Gravitational Magnus effect

L. Filipe O. Costa1, ∗, Rita Franco2, Vitor Cardoso2,3 1 GAMGSD, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal∗ 2 CENTRA, Departamento de Física, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal and 3 Perimeter Institute for Theoretical Physics, 31 Caroline Street North Waterloo, Ontario N2L 2Y5, Canada

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 mag- nitude 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, nonparallel to a body’s spin. We compute the effect explic- itly 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 match- ing with its reciprocal, and clarifying common misconceptions in the literature regarding the action-reaction law in post-Newtonian gravity.

CONTENTS Quasi-circular orbits 18

I. Introduction1 VI. Magnus effect in cosmology: the FLRW metric 20 A. Notation and conventions2 B. Executive summary3 VII. Conclusions 21

II. Electromagnetic (anti) Magnus effect3 Acknowledgments 21 A. Example: The force exerted by a current slab on a dipole4 A. Infinite clouds and Fubini’s theorem 22 B. Reciprocal problem: The force exerted by the dipole on the slab5 B. Action-reaction law and magnetic and gravitomagnetic interactions 22 III. Gravitational Magnus effect5 1. Magnetism 23 A. Post-Newtonian approximation7 a. Simplest example: Two moving point charges B. A cloud “slab”8 (Feynman paradox) 23 1. The Magnus force on spinning objects8 b. Interaction of a magnetic dipole with 2. The force exerted by the body on the cloud9 individual particles of the cloud 23 3. Infinite clouds 10 c. A magnetic dipole and an infinite straight IV. Magnus effect in dark matter halos 10 wire 24 A. Spherical, uniform dark matter halo 11 2. Gravitomagnetism 24 B. Realistic halos 11 a. Interaction of a spinning body with individual arXiv:1805.01097v2 [gr-qc] 25 Jul 2018 C. Objects on quasi-circular orbits 12 particles of the cloud 24 1. Spin orthogonal to the orbital plane z (S = S ez) 12 References 26 2. Spin parallel to the orbital plane (Sz = 0) 12 3. Particular examples in the Milky Way DM Halo 14 I. INTRODUCTION V. Magnus effect in accretion disks 16 A. Orders of magnitude for a realistic density The Magnus effect is well known in classical fluid dynam- profile 16 ics: when a spinning body moves in a fluid, a force orthogonal B. Miyamoto-Nagai disks 17 to the body’s velocity and spin acts on it. If the body spins with angular velocity ω, moves with velocity v, and the fluid density is ρ, such force has the form (see e.g. [1,2])

∗ [email protected] FMag = αρω × v . (1) 2 a) b) v x H z tion cross-section being larger for counter- than for corotating Low velocity, y particles) — which is but another consequence of the grav- high pressure v H x itomagnetic “forces”: these are attractive for counterrotating particles, and repulsive for corotating ones, as illustrated in

v x H v x H Fig.1b (for particles in the equatorial plane). Such argument FMag v leads however to the prediction of an effect in the direction op- v posite to the Magnus effect (“anti-Magnus”), thus seemingly at odds with the results in Ref. [6]. Very recently, and while

High velocity, v x H our work was being completed, there was also an attempt to low pressure v demonstrate the existence of what, in practice, would amount to such an effect, based both on particle’s absorption and on FIG. 1. a) Magnus effect in fluid dynamics, as viewed from a spin- orbital precessions around a spinning BH [8] (which, again, ning body’s frame: the body’s rotation slows down the flow which are gravitomagnetic effects); a force in the direction opposite opposes the body’s surface velocity, while speeding it up otherwise, to the Magnus effect was again suggested. These (conflict- generating a pressure gradient and a net force FMag on the body. b) A gravitational analogue of the Magnus effect? — due to its gravito- ing) treatments are however based on loose estimates, not on a magnetic field H, a spinning body deflects particles of a cloud flow- concrete computation of the overall gravitomagnetic force ex- ing around it, via the gravitomagnetic “force” FGM = Mv ×H. By erted by the spinning body on the surrounding matter. More- naive application of an action-reaction principle, a force on the body over, these are all indirect methods, in which one infers the orthogonal to its spin and velocity (like the Magnus force) might be motion of the body by observing its effect on the cloud, try- expected. ing then to figure out the backreaction on the body (which, as we shall see, is problematic, since the gravitomagnetic inter- actions, analogously to the magnetic interactions, do not obey (Here α is a a factor that differs according to the flow in general an action-reaction law). 1 regime. ) This effect is illustrated in Fig.1. It can, in simple One of the purposes of this work is to perform the first con- terms, be understood from the fact that the fluid circulation crete and rigorous calculation of this effect. We first take a di- induced by the body’s rotation decreases the flow velocity on rect approach — that is, we investigate this effect from the ac- one side of the body while increasing it on the opposite side. tual equations of motion for spinning bodies in general relativ- The Bernoulli equation then implies that a pressure differen- ity. These are well established, and known as the Mathisson- tial occurs, leading to a net force on the body [4]. Papapetrou (or Mathisson-Papapetrou-Dixon) equations [9– By its very own nature, the fluid-dynamical Magnus force 14]. We will show that a Magnus-type force is a fundamental hinges on contact interactions between the spinning body and part of the spin-curvature force, which arises whenever a spin- the fluid. Thus, ordinary Magnus forces cannot exist in the ning body moves in a medium with a relative velocity not par- interaction between (i) a fluid and a spinning black hole (BH), allel to its spin axis; it has the same direction as the Magnus or (ii) an ordinary star and dark matter (DM) which only in- force in fluid dynamics, and depends only on the mass-energy teracts with it via gravity. However, in general relativity any current relative to the body, and on the body’s spin angular form of energy gravitates and contributes to the gravitational momentum. Then we also consider the reciprocal problem, field of bodies. In particular, a spinning body produces a rigorously computing the force that the body exerts on the “gravitomagnetic field” [5]; if the spinning body is immersed surrounding matter (in the regime where such “force” is de- in a fluid, such field deflects the fluid-particles in a direction fined), correcting and clarifying the earlier results in the liter- orthogonal to their velocity, as illustrated in Fig.1b, seem- ature. These effects have a close parallel in electromagnetism, ingly leading to a nonzero “momentum transfer” to the fluid. where an analogous (anti) Magnus effect also arises. For this The question then arises if some backreaction on the body, in reason we will start by electromagnetism — and by the clas- the form of a Magnus- (or anti-Magnus)-like force — in the sical problem of a magnetic dipole inside a current slab — sense of being orthogonal to the flow and to the body’s spin — which will give us insight into the gravitational case. might arise. Indeed, the existence of such a force, in the same direction of the Magnus effect of fluid dynamics, is strongly suggested by numerical studies of nonaxisymmetric relativis- tic Bondi-Hoyle accretion onto a Kerr BH [6]. These studies A. Notation and conventions focused on a fixed background geometry and studied the mo- mentum imparted to the fluid as it accretes or scatters from the √ We use the signature (− + ++); αβσγ ≡ −g[αβγδ] is BH. A theoretical argument for the existence of such an effect the Levi-Civita tensor, with the orientation [1230] = 1 (i.e., has also been put forth in Ref. [7], based on the asymmet- in flat spacetime, 1230 = 1); ijk ≡ ijk0. Greek letters α, ric accretion of matter around a spinning BH (i.e, the absorp- β, γ, ... denote 4D spacetime indices, running 0-3; Roman letters i, j, k, ... denote spatial indices, running 1-3. The con- α α α vention for the Riemann tensor is R βµν = Γβν,µ −Γβµ,ν +... µν 1 Its value is not generically established. According to theoretical and exper- . ? denotes the Hodge dual: ?Fαβ ≡ αβ Fµν /2 for an anti- imental results, it is nearly a constant at low Reynolds numbers [1–3], but symmetric tensor Fαβ = F[αβ]. Ordinary time derivatives are seemingly velocity dependent at higher Reynolds numbers [3]. sometimes denoted by dot: X˙ ≡ ∂X/∂t. 3

B. Executive summary It acts on any celestial body that moves with respect to the background fluid with a velocity v ∦ S, and might possibly For the busy reader, we briefly outline here the main results be observed in the motion of galaxies with large peculiar ve- of our paper. A spinning body in a gravitational field is acted, locities v. Due to the occurrence of the factor (ρ + p), this in general, by a covariant force DP α/dτ (the spin-curvature force acts as a probe for the matter/energy content of the uni- force), deviating it from geodesic motion. Such force can be verse (namely for the ratio ρ/p, and for the different dark en- can be split into the two components ergy candidates). Any mater/energy content gives rise to such gravitational Magnus force, except for dark energy if modeled DP α = F α + F α , (2) with a cosmological constant (ρ = −p). dτ Weyl Mag α αβ α α β σ γ FWeyl ≡ −H Sβ; FMag ≡ 4π βσγ J S U , (3) II. ELECTROMAGNETIC (ANTI) MAGNUS EFFECT where U α is the body’s 4-velocity, Sα its spin angular mo- mentum 4-vector, and J α = −T αβU the mass-energy 4- β We start with a toy problem borrowed from the electromag- current relative to the body. The force F α is due to the Weyl netic interaction. Consider a magnetic dipole within a cloud magnetic part of the Weyl tensor, H = ?C U µU ν , de- αβ αµβν of charged particles. Is there a Magnus-type force? termined by the details of the system (boundary conditions, The relativistic expression for the force exerted on a mag- etc). The force F α , which, in the body’s rest frame reads Mag netic dipole, of magnetic moment 4-vector µα, placed in F = 4πJ × S, is what we call a gravitational analogue to Mag a electromagnetic field described by a Faraday tensor F αβ, the Magnus force of fluid dynamics; it arises whenever, rela- is [11–13, 15] tive to the body, there is a spatial mass-energy current J not parallel to S. We argue that (2) is the force that has been at- DP α = Bβαµ ≡ F α ; B ≡ ?F U µ , (4) tempted to be indirectly computed in the literature [6–8], from dτ β EM αβ αµ;β the effect of a moving BH (or spinning body) on the surround- α α ing matter. We base our claim on a rigorous computation of where P is the particle’s 4-momentum, U its 4-velocity, the reciprocal force exerted by the body on the medium, in the and Bαβ is the “magnetic tidal tensor” [16, 17] as measured in the particle’s rest frame. In the inertial frame momentar- cases where the problem is well posed, and where an action- α α α ily comoving with the particle, the space components of FEM reaction law can be applied. FMag and FWeyl are also seen to have direct analogues in the force that an electromagnetic yield the textbook expression field exerts on a magnetic dipole. FEM = ∇(B · µ) . (5) The two components of the force are studied for spinning bodies in (“slab”) toy models, and in some astrophysical se- Taking the projection orthogonal to U α of the Maxwell field tups. For quasi-circular orbits around stationary axisymmetric αβ α γ equations F ;β = 4πj , leads to B[αβ] = ?Fαβ;γ U /2 − σ γ spacetimes studied — spherical DM halos, BH accretion disks 2παβσγ j U (cf. Eq. (I.3a) in Table I of Ref. [15]), where — when S lies in the orbital plane, the spin-curvature force jα is the current density 4-vector. Therefore takes the form F = A(r)S × v, where the function A(r) is specific to the system. Its Magnus component is similar for 1 γ σ γ Bαβ = B + ?Fαβ;γ U − 2παβσγ j U . (6) all systems, whereas the Weyl component greatly differs. The (αβ) 2 force F causes the orbits to oscillate, and to undergo a secular Thus, the magnetic tidal tensor decomposes into three parts: precession, given by its symmetric part B(αβ), plus two antisymmetric contri- σ γ dL A(r) butions: the current term −2παβσγ j U , and the term = Ω × L; Ω = S . γ dt 2m ?Fαβ;γ U /2, which arises when the fields are not covariantly constant along the particle’s worldline (it is related to the laws The effect might be detectable in some astrophysical settings, of electromagnetic induction, as discussed in detail in [15]). likely candidates are: i) signature in the Milky Way galactic The force (4) can then be decomposed as disk: stars or BHs with spin axes nearly parallel to the galactic α α α α plane, should be in average more distant from the plane than FEM = FSym + FMag + Find , (7) other bodies; ii) BH binaries where one of the BHs moves 1 F α ≡ B(αβ)µ ,F α ≡ − ?F αβ U γ µ , (8) in the others’ accretion disk, the secular precession might be Sym β ind 2 ;γ β detected in gravitational wave measurements in the future, F α ≡ 2πα U γ jσµβ . (9) through its impact on the waveforms and emission directions. Mag βσγ In an universe filled with an homogeneous isotropic fluid, α α Let h β denote the space projector with respect to U (projec- described by the FLRW spacetime, representing the large tor orthogonal to U α), scale structure of the universe, which is conformally flat, we αβ α hα ≡ U αU + δα . (10) have that H = 0 ⇒ FWeyl = 0, and so the Magnus force β β β α FMag is the only force that acts on a spinning body. It reads, γ exactly, Since the tensor αβσγ U automatically projects spatially, in any of its indices, in fact only the projection of jσ orthogo- 0 2 γ σ µ α σ µ F = −4π(ρ + p)(U ) v × S nal to U , h µj , contributes to FMag. Physically, h µj is 4

