Highly relativistic spinning particle in the Schwarzschild field: Circular and other orbits

The Mathisson-Papapetrou equations in Schwarzschild’s background both at the Mathisson-Pirani and Tulczyjew-Dixon supplementary condition are considered. The region of existence of highly relativistic planar circular orbits of a spinning particle in this background and dependence of the particle’s orbital velocity on its spin and radial coordinate are investigated. It is shown that in contrast to the highly relativistic circular orbits of a spinless particle, which exist only for $r=1.5r_g(1+\delta )$, $0<\delta \ll 1$, the corresponding orbits of a spinning particle are allowed in a wider space region, and the dimension of this region essentially depends on the supplementary condition. At the Mathisson-Pirani condition new numerical results which describe some typical cases of noncircular highly relativistic orbits of a spinning particle starting from $r>1.5r_g$ are presented.

Figure 1

Dependence of the Lorentz factor on $\delta >0$ for the highly relativistic circular orbits with $d\phi /ds>0$ of the spinning particle in the small neighborhood of $r=3M$ according to the exact MP equations under the Mathisson-Pirani condition (solid line) and under the Tulczyjew-Dixon one (dashed line). The dotted line corresponds to the geodesic circular orbits.

Figure 2

Dependence of the Lorentz factor on $r$ for the highly relativistic circular orbits with $d\phi /ds>0$ of the spinning particle beyond the small neighborhood of $r=3M$ by the exact MP equations under the Mathisson-Pirani condition (solid line). The dotted line corresponds to the geodesic circular orbits.

Figure 3

Lorentz factor vs $\delta$ for the highly relativistic circular orbits with $d\phi /ds<0$. In the main spin approximation the solid line is common for the exact MP equations under the Mathisson-Pirani and Tulczyjew-Dixon conditions. The dotted line corresponds to the geodesic circular orbits.

Figure 4

Lorentz factor vs $\delta$ for the highly relativistic circular orbits with $d\phi /ds<0$ of the spinning particle in the small neighborhood of $r=3M$ according to the exact MP equations under the Mathisson-Pirani condition (solid line) and under the Tulczyjew-Dixon one (dashed line).

Figure 5

Lorentz factor vs $r$ for the highly relativistic circular orbits with $d\phi /ds<0$ of the spinning particle beyond the small neighborhood of $r=3M$ according to the exact MP equations under the Mathisson-Pirani condition.

Figure 6

Radial coordinate vs proper time for the spinning particle with the initial values of the tangential and radial velocities which are equal to $\approx 35$ and $-{10}^{-2}$ correspondingly (solid line) and for the geodesic motion (dashed line).

Figure 7

Trajectories in the polar coordinates of the spinning (solid line) and the spinless particle (dashed line) with the same initial values of the coordinates and velocity. The circle with the radius 2 corresponds to the horizon line.

Figure 8

Radial coordinate vs proper time for the spinning particle with the initial values of the tangential and radial velocities which are equal to $\approx 35$ and ${10}^{-2}$ correspondingly (solid line) and for the geodesic motion (dashed line).

Figure 9

Trajectories in the polar coordinates of the spinning (solid line) and the spinless particle (dashed line) with the same initial values of the coordinates and velocity as for Fig. 8.

Figure 10

Radial coordinate vs proper time for an oscillatory solution of the MP equations (solid line) and for the geodesic motion with the same initial values of the coordinate and velocity (dashed line).

Figure 11

The graphs, corresponding to Fig. 10, in the polar coordinates.