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Time Dilation in Relativistic Two-Particle Interactions B T Illinois State University ISU ReD: Research and eData Faculty publications – Physics Physics 11-2010 Time dilation in relativistic two-particle interactions B T. Shields Illinois State University Rainer Grobe Illinois State University E V. Stefanovich Weidlinger Association M R. Ware Illinois State University Qichang Su Illinois State University See next page for additional authors Follow this and additional works at: https://ir.library.illinoisstate.edu/fpphys Part of the Atomic, Molecular and Optical Physics Commons Recommended Citation Shields, B T.; Grobe, Rainer; Stefanovich, E V.; Ware, M R.; Su, Qichang; and Morris, M C., "Time dilation in relativistic two-particle interactions" (2010). Faculty publications – Physics. 16. https://ir.library.illinoisstate.edu/fpphys/16 This Article is brought to you for free and open access by the Physics at ISU ReD: Research and eData. It has been accepted for inclusion in Faculty publications – Physics by an authorized administrator of ISU ReD: Research and eData. For more information, please contact [email protected]. Authors B T. Shields, Rainer Grobe, E V. Stefanovich, M R. Ware, Qichang Su, and M C. Morris This article is available at ISU ReD: Research and eData: https://ir.library.illinoisstate.edu/fpphys/16 PHYSICAL REVIEW A 82, 052116 (2010) Time dilation in relativistic two-particle interactions B. T. Shields,1 M. C. Morris,1 M. R. Ware,1 Q. Su,1 E. V. Stefanovich,2 and R. Grobe1 1Intense Laser Physics Theory Unit and Department of Physics, Illinois State University, Normal, Illinois 61790-4560, USA 22255 Showers Drive, Unit 153, Mountain View, California 94040, USA (Received 6 July 2010; published 29 November 2010) We study the orbits of two interacting particles described by a fully relativistic classical mechanical Hamiltonian. We use two sets of initial conditions. In the first set (dynamics 1) the system’s center of mass is at rest. In the second set (dynamics 2) the center of mass evolves with velocity V . If dynamics 1 is observed from a reference frame moving with velocity −V , the principle of relativity requires that all observables must be identical to those of dynamics 2 seen from the laboratory frame. Our numerical simulations demonstrate that kinematic Lorentz space-time transformations fail to transform particle observables between the two frames. This is explained as a result of the inevitable interaction dependence of the boost generator in the instant form of relativistic dynamics. Despite general inaccuracies of the Lorentz formulas, the orbital periods are correctly predicted by the Einstein’s time dilation factor for all interaction strengths. DOI: 10.1103/PhysRevA.82.052116 PACS number(s): 03.30.+p, 45.50.Jf, 45.20.Jj I. INTRODUCTION To study potential deviations from predictions of Eqs. (1.3), several works [8–11] have investigated the decay law of an The fundamental principle of relativity requires that the unstable particle acting like a moving clock. Using relativistic laws of physics should be invariant under changes of inertial quantum mechanics, they have suggested that the true decay reference frames. This property is formulated in terms of law for a fast particle is not provided by the Einstein’s time symmetry of the theory under the Poincare´ group of iner- dilation factor. Unfortunately, the predicted deviations are tial transformations [1–3]. For simplicity, in this work we several orders of magnitude smaller than the experimental error limit ourselves to one spatial dimension only. Then inertial of the most accurate experiments to date [12,13]. observers are related to each other by space, time, and velocity In this work we consider another type of physical system, translations associated with the three generators P , H , and K, where deviations from Eqs. (1.3) can be, in principle, observed. respectively. In classical mechanics, they have to satisfy the We give a concrete example for a system of two mutually at- three Poisson brackets (Lie algebra) tracting classical particles. As these two particles oscillate with {P,H}=0, (1.1a) respect to each other, there are distinct periodic moments in time when their trajectories cross each other. The occurrences {H,K}=P, (1.1b) of these unambiguous events serve as a clock, and we can test how Einstein’s time dilation formula would describe the obser- {P,K}=H/c2. (1.1c) vations by a moving observer. In agreement with theoretical If we would like to predict how an observable A(X,P )(asa predictions [6], our numerical simulations demonstrate that