B Stokes theorem B · dl = ∇ × B · ndA = 4π j · ndA = 4π∆z∆yj , y ∆z ∆ F z z z y EM ∂A A A B (13) h A where we took, for the surface A, the orientation n k j. By Magnetic B = 0 x dipole the right-hand-rule and symmetry arguments, B is parallel to z j the slab and orthogonal to j, pointing in the positive z direc- tion for y > 0, in the negative z direction for y < 0, and vanishing at y = 0. Therefore B · dl = B| ∆z, and B u∂A y=∆y so

FIG. 2. A magnetic dipole µ = µez inside a semi-infinite cloud of Bz(y) = 4π∆yj = 4πyj . (14) charged particles flowing in the ex direction. The cloud is infinite along the x and z directions, but of finite thickness in the y direction, Consider now a magnetic dipole at rest inside the cloud (for contained within −h/2 ≤ y ≤ +h/2. The magnetic field B gener- instance, the magnetic dipole moment of a spinning charged ated by the cloud points in the positive z direction for y > 0, and in body), as depicted in Fig.2. The magnetic field (14) has a gra- the negative z direction for y < 0. B has a gradient inside the cloud, αβ whose only nonvanishing component is Bz,y = 4πj. Due to that, dient inside the cloud, leading to a magnetic tidal tensor B z,y a force FEM = B µey = 4πjµey, pointing upwards, is exerted (as measured by the dipole) whose only nonvanishing compo- zy z,y on the dipole. In this case FSym = FMag, so the force is twice the nent is B = B = 4πj. Therefore, the force exerted on Magnus force: FEM = FSym + FMag = 2FMag. Considering in- the dipole is, cf. Eq. (4), stead a cloud finite along z, infinite along x and y, FMag = 2πjµey ji zy remains the same, but FSym inverts its direction: FSym = −FMag, FEM = B µjei = B µzey = 4πjµzey . (15) causing the total force to vanish: FEM = 0. It consists of the sum of the Magnus force plus the force FSym (Find = 0 since the configuration is stationary): FEM = the spatial charge current density as measured in the parti- FMag + FSym, α cle’s rest frame. In such frame, the time component of FMag vanishes, and the space components read FMag = 2πµ × j = 2πj(µzey − µyez) (16) (ji) FSym = B µjei = 2πj(µzey + µyez) (17) FMag = 2πµ × j . (11) This is a force orthogonal to µ and to the spatial current den- Equations (15)-(17) yield the forces for a fixed orientation of sity j, which we dub electromagnetic “Magnus” force. If the the slab (orthogonal to the y-axis), and an arbitrary µ. This is magnetic dipole consists of a spinning, positively (and uni- of course physically equivalent to considering instead a mag- netic dipole µ with fixed direction, and varying the orientation formly) charged body, so that µ k S, the force FMag has a direction opposite to the Magnus force of fluid dynamics (so it of the slab; in this framework, taking µ = µez, two notable is actually “anti-Magnus”). If the body is negatively charged, cases stand out: so that µ k −S, the force points in the same direction of a 1. Slab finite along y axis, infinite along x and z (see Fig. Magnus force. 2). The Magnus and the “symmetric” forces are equal: FMag = FSym = 2πjµey, so there is a total force along the y direction equaling twice the Magnus force: A. Example: The force exerted by a current slab on a dipole FEM = 2FMag = 4πjµey.

α The induction component Find has no gravitational coun- 2. Slab finite along z axis, infinite along x and y (slab or- terpart, as we shall see. Therefore, from now onwards we will thogonal to µ). The Magnus force remains the same not consider it any further. To shed light on the components as in case1; but FSym changes to the exact opposite, α α FMag and FSym, we consider a simple stationary setup (Ex- FSym = −2πjµey = −FMag. The total force on the ercise 5.14 of Ref. [18]): a semi-infinite cloud of charged gas dipole now vanishes: FEM = 0. which is infinitely long (x direction) and wide (z direction), 2 but of finite thickness h in the y direction, contained between The results in case2 follow from noting that, for a slab or- the planes y = h/2 and y = −h/2, see Fig.2. thogonal to the z axis, B is along y and given by B = Outside the slab, the field is uniform and has opposite di- −4πjzey, leading to a magnetic tidal tensor with only non- vanishing component Byz = By,z = −4πj, thus causing rections in either side [18]. The field at any point inside the (ij) cloud is readily obtained by application of the Stokes theorem B to globally change sign comparing to case1. For other to the stationary Maxwell-Ampère equation ∇ × B = 4πj . (12) 2 Equivalently, they follow from rotating the frame in Fig.2 by −π/2 about That is, let A be a rectangle in the z − y plane, as illustrated ex [which amounts to swapping y ↔ z and changing the signs of the in Fig.2, with boundary ∂A and normal unit vector n. By the right-hand members of Eqs. (15)-(17)] while still demanding µ = µez. 5 orientations of the slab/dipole, the forces FSym and FMag as explained in detail in pp. 187-188 of [19]. The magnetic are not collinear. When µ coincides with an eigenvector of field in any region exterior to the dipole is (e.g. [18, 19]) the matrix B(ij), they are actually orthogonal; in the slab µ 3(µ · r)r in Fig.2 (orthogonal√ to the y-axis), that is√ the case for Bdip|r>R = − 3 + 5 . (21) µ = µ(ey + ez)/ 2 and µ = µ(ez − ey)/ 2 (the third r r eigenvector of B(ij), µ = µe , has zero eigenvalue and leads x For the setup in Fig.2 (slab orthogonal to the y-axis, µ = to FMag = FSym = 0). µez), and considering a spherical coordinate system where Notice that neither the field B at any point inside the cloud, z2/r2 = cos2 θ and the plane y = h/2 is given by the equa- nor its gradient, or the force (15), depend on the precise width tion r = h/(2 sin θ sin φ), the exterior integral becomes h of the cloud; in particular, they remain the same in the limit h → ∞. The role of considering (at least in a first moment) 3 a finite h is to fix the direction of B. Equation (12), together Bdipd x = 2µez ˆr>R with the problem’s symmetries, then fully fix B via Eq. (13) π π β 3 cos2 θ − 1 4 and, therefore, FSym. Taking the limit h → ∞ in cases1- × dθ dφ dr sin θ = πµez ,(22) 2 above yields two different ways of constructing an infinite ˆ0 ˆ0 ˆR r 3 leading to a different B current cloud, each of them and force with β = h/(2 sin θ sin φ). Substituting Eqs. (20) and (22) on the dipole (the situation is analogous to the “paradoxes” into (19), and comparing to Eq. (15), we see that of the electric field of a uniform, infinite charge distribution, or of the Newtonian gravitational field of a uniform, infinite Fdip,cloud = −4πjµey = −FEM ≡ −Fcloud,dip , (23) mass distribution, see Sec. III B 3). Had one started with a cloud about which all one is told is that it is infinite in all di- i.e., the force exerted by the dipole on the cloud indeed equals rections, it would not be possible to set up the boundary con- minus the force exerted by the cloud on the dipole. It is how- ditions needed to solve the Maxwell-Ampère equation (12), ever important to note that this occurs because one is dealing so the question of which is the magnetic field (thus the force here with magnetostatics; for general electromagnetic inter- 3 the dipole) would have no answer. actions do not obey the action-reaction law (in the sense of a reaction force equaling minus the action). This is exemplified in AppendixB1. In particular it is so for the interaction of the B. Reciprocal problem: The force exerted by the dipole on the dipole with individual particles of the cloud. slab If one considers instead a slab orthogonal to the z axis (contained within −h/2 ≤ z ≤ +h/2), and noting that the There have been attempts at understanding and quantifying plane z = h/2 is given by r = h/(2 cos θ), one obtains 3 the gravitational analogue of the Magnus effect [7,8]. How- r>R Bdipd x = −(8π/3)µez, which exactly cancels out ever, in these works, the force on the spinning body was in- the´ interior integral (20), leading to a zero force on the cloud: ferred from its effect on the cloud, by guessing its back reac- Fdip,cloud = 0 (matching, again, its reciprocal). tion on the body. Here we will start by computing it rigor- Just like in the reciprocal problem, the results do not de- ously in the electromagnetic analogue, i.e., the reciprocal of pend on the width h of the slabs, so taking the limit h → ∞ the problem considered above: the force exerted by the mag- of the slabs orthogonal to y and to z are two different ways netic dipole on the cloud. It is given by the integral of obtaining equally infinite clouds, but on which very differ- ent forces are exerted. Here the issue does not boil down to a 3 3 problem of boundary conditions for PDE’s (as was the case for Fdip,cloud = j × Bdipd x = j × Bdipd x . B in Sec.IIA); it comes about instead in another fundamental ˆcloud ˆcloud (18) mathematical principle (Fubini’s theorem [20, 21]): the mul- Consider a sphere completely enclosing the magnetic dipole, tiple integral of a function which is not absolutely convergent, and let R be its radius; we may then split depends in general on the way the integration is performed. This is discussed in detail in AppendixA. It tells us that, like 3 3 its reciprocal, Fdip,cloud is not a well defined quantity for an Fdip,cloud = j × Bdipd x + j × Bdipd x (19) ˆr≤R ˆr>R infinite cloud.

