On the Apparent Superluminal Motion of a Damped Gaussian Pulse

arXiv:1112.1324v1 [physics.gen-ph] 4 Dec 2011 On the Apparent Superluminal Motion of a Damped Gaussian Pulse N. Redington Net Advance of Physics redingtn@mit.edu Alicki has demonstrated that a travelling Gaussian pulse subject to damping is indistinguishable from an undamped pulse moving with greater speed; such an effect could create the illusion of a pulse moving faster than light. In this note, an alternative derivation of the same result is presented. However, it is unlikely that this particular illusion could explain the superluminal neutrino-velocities reported by OPERA. The recent claim that the measured velocity of mu neutrinos exceeds that of light [1] has rekindled interest in the old subject of illusory superluminal motion. Of the various proposals advanced to explain the OPERA results without jeop- ardy to established physics [2], that of Alicki [3] is among the most intriguing. Alicki argues that a stationary observer cannot tell a moving normal distribution subject to global exponential damping from an undamped normal distribution with different characteristic parameters, including higher (even apparently su- perluminal) velocity. This result, which may be of some importance in optics and other fields unrelated to neutrino physics, is re-derived in the present note by very elementary means. Unfortunately (or otherwise, if one wishes to break the light barrier in fact rather than appearance), such an effect seems unlikely to be the solution to the OPERA puzzle: the CNGS-beam pulse lacks the “fat tail” needed for the argument to apply. To derive Alicki’s result, it is first helpful to consider the case of a Gaussian pulse with no damping at all. Let ρ(x, t) denote the intensity, assumed non- negative, of a signal moving at constant speed v > 0 in one dimension without dispersion or distortion toward an observer at some fixed location x = L. By hypothesis, observer and source are in a state of mutual rest. Let the profile of the signal as initially transmitted be Gaussian with maximum height ρ0 and full width σ at half maximum. Since transmission is perfect, the observer at L will see: ρ(t) = ρ0 exp −(L −vt)2 2σ2 1 The time derivative of ρ is: ˙ρ = ρ v σ2 (L −vt) so the observer will notice a peak at time t = L/v Prior to that time, ρ is increasing; afterwards, decreasing. Knowing L and measuring t, the observer easily determines the velocity of the pulse to be v, as expected. Now introduce a simple damping: ρ(t) = (ρ0 exp −(L −vt)2 2σ2 ) exp −γt (1) where γ is a positive constant. The time derivative of ρ becomes: ˙ρ = ρ v σ2 (L −vt −δL) where δL ≡γσ2 v ≥0 (2) The observer will now notice a peak at time t = T ≡(L −δL)/v Prior to that time, ρ is increasing; afterwards, decreasing. It is therefore rea- sonable to define v′ ≡L/T = Lv L −δL (3) Note that v′ ≥v provided 0 ≤δL ≤L. To shew that v′ could in fact be interpreted as the velocity of a moving, undamped Gaussian pulse, note that according to Eq. (2): σ2γt = vtδL (4) Therefore, we may rewrite Eq. (1) in terms of δL as: ρ(t) = ρ0 exp −F 2σ2 (5) where F ≡L2 + (vt)2 −2(L −δL)vt (6) Furthermore, define two new constants σ′ ≡ σ 1 −(δL/L) (7) 2 and ρ′ 0 ≡ρ0 exp −δL 2σ2 (2L −δL) (8) With the help of Eqs. (2), (3), (6), (7), and (8), Eq. (5) becomes: ρ(t) = ρ′ 0 exp −(L −v′t)2 2σ′2 (9) quod erat demonstrandum. Less formally, the Alicki effect can be understood in terms of the damping acting on the signal over time. At t = 0, the value of ρ in the neighbourhood of L is ρ0 exp −(L/σ)2/2. A few moments later, the part of the distribution originally just a bit to the left of L will pass the observer; the damping force will have acted upon it briefly, so its magnitude will be somewhat diminished. By contrast, the regions of the distribution which at t = 0 were far to the left, though they may have had initially large values of ρ, will arrive at the observer much later, and the exponential decay induced by the damping force will have acted on them for a long time. Thus, the magnitude of parts of a sufficiently wide Gaussian may be greatly diminished by even a small damping, and the peak of the received signal will be shifted to the right relative to the peak in the signal as originally sent. The constraint δL ≤L for positive v′ can also be given a qualitative in- terpretation. Consider the three factors which define δL according to Eq. (2). Should the damping constant γ be too large, or the initial velocity v too small, the magnitude of the signal reaching the observer will diminish from the very first observation at t = 0. The observer might conclude from this that the pulse is moving to the left (in agreement with Eq. (3) for this case), and that only its receding tail is being observed. The same is true, if perhaps less self-evidently, for a very broad pulse: it rises too slowly to keep up with the decay, and may also be construed as moving leftwards. Since both source and observer are stationary in a common frame of refer- ence, the initial velocity v may be given a

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