Aschenbach effect: Unexpected topology changes in the motion of particles and fluids orbiting rapidly rotating Kerr black holes

Newtonian theory predicts that the velocity $\mathcal{V}$ of free test particles on circular orbits around a spherical gravity center is a decreasing function of the orbital radius $r$, $\mathrm{d}\mathcal{V}/\mathrm{d}r<0$. Only very recently, Aschenbach [B. Aschenbach, Astronomy and Astrophysics, 425, 1075 (2004)] has shown that, unexpectedly, the same is not true for particles orbiting black holes: for Kerr black holes with the spin parameter $a>0.9953$, the velocity has a positive radial gradient for geodesic, stable, circular orbits in a small radial range close to the black-hole horizon. We show here that the Aschenbach effect occurs also for nongeodesic circular orbits with constant specific angular momentum $\ell ={\ell }_0=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$. In Newtonian theory it is $\mathcal{V}={\ell }_0/\mathcal{R}$, with $\mathcal{R}$ being the cylindrical radius. The equivelocity surfaces coincide with the $\mathcal{R}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ surfaces which, of course, are just coaxial cylinders. It was previously known that in the black-hole case this simple topology changes because one of the “cylinders” self-crosses. The results indicate that the Aschenbach effect is connected to a second topology change that for the $\ell =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ tori occurs only for very highly spinning black holes, $a>0.99979$.

Figure 1

Reality condition for the existence of local extrema of the function ${\ell }_{\mathrm{e}\mathrm{x}}(r;a)$. The extrema are allowed, if $\mathcal{D}(r)>0$. Clearly, the physically relevant extrema, located above the outer horizon, can exist in the interval $r\in (1,r_4)$.

Figure 2

Loci of local extrema of the function ${\ell }_{\mathrm{e}\mathrm{x}}(r;a)$. They are determined by the functions $a_{\mathrm{e}\mathrm{x}\pm }^2(r)$. (a) The function $a_{\mathrm{e}\mathrm{x}-}^2(r)$ is relevant for both black holes and naked singularities; its local minimum is denoted I. (b) The function $a_{\mathrm{e}\mathrm{x}+}^2(r)$ is relevant for naked singularities only; its local maximum is denoted II.

Figure 3

Kerr spacetimes with the change of sign of the gradient of LNRF velocity. In the $\ell$–$a$ plane, the functions ${\ell }_{\mathrm{e}\mathrm{x}(\mathrm{m}\mathrm{a}\mathrm{x})}(a)$ (upper solid curve), ${\ell }_{\mathrm{e}\mathrm{x}(\mathrm{m}\mathrm{i}\mathrm{n})}(a)$ (lower solid curve), ${\ell }_{\mathrm{m}\mathrm{s}}(a)$ (dashed curve) and ${\ell }_{\mathrm{m}\mathrm{b}}(a)$ (dashed-dotted curve) are given. For pairs of $(a,\ell )$ from the shaded region, the gradient of the orbital velocity of tori changes its sign twice inside the tori. Between the points A, B, the ${\ell }_{\mathrm{e}\mathrm{x}(\mathrm{m}\mathrm{a}\mathrm{x})}(a)$ curve determines an inflex point of ${\mathcal{V}}^{(\varphi )}(r;a,\ell )$. The inflex points determined by the curve ${\ell }_{\mathrm{e}\mathrm{x}(\mathrm{m}\mathrm{i}\mathrm{n})}(a)$ are irrelevant being outside of the definition region for constant specific angular momentum tori, $\ell \in [{\ell }_{\mathrm{m}\mathrm{s}}(a),{\ell }_{\mathrm{m}\mathrm{b}}(a)]$. The point I corresponds to the inflex point of ${\ell }_{\mathrm{e}\mathrm{x}}(r;a)$; cf. Figure 2a.