The interior integral yields III. GRAVITATIONAL MAGNUS EFFECT 3 8π Bdipd x = µ , (20) ˆr≤R 3 Contrary to idealized point (“monopole”) particles, “real,” extended bodies, endowed with a multipole structure, do not move along geodesics in a gravitational field. This is be- cause the curvature tensor couples to the multipole moments αβ 3 This indeterminacy is readily seen noting that, given a solution of Eq. (12), of the body’s energy momentum tensor T (much like in the adding to it any solution of the homogeneous equation ∇ × B = 0 yields way the electromagnetic field couples to the multipole mo- another solution of Eq. (12). ments of the current 4-vector jα). In a multiple scheme, the 6

first correction to geodesic motion arises when one consid- where ers pole-dipole spinning particles, i.e., particles whose only α α γ β σ multipole moments of T αβ relevant to the equations of mo- FMag ≡ 4π βσγ U J S , (32) tion are the momentum P α and the spin tensor Sαβ (see e.g. α αβ FWeyl ≡ −H Sβ . (33) [11, 12, 22] for their definitions in a curved spacetime). In γ this case the equations of motion that follow from the conser- Since the tensor αβσγ U automatically projects spatially (in β vation laws T αβ = 0 are the so-called Mathisson-Papapetrou any of its free indices), only the projection of J orthogonal ;β γ β µ α (or Mathisson-Papapetrou-Dixon) equations [9–14]. Accord- to U , h µJ [see Eq. (10)], contributes to FMag. ing to these equations, a spinning body experiences a force, Equations (31)-(33) thus tell us that the spin-curvature force α the so called spin-curvature force, when placed in a gravita- splits into two parts: FWeyl, which is due to the magnetic part tional field. It is described by of the Weyl tensor, and is analogous (to some extent) to the “symmetric force” F α of electromagnetism, Eq. (8). The α Sym DP 1 α µν β α second part is F α which is nonvanishing whenever, relative = − R βµν S U ≡ F , (24) Mag dτ 2 β µ to the body, there is a spatial mass-energy current h µJ not where U α = dxα/dτ is the body’s 4-velocity (that is, the parallel to Sα. In the body’s rest frame, we have tangent vector to its center of mass worldline). This is a phys- ical, covariant force (as manifest in the covariant derivative FMag = 4πJ × S , (34) α operator D/dτ ≡ U ∇α), which causes the body to devi- α thus FMag is what one would call a gravitational analogue of ate from geodesic motion. Under the so-called Mathisson- the Magnus effect in fluid dynamics, since αβ Pirani [9, 23] spin condition S Uβ = 0, one may write i) it arises whenever the body rotates and moves in a medium µν µντλ α α β µν S =  Sτ Uλ, where S ≡  βµν U S /2 is the spin with a relative velocity not parallel to its spin axis (that is, 4-vector [whose components in an orthonormal frame comov- when there is a spatial mass-energy current density J relative ing with the body are Sα = (0, S)]. Substituting in (24), leads to the body, such that S ∦ J); to [15, 16], ii) the force is orthogonal to both the axis of rotation of the α body (i.e. to S) and to the current density J, like in an ordi- α DP βα F = = − Sβ , (25) nary Magnus effect; moreover, it points precisely in the same dτ H direction of the latter.4 where Notice that Eq. (31) is a fully general equation that can α 1 be applied to any system, and that the “Magnus force” FMag µ ν λτ µ ν α α α Hαβ ≡ ?Rαµβν U U = αµ Rλτβν U U , (26) depends only on U , S , and the local J , and not on any 2 α further detail of the system. The force FWeyl, by contrast, is the “gravitomagnetic tidal tensor” (or “magnetic part” of strongly depends on the details of the system (as exemplified the Riemann tensor, e.g. [24]) as measured by an observer in Sec. III B below). This can be traced back to the fact that α comoving with the particle. Using the decomposition of the FMag comes from the Ricci part of the curvature, totally fixed Riemann tensor in terms of the Weyl (Cαβγδ) and Ricci ten- by the energy-momentum tensor T αβ of the local sources via sors (e.g. Eq. (2.79) of Ref. [25]), the Einstein Eqs. (29), whereas the Weyl tensor describes the “free gravitational field,” which does not couple to the sources [α 1 Rαβ = Cαβ + 2δ Rβ] − Rδα δβ , (27) via algebraic equations, only through differential ones (the γδ γδ [γ δ] 3 [γ δ] differential Bianchi identities [25, 26]), being thus determined αβ we may decompose Hαβ as not by the value of T at a point, but by conditions elsewhere [25]. 1 γ σλ Note also that, in general, F is not the total force in αβ = + = Hαβ + αβσγ U R Uλ , (28) Mag H H(αβ) H[αβ] 2 α the direction orthogonal to S and J; FWeyl may also have a 5 where the symmetric tensor Hαβ = H(αβ) is the magnetic component along it . µ ν part of the Weyl tensor, Hαβ ≡ ?Cαµβν U U . Using the Einstein field equations

4 This is always so if the parallelism drawn is between J and the flux vector 1 α Rµν = 8π(Tµν − gµν T α) + Λgαβ , (29) of fluid dynamics. If the analogy is based instead on the velocity of the fluid 2 relative to the body, the gravitational and ordinary Magnus effects have the this becomes (cf. e.g. Eq. (I.3b) of Table I of [15]) same direction for a perfect fluid obeying the weak energy condition (cf. Sec.VI and Eq. (110) below), but otherwise it is not necessarily so (e.g., = + = H − 4π U γ J σ , (30) an imperfect fluid conducting heat gives rise to mass currents nonparallel Hαβ H(αβ) H[αβ] αβ αβσγ to the fluid’s velocity). 5 α α αβ Its behavior however (unlike the part FMag) is not what one would expect where J ≡ −T Uβ is the mass/energy current 4-vector as from a gravitational analogue of the Magnus effect, namely (a) it is not measured by an observer of 4-velocity U α (comoving with the determined, nor does it depend on J and S in the way one would expect particle, in this case). We thus can write from a Magnus effect and (b) it is not necessarily nonzero when J×S 6= 0. α For this reason we argue that only FMag should be cast as a gravitational α αβ α β σ γ α α “Magnus effect.” F = −H Sβ +4π βσγ J S U = FWeyl +FMag , (31) 7

A. Post-Newtonian approximation One has then 1 = − H −  G˙k + 2 kmv G Up to here, we used no approximations in the description Hij 2 i,j ijk i k j,m of the gravitational forces. For most astrophysical systems, −  mG vk + O(5) . (39) however, no exact solutions of the Einstein field equations ij k,m α are known; in these cases we use the post-Newtonian (PN) Noting that the orthogonality relation SαU = 0 implies i approximation to general relativity. This expansion can be S0 = −Siv = O(1), it follows, from Eqs. (25) and (37), j ij 0j ij framed in different — equivalent — ways; namely, by count- that F = −H Si − H S0 = −H Si + O(5), and so the ing powers of c [27, 28] or in terms of a dimensionless pa- 1PN spin-curvature force reads rameter [17, 29–32]. Here we will follow the latter, which consists of making an expansion in terms of a small dimen- j 1 i,j j ikm j F = H Si − (S × G˙ ) − 2 vkG Si 2 ,m sionless parameter , such that U ∼  and v . , where U is 2 jim k (minus) the Newtonian potential, and v is the velocity of√ the −  Gk,mv Si + O(5) . (40) bodies (notice that, for bodies in bounded orbits, v ∼ U). In terms of “forces,” the Newtonian force m∇U is taken to Its Magnus and Weyl components, Eqs. (32)-(33), are 2 be of zeroth PN order (0PN), and each factor  amounts to a F i = 4πi Sk(T 0j − ρvj) + O(5) , (41) unity increase of the PN order. Time derivatives increase the Mag jk i ij degree of smallness of a quantity by a factor ; for example, FWeyl = − H Sj + O(5) (42)

∂U/∂t ∼ Uv ∼ U. The 1PN expansion consists of keeping 1 (i,j) (i j),m k 4 = H S − 2 G v S + O(5) ; (43) terms up to O( ) ≡ O(4) in the equations of motion [30]. 2 j km j This amounts to considering a metric of the form [27, 31] where for ρ one can take the mass/energy density as measured 2 either in the body’s rest frame, or in the PN background frame g00 = −1 + 2w − 2w + O(6) (the distinction is immaterial in Eq. (41), to the accuracy at gi0 = Ai + O(5); gij = δij (1 + 2U) + O(4) , (35) 0j j j β hand). Notice that T −ρv = h βJ [see Eq. (10)] is indeed where A is the “gravitomagnetic vector potential” and the the spatial mass-energy current with respect to the body’s rest T 0j scalar w consists of the sum of U plus nonlinear terms of or- frame ( , in turn, yields the spatial mass-energy current as der 4, w = U + O(4). For the computation of the space part measured in the PN frame). To obtain the coordinate acceleration of a spinning test of the force (25), the components Hij, H0i, of the gravitomag- α 0 body, we first note that, under the Mathisson-Pirani spin con- netic tidal tensor (26) are needed. Using U = U (1, v) and α 6 dition, the relation between the particle’s 4-momentum P the 1PN Christoffel symbols in e.g. Eqs. (8.15) of Ref. [28], α α αβ and its 4-velocity is (e.g. [22]) P = mU + S aβ, where they read α α m ≡ −P Uα is the proper mass, which is a constant, a ≡ α αβ 1 DU /dτ the covariant acceleration, and the term S aβ is = −  lkA −  kU˙ + 2 kmv U Hij 2 i k,lj ij ,k i k ,jm the so-called “hidden momentum” [13, 15, 22]. In the post- Newtonian regime one can neglect8 the hidden momentum, −  mU vk + O(5) , (36) ij ,km leading to the acceleration equation maα ' DP α/dτ ≡ F α. 1 α 2 α 2 α β γ l ˙ j lk j l j k Using DU /dτ = d x /dτ + Γβγ U U and d/dτ = H0i =ij U,lv + j Ak,liv +  jiU,lkv v (= O(4)) , 2 (dt/dτ)d/dt, where t is the coordinate time, one gets, after (37) some algebra, where dot denotes ordinary time derivative, ∂/∂t. Equation d2xi m = F i + F i + O(5) , (44) (36) is a generalization of Eq. (3.41) of Ref. [27] for nonva- dt2 I cuum, and for the general case that the observer measuring where the tensor Hαβ moves (i.e., v 6= 0). It is useful to write Hij in terms of the gravitoelectric (G) and gravitomagnetic (H) dxi  dxβ dxγ F i = m Γ0 − Γi (45) fields, defined by [17, 27, 31] I dt βγ βγ dt dt G = ∇w − A˙ + O(6) , H = ∇ × A + O(5) . (38) is the inertial “force” already present in the geodesic equation 2 i 2 i for a nonspinning point particle: d x /dt = FI /m (cf. e.g. The reason for these denominations is that these fields play in gravity a role analogous to the electric and magnetic fields.7

8 That actually amounts to pick, among the infinite solutions allowed by the (degenerate) Mathisson-Pirani spin condition, the “non-helical” one (avoiding the spurious helical solutions) [15, 33]. For such solution, the 6 α Identifying, in the notation therein, w → U + Ψ, Ai → −4Ui. acceleration comes, at leading order, from the force F ; and so the term 7 2 i 2 i i αβ α Namely comparing the geodesic equation d x /dt = FI /m [FI given D(S aβ )/dτ is always of higher PN order than F (for details, see Sec. by Eq. (46)] with the Lorentz force, and comparing Einstein’s field equa- 3.1 of the Supplement in [15]). It is also quadratic in spin, and, as such, tions in e.g. Eqs. (3.22) of Ref. [27] with the Maxwell equations. arguably to be neglected at pole-dipole order [14, 15, 33]. 8

