Time parameterizations and spin supplementary conditions of the Mathisson-Papapetrou-Dixon equations

The implications of two different time constraints on the Mathisson-Papapetrou-Dixon (MPD) equations are discussed under three spin supplementary conditions (SSCs). For this reason the MPD equations are revisited without specifying the affine parameter and several relations are reintroduced in their general form. The latter allows one to investigate the consequences of combining the Mathisson-Pirani (MP) SSC, the Tulczyjew-Dixon (TD) SSC and the Ohashi-Kyrian-Semerák (OKS) SSC with two affine parameter types: the proper time on one hand and the parameterizations introduced in [Gen. Relativ. Gravit. 8, 197 (1977)] on the other. For the MP SSC and the TD SSC it is shown that quantities that are constant of motion for the one affine parameter are not for the other, while for the OKS SSC it is shown that the two affine parameters are the same. To clarify the relation between the two affine parameters in the case of the TD SSC the MPD equations are evolved and discussed.

Figure 1

Two orbits following the MPD equations with TD SSC are depicted in the configuration space. The orbits share the same initial conditions $r=10$, $\theta =\pi /2$, $u^r=0.1$, $S^r=0.1S$, $S^{\theta }=0.01S$ and constants of motion $m=1$, $S=0.9$, $E=0.97$ $J_z=3$. The black curve shows the orbit for which the affine parameter is the proper time, while the gray curve shows the orbit for which the affine parameter is defined by the constraint (17). The left panel shows the evolution of the orbits for $0\le \chi \le {10}^5$, while the right for $0.99\times {10}^5\le \chi \le {10}^5$.

Figure 2

Top: relative radial difference $\mathrm{\Delta }r$ of the two orbits shown in Fig. 1 as a function of the coordinate time $t$. Bottom: relative error in the spin measure (15) $\mathrm{\Delta }{S}^2$ of the two orbits shown in Fig. 1 as a function of $t$. The colors denote the same as in Fig. 1, i.e. black denotes the evolution using the proper time, while gray using the time constraint (17).

Figure 3

Top: relative error/difference (black/gray) $\mathrm{\Delta }{v}^2=|1-\frac{v^2(t)}{v^2}|$ in the value of Eq. (3) as the two orbits evolve with respect to the coordinate time $t$, where $v^2(t)$ is the numerical value of Eq. (3) calculated at time $t$ and $v^2$ is the initial value. Bottom: relative difference/error (black/gray) $\mathrm{\Delta }{m}_v=|1-\frac{m_v(t)}{m_v}|$ in the value of Eq. (5) as the two orbits evolve with respect the coordinate time $t$, where $m_v(t)$ is the numerical value of Eq. (5) calculated at time $t$ and $m_v$ is the initial value. The colors denote the same as in Fig. 1, i.e. black denotes the evolution using the proper time, while gray using the time constraint (17).