Figure 4

Classification of the Kerr black-hole spacetimes according to the properties of the functions ${\ell }_{\mathrm{e}\mathrm{x}}(r;a)$ (solid curves) and ${\ell }_{\mathrm{K}}(r;a)$ (dashed curves). The functions are plotted for six cases corresponding to the classification. The constant specific angular momentum tori can exist in the shaded region only along $\ell =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ lines. Their inner edge (center) is determined by the decreasing (increasing) part of ${\ell }_{\mathrm{K}}(r;a)$. The local extrema of the orbital velocity relative to LNRF relevant for tori are given by the intersections of $\ell =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ line with the curve of ${\ell }_{\mathrm{e}\mathrm{x}}(r;a)$ in the shaded region. Notice that the region corresponding to the allowed values of $\ell$ for the discs is narrowing with $a\to 1$, it is degenerated into the $\ell =2$ line for $a=1$ as ${\ell }_{\mathrm{m}\mathrm{s}}={\ell }_{\mathrm{m}\mathrm{b}}=2$ in this case. In the case (e), the gradient $\partial {\mathcal{V}}^{(\varphi )}/\partial r$ changes sign for all values of $\ell \in ({\ell }_{\mathrm{m}\mathrm{s}},{\ell }_{\mathrm{m}\mathrm{b}})$ allowed for the tori, while in the case (d), it is allowed for a region restricted from above by the value ${\ell }_{\mathrm{e}\mathrm{x}(\mathrm{m}\mathrm{a}\mathrm{x})}(a)$. In the cases (a)–(c), the change of sign of $\partial {\mathcal{V}}^{(\varphi )}/\partial r$ cannot occur in the disc. It is directly seen from cases (d)–(f) that the gradient $\partial {\mathcal{V}}^{(\varphi )}/\partial r$ changes the sign closely above the center of the disc.

Figure 5

Von Zeipel surfaces (meridional sections). (a) For $a<a_{\mathrm{c}(\mathrm{b}\mathrm{h})}$ and any $\ell$, only one surface with a cusp in the equatorial plane and no closed (toroidal) surfaces exist. The cusp is, however, located outside the toroidal equilibrium configurations of perfect fluid. (b)–(f) For $a>a_{\mathrm{c}(\mathrm{b}\mathrm{h})}$ and $\ell$ appropriately chosen, two surfaces with a cusp ((c), (e)), or one surface with both the cusps (d), together with closed (toroidal) surfaces, exist. Moreover, if $a>a_{\mathrm{c}(\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{i})}$, both the outer cusp and the central ring of closed surfaces are located inside the toroidal equilibrium configurations corresponding to constant specific angular momentum discs (cases (c)–(e)). If $\ell$ is sufficiently low/high ((b)/(f)), there is only one surface with a cusp outside the configuration. Shaded region corresponds to the black hole.

Figure 6

Profiles of the equatorial orbital velocity of constant specific angular momentum tori in LNRF in terms of the radial Boyer-Lindquist coordinate. The profiles are given for typical values of $a$ corresponding to the classification of the Kerr black-hole spacetimes. For comparison, the profiles are given for the orbital velocity of Keplerian discs in Kerr spacetimes with the same rotational parameter $a$. For tori, values of $\ell =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ are appropriately chosen; commonly, $\ell ={\ell }_{\mathrm{m}\mathrm{s}}$ is used giving the maximal value of the velocity difference in between the local extrema, and representing the limiting case of constant specific angular momentum tori.

Figure 7

(a) Positions of local extrema of ${\mathcal{V}}_{\mathrm{L}\mathrm{N}\mathrm{R}\mathrm{F}}^{(\varphi )}$ (in B-L coordinates) for the constant specific angular momentum tori with $\ell ={\ell }_{\mathrm{m}\mathrm{s}}$ in dependence on the rotational parameter $a$ of the black-hole. (b) Velocity difference $\Delta {\mathcal{V}}^{(\varphi )}={\mathcal{V}}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{(\varphi )}-{\mathcal{V}}_{\mathrm{m}\mathrm{i}\mathrm{n}}^{(\varphi )}$ as a function of the rotational parameter $a$ of the black-hole for both the Keplerian disc and the constant specific angular momentum (non-Keplerian) disc with $\ell ={\ell }_{\mathrm{m}\mathrm{s}}$. (c) orbital-velocity curves in the limiting case of the extreme black-hole. At $r=1$, the Keplerian orbital velocity ${\mathcal{V}}_{\mathrm{K}}^{(\varphi )}$ has a local minimum, whereas the orbital velocity ${\mathcal{V}}_{{\ell }_{\mathrm{m}\mathrm{s}}}^{(\varphi )}$ of the constant specific angular momentum disc has an inflex point. In both cases, the velocity difference $\Delta {\mathcal{V}}^{(\varphi )}$ reaches its maximal values: $\Delta {\mathcal{V}}_{\mathrm{K}}^{(\varphi )}\dot{=}0.06986,\Delta {\mathcal{V}}_{{\ell }_{\mathrm{m}\mathrm{s}}}^{(\varphi )}\dot{=}0.02241$.