (3)and(4)prescribed by as sumptions(a)-(c), can expand or contract no faster than c non-negative, The velocity of the support is called the front velocity. We E认)≥0 (61) begin by assuming that the total dynamical energy density u as given by Eq (50)is zero in some spherical region of space Now from Eq (57) we learn that E(t)has the initial data at a time ti. We then demonstrate that this initial condition guarantees that u is also zero on the space-time"cone''of E认(t1)=0. slope c with this initial sphere as its base(see Fig. 1). In other words, no energy(and hence no signal) can enter the We now show that Ein does not differ from this initial initial sphere with a speed greater than c.(For a relevant value for as long as it is defined, i.e., for all time t in similar derivation see [16].) [ti, taper]. Differentiating Et) Using Eq.(60)] we get 046610-7nuity in the truncated signal ~56!. In this sense, we may say that the medium responds dynamically to ‘‘virtual’’ frequen￾cies, i.e., to frequencies that would be produced if the signal were suddenly turned off. It is as if the causal medium must be prepared for this possibility and responds accordingly. The instantaneous spectra contribute in Eq. ~54! to the total energy density of the medium-field system through summation over all frequency contributions. Of course, the imaginary parts of the permittivity and permeability are also present in the integrand giving the energy density stored in the medium. The energy reactively stored in the dissipative medium is greatest, then, when the instantaneous spectrum produces the most overlap with the medium resonances, which resonances are given by peaks in the imaginary parts of ~the eigenvalues of! the permittivity eˆ and permeability mˆ . Depending on the detuning of the incident radiation from these resonances ~i.e., depending on the asymptotic t→` value taken on by its instantaneous spectrum! this time of greatest energy storage can be before or after the peak of the propagating components of the pulse ~which are given solely by the fields E and H) have arrived at a specific position x. This ‘‘temporal’’ disparity of energy storage in the medium ~and subsequent retrieval from the medium! caused by the medium’s response to virtual frequencies then leads to spa￾tial redistribution of the field energy, giving rise to a ~poten￾tially anomalous! global energy transport mechanism. It can be shown, though, that when this spatial redistribu￾tion of energy makes the pulse appear to move superlumi￾nally the redistribution does not constitute a signal in the direction of energy transport. Rather the redistribution is due to a change in the form of the energy—a change from me￾dium to field energy, for example. Thus no matter how fast the pulse may appear to move in a global sense ~e.g., in the sense of center of mass!, the associated signal velocities are always luminal @3,9#. In this sense, the anomalous speeds apparently produced by these spatial redistributions are com￾pletely analogous to the phenomena in which two detectors can be made to ‘‘click’’ simultaneously regardless of their separation simply by irradiating them simultaneously with the same source. The clicking of the two detectors in this example does not, of course, constitute superluminal com￾munication between those detectors, rather it merely consti￾tutes simultaneous luminal communication between the source and the detectors. 2. A dynamical energy density implies a maximum front speed In this section we show by looking at energy flow that the support of fields satisfying the Maxwell equations ~1! and ~2!, with constitutive relations ~3! and ~4! prescribed by as￾sumptions ~a!–~c!, can expand or contract no faster than c. The velocity of the support is called the front velocity. We begin by assuming that the total dynamical energy density u as given by Eq. ~50! is zero in some spherical region of space at a time ti . We then demonstrate that this initial condition guarantees that u is also zero on the space-time ‘‘cone’’ of slope c with this initial sphere as its base ~see Fig. 1!. In other words, no energy ~and hence no signal! can enter the initial sphere with a speed greater than c. ~For a relevant similar derivation see @16#.! Given some final time tapex , we prescribe an initial time ti(ti,tapex) at which u vanishes inside an x ball of radius c(tapex2ti) centered at position xapex : u~x,ti!50, xPB„xapex ,c~tapex2ti!…. ~57! Here the notation is defined by B~x0 ,r0!ª$xuix2x0i<r0%. ~58! @Note that in Fig. 1 the coordinates of the cone’s apex are (xapex ,tapex).# Given this initial state, we can now show that the energy density u, and thus the fields, vanish in the cone depicted in Fig. 1, i.e., in the forward light cone defined by V~xapex ,tapex !ª$~x,t!uix2xapexi <c~tapex2t!,ti<t<tapex%, ~59! thereby establishing luminal front velocity. To this end consider the energy in the various x balls comprising the cone, one for each time t in the cone. Let EV(t) denote these energies and note they are defined by EV~t!ª E B„xapex ,c(tapex2t)… u~x,t!d3x; ti<t<tapex . ~60! Note also that since u is positive definite, EV(t) is always non-negative, EV~t!>0. ~61! Now from Eq. ~57! we learn that EV(t) has the initial data EV~ti!50. ~62! We now show that EV(t) does not differ from this initial value for as long as it is defined, i.e., for all time t in @ti ,tapex#. Differentiating EV(t) @using Eq. ~60!# we get FIG. 1. The space-time ‘‘cone’’ of a spherical region of space that is initially free of energy. Three-dimensional space is repre￾sented by the horizontal dimensions and time proceeds vertically. POYNTING’S THEOREM AND LUMINAL TOTAL ENERGY . . . PHYSICAL REVIEW E 64 046610 046610-7