Gravitational Magnus effect

It is well known that a spinning body moving in a fluid suffers a force orthogonal to its velocity and rotation axis—it is called the Magnus effect. Recent simulations of spinning black holes and (indirect) theoretical predictions, suggest that a somewhat analogous effect may occur for purely gravitational phenomena. The magnitude and precise direction of this “gravitational Magnus effect” is still the subject of debate. Starting from the rigorous equations of motion for spinning bodies in general relativity (Mathisson-Papapetrou equations), we show that indeed such an effect takes place and is a fundamental part of the spin-curvature force. The effect arises whenever there is a current of mass/energy, non-parallel to a body’s spin. We compute the effect explicitly for some astrophysical systems of interest: a galactic dark matter halo, a black hole accretion disk, and the Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime. It is seen to lead to secular orbital precessions potentially observable by future astrometric experiments and gravitational-wave detectors. Finally, we consider also the reciprocal problem: the “force” exerted by the body on the surrounding matter, and show that (from this perspective) the effect is due to the body’s gravitomagnetic field. We compute it rigorously, showing the matching with its reciprocal, and clarifying common misconceptions in the literature regarding the action-reaction law in post-Newtonian gravity.

Figure 1

(a) Magnus effect in fluid dynamics, as viewed from a spinning body’s frame: the body’s rotation slows down the flow which opposes the body’s surface velocity, while speeding it up otherwise, generating a pressure gradient and a net force ${\mathit{F}}_{\text{Mag}}$ on the body. (b) A gravitational analogue of the Magnus effect?—due to its gravitomagnetic field $\mathit{H}$, a spinning body deflects particles of a cloud flowing around it, via the gravitomagnetic “force” ${\mathit{F}}_{\mathrm{GM}}=M\mathit{v}\times \mathit{H}$. By naive application of an action-reaction principle, a force on the body orthogonal to its spin and velocity (like the Magnus force) might be expected.

Figure 2

A magnetic dipole $\mathit{\mu }=\mu {\mathit{e}}_z$ inside a semi-infinite cloud of charged particles flowing in the ${\mathit{e}}_x$ direction. The cloud is infinite along the $x$ and $z$ directions, but of finite thickness in the $y$ direction, contained within $-h/2\le y\le +h/2$. The magnetic field $\mathit{B}$ generated by the cloud points in the positive $z$ direction for $y>0$, and in the negative $z$ direction for $y<0$. $\mathit{B}$ has a gradient inside the cloud, whose only nonvanishing component is $B^{z,y}=4\pi j$. Due to that, a force ${\mathit{F}}_{\mathrm{EM}}=B^{z,y}\mu {\mathit{e}}_y=4\pi j\mu {\mathit{e}}_y$, pointing upwards, is exerted on the dipole. In this case ${\mathit{F}}_{\text{Sym}}={\mathit{F}}_{\text{Mag}}$, so the force is twice the Magnus force: ${\mathit{F}}_{\mathrm{EM}}={\mathit{F}}_{\text{Sym}}+{\mathit{F}}_{\text{Mag}}=2{\mathit{F}}_{\text{Mag}}$. Considering instead a cloud finite along $z$, infinite along $x$ and $y$, ${\mathit{F}}_{\text{Mag}}=2\pi j\mu {\mathit{e}}_y$ remains the same, but ${\mathit{F}}_{\text{Sym}}$ inverts its direction: ${\mathit{F}}_{\text{Sym}}=-{\mathit{F}}_{\text{Mag}}$, causing the total force to vanish: ${\mathit{F}}_{\mathrm{EM}}=0$.

Figure 3

A spinning body (e.g., a BH), with $\mathit{S}=S{\mathit{e}}_z$, inside a massive cloud flowing in the ${\mathit{e}}_x$ direction. The cloud is infinite along $x$ and $z$, and finite along the $y$-axis, contained within $-h/2\le y\le h/2$. The vector $\mathit{H}$ is the gravitomagnetic field generated by the cloud; it points in the negative (positive) $z$ direction for $y>0$ ($<0$). It has a gradient inside the cloud, whose only non-vanishing component is $H^{z,y}=-16\pi J$; due to that, a spin-curvature force $\mathit{F}=(1/2)H^{z,y}S{\mathit{e}}_y=-8\pi JS{\mathit{e}}_y$, pointing downwards, is exerted on the body. In this case ${\mathit{F}}_{\text{Mag}}={\mathit{F}}_{\text{Weyl}}$, so the total force is twice the Magnus force: $\mathit{F}={\mathit{F}}_{\text{Mag}}+{\mathit{F}}_{\text{Weyl}}=2{\mathit{F}}_{\text{Mag}}$. Considering instead a cloud finite along $z$, and infinite along $x$ and $y$, ${\mathit{F}}_{\text{Mag}}=-4\pi JS{\mathit{e}}_y$ remains the same, but ${\mathit{F}}_{\text{Weyl}}$ changes to the exact opposite: ${\mathit{F}}_{\text{Weyl}}=-{\mathit{F}}_{\text{Mag}}$, causing the total spin-curvature force to vanish: $\mathit{F}=0$.

