Investigating spinning test particles: Spin supplementary conditions and the Hamiltonian formalism

In this paper we report the results of a thorough numerical study of the motion of spinning particles in Kerr spacetime with different prescriptions. We first evaluate the Mathisson-Papapetrou equations with two different spin supplementary conditions, namely, the Tulczyjew and the Newton-Wigner, and make a comparison of these two cases. We then use the Hamiltonian formalism given by Barausse, Racine, and Buonanno [Phys. Rev. D 80, 104025 (2009)] to evolve the orbits and compare them with the corresponding orbits provided by the Mathisson-Papapetrou equations. We include a full description of how to treat the issues arising in the numerical implementation.

Figure 1

The left panel shows a MP orbit with T SSC (black dots) and a MP orbit with NW SSC (gray dots) in the configuration space $x,y,z$ (Cartesian coordinates). The common parameters for these orbits are $a=0.5$, $r=11.7$, $\theta =\pi /2$, $p^r=0.1$, $S=1$, $S^r=0.1S$, $S^{\theta }=0.01S$, $E=0.97$, $J_z=3$, and $\mu =1$. The central panel shows the logarithm of the Euclidean distance in the configuration space between these two orbits as a function of the proper time. The right panel shows the logarithm of the difference $\mathrm{\Delta }{S}_{4\mathrm{x}4}$ between the spin tensors of these two orbits as a function of the proper time.

Figure 2

The left panel shows the logarithm of the Euclidean distance in the configuration space between a MP orbit with T SSC and a MP orbit with NW SSC as a function of the proper time. The common parameters for these orbits read $a=0.5$, $r=11.7$, $\theta =\pi /2$, $p^r=0.1$, $S={10}^{-8}$, $S^r=0.1S$, $S^{\theta }=0.01S$, $E=0.97$, $J_z=3$, and $\mu =1$. The right panel shows the logarithm of the difference $\mathrm{\Delta }{S}_{4\mathrm{x}4}$ between the spin tensors of these two orbits as a function of the proper time.

Figure 3

The top row of panels corresponds to the orbits of Fig. 1, while the bottom row of panels corresponds to the orbits of Fig. 2. The black lines represent the evolution of the MP equations with T SSC, while the gray lines represent the NW SSC. The left column of panels shows the relative error in the preservation of the four-momentum, while the right depicts the preservation of the spin.

Figure 4

The relative error of the four-momentum $\mathrm{\Delta }{\mu }^2$ as a function of the spin measure $S$ for the NW SSC. The black dots correspond to the maximum values of $\mathrm{\Delta }{\mu }^2$ during the evolution for each $S$. The dashed line is a linear fit of the form ${\log }_{10}\mathrm{\Delta }{\mu }^2=a{\log }_{10}S+b$ for data with $S>{10}^{-6}$, where $a=1.995\pm 0.004,b=-4.136\pm 0.013$.

Figure 5

The left panel shows how the orbit evolves through the MP equations (gray dots) and through the Hamiltonian equations (black dots) in the configuration space $x,y,z$, when we use the initial conditions given in Fig. 1. The central panel shows the logarithm of the Euclidean distance in the configuration space between these two orbits as a function of the coordinate time. The right panel shows the logarithm of the Euclidean norm of the difference between the spin vectors of these two orbits as a function of the coordinate time.

Figure 6

The left panel shows the logarithm of the Euclidean distance in the configuration space between an orbit calculated with the MP equations and an orbit calculated with the Hamilton equations as a function of the coordinate time. For the orbits we have used the initial conditions given in Fig. 2. The right panel shows the logarithm of the Euclidean norm of the difference between the spin vectors of these two orbits as a function of the coordinate time.

Figure 7

The top row of panels corresponds to the orbits of Fig. 5, while the bottom row of panels corresponds to the orbits of Fig. 6. The middle row of panels corresponds to initial conditions similar to Fig. 1 only instead of spin measure $S=1$ we set $S={10}^{-4}$. The gray lines represent the evolution of the MP equations, while the black lines represent the evolution of the Hamilton equations. The left column of panels shows the relative error in the preservation of the Hamiltonian function, while the right shows the preservation of the spin.

Figure 8

The left panel shows the relative error of the Hamiltonian $\mathrm{\Delta }H$ of orbits evolved through the MP equations for different spin measures $S$ of the particle, while the right panel shows the corresponding preservations of the measure of the three-vector $\mathrm{\Delta }{S}^2$. The black dots correspond to the maximum values of $\mathrm{\Delta }H$, $\mathrm{\Delta }{S}^2$, respectively, for each $S$. The dashed lines are linear fits of the form ${\log }_{10}\mathrm{\Delta }H=a{\log }_{10}S+b$, and ${\log }_{10}\mathrm{\Delta }{S}^2=c{\log }_{10}S+d$, respectively, for data with $S>{10}^{-6}$, where $a=1.9968\pm 0.0015,b=-2.644\pm 0.004$, and $c=1.031\pm 0.015,d=-2.46\pm 0.06$.

Figure 9

The relative error of the $z$ angular momentum, $\mathrm{\Delta }{J}_z$, (top panel) and the relative error of the energy, $\mathrm{\Delta }E$, (bottom panel) against integration time $\tau$ for the 4-stage Gauss scheme with step size $h=1$ and the fifth order Cash-Karp scheme with step size $h=0.1$ applied to the initial value problem (56) with initial data as stated in the text. CPU time was 214.1 s for the Gauss Runge-Kutta scheme and 422.7 s for the Cash-Karp scheme.

Figure 10

The relative error of the Hamiltonian, $\mathrm{\Delta }H$, against integration time $t$ for the 4-stage Gauss scheme with step size $h=2$ and the fifth order Cash-Karp scheme with step size $h=0.2$ applied to the initial value problem (b1) with initial data as stated in the text. CPU time was 7.83 s for the Gauss Runge-Kutta scheme and 24,7 s for the Cash-Karp scheme.

Figure 11

The relative difference, $\mathrm{\Delta }r$, between the radial distance calculated with the interpolation method and the radial distance calculated via the cumbersome method with extra integration steps plotted against output time $t$.