Perturbation method for classical spinning particle motion. II. Vaidya space-time

This paper describes an application of the Mathisson-Papapetrou-Dixon (MPD) equations in analytic perturbation form to the case of circular motion around a radially accreting or radiating black hole described by the Vaidya metric. Based on the formalism presented earlier, this paper explores the effects of mass accretion or loss of the central body on the overall dynamics of the orbiting spinning particle. This includes changes to its squared mass and spin magnitude due to the classical analog of radiative corrections from spin-curvature coupling. Various quantitative consequences are explored when considering orbital motion near the black hole’s event horizon. An analysis on the orbital stability properties due to spin-curvature interactions is examined briefly, with conclusions in general agreement with previous work performed for the case of circular motion around a Kerr black hole.

Figure 1

Coordinate speed $v(\tau )$ of the spinning particle for circular motion around a black hole described by the Vaidya metric, for $r=6M$ and $\widehat{\theta }=\widehat{\varphi }=\pi /4$. (a) shows a gradual increase in amplitude for $s_0/(m_{0}r)={10}^{-2}$ with no significant difference due to second-order contributions in $\epsilon$. For $\alpha =-1$, (b) shows a corresponding gradual decrease in amplitude. In contrast, (c) and (d) indicate the existence of an instability in the orbit when $s_0/(m_{0}r)={10}^{-1}$, due to the rise of $v(\tau )$ from the second-order contribution in $\epsilon$.

Figure 2

Møller radius $\rho (\tau )=(s/m)(\tau )$ in the Vaidya background for $r=6M$ and $\widehat{\theta }=\widehat{\varphi }=\pi /4$, in units of $s_0/m_0$. (a) and (b) show that while the higher-order contributions in $\epsilon$ lead to a slowly increasing amplitude in $\rho$, (c) and (d) indicate a much larger amplitude increase as $s_0/(m_{0}r)={10}^{-1}$, where the second- and third-order contributions in $\epsilon$ become distinctive.

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 $\alpha =1$. (a) shows a complicated peak and valley structure to $⟨\rho ⟩$ that simplifies somewhat in (b).

Figure 4

Time-averaged Møller radius as a function of $\widehat{\theta }$ and $\widehat{\varphi }$ for $r=6M$ and $\alpha =-1$. The plots are essentially indistinguishable when compared to Fig. 3, as $\alpha =1$ goes to $\alpha =-1$.

Figure 5

Radial component $P^1(\tau )$ of the linear momentum in the Vaidya background for $r=6M$ and $\widehat{\theta }=\widehat{\varphi }=\pi /4$. When adding the second-order contribution in $\epsilon$ for $s_0/(m_{0}r)={10}^{-2}$, (a) and (b) indicate a slight increase in the amplitude, while (c) and (d) show a much stronger amplitude increase as $s_0/(m_{0}r)={10}^{-1}$.

Figure 6

Polar component $P^2(\tau )$ of the linear momentum for $r=6M$ and $\widehat{\theta }=\widehat{\varphi }=\pi /4$ in the Vaidya background. All three plots show that the second-order contribution in $\epsilon$ results in a net nonzero magnitude for the polar component. While (a) and (b) yield a modest nonzero effect for $s_0/(m_{0}r)={10}^{-2}$, (c) and (d) show a significantly more pronounced effect for $s_0/(m_{0}r)={10}^{-1}$.

Figure 7

Azimuthal component $P^3(\tau )$ of the linear momentum for $r=6M$ and $\widehat{\theta }=\widehat{\varphi }=\pi /4$ in the Vaidya background. For (a) and (b), the change in magnitude is almost totally dominated by the first-order contribution in $\epsilon$. (c) and (d), in contrast, indicate that the second-order contribution in $\epsilon$ dominates with a strongly unbounded increase in the amplitude as $s_0/(m_{0}r)={10}^{-1}$.

Figure 8

Ratio of $P^3(\tau )$ to $P^0(\tau )$ in the Vaidya background for $r=6M$ and $\widehat{\theta }=\widehat{\varphi }=\pi /4$. (a) and (b) indicate that the second-order expression in $\epsilon$ leads to a wider variation in the ratio than found in the first-order expression only, irrespective of the given choice for $\alpha$.

Figure 9

The $S^{02}(\tau )$ component of the spin tensor for $r=6M$ and $\widehat{\theta }=\widehat{\varphi }=\pi /4$ in the Vaidya background. (a) shows a modest decrease in amplitude for $s_0/(m_{0}r)={10}^{-2}$ and $\alpha =1$, with a corresponding modest increase in amplitude in (b) for $\alpha =-1$. In contrast, both (c) and (d) show a significantly more pronounced amplitude increase for $s_0/(m_{0}r)={10}^{-1}$.