Perturbation method for classical spinning particle motion. I. Kerr space-time

This paper presents an analytic perturbation approach to the dynamics of a classical spinning particle, according to the Mathisson-Papapetrou-Dixon (MPD) equations of motion, with a direct application to circular motion around a Kerr black hole. The formalism is established in terms of a power series expansion with respect to the particle’s spin magnitude, where the particle’s kinematic and dynamical degrees are expressed in a completely general form that can be constructed to infinite order in the expansion parameter. It is further shown that the particle’s squared mass and spin magnitude can shift due to a classical analogue of radiative corrections that arise from spin-curvature coupling. Explicit expressions are determined for the case of circular motion near the event horizon a Kerr black hole, where the mass and spin shift contributions are dependent on the initial conditions of the particle’s spin orientation. A preliminary analysis of the stability properties of the orbital motion in the Kerr background due to spin-curvature interactions is explored and briefly discussed.

Figure 1

Coordinate speed $v(\tau )$ of the spinning particle while in circular orbit around a Kerr black hole, for $r=6M$ and $\widehat{\theta }=\widehat{\varphi }=\pi /4$. It is clear from (a) that $v(\tau )$ to second-order contribution in $\epsilon$ has no significant impact on altering the particle’s orbital speed for $s_0/(m_{0}r)={10}^{-2}$, though (b) indicates a slight increase over time. In contrast, (c) and (d) show that an instability occurs as $s_0/(m_{0}r)={10}^{-1}$ when considering the second-order contribution of $\epsilon$ in $v(\tau )$.

Figure 2

Møller radius $\rho (\tau )=(s/m)(\tau )$ for $r=6M$ and $\widehat{\theta }=\widehat{\varphi }=\pi /4$, in units of $s_0/m_0$. (a) shows that while the higher-order contributions in $\epsilon$ lead to a slowly increasing amplitude in $\rho$, with a moderate increase found in (b). As $s_0/(m_{0}r)={10}^{-1}$, the amplitude increase becomes more pronounced for (c) and (d), where the second- and third-order contributions in $\epsilon$ are noticeable.

Figure 3

Three-dimensional plot of the time-averaged Møller radius $⟨\rho ⟩=⟨s/m⟩$ in units of $s_0/m_0$ as a function of $\widehat{\theta }$ and $\widehat{\varphi }$ for $r=6M$ and $a=M$. (a) shows a complicated peak and valley structure to $⟨\rho ⟩$ that simplifies somewhat in (b), with two peaks present.

Figure 4

Time-averaged Møller radius as a function of $\widehat{\theta }$ and $\widehat{\varphi }$ for $r=6M$ and $a=-M$. It is clear that while the peak structure is essentially unchanged when compared to Fig. 3, the magnitude increases tenfold when going from $a=M$ to $a=-M$.

Figure 5

Radial component $P^1(\tau )$ of the linear momentum for $r=6M$ and $\widehat{\theta }=\widehat{\varphi }=\pi /4$. (a) shows a slightly growing amplitude due to the second-order contribution in $\epsilon$ for $s_0/(m_{0}r)={10}^{-2}$, with a more moderate growth in (b). The amplitude grows much more rapidly for (c) and (d) as $s_0/(m_{0}r)={10}^{-1}$.

Figure 6

Polar component $P^2(\tau )$ of the linear momentum for $r=6M$ and $\widehat{\theta }=\widehat{\varphi }=\pi /4$. The second-order contribution in $\epsilon$ introduces a slight nonzero value in the net magnitude for (a) and (b) with $s_0/(m_{0}r)={10}^{-2}$, whose average slope becomes more pronounced for (c) and (d) as $s_0/(m_{0}r)={10}^{-1}$.

Figure 7

Azimuthal component $P^3(\tau )$ of the linear momentum for $r=6M$ and $\widehat{\theta }=\widehat{\varphi }=\pi /4$. The second-order contribution in $\epsilon$ introduces a slight nonzero change in the amplitude for (a) with $s_0/(m_{0}r)={10}^{-2}$, while (b) shows a more moderate growth in amplitude. This growth becomes strongly unbounded for (c) and (d) as $s_0/(m_{0}r)={10}^{-1}$.

Figure 8

Ratio of $P^3(\tau )$ to $P^0(\tau )$ for $r=6M$ and $\widehat{\theta }=\widehat{\varphi }=\pi /4$. When considering the expression to second order in $\epsilon$, it is evident from (a) and (b) that the higher-order contribution becomes gradually unbounded compared to the constant ratio given by the first-order contribution in $\epsilon$.

Figure 9

The $S^{02}(\tau )$ component of the spin tensor for $r=6M$ and $\widehat{\theta }=\widehat{\varphi }=\pi /4$. In (a), there is virtually no contribution of the second-order and third-order contributions in $\epsilon$ to the amplitude, while there appears a slight growth in amplitude for (b) due to these contributions. In contrast, (c) and (d) show a noticeable increase in the amplitude as $s_0/(m_{0}r)={10}^{-1}$ due to the higher-order contributions in $\epsilon$.