function of the phase-space variables X and P ) is measured space-time Lorentz transformations Eqs. (1.3) do not hold for from a different reference frame, we have to solve the equation particle trajectories. Moreover, strong interaction potentials allow for particle velocities higher than the speed of light. ∂A(s)/∂s ={G,A(s)}, (1.2) Nevertheless, Einstein’s time dilation formula remains valid with the initial condition A(s = 0) = A(X,P ), where G is for the orbital periods. either P , K,orH. Each generator is associated with its cor- responding group parameter s, which can be the displacement II. NONINTERACTING PARTICLES = −1 d, the rapidity cθ (where θ tanh (V/c)isafunctionofthe Let us consider a system of two classical particles. We velocity V ), or the time t of the new reference frame relative assume that they have the same mass m1 = m2 ≡ m, which to the laboratory frame. will make the expressions below a little bit more transparent. In the instant form of relativistic dynamics, both the The phase space associated with positions xi and momenta pi Hamiltonian H and the velocity boost operator K must depend for i = 1 and 2 is four-dimensional and we define the usual on the interaction in order to satisfy the Poincare´ relations Poisson brackets as {A,B}=i ∂xiA∂piB − ∂xiB∂piA.Ifthe [2–5]. Therefore one can expect that boost transformations particles are noninteracting, then Poincare´ relations (1.1) are of particle trajectories are interaction dependent and system easily satisfied by choosing specific [6,7]. This conclusion is in obvious disagreement with the traditionally assumed interaction-independent and P = p1 + p2, (2.1a) universal form of the space-time Lorentz transformation H = h + h , (2.1b) formulas (x,t) → (x,t ): 0 1 2 K0 = k1 + k2, (2.1c) x = x cosh(θ) − ct sinh(θ), (1.3a) ≡ 2 4 + 2 2 1/2 =− 2 where hi (m c c pi ) and ki xi hi /c and the t = x sinh(θ)/c − t cosh(θ). (1.3b) subscript “0” indicates the absence of interactions. It is 1050-2947/2010/82(5)/052116(6) 052116-1 ©2010 The American Physical Society SHIELDS, MORRIS, WARE, SU, STEFANOVICH, AND GROBE PHYSICAL REVIEW A 82, 052116 (2010) convenient to introduce the total mass observable M (which noting that c tanh(θ) is the relative velocity of the moving has vanishing Poisson brackets with all three generators P , frame. Let us now calculate boost transformations for posi- H0, and K0) and the center of mass position R: tions. From Eq. (1.2) with A = xi and G = K0, s = cθ,we / obtain M = H 2 − c2P 2 1 2 c2, (2.2) 0 2 ∂xi (θ)/∂(cθ) ={−x1(θ)h1(θ) − x2(θ)h2(θ),xi (θ)}/c R = (x1h1 + x2h2)/(h1 + h2). (2.3) = x (θ)v (θ)/c2, (3.5a) Then Eqs. (2.1b) and (2.1c) can be rewritten as i i which, together with the initial condition x (θ = 0) = x (0), = 2 4 + 2 2 1/2 i i H0 [M c c P ] , (2.4) results in the familiar length contraction 2 K0 =−RH0/c . (2.5) xi (θ) = xi (0)hi (0)/hi (θ). (3.5b) The time evolution of the particle position in the moving III. TRANSFORMATIONS OF OBSERVABLES BETWEEN frame is a function of time t measured by the moving clock: DIFFERENT FRAMES = + In this section we provide two examples for using the xi (θ,t ) xi (θ) vi (θ)t 2 Poincare´ generators (2.1) constructed previously to connect = xi (0)hi (0)/hi (θ) + c pi (θ)/hi(θ)t . (3.6) observables measured in different inertial frames, correspond- ingtoapassive coordinate transformation. These examples Suppose that the observer at rest sees the ith particle at location xi (t) at (lab time) t. Let us define a specific are combined to derive the usual Lorentz formulas (1.3) for noninteracting particle systems. The corresponding equivalent time t (measured by the clock in the moving frame) by the active transformations, where the coordinates are shifted by requirement d, cθ,ort can be obtained by reversing the sign of the t ≡ t cosh(θ) − xi (t)sinh(θ)/c. (3.7a) parameters. For example, if in Eq. (1.2) we choose A = xi or pi , with G = H0 and s =−t, then we obtain the familiar Can we find the associated position where the particle is Hamilton equations of motion for the time dependence of located from the point of view of the moving observer at particle positions and momenta: this time? To do that, we replace on the right-hand side of Eq. (3.6) xi (0) by xi (t) − vi (0)t and t by Eq. (3.7a), and we ∂x (t)/∂t =−{H ,x (t)}=∂H /∂p 2 i 0 i 0 i use c pi (0) = vi (0)hi(0) from Eq. (3.1a). The right-hand side 2 = c pi (t)/hi(t) = vi (t), (3.1a) of Eq. (3.6) can then be expressed in terms of the original x ,t ∂pi (t)/∂t =−{H0,pi (t)}=−∂H0/∂xi = 0, (3.1b) observable i (0 )as where velocities are defined as vi = ∂xi (t)/∂t.
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