S. GLASGOW. M. WARE AND J PEATROSS PHYSICAL REVIEW E 64 046610 (x,) IV SUPERLUMINAL GLOBAL TRANSPORT OF v2(1)=-1d3xx-at SUBSETS OF THE TOTAL ENERGY THE LORENTZ MODEL xxv. s( While the global notion of energy transport defined by center-of-mass motion of the total dynamical energy in a passive media is always luminal, global energy transport is not so constrained when only a subset of the total dynamical energy is considered. This indicates that in the global sense, (O=ce-l d'x S(x, n) 7) the root of superluminal behavior is associated with incom- nergy accounting.(Note that via the Sommerfeld- The fact that the magnitude of the velocity so defined Brillouin theorems, it is only in a nonlocal sense that super- al ways bounded by c is now straightforward. Ostensibly it luminal phenomena are not strictly prohibited. In order to simplify the discussion, we consider the amounts to no more than a statement of the fact that the Abraham-Lorentz model of a nonmagnetic [H(x, 1) magnitude of the Poynting vector S(x, 1)=E(X, IXH(x, D)Is =B(x, /] homogeneous, isotropic dielectric with a single al ways less than or equal to the energy density u(x, r) resonance frequency, and consider only one-dimensional so- lutions of the original three-dimensional system. In one vI=ce-1dxE(x,1)×H(x (88) space dimension, we can write the equations as a system of first order partial differential equations, E1dxE(x,1)×H(x,)‖ d-c000 B 2|E(x|2+2|Hx at P (x,D) dxu(x, t) (91) 000 0000|B (92) (x,) In passing from Eq.(89)to Eq.(90) we used Lagrange identity, and in passing from Eq (90)to Eq (91) we used the definition of the total dynamical energy density u(x, t), Eq (50) astly we show that the total dynamical energy's center We note in passing that since the eigenvalues of the first of-mass velocity just derived is a spatial average of the tra- matrix on the right of equation(96)(less the spatial deriva- ditional energy transport velocity. Denote and define the tive) are real, the system is hyperbolic. Furthermore, the u-average of a measurable O(x, t)by theory of hyperbolic partial differential equations dictates that these eigenvalues give the limiting speeds at which sin- gularities propagate so that for this model we already have dxO(x, t)u(x, t) the(luminal) Sommerfeld-Brillouin result for the front ve- O(x,1):= (93) locity[18] dxu(x, t) The scalar permittivity E(o) for this model can be calcu- lated to be the usual prototypical example [10] possessing all of the relevant requirements of causality and passivity Then, with this notation, we see that =(ca/(x)=(e(xD)9 (97) where vd, t) is the traditional energy transport velocity Note that in Eqs. (88)through(91)we also effectively dem- Using the fact that the operator on the right of Eq (96)is onstrated that the traditional energy transport velocity is lu- already in a form in which it can be written as a sum of an minal for passive dielectrics, operator that is skew symmetric and one that is negative definite with respect to the usual inner product, we see that vx,l川≤c (95) Eq.(96)dictates a law of dissipation [similar to the law of conservation(48) simply by expressing the time evolution By more complicated arguments, in Ref [3] we also show of the particular positive definite quadratic form associated hat the same is true for active dielectrics with the(relevant) identity matrix 046610-10vu~t!5E 21 E d3x x ]u~x,t! ]t ~85! 52c E 21 E d3x x “•S~x,t!. ~86! Integration by parts then gives vu~t!5c E 21 E d3x S~x,t!. ~87! The fact that the magnitude of the velocity so defined is always bounded by c is now straightforward. Ostensibly it amounts to no more than a statement of the fact that the magnitude of the Poynting vector S(x,t)5E(x,t)3H(x,t) is always less than or equal to the energy density u(x,t), ivu~t!i5c E 21 IE d3x E~x,t!3H~x,t!I ~88! <c E 21 E d3xiE~x,t!3H~x,t!i ~89! <c E 21 E d3x H 1 2 iE~x,t!i 21 1 2 iH~x,t!i 2 J ~90! <c E 21 E d3x u~x,t! ~91! 5c E 21 E5c. ~92! In passing from Eq. ~89! to Eq. ~90! we used Lagrange’s identity, and in passing from Eq. ~90! to Eq. ~91! we used the definition of the total dynamical energy density u(x,t), Eq. ~50!. Lastly we show that the total dynamical energy’s center￾of-mass velocity just derived is a spatial average of the tra￾ditional energy transport velocity. Denote and define the ‘‘u-average’’ of a measurable O(x,t) by ^O~x,t!&uª E d3xO~x,t!u~x,t! E d3xu~x,t! . ~93! Then, with this notation, we see that vu~t!5K S c S u D ~x,t!L u 5^vE~x,t!&u , ~94! where vE(x,t) is the traditional energy transport velocity. Note that in Eqs. ~88! through ~91! we also effectively dem￾onstrated that the traditional energy transport velocity is lu￾minal for passive dielectrics, ivE~x,t!i<c. ~95! By more complicated arguments, in Ref. @3# we also show that the same is true for active dielectrics. IV. SUPERLUMINAL GLOBAL TRANSPORT OF SUBSETS OF THE TOTAL ENERGY: THE LORENTZ MODEL While the global notion of energy transport defined by center-of-mass motion of the total dynamical energy in a passive media is always luminal, global energy transport is not so constrained when only a subset of the total dynamical energy is considered. This indicates that in the global sense, the root of superluminal behavior is associated with incom￾plete energy accounting. ~Note that via the Sommerfeld￾Brillouin theorems, it is only in a nonlocal sense that super￾luminal phenomena are not strictly prohibited.! In order to simplify the discussion, we consider the Abraham-Lorentz model of a nonmagnetic @H(x,t) 5B(x,t)# homogeneous, isotropic dielectric with a single resonance frequency, and consider only one-dimensional so￾lutions of the original three-dimensional system. In one space dimension, we can write the equations as a system of first order partial differential equations, ] ]t S E B P Q D ~x,t!5 ] ]x S 0 2c 0 0 2c 0 00 0 0 00 0 0 00 D S E B P Q D ~x,t! 1S 00 0 2vp 00 0 0 00 0 v0 vp 0 2v0 2g D S E B P Q D ~x,t!. ~96! We note in passing that since the eigenvalues of the first matrix on the right of equation ~96! ~less the spatial deriva￾tive! are real, the system is hyperbolic. Furthermore, the theory of hyperbolic partial differential equations dictates that these eigenvalues give the limiting speeds at which sin￾gularities propagate so that for this model we already have the ~luminal! Sommerfeld-Brillouin result for the front ve￾locity @18#. The scalar permittivity e(v) for this model can be calcu￾lated to be the usual prototypical example @10# possessing all of the relevant requirements of causality and passivity, e~v!511 vp 2 2v22igv1v0 2 . ~97! Using the fact that the operator on the right of Eq. ~96! is already in a form in which it can be written as a sum of an operator that is skew symmetric and one that is negative definite with respect to the usual inner product, we see that Eq. ~96! dictates a law of dissipation @similar to the law of conservation ~48!# simply by expressing the time evolution of the particular positive definite quadratic form associated with the ~relevant! identity matrix, S. GLASGOW, M. WARE, AND J. PEATROSS PHYSICAL REVIEW E 64 046610 046610-10