H we have (cf. e.g. Eq. (2.6d) of Ref. [31]) ∇ × H = −16πJ , (47) y ∆z ∆y 0i i α αβ where we noted that T = J + O(5), and J = −T uβ h A H Black = 0 is the mass/energy current as measured by the reference ob- Hole H α 0 α x servers u = u δ0 [at rest in the coordinate system of (35)]. z J This equation resembles very closely Eq. (12). For a sys- tem analogous to that in Fig.2 — a cloud of matter passing H F through a spinning body — an entirely analogous reasoning to that leading to Eq. (14) applies here to obtain the gravito- magnetic field FIG. 3. A spinning body (e.g., a BH), with S = Sez, inside a massive cloud flowing in the e direction. The cloud is infi- x H = Hz(y)e = −16πyJe . (48) nite along x and z, and finite along the y-axis, contained within z z −h/2 ≤ y ≤ h/2. The vector H is the gravitomagnetic field gen- erated by the cloud; it points in the negative (positive) z direction for This solution is formally similar to the magnetic field in Eq. y > 0 (< 0). It has a gradient inside the cloud, whose only nonva- (14), apart from the different factor and sign. For a spin- nishing component is Hz,y = −16πJ; due to that, a spin-curvature ning body at rest in a stationary gravitational field, the spin- z,y force F = (1/2)H Sey = −8πJSey, pointing downwards, is curvature force, Eq. (40), reduces to exerted on the body. In this case FMag = FWeyl, so the total force 1 1 is twice the Magnus force: F = FMag + FWeyl = 2FMag. Con- F i = Hj,iS ⇔ F = ∇(H · S) , (49) sidering instead a cloud finite along z, and infinite along x and y, 2 j 2 FMag = −4πJSey remains the same, but FWeyl changes to the exact opposite: FWeyl = −FMag, causing the total spin-curvature (cf. e.g. Eq. (4) of [34]), similar to the dipole force (15). force to vanish: F = 0. Hence, due to the gradient of H, whose only nonvanishing component Hi,j is Hz,y = −16πJ, a force F is exerted on a spinning body at rest inside the cloud, given by Eq. (8.14) of [28]). Using, again, the 1PN Christoffel symbols in Eqs. (8.15) of [28], yields F = −8πJSzey . (50)

F It is thus along the y direction, pointing downwards, in the I = (1 + v2 − 2U)G + v × H − 3U˙ v − 4(G · v)v + O(6) . same direction of an ordinary Magnus effect (and opposite to m (46) the electromagnetic analogue). This force consists of the sum Equation (44) is a general expression for the coordinate accel- of the Magnus force plus the Weyl force: F = FMag +FWeyl, eration of a spinning particle in a gravitational field, accurate to 1PN order. FMag = 4πJ × S = 4πJ(Syez − Szey) , (51) ij FWeyl = −H µjei = −4πJ(Szey + Syez) . (52)

Here, Hij = H(ij) = −H(i,j)/2, and its nonvanishing com- B. A cloud “slab” ponents are Hzy = Hyz = 4πJ. Again, Eqs. (50)-(52) yield the forces for a fixed orientation of the slab (orthogonal to the 1. The Magnus force on spinning objects y-axis), and an arbitrary S. Of course, this is physically equiv- alent to considering instead a body with fixed spin direction, Before moving on to more realistic scenarios, we start by and varying the orientation of the slab; choosing S = Sez, investigating the gravitational Magnus force, in the PN ap- one can make formally similar statements to those in Sec.IIA, proximation, for the gravitational analogue of the electromag- by replacing FSym by FWeyl. Namely, the two notable cases netic system in Sec.IIA. In particular, we consider a spinning arise: body (for example, a BH) inside a medium flowing in the e x 1. Cloud finite along the y-axis, infinite along x and z direction, that we assume to be infinitely long and wide (in (Fig.3). The Magnus force F equals the Weyl the x and z directions), but of finite thickness h (y direction), Mag force: F = F = −4πJSe , so there is a to- contained within the planes −h/2 ≤ y ≤ h/2. The system is Mag Weyl y tal force downwards which is twice the Magnus force: depicted in Fig.3. F = 2F = −8πJSe . The Einstein field equations yield a gravitational analogue Mag y to the Maxwell-Ampère law (12), as we shall now see. For the 2. Cloud finite along z, infinite along x and y (i.e., slab metric (35), the Ricci tensor component R0i = (∇×H)i/2− orthogonal to S). The Magnus force FMag remains the 2G˙ + O(5), where H is the gravitomagnetic field as defined same as in case1; the Weyl force is now exactly oppo- by Eq. (38). On the other hand, from the Einstein equations site to the Magnus force: FWeyl = 4πJSey = −FMag, (29), we have that R0i = 8πT0i + O(5). Equating the two so the total spin-curvature force vanishes: F = FMag + expressions, and taking the special case of stationary setups, FWeyl = 0. 9

In case2 we noted that, for a slab orthogonal to the z axis, integral H = 16πJzey, and so the magnetic part of the Weyl ten- sor Hij changes sign comparing to the setup in Fig.3: F = J × H d3x (53) body,cloud ˆ body Hzy = Hyz = −4πJ. For other orientations of the slab/S, cloud the Weyl and Magnus forces are not parallel. When S co- 3 3 = J × Hbodyd x + J × Hbodyd x incides with an eigenvector of the magnetic part of the Weyl ˆr≤R ˆr>R ij tensor H , FWeyl ∝ S, being therefore orthogonal to FMag. where Hbody is the gravitomagnetic field generated by the For the cloud in Fig.3 (orthogonal√ to the y-axis), this√ is the spinning body. Equation (47), formally similar to (12) up to a case for S = S(ey + ez)/ 2 and S = S(ez − ey)/ 2 (the 9 ij factor −4, implies that third eigenvector of H , S = Sex, has zero eigenvalue and leads to FMag = FWeyl = 0). Cases1-2 sharply illustrate the 3 16π Hbodyd x = − S (54) contrast between the two parts of the spin curvature force: on ˆr≤R 3 the one hand the Magnus force F , which depends only on Mag and that the exterior gravitomagnetic field is (cf. e.g. [5, 35]) S and on the local mass-density current J, and is therefore the same regardless of the boundary; and, on the other hand, S (S · r)r H | = 2 − 6 , (55) the Weyl force, which is determined by the details of the sys- body r>R r3 r5 tem, namely the direction along which this cloud model has analogous, up to a factor -2, to (20) and (21), respectively. a finite width h. Similarly to the electromagnetic case, nei- i,j Therefore, for S = Sez, and a slab finite in the y direction ther H at any point inside the cloud (or its gradient H ), (contained within −h/2 < y < h/2), as depicted in Fig.3, nor FWeyl, depend on the precise value of h; the role of its an integration analogous to (22) leads to finiteness boils down to fixing the direction of H. Equation (47), together with the problem’s symmetries, then fully fix H Fbody,cloud = 8πJSey = −F ≡ −Fcloud,body , (analogously to the situation for B in Sec.IIA). One can then i.e., minus the force exerted by the slab on the body, Eq. (50), say that, in this example, the magnetic part of the Weyl tensor, satisfying an action-reaction law. For a slab finite in the z Hij = H(ij) (and therefore FWeyl), is fixed by the boundary, direction (contained within −h/2 < z < h/2), like in the whereas antisymmetric part of the gravitomagnetic tidal ten- electromagnetic analogue the force vanishes: Fbody,cloud = sor, H[ij], depends only on the local mass current density J, 0, matching its reciprocal. cf. Eq. (30). Several remarks must however be made on this result. First In general one is interested in the total force F = FMag + we note that, unlike the spin-curvature force F ≡ Fcloud,body FWeyl (for it is what determines the body’s motion); the de- exerted by the cloud on the body (which is a physical, covari- pendence of FWeyl on the details/boundary conditions of the ant force), the gravitomagnetic “force” mv × H, that (when system shown by the results above hints at the importance of summed over all particles of the cloud) leads to Fbody,cloud, is appropriately modeling the astrophysical systems of interest. an inertial force, i.e., a fictitious force (in fact H is but twice the vorticity of the reference observers, see e.g. [17, 35]). Moreover, an integration in the likes of Eq. (53) is not possi- ble in a strong field region, for the sum of vectors at different 2. The force exerted by the body on the cloud points is not well defined. Such integrations make sense only in the context of a PN approximation, which requires a Newto- nian potential such that U  1 everywhere within the region Previous approaches in the literature attempted to compute of integration. This requires a body with a radius such that the gravitational Magnus force by inferring it from its recip- R  m (and spinning slowly), so that the field is weak even rocal – the force exerted by the body on the cloud [7,8]. Un- in its interior regions, which precludes in particular the case of fortunately, these attempts were not based on concrete com- BHs or compact bodies. (It does not even make sense to talk putations of such force, but on estimates which are either not about an overall force on the cloud in these cases). In addition complete (and thereby misleading) or rigorous, and turn out 3 to that, the interior integral r≤R J × Hbodyd x obviously in fact to yield incorrect conclusions (see Sec. III B 3 and Ap- only makes sense if the cloud´ is made of dark matter or some pendixB2a below for details). In this section we shall rig- other exotic matter that is able to permeate the body; other- orously compute, in the framework of the PN approximation, wise J = 0 for r ≤ R, and so such integral would be zero.10 the “force” exerted by the spinning body on the cloud for the setups considered above. In the first PN approximation, the geodesic equation for a 9 This can be shown by steps analogous to those in pp. 187-188 of Ref. [19], point particle of coordinate velocity v = dx/dt can be writ- replacing therein the magnetic vector potential A by the gravitomagnetic 0 0 3 0 dv/dt = F /m F vector potential Abody(x) = −4 Jbody(x )/|x − x |d x . ten as I , with I given by Eq. (46). This equa- 10 tion exhibits formal similarities with the Lorentz force law; Still that will not lead to a mismatch´ between action and reaction [com- paring to F ≡ Fcloud,body as given by Eq. (50)], because in that case namely the gravitomagnetic “force” mv × H, analogous to the mass current around the body would not be uniform and along x [that the magnetic force qv × B. The total gravitomagnetic force would violate the PN continuity equation ∂ρ/∂t = −∇ · J + O(5)], but exerted by the spinning body on the cloud is the sum of the instead one would have a continuous flow around the body, as described by fluid dynamics, accordingly changing J × H d3x. force exerted in each of its individual particles, given by the r>R body ´ 10

Finally, it should be noted that, although for these station- spherical boundary is considered, with arbitrarily large ra- ary setups the action Fcloud,body equals minus the reaction dius R. The limit R → ∞ yields yet another way of con- Fbody,cloud, in general dynamics the gravitomagnetic interac- structing an infinite cloud. The force exerted by the body 3 tions, just like magnetism, do not obey the action-reaction law on such cloud, Fbody,cloud = J × r

FMag A. Spherical, uniform dark matter halo FIG. 4. Spinning bodies moving in a “pseudo-isothermal” DM halo. For a body in quasi-circular orbits, with spin lying in the orbital Let us start by considering a spherical DM halo of constant plane, the Magnus (FMag) and Weyl (FWeyl) forces are parallel. density ρ = ρ0, which, although unrealistic, is useful as a The total force is of the form F = A(r)S × v, pointing out- toy model. It follows from Eqs. (57) and (61) that M(r) = wards the orbital plane on one half of the orbit, and inwards the 3 4πρ0r /3 and A(r) = 4πρ0, therefore, by Eqs. (59) and other half; this “torques” the orbit, leading to a secular orbital pre- (63), the magnetic part of the Weyl tensor, and the Weyl force, cession Ω. For a body moving radially towards the center of the halo, F = 0, and so the total force exerted on it reduces to the vanish for all v: Hij = H(ij) = 0 ⇒ FWeyl = 0. The Weyl gravitomagnetic tidal tensor reduces to its antisymmetric part, Magnus force: F = FMag = 4πρS × v1. Generically FWeyl and = , and the total force reduces to the Magnus force, FMag have different directions. If the halo’s density was uniform, Hij H[ij] F = 0 ⇒ F = F for all particles. cf. Eq. (62), Weyl Mag

