Time dilation in relativistic two-particle interactions

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.

Figure 1

Two-particle trajectories $x_1(t)$ and $x_2(t)$ as a function of time: (a) for $p_1(t=0)=-400$ and $p_2(t=0)=400$ [corresponding to $v_1(t=0)=-129$.64 and $v_2(t=0)=129.64$]; (b) for $p_1(t=0)=-83.82$ and $p_2(t=0)=1381.39$ [corresponding to $v_1(t=0)=94.12$ and $v_2(t=0)=140.85$], the drift term $\mathit{Vt}$ has been subtracted out; and (c) the Lorentz-transformed trajectories based on the solutions displayed in (a). The drift term ${\mathit{Vt}}^{\prime }$ has been subtracted out. [Parameters in all simulations are in atomic units: $x_1(t=0)=x_2(t=0)=0$, $m_1=m_2=1$, $c=137.036$, $U_0=-5c^2$, $a=1$].