Figure 4

Spinning bodies moving in a “pseudo-isothermal” DM halo. For a body in quasicircular orbits, with spin lying in the orbital plane, the Magnus (${\mathit{F}}_{\text{Mag}}$) and Weyl (${\mathit{F}}_{\text{Weyl}}$) forces are parallel. The total force is of the form $\mathit{F}=A(r)\mathit{S}\times \mathit{v}$, pointing outwards the orbital plane on one half of the orbit, and inwards the other half; this “torques” the orbit, leading to a secular orbital precession $\mathit{\Omega }$. For a body moving radially towards the center of the halo, ${\mathit{F}}_{\text{Weyl}}=0$, and so the total force exerted on it reduces to the Magnus force: $\mathit{F}={\mathit{F}}_{\text{Mag}}=4\pi \rho \mathit{S}\times {\mathit{v}}_1$. Generically ${\mathit{F}}_{\text{Weyl}}$ and ${\mathit{F}}_{\text{Mag}}$ have different directions. If the halo’s density was uniform, ${\mathit{F}}_{\text{Weyl}}=0\Rightarrow \mathit{F}={\mathit{F}}_{\text{Mag}}$ for all particles.

Figure 5

Numerical 1PN results for quasicircular orbits in a pseudo-isothermal DM halo, with ${\rho }_0={10}^8{M}_{\odot }{\mathrm{pc}}^{-3}$, $r_{\mathrm{c}}=0.02\mathrm{kpc}$ (typical of satellite galaxies [46]), and $r=8r_{\mathrm{c}}$, case in which $v=0.15c$. The test body has the Sun’s mass $m=M_{\odot }$, and an initial spin vector $\mathit{S}{|}_{\mathrm{in}}=S{\mathit{e}}_x$, with $S=0.5m^2$. Left panel: three-dimensional plot of the orbit, showing the orbital precession $\mathbf{\Omega }\propto \mathit{S}$ in Eq. (80) (i.e., about ${\mathit{e}}_x$, initially). Right panel: plot of $z(t)/2Z$, for $t\in [0,2\pi /{\mathrm{\Omega }}_{\mathrm{s}}]$; the numerical result agrees well with the (simplified) analytical result (77). It shows clearly the modulation by the spin precession ${\mathbf{\Omega }}_{\mathrm{s}}$: the orbital precession $\mathbf{\Omega }$ causes $z$ oscillations of initially increasing amplitude, reaching its peak $z=2Z$ at $t=\pi /{\mathrm{\Omega }}_{\mathrm{s}}$, corresponding to the maximum inclination of the orbital plane. At that point the direction of $\mathit{S}$ (thus of $\mathbf{\Omega }$) becomes inverted relative to the initial one, so the orbital inclination (and the oscillation amplitude) starts decreasing.

Figure 6

Orbits of a satellite galaxy in the Milky way DM halo (numerical 1PN results). Due to their disk-like shape, galaxies have a large Møller radius $S/m$, making them especially suitable test bodies for the effects under study. The satellite is assumed to be a scale reduced version of the Milky Way, of diameter 2.5 kpc, and orbiting at a distance 8 kpc (= Sun’s distance) from the Halo’s center. The peak amplitude predicted by Eqs. (74, 75, 76, 77) is $2Z\approx 60\mathrm{pc}$; but it is never reached, due to the very strong dynamical friction. Still a peak of about ${10}^{-3}\mathrm{pc}$ is reached after about 2.2 Gyr ($\simeq 1/5$ the age of the universe).

Figure 7

Range of frequencies of the emitted gravitational radiation for binary systems in which the Magnus force $F_{\text{Mag}}$ is larger than $F_{\mathrm{SO}}$ and $F_{\mathrm{SS}}$, for $\alpha =0.01$, $f_{\text{Edd}}=0.2$ [see Eq. (89)] and $S_{\mathrm{BH}}=0.1M_{\mathrm{BH}}^2$. The solid curves represent, as functions of the central BH’s mass ($M_{\mathrm{BH}}$), the frequencies for which $F_{\text{Mag}}\sim F_{\mathrm{SO}}$ and $F_{\text{Mag}}\sim F_{\mathrm{SS}}$. In the shadowed regions it holds, respectively, $F_{\text{Mag}}\gtrsim F_{\mathrm{SO}}$ and $F_{\text{Mag}}\gtrsim F_{\mathrm{SS}}$. The Magnus force tends to be the leading spin dependent force at low frequencies and for SMBHs, namely within the band of pulsar timing arrays (${10}^{-9}\mathrm{Hz}\lesssim f_{\mathrm{GW}}\lesssim {10}^{-7}\mathrm{Hz}$). The frequency for which $F_{\text{Mag}}\sim F_{\mathrm{SS}}$ lies moreover just below (or within, depending on $\alpha$, $f_{\text{Edd}}$, and $S_{\mathrm{BH}}$) the LISA band.

Figure 8

Two situations where action-reaction law is not obeyed in electrodynamics: (a) two orthogonality moving point charges (Feynman paradox); (b) interaction of a magnetic dipole with a particle of the cloud. In (a), the electric forces each particle exerts on the other are nearly opposite, but particle 2 exerts a nonvanishing magnetic force $q{\mathit{v}}_1\times {\mathit{B}}_2$ on particle 1, whereas particle 1 does not exert any magnetic force on particle 2. In (b), the force exerted by the dipole (particle 2) on the moving particle 1 is only minus half the force that the latter exerts on the former, ${\mathit{F}}_{1,\mathrm{dip}}=-2{\mathit{F}}_{\mathrm{dip},1}$.

Figure 9

A situation where the action reaction law is not obeyed in PN gravity [analogue of Fig. 8]: the spin-orbit force ${\mathit{F}}_{\mathrm{SO}2,1}$ exerted by the spinning body (e.g., a BH) on a moving particle is not the same as the spin-orbit force ${\mathit{F}}_{\mathrm{SO}1,2}$ that the latter exerts on the former: ${\mathit{F}}_{\mathrm{SO}1,2}=-3{\mathit{F}}_{\mathrm{SO}2,1}/2$.