F = FMag = 4πρ0S × v . (64) This equation tells us that any spinning body moving inside ρ such halo suffers a Magnus force. It is (to dipole order) ρ(r) = 0 , r2 (66) the only physical force acting on the body, deviating it from 1 + r2 geodesic motion. It can also be seen from Eqs. (44)-(46) that c

FMag/m is, to leading PN order, the total coordinate acceler- where rc is the core radius. For r  rc, the velocity of the cir- ation in the direction orthogonal to v . cular orbits becomes nearly constant, whilst at the same time not diverging at r = 0 (as is the case for the isothermal profile, ρ(r) ∝ r−2). From Eqs. (66), (57), and (61), we have B. Realistic halos r2  r  r  A(r) = 12πρ c 1 − c arctan . (67) The simplistic model above can be improved to include 0 r2 r r more realistic density profiles. c Power law profiles (ρ ∝ r−γ )—In some literature (e.g. [41, Notice that A(r) > 0 for all r. 42]) models of the form ρ(r) = Kr−γ are proposed, where ρ(r) A(r) K is a r independent factor. The condition that the mass (57) Substitution of the expressions for and in (59)- inside a sphere of radius r be finite requires γ < 3; in this case (60), (63), yield, for each model, the gravitomagnetic tidal v we have tensor Hij as measured by the body moving with velocity , and the spin-curvature force exerted on it. Comparing with Kr−γ ρ(r) the situation for the uniform halo highlights the contrast be- A(r) = 12π = 12π . (65) 3 − γ 3 − γ tween the two components of the spin-curvature force (and the dependence of the Weyl force on the details of the system): 2 For γ = 2, this yields the isothermal profile ρ(r) = K/r , FMag remains formally the same (for it depends only on the leading√ to A(r√) = 12πρ(r), and to a constant orbital velocity local density ρ and on v), whereas FWeyl is now generically v = Gr = 2 Kπ. This is consistent with the observed flat nonzero. It is different for each model, and has generically a rotation curves of some galaxies, and is known to accurately different direction from FMag. The Weyl force vanishes re- describe at least an intermediate region of the Milky Way DM markably when (at some instant) v k r. Hence, if one takes halo [41]. Values 1 ≤ γ ≤ 1.5 have also been suggested a particle with initial radial velocity, initially one has, exactly, [42, 43], based on numerical simulations, for the inner regions F = FMag; and afterwards the spin-curvature force will con- of spiral galaxies like the Milky Way. sist on FMag plus a smaller correction FWeyl due to the non- Pseudo-isothermal density profile.—Consider a density radial component of the velocity that the particle gains due to profile ρ(r) given by [43] the force’s own action. 12

C. Objects on quasi-circular orbits The Magnus, Weyl, and total forces then read

We shall now consider the effect of the spin-curvature force FMag = 4πρSv cos[(ω − Ωs)t]ez , (FMag + FWeyl) exerted on test bodies on (quasi-) circular FWeyl = [A(r) − 4πρ]Sv cos[(ω − Ωs)t]ez , orbits within the DM halo. The evolution equation for the spin F = A(r)S × v = A(r)Sv cos[(ω − Ωs)t]ez . (72) vector of a spinning body reads, in an orthonormal system of axes tied to the PN background frame (i.e., to the basis vectors All these forces are thus along the direction orthogonal to the of the coordinate system in (35); this is a frame anchored to orbital plane. The situation is inverted comparing to the case the “distant stars”) [5, 29, 35] in Sec. IV C 1 above: FWeyl points in the same direction of dS 1 3 FMag for A(r) > 4πρ, and in opposite direction for A(r) < = Ωs × S ; Ωs = − v × a + v × G , (68) 4πρ. For all the models considered in Sec.IVB (the pseudo- dt 2 2 isothermal, and those of the form ρ ∝ r−γ , with 0 ≤ γ < 3), where the first term is the Thomas precession and the second we have A(r) > 4πρ, cf. Eqs. (67), (65); so both FWeyl the geodetic (or de Sitter) precession. Since the only force and the total force F point in the same direction as FMag (the present is the spin-curvature force, then a = F /m, and the latter condition requiring only A(r) > 0), see Fig.4. Thomas precession is negligible to first PN order. So, in what The force F causes the spinning body to oscillate (in the ez follows, Ωs ≈ 3v × G/2. Without loss of generality, let us direction) along the orbit, perturbing the circular motion. The assume the orbit to lie in the xy-plane. Two notable cases to coordinate acceleration orthogonal to the orbital plane is, from consider are the following. Eqs. (44)-(46), z¨ = F z/m + Gz. Gz is the component of the gravitational field along z, that is acquired when the body os- cillates out of the plane. Making a first order Taylor expansion 1. Spin orthogonal to the orbital plane (S = Sze ) z z about z = 0, we have G ' Gz,z|z=0z ≡ Gz,zz. The general 2 solution, for Gz,z < 0 and Gz,z 6= −∆ω , is In this case Ωs k S, and so dS/dt = 0, i.e., the components p p of the spin vector are constant along the orbit (so it remains z(t) = c1 cos( −Gz,zt)+c2 sin( −Gz,zt)+Z cos(∆ωt) , along ez). The Magnus and Weyl forces are (73) where (S · L) (S · L) F = −4πρ e , F = F +A(r) e , Mag mr r Weyl Mag mr r S A(r)v S A(r)v (69) Z ≡ − = , (74) m G + ∆ω2 m Ω (2ω − Ω ) where L = mr × v is (to lowest order) the orbital an- z,z s s gular momentum (see e.g. [5]). All the forces are radial. c1 and c2 are arbitrary integration constants, ∆ω ≡ ω − Ωs For A(r) < 4πρ, FWeyl points in the same direction of and r is the radius of the fiducial circular geodesic. In the F , which resembles case1 of the slab in Sec. III B. For Mag second equality in (74) we noted, from Eq. (58), that Gz,z = A(r) > 4πρ, which is the case in all the models considered in −G/r = −ω2. Noticing, moreover, from Eqs. (56), (61), that Sec.IVB, FWeyl points in the direction opposite to FMag, re- sembling case2 of the slab. As for the total force F , it points v2 A(r) = 3ω2 = 3 (75) in the same direction of FMag for A(r) < 8πρ, which, for r2 the power law profiles ρ ∝ r−γ in Sec.IVB, is the case for 3 2 γ < 1.5; it vanishes when A(r) = 8πρ; and it points in op- and Ωs = 3ω r /2, we can re-write (74) as a function of the posite direction to FMag when A(r) > 8πρ, which is the case orbital velocity (v = ωr) only, for γ > 1.5. The pseudo-isothermal profile (66) realizes all S 1 the three cases, having an interior region where F k FMag, Z = . (76)  3 2 whereas F k −FMag for large r. The orbital effect of F m v 1 − 4 v amounts to a change in the effective gravitational attraction. The first two terms of Eq. (73) are independent of the spin- curvature force (if F = 0, they simply describe the z oscilla- z 2. Spin parallel to the orbital plane (S = 0) tions of a circular orbit lying off the xy-plane), so c1 and c2 essentially set up the initial inclination of the orbit. Two nat- In this case Eq. (68) tells us that S precesses, but remains ural choices of these constants stand out (analogous to those always in the plane; since Ωs is constant, this equation yields first found in Ref. [44], for orbits around BHs). (taking, initially, S = Sex) Constant amplitude regime: c1 = c2 = 0. In this case z(t) = Z cos(∆ωt), yielding a “bobbing” motion of fre- S = S cos(Ωst)ex + S sin(Ωst)ey . (70) quency ∆ω and constant amplitude Z. They may be seen as To the accuracy needed for Eqs. (63), v ≈ v(− sin φe + an orbit which is inclined relative to the fiducial geodesic, and x whose plane precesses with the frequency of the geodetic pre- cos φey), with φ = ωt, where ω is the orbital angular velocity. Therefore cession (Ωs). “Beating” regime: one starts with the same initial data of S × v = vS cos(φ − Ωst)ez = vS cos[(ω − Ωs)t]ez . (71) a circular orbit in the xy-plane: z(0) =z ˙(0) = 0, implying 13

First 5 cycles z-t plot z 2 Z 1.0

0.5 z 2 2 ´ 1015 m 0 t -2 0.5 1.0 1.5 2.0 Π Ws 0 y m -2 ´ 1015 -0.5  0  x m -2 ´ 1015 15 2 ´ 10 -1.0 

8 −3 FIG. 5. Numerical 1PN results for quasi-circular orbits in a pseudo-isothermal DM halo, with ρ0 = 10 M pc , rc = 0.02 kpc (typical of satellite galaxies [45]), and r = 8rc, case in which v = 0.15c. The test body has the Sun’s mass m = M , and an initial spin vector 2 S|in = Sex, with S = 0.5m . Left panel: three-dimensional plot of the orbit, showing the orbital precession Ω ∝ S in Eq. (80) (i.e., about ex, initially). Right panel: plot of z(t)/2Z, for t ∈ [0, 2π/Ωs]; the numerical result agrees well with the (simplified) analytical result (77). It shows clearly the modulation by the spin precession Ωs: the orbital precession Ω causes z oscillations of initially increasing amplitude, reaching its peak z = 2Z at t = π/Ωs, corresponding to the maximum inclination of the orbital plane. At that point the direction of S (thus of Ω) becomes inverted relative to the initial one, so the orbital inclination (and the oscillation amplitude) starts decreasing.

c2 = 0, c1 = −Z. Using the trigonometric identity cos(b) − orbit, in order to extract the secular effect. First we note, from  a+b   a−b  2 cos(a) = 2 sin 2 sin 2 , Eq. (73) becomes (68), that Ωs = 3vG/2 ∼ ω , so typically Ωs  ω, and, therefore, along one orbit, the spin vector is nearly constant. 2ω − Ω  Ω  z(t) = 2Z sin s t sin s t . (77) So, for averaging along an orbit, we may approximate S ' 2 2 2 Sex. It follows that h(S × r) × vi = −Srv sin φ ey = This corresponds to a rapid oscillatory motion of frequency S × L/(2m), leading to the secular orbital precession (2ω − Ωs)/2 (close to the orbital frequency ω), modulated by dL A(r) a sinusoid of frequency Ωs/2 (half the frequency of the spin = Ω × L; Ω = S . (80) precession), and of peak amplitude 2Z. In spite of the simpli- dt 2m fying approximations made in its derivation, Eq. (77) shows very good agreement with the numerical results plotted in the So we are led to the interesting result that the orbit precesses right panel of Fig.5. As shown by Eqs. (74)-(76), Z is pro- about the direction of the spin vector S. This can be sim- portional to the ratio S/m, known as the test body’s “Møller ply understood from Fig.4: since S is nearly constant along radius” [46]; it is the minimum size an extended body can one orbit, the force (72) points in the positive ez direction for have in order to have finite spin without violating the domi- nearly half of the orbit, and in the opposite direction in the nant energy condition [22, 46]. Since v < 1, we see from Eq. other half; this “torques” the orbit, causing it to precess. The (76) that Z is always larger than such radius. effect is clear in the numerical results in the left panel of Fig. The force (72) originates also a precession of the orbital 5. This precession is, of course, not independent from the os- plane. Recalling that (to lowest order) L = mr × v, cillations studied above; in fact, it is the origin of the beating regime of Eq. (77), which may be seen as follows. Multiply- dL dv = mr × = r × F = −A(r)r × (v × S) , (78) ing the angular velocity Ω of rotation of the orbital plane by dt dt r, yields the “rotational velocity” of the orbit; this precisely 2 2 where we substituted dv/dt ≡ d x/dt from Eqs. (44)-(46) matches [under the same assumption ω  Ωs that leads to (noting that G×r = 0), and A(r) is given by Eq. (61). Using Eq. (80)] the initial slope of the function 2Z sin(Ωst/2) that the vector identity r × (v × S) = (r × v) × S + (S × r) × v, modulates (77): we have dL  1  ΩsZ ' Ωr . (81) = A(r) S × L − (S × r) × v . (79) dt m So, the increase in the amplitude of the oscillations in Fig.5 The first term is fixed along the orbit, and is already in a pre- (these rapid oscillations are the variation of z along each or- cession form. The second term must be averaged along the bit, notice) is the reflex of the orbital precession Ω. Now, such 14 orbital rotation does not go on forever in the same sense, be- Both Z and the angular velocity Ω of orbital precession, Eq. cause S itself undergoes the precession in Eq. (68), which (80), are proportional to the body’s Møller radius S/m. For a 2 2 means that after a time t = π/Ωs the direction of the spin Sun-like star (m = M , S ≈ 0.2m [50], S/m = 3×10 m), vector is reversed. Likewise Ω and hdL/dti are reversed (be- the effect is very small: the secular orbital precession (80) in- fore one full revolution about S is completed if Ω < Ωs, as is clines the orbit by about 1.6 m per lap, with peak amplitude usually the case), and this is why the amplitude in Eq. (77) is 2Z = 106 m (about 1400 times smaller that the Sun’s diam- 2 modulated by the geodetic spin precession Ωs. eter), and 2Ztoday ∼ 10 m. Larger or more massive bod- In Fig.5 numerical results are plotted for a test body with ies will typically have a larger Møller radius, thereby yielding 2 the Sun’s mass m = M and S = 0.5m , in a pseudo- more interesting numbers; but, on the other hand, for large m, isothermal DM halo typical of a satellite galaxy (correspond- the dynamical friction force FD, Eq. (82) (which is propor- ing to much larger DM densities than those typical of the tional to m2), becomes also important. From the known ob- Milky Way, which makes them more suitable to illustrate the jects moving in the Milky Way’s DM halo, those with largest effects described above). Such results are obtained by numeri- Møller radius are (due to their size and flattened shape) Milky cally solving the system of equations formed by md2x/dt2 = Way’s satellite galaxies. Consider first, for a comparison (still 12 mG + F + FD together with Eq. (68), with F as given by at r ' r ), a hypothetical satellite galaxy with diameter Eqs. (62)-(63), (59), (67), and G given by Eqs. (56), (57), ' 2.5 kpc, and assume it to be a “scale reduced” version of 12 (67). The term FD is the dynamical friction force the Milky Way (diameter 55 kpc, mass mMW = 10 M , an- S = 2.6 × 1031 m2   gular momentum MW ), rotating with the 2X 2 v v F = −40πρm2 erf(X) − √ e−X ,X = , same velocity. Since S ∝ mvrotR, this yields Ssat/SMW ∼ D 3 4 3 π v vcirc (2.5/55) , msat/mMW ∼ (2.5/55) , leading to a Møller ra- (82) dius Ssat/msat ∼ 0.02 pc. In this case the orbit inclines at which is here included. It follows from√ Eq. (8.6) of [41], or an initial rate of 4 × 1012 m per orbit (2 × 104 m per year), Eq. (3) of [47], by taking λ = 10 and 2σ = vcirc ≡ velocity and the peak amplitude, as predicted by Eqs. (74)-(77), (83) of the circular orbit at r, cf. [41]. (The impact of FD, in the would now be 2Z ' 60pc. Such large peak value however case of the motion in Fig.5, turns out to be unnoticeable.) is never reached, due to the damping action of FD; the nu- merical results shown in Fig.6 show that a peak of about 0.001 pc (i.e., about 10 times the radius of the solar system), 3. Particular examples in the Milky Way DM Halo is reached within about 2.2Gyr (a fifth of the age of the uni- verse), after which the orbit and its oscillations pronouncedly Pseudo-isothermal profile.—The DM density at the solar decay. As a concrete example in the Milky Way DM Halo, we 3 −21 −3 system position is about 0.01M /pc = 10 kg m [48, take the Large Magellanic Cloud (the largest satellite galaxy), 49]; the Sun’s distance from the center of the Milky Way is located at r = 48 kpc from the MW center. It has mass 10 r = 8 kpc. The core radius rc of the Milky Way DM halo is mLMC ≈ 10 M , diameter ≈ 4.3 kpc, and rotational ve- 4 −1 about 1 kpc. Assuming the pseudo-isothermal density profile locity vrot ≈ 9 × 10 m s [51], from which we estimate −3 in Eq. (66), this means that ρ0 = 0.7M pc . The velocity a Møller radius SLMC/mLMC ∼ 0.3 pc. We find a gradual v of the quasi-circular orbits is obtained from Eqs. (67), (75). inclination of the orbit of about ∼ 4 × 10−9 masyr−1 (or 4 −1 For a body orbiting at r = r , the peak amplitude in Eqs. 3 × 10 m yr ); this is far beyond the current observational (76)-(77) then reads accuracy, since the uncertainty in the LMC’s proper motion is presently much larger (∼ 10−2masyr−1 [51]). The peak S 2Z = 3.5 × 103 . (83) amplitude predicted by Eqs. (74)-(77), (83) is 2Z ≈ 1 kpc m (which, again, is not reached due to dynamical friction). Nu- The time to reach it (“beating” half period) is however very merical simulations (similar to those in Fig.6) show that an 14 4 10−3 pc 2.5 Gyr long: tpeak ≈ π/Ωs = 10 yr (10 × age of the universe); effective peak of about is reached within . 6 Power law profiles.— ρ(r) = this corresponds to ω/(2Ωs) = 10 laps around the center. For the models of the form Kr−γ Noting that the initial of slope of the function 2Z sin(Ωst/2) in Sec.IVB, substituting (65) in (75), (80), it follows that modulates Eq. (77) is ZΩs, the maximum amplitude ac- from Eqs. (76) and (85) that tually reached within the age of the universe (≡ t0) is  1/2 2S S 3 − γ γ/2−1   2Z ≈ = r , (87) Ωst0 mv m Kπ 2Ztoday ≡ 2Z sin ≈ ZΩst0 (84) 2 S 6πt0K 1−γ 2Ztoday ≈ Ωrt0 = r , (88) 3v2t S m 3 − γ ≈ Ωrt = 0 (85) 0 2r m where K is determined from the value of ρ(r ) which we −3 where in the second approximate equality we used (81), and assume, for all models, ρ(r ) ≈ 0.01M pc [42, 48, 49]. in the last equality we used (80), (75). Hence, for the setting above, 12 It is nearly the case for the Canis Major Dwarf, and nearly twice that value −4 S 2Z = 1.4 × 10 Z = 0.25 . (86) for the Sagittarius dwarf. today m 15

First 5 cycles

z-t plot z pc

0.0010H L

0.0005 z pc0.0005 0.0000 t Gyr 5000 1 2 3 4 H L -0.0005 -0.0005 H L 0 y pc -5000 -0.0010 0 -5000 H L x pc 5000

H L

FIG. 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 2.5kpc, and orbiting at a distance 8kpc (= Sun’s distance) from the Halo’s center. The peak amplitude predicted by Eqs. (74)-(77) is 2Z ≈ 60 pc; but it is never reached, due to the very strong dynamical friction. Still a peak of about 10−3 pc is reached after about 2.2 Gyr (' 1/5 the age of the universe).

For γ = 1, Ztoday ≈ 0.08S/m is approximately constant, tant from the plane than other bodies, by a distance of or- −1/2 13 and Z decreases with r as Z ∝ r ; for bodies orbit- der ∼ 4Ztoday/π. This effect might be observable. The 3 ing at r = r , one has Z ≈ 2.3 × 10 S/m. That is, the most precise map of the sky is expected to be given by the −11 peak/present time amplitudes are, respectively (at r = r ), Gaia mission [52], able to measure angles of about 2 × 10 somewhat larger/smaller than those for the pseudo-isothermal rads. Therefore, on test bodies whose distance d from Gaia −11 profile, Eqs. (83), (86). For 1 < γ < 2, it follows from Eqs. (i.e., from the Earth) is such that d . 4Ztoday/(π2 × 10 ), (88) that both Z and Ztoday decrease with r. The isother- the effect would be within the angular resolution. To be mal case, γ = 2, yields a Z = 1.6 × 103S/m approxi- concrete, consider a giant star like Antares; it has radius −1 3 mately independent of r, and Ztoday ∝ r . At r = r , Rant ∼ 10 R , mass mant ∼ 12M , surface rotational ve- −5 Ztoday = 0.15S/m; thus Z is slightly smaller and Ztoday locity vrot = 7×10 . For simplicity, assume it to be uniform slightly larger that in the pseudo-isothermal profile. However, and rotate rigidly, leading to a Møller radius Sant/mant = 7 contrary to the pseudo-isothermal case (where Ztoday reaches 2 × 10 m (five orders of magnitude larger than that of the a maximum ≈ 0.26S/m at r ≈ 1.5kpc), Ztoday increases Sun). Assuming the pseudo-isothermal profile, this yields a 7 steeply as one approaches the halo center, approaching the present time amplitude 2Ztoday ∼ 2S/m = 4 × 10 m. Gi- peak value Z [cf. Eq. (85)]. ants of this type, with spin axes nearly parallel to the galactic plane, should on average be farther from the plane than others, Inside the galactic disk.—The above are results taking into by about ∼ 3 × 107 m (∼ 10−5× Antares’s diameter). Con- account DM only; so they apply to orbits outside the galactic sidering the density value at r = r , their maximum allowed disk. Within the disk, the density of baryonic matter, in the distance from the Earth (so that the effect can be detected) vicinity of the Sun, is about ρ ≈ 0.1M pc−3 [41, 48], i.e. b is then d ≈ 0.04 kpc, which is not far from the order of one order of magnitude larger than that of DM. This leads to max magnitude of Antares’ actual distance (d = 0.17 kpc), and of an enlarged effect. The field produced by the disk is a com- other large stars. Thus, albeit small, the effect on such stars is plicated problem (see Sec.V below). The analysis of a sim- close to the angular resolution. The matter density (baryonic ple model in Sec.VB reveals however that, just for an order and DM) increases however as one approaches the galaxy cen- of magnitude estimate, the force caused by the disk can be ter; for stars along the line connecting the solar system to the taken as the corresponding Magnus force F , and its con- Mag center, the angle that the effect subtends on the GAIA space- tribution to the orbital precession as Ωb ∼ FMag/(vm) ∼ craft is θ ≈ Ztoday/d, with d = r − r. For DM models 4πρbS/m. Assuming, for DM, the pseudo-isothermal profile −γ of the type ρDM ∝ r with γ > 1, the angle θ increases (66), leads to 2Ztoday ≈ Ωrt0 ∼ 2S/m, cf. Eq. (85) [here Ω ≡ Ωb + ΩDM, with ΩDM given by Eqs. (80), (67)]. More importantly, the galactic disk might reveal a signa- ture of the orbital precession (80): BHs or stars with spin axes 13 Note that the time scale for formation and flattening of the galactic disk is nearly parallel to the galactic plane are, on average, more dis- much shorter than that of the orbital precession (2π/Ω). 16 with decreasing r (after initially decreasing, and bouncing), to geometrized units, and using MBH = M (MBH/M )], cf. Eq. (88). Considering the isothermal profile (γ = 2), and 11 taking into account DM only, θ enters GAIA’s resolution for " r # 20 FMag ˜ 6 −3/8 r 4pc. The Magnus signature on the galactic disk might ∼ SF(MBH) 1 − r˜ (90) . mkG k r˜ thus serve as a test for such models. Independently of such BH DM models, the baryonic matter is known to reach high den- where sities in the galaxy’s inner regions; using ρb(r) as given in 13 Eq. (2) of [53], we have that, for r 1pc, the baryonic matter   10 . −15 11/20 −7/10 MBH alone is sufficient for θ to enter GAIA’s resolution. F(MBH) ≡ 1.6 × 10 fEdd α . M

Let us now compare the magnitude of FMag with the spin- V. MAGNUS EFFECT IN ACCRETION DISKS orbit force FSO exerted on the “test” body due to its spin S (given by Eq. (97) below). Assume it to move, relative to the The gravitational Magnus effect due to DM is limited by central BH, with velocity ∼ v (i.e, of the same order of magni- its typically very low density. Accretion disks around BHs or tude of that relative to the matter in the disk, see footnote 14) 3 stars provide mediums with relatively much higher densities, so that FSO ∼ MBHvS/r . It follows that where the effect can be more significant, possibly within the 11 reach of near future observational accuracy. 3 " r # 20 FMag ρr 6 9/8 ∼ = F(MBH) 1 − r˜ . (91) FSO MBH r˜ A. Orders of magnitude for a realistic density profile Comparing with the magnitude of the spin-spin force 4 FSS ∼ SBHS/r [34], The standard model for relativistic thin disks is the Novikov-Thorne model [54], which generalizes the Shakura- 11 " # 20 Sunyaev [55] model to include relativistic corrections. Ac- F ρvr4 ρvr4 F(M ) r6 cording to such model the disk is divided into different re- Mag ∼ = = BH 1 − r˜13/8 , F S ˜ 2 ˜ r˜ gions, the outer and more extensive of them being well ap- SS BH SBHMBH SBH proximated by the Newtonian counterpart. The density of the (92) ˜ 2 later reads (in the equatorial plane) [55] where SBH ≡ SBH/MBH. Finally, let us compare the magnitude of FMag with that of 11 11/20 7 " r # 20 the “orbit-orbit” gravitomagnetic forces F = v ×H f  M  10 6 OO 1 1trans ρ = Edd α−7/10 1 − 1×106kg m−3 , in the binary; that is, the force exerted on the “test” body 15/8 r˜ MBH r˜ (dub it body 2) due to the gravitomagnetic field H1trans gen- (89) erated by the translational motion of the “source” (body 1), where r˜ ≡ r/MBH, MBH is the mass of the central BH, r˜in ≡ with respect to the binary center of mass frame. This is of rin/MBH, α the “viscosity parameter” and fEdd Eddington’s interest in this context for being an effect that has already ratio for mass accretion (e.g. [45]). The density (89) leads to a been detected to very high accuracy in binaries (relative un- Magnus force FMag = 4πρS × v (cf. Eq. (41)) of interesting certainty of about 10−3, in observations of the Hulse-Taylor magnitude, compared with other relevant forces. pulsar [56]). It is moreover typically larger than its spin-spin Comparing with the Newtonian gravitational force mGBH and spin-orbit counterparts. The translational gravitomagnetic exerted on the body by the central BH, we have field is given by Eq. (B6)(H1 = H1trans therein); so FOO ∼ 2 2 2 2 FMag/(mkGBHk) ∼ r ρvS/(mMBH). Different estimates mMBHv1v2/r , which we may take as FOO ∼ mMBHv /r can be made. Let us consider the case of a binary of BHs with (see footnote 14). Considering moreover m ∼ MBH, we have similar masses m ∼ MBH; in this case FMag/(mkGBHk) ∼ 2 2 11 Sr˜ ρv, where S˜ ≡ S/m (for a fast spinning BH S˜ 1; for ˜ 2 " r # 20 . FMag ρSr 6 5/8 extended bodies it could be much larger). Now we need an es- ∼ = S˜F(MBH) 1 − r˜ , (93) F v r˜ timate for v (the velocity of the “test” body with respect to the OO disk of the “source” BH); it can be taken has having the mag- p 14 p −1/2 −3 where, again, we used v ∼ MBH/r. nitude v ∼ MBH/r =r ˜ . Then [converting kg m All the four ratios (90)-(93) increase with MBH, and depend also on r; the ratio to the Newtonian force decreases with r, whereas all the others increase with r. Choosing, from the 14 Except for the case that the “test” body is much smaller and as such can range of values in [45], α ∼ 0.01, fEdd ∼ 0.2, and consid- 9 be in a circular orbit corotating with the source’s disk, and the latter is ering supermassive BHs with MBH ∼ 10 M , FMag starts moreover mostly gravitationally driven (i.e., not very affected by hydro- being larger than F for r 81M . Assuming the cen- dynamics), the test body will not comove with the matter on the disk. In SO & BH general the orbit will be eccentric relative to the center of the disk; so it tral black hole to be fast spinning (unfavorable case) with e.g. ˜ will have a velocity v relative to the matter in the disk typically within the SBH ∼ 0.1, we have that FMag & FSS for r & 17MBH. As- same order of magnitude of its velocity relative to the central BH. It is also suming moreover that the “test” black hole is also fast spin- so for counterrotating, or for unbound orbits. ˜ ning (favorable scenario) with e.g. S = 0.5, FMag & FOO for 17

GW frequencies for FMag FSOt1 GW frequencies for FMag FSSt1

fGW Hz fGW Hz 2. ´ 10-7   H L 3. ´ 10-6 H L

´ -6 1.5 ´ 10-7 2.5 10

2. ´ 10-6

1. ´ 10-7 1.5 ´ 10-6

1. ´ 10-6 5. ´ 10-8 5. ´ 10-7

MBH MBH Log10 Log10 7 8 9 10 MŸ 2 4 6 8 MŸ

FIG. 7. Range of frequencies of the emitted gravitational radiation for binary systems in which the Magnus force FMag is larger than FSO and 2 FSS, for α = 0.01, fEdd = 0.2 [see Eq. (89)] and SBH = 0.1MBH. The solid curves represent, as functions of the central BH’s mass (MBH), the frequencies for which FMag ∼ FSO and FMag ∼ FSS. In the shadowed regions it holds, respectively, FMag & FSO and FMag & FSS. 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 −9 −7 (10 Hz . fGW . 10 Hz). The frequency for which FMag ∼ FSS lies moreover just below (or within, depending on α, fEdd, and SBH) the LISA band.

3 5 r & 6×10 MBH. Even comparing with the Newtonian force, LISA’s band (for fGW ∼ 10 Hz, FMag reaches a maximum 4 7.5 the magnitude of FMag is interesting: for r . 2 × 10 MBH, FMag ∼ 0.2FSS for M ∼ 10 M ). Moreover, mild de- −4 we have FMag/(mkGk) & 10 ; this is the same order of viations in the disk parameters from the conservative choice magnitude of the leading 1PN corrections (which are of frac- above (e.g., α ∼ 0.005, fEdd ∼ 0.8 [55]), or simply consider- 2 −4 tional order  ∼ MBH/r ∼ 10 , see Sec. III A). ing a central black hole with smaller spin (e.g. S˜BH ∼ 0.03) These comparisons are relevant for binary systems, due are sufficient to make FMag of the same order of magnitude as to the impact of spin effects (both spin-orbit and spin-spin) FSS within such band. Since LISA is expected to be sensitive in the emitted gravitational radiation, which is expected to to spin-spin effects [58, 63], this suggests that there might be be observed in the near future [57–63]. There are different prospects of detecting as well the Magnus effect. The lowest existing/proposed detectors, operating at different frequen- frequency planned detectors are pulsar timing arrays (PTAs, −9 −7 cies (see e.g. [64]). The frequency fGW of the emitted see e.g. [64, 66–70]), of band 10 Hz . fGW . 10 Hz, gravitational radiation is approximately related to the Kepler and which in the future are expected to detect waves from in- orbital angular velocity ω by fGW = ω/π. Since ω ≈ dividual SMBH binary sources [66, 69]. Within this band, 1/2 −3/2 −1 −3/2 Fig.7 shows that FMag can be the leading spin effect. MBH r = MBHr˜ , one may eliminate either MBH or r˜ from Eqs. (89)-(93) above. Eliminating r˜ [by substi- −2/3 tuting r˜ → (ωMBH) ] one obtains ρ(MBH, ω), and all B. Miyamoto-Nagai disks the ratios above, as functions of MBH and ω. We are espe- cially interested in the ratios to the spin-spin and spin-orbit forces; they both decrease with ω; hence, solving for ω the Although to compute the Magnus force (41) all one needs equalities FMag/FSO = 1 and FMag/FSS = 1 yields, as to know is the disk’s local density ρ and the relative velocity a function of MBH, the maximum orbital angular velocity v of the test body, in order to determine the body’s motion, ωmax(MBH) [and thus the maximum gravitational wave fre- one needs the total spin-curvature force F = FMag + FWeyl; quency fGWmax(MBH)] allowed in order to have FMag larger that requires knowledge of the gravitational field produced by than FSS or FSO. These curves are plotted in Fig.7, for the sources (disk plus central BH), since FWeyl depends on it. α ∼ 0.01, fEdd ∼ 0.2, and S˜BH = 0.1. They tell us that This is however a complicated problem. There is an extensive the Magnus force is more important at low frequencies, and literature on the fields of disks, from exact solutions [71–78]), for supermassive black holes. Within the band of ground- to perturbative [79, 80], PN [81], and Newtonian [82–85] so- 3 based detectors such as LIGO (10Hz . fGW . 10 Hz) it lutions. Even though the formalism in Sec. III (by being exact) is much smaller than FSO and FSS. Within the band of the could in principle be used to treat the exact problem, most ex- −5 spacebased LISA [65](10 Hz . fGW . 1Hz), we have act solutions in the literature are not practical or suitable for that FMag  FSO. Fixing the frequency at the most favorable our problem, since they are either nonanalytical [78, 80], or −5 −5 −1 value fGW = 10 Hz (i.e., fixing ω ∼ 3 × 10 s ), and describe the field only outside the disk [71], or are not realis- 2 plotting the corresponding ratio FMag/FSO = ρ(MBH)/ω tic models of astrophysical systems [71–77]. In this context −2 (not shown in Fig.7), one sees that FMag/FSO . 10 . the Newtonian solutions provide the more treatable examples On the other hand, Fig.7 shows also that the frequency for for us. According to Eq. (43), to compute FWeyl (and thus which FMag ∼ FSS lies just below the LISA band. In fact, F ) to leading PN order, only the Newtonian and gravitomag- the magnitude of FMag is already comparable to FSS within netic (H) fields of the source are needed. By considering 18 a Newtonian field, one is ignoring the gravitomagnetic fields Quasi-circular orbits (frame-dragging) produced by the rotation of the disk (and of the central BH); this would be accurate if either the disk We shall now consider the effect of the spin-curvature force was static, or composed of counterrotating streams of mat- (FMag +FWeyl) produced by the disk on test bodies on (quasi- ter, or the test body moves considerably faster relative to the ) equatorial circular orbits around the central object. This de- disk than the disk’s average rotational velocity (so that one mands the central object to be much more massive than the vG  H can have ). One might argue that none of these is a test body, MBH  m. We also consider the test body to be a realistic assumption — the disk is (at least in part) gravitation- BH, in order to preclude surface effects (such as an ordinary ally driven, so it must rotate, with a velocity of the same order Magnus effect), and ensure that the motion is gravitationally of magnitude of the velocity of orbiting test bodies. But still driven. In the equatorial plane % = r, thus the disk’s density it is no less realistic than most exact solutions — which are ρ = −∇2U/4π, that follows from (94), is precisely static [71–76] and/or composed of streams of matter " # flowing in opposite directions [72–77]. More realistic solu- M 3r2 a ρ = disk 3 − + tions, where H is taken into account, are found in PN theory; 3/2 2 h 2i r2 + (a + b) b the field is however very complicated already at 1PN, and not 4π r2 + (a + b) obtained analytically (e.g. [81]). So, here, just to illustrate (98) the basic features of the spin-curvature force produced by the (cf. Eq. (5) of [82]). As in Sec.IVB, there are two notable disk, we consider one of the simplest Newtonian 3D models,15 cases to consider. z the Miyamoto-Nagai disks [82–84], also called the “inflated” Spin parallel to the symmetry axis (S = S ez). Equa- Kusmin model [84]. The Newtonian potential is tion (68) tells us that, in this case, the components of S are M M constant. The Magnus, Weyl and total force due to the disk, U(%, z) = disk + BH ≡ U + U q √ r disk BH Fdisk ≡ FMag + FWeyldisk, are %2 + ( z2 + b2 + a)2 (94) (S · L) 2 2 2 FMag = −4πρ er , where % = x + y , Mdisk is the disk’s total mass, and b and mr " 2 # a are constants with dimensions of length. The ratio b/a is a (S · L)Mdisk 3r a FWeyldisk = + er, measure of the flatness of the disk [82]. The Laplace equation h i3/2 2 2 2 2 2 2 r + (a + b) b ∇ U = ∇ Udisk = −4πρ yields the disk’s density, Eq. (5) of mr r + (a + b) [82]. The Weyl force is obtained from Eq. (43), which reads " 2 # i (i j),m k 3(S · L)Mdisk 2r here FWeyl = 2 kmG v Sj, where G is, to the accuracy Fdisk = − 1 er, h i3/2 2 2 needed for this expression, the sum of the Newtonian fields mr r2 + (a + b)2 r + (a + b) produced by the disk and the central BH, G ' ∇Udisk + ∇UBH. It can be split into the Weyl forces due to the disk and with ρ as given by Eq. (98). So the Magnus and Weyl forces due to the central BH, which read explicitly, in the equatorial are both radial, but have opposite directions. This resem- plane bles case2 of the slab model of Sec. III B, but now the re- 2 2 FWeyl = FWeyldisk + FBH (95) sulting force Fdisk is not zero. It has, for r < (a + b) , the same direction of the Magnus force, and opposite direc- M h tion for r > (a + b)2; in any case it is of qualitative differ- F j = disk (S × v)j| − j Sivk| Weyldisk 3 j6=z ik i6=z ent nature from F or F in that it lacks the important [r2 + (a + b)2] 2 Mag Weyl # a/b term (that can be very large for highly flattened disks). 3rj(S × v) · r + 3(r · S)(v × r)j Since the forces are radial, the orbital effect amounts to a − 2 r2 + (a + b)2 change in the gravitational attraction — for r > (a + b) , Fdisk is repulsive (attractive) when S is parallel (antiparal- (a + b)Mdisk  j z z jkz  lel) to the orbital angular momentum L; and the other way + 3 δz(S × v) + S  vk b [r2 + (a + b)2] 2 around for r < (a + b)2. Its relative magnitude compared (96) to the Newtonian gravitational force produced by the disk is Fdisk/(mG) ∼ vS/(rm). 3M  2r[(v × r) · S] (v · r)S × r  The effect is important in connection to the measurements F = − BH v × S + + BH r3 r2 r2 of the gravitation radiation emitted by binary systems, namely (97) in mass estimates. These are affected [63] by the spin-orbit (Notice that FBH is the well-known expression for the spin- (FSO ≡ FBH) and spin-spin (FSS) forces. FSO ≡ FBH is orbit part of the spin-curvature force exerted by a BH on a given by Eq. (97), and like Fdisk it is parallel to the symmetry spinning body, e.g. Eq. (44) of [34]). axis; FSS [not taken into account in Eq. (97)] is given by e.g. Eq. (24) of [34] (it is parallel to the symmetry axis if the spin of the central BH is along ez). As we have seen in 15 There are also 2D models of thin-disks such as those by Kusmin-Toomre Sec.VA using a realistic density profile, the Magnus force [82–84]; they are however unsuitable for studying the spin-curvature force, FMag is generically larger than both FSO and FSS in systems for having singular tidal tensors along the disk. emitting GW’s within the band of pulsar timing arrays, and 19 is comparable to FSS in the lowest part of LISA’s band. the error in approximating the peak of a sinusoidal function In the latter, in particular, the impact of FSS in the mass by a first order Taylor expansion at t = 0). For the present measurement accuracy is significant [63]; hence that of FMag problem, these estimates are validated by numerical results (and Fdisk) might likewise be. assuming the force expressions (95)-(97). It should be stressed that Eq. (80), with A(r) as given by Spin parallel to the orbital plane (Sz = 0). In this case Eqs. (101), assumes the orbit to lie near the equator, since the force (68)-(70) tell us that the spin vector S precesses, but remains expressions (95)-(97), (99)-(101), are for the equatorial plane. in the plane. The Magnus, Weyl (FWeyldisk + FBH = FWeyl), The precession Ω will however gradually incline the orbit; as and total spin-curvature force, F = FMag + FWeyl, are, from the inclination increases, the body will be in contact with the Eqs. (41), (95)-(98), disk’s higher density regions for shorter periods of time, so Eq. (80) will gradually become a worse approximation (it is " 2 # Mdisk 3r a a peak value). From relations (103) we see that, when Ω  FWeyldisk = 2 + S × v, [r2 + (a + b)2]3/2 r2 + (a + b) b Ωs, the peak inclination angle is small, so the orbit remains, (99) on the whole, close to the equatorial plane. Otherwise, the approximation remains acceptable after several orbits if Ω  MBH 3/2 5/2 FMag = 4πρS × v, FBH = 3 3 S × v, (100) ω. Noting that Ωs = 3Gv/2 ≈ 3MBH /(2r ) and ω = r p 1/2 −3/2 2 G/r = M r , and since S < m (for BHs), Mdisk < (3 + 2a/b)Mdisk MBH BH F = A(r)S × v; A(r) ≡ + 3 , MBH, r > 2MBH, and we are assuming m  MBH, we [r2 + (a + b)2]3/2 r3 have that, for not too large a/b, both Ω  Ωs and Ω  ω (101) are satisfied. The computation of the precise precession for an with ρ given by (98). It is remarkable that all the compo- arbitrary inclination can be done using the general expression nents of the force are parallel. In particular, for large a/b for the force as given in Eq. (43), with G = ∇U given by Eq. (94). (thin disks), FMag and FWeyldisk are qualitatively similar. This resembles case1 of the slab model in Sec. III B. Since An important conclusion that can directly be extrapolated to more realistic models, is that the orbital precession caused S × v = vS cos[(ω − Ωs)t]ez, cf. Eq. (71), the force (101), similarly to its counterpart in the DM halo of Sec.IV, causes by the disk has the order of magnitude Ω ∼ FMag/(vm) ∼ ρS/m, which might possibly be measurable in a not too dis- the spinning body to bob up and down (in the ez direction) along the orbit. It leads also to a secular orbital precession, tant future: the secular precession of the orbital plane of bi- which is again of the form (80), nary systems affects the principal directions and waveforms of the emitted gravitational radiation [57–59, 61, 62, 86, 87]. A(r) In the absence of disk (thus of Magnus force), such preces- Ω = S , (102) 2m sion reduces to that caused by the spin-orbit and spin-spin couplings. Both are expected to be detected in gravitational where A(r) is now given by Eq. (101), leading to a much wave measurements in the near future [57–59, 61, 62]. The larger precession. Unfortunately here we are unable to derive former is is the leading one, and has magnitude of the form an analytical expression for the oscillations along z caused by 3 ΩSO ∝ S/r [59, 62, 86]; in particular, for the precession Ω in the likes of Eq. (77) of Sec.IVB. This because the first 3 z z caused by the force (97), ΩSO ∼ (MBH/m)S/r , cf. Eqs. order Taylor expansion G ' G ,z|z=0z made therein is here z (101)-(102). Comparing with the magnitude of the Magnus a bad approximation to the true value of G when the body is precession, outside the equatorial plane, due to the rapidly varying deriva- 3 tive Gz,z at the equatorial plane. This causes Eq. (77) to fail, Ω ρr ρ FMag which is made clear by numerical simulations. Still one can ∼ = 2 ∼ , (104) ΩSO MBH ω FSO devise rough, but robust estimates of the peak orbital inclina- tion and oscillation amplitude. As explained in Sec.IVB and cf. Eq. (91). The the orbital precession originated by caption of Fig.5, since the orbital precession Ω is propor- the spin-spin couplings has approximate magnitude ΩSS ∼ 4 tional to S, it is constrained by the spin precession Ωs (Eq. SBHS/(mvr ) (cf. e.g. Eq. (11.a) of [86]); hence (68)), because after a time interval t = π/Ωs ≡ tpeak the Ω ρvr4 F direction of S, and thus of Ω and hdL/dti, become inverted ∼ ∼ Mag , (105) relative to the initial ones, so the inclination stops increasing ΩSS SBH FSS and starts decreasing. Approximating the inclination angle α cf. Eq. (92). So, the ratios amount to those of the corre- by Ωt, one may estimate the peak inclination angle and oscil- sponding forces. This means that what is said in Sec.VA lation amplitude by and Fig.7, concerning the relative orders of magnitude of πΩ πrΩ the forces, applies here to the precessions. Namely, in SMBH αpeak ∼ Ωtpeak = ; zpeak ∼ rαpeak = . Ωs Ωs binaries emitting low frequency GWs, such as those within (103) the pulsar timing arrays band, the plots in Fig.7 show that Testing first the validity of these estimates in the problem of the Magnus precession Ω can be the leading spin-induced or- Sec.IVB, we notice that therein zpeak differs from the pre- bital precession, larger than both ΩSS and ΩSO. As an exam- 10 cise result 2Z ' 2Ωr/Ωs by a factor π/2 (corresponding to ple, taking MBH ∼ 10 M , we have that Ω/ΩSO & 1 for 20

−7 −1 fGW / 2 × 10 Hz(≈ 6yr ), cf. Fig.7, the approximate where equalities corresponding to r˜ ∼ 10, ρ ∼ 6 × 10−3kg m−3 (U 0)2 [cf. Eqs. (91) and (89)]. Taking a “test” companion of mass A(t, r, θ) ≡ k +a ˙(t)2 − a(t)¨a(t) . 9 2 a2(t) m ∼ 0.1MBH = 10 M and spin S = 0.5m , the Magnus −9 precession for such setting, Ω ∼ 0.5ρm = 10 Hz, amounts It thus reduces to the current term (responsible for the Magnus to an orbital inclination of ∼ 0.6 degrees per cycle, reach- γ σ o effect): Hαβ = H[αβ] = −4παβσγ U J , cf. Eq. (30). ing a peak angle αpeak ∼ 2 [cf. Eq. (103)] after a time For the observers at rest (v = 0) in the coordinate system interval tpeak = π/Ωs = 1yr (≈ 3.4 cycles). If (as above) 2 of (106), the gravitomagnetic tidal tensor vanishes, Hαβ = the central black hole has spin SBH = 0.1MBH, we have 0. Therefore, the spin-curvature force on a spinning body at Ω ∼ 32ΩSS. Considering instead, for the same binary, a fre- α βα −8 rest vanishes: F = −H Sβ = 0. This is the expected quency one order of magnitude smaller, fGW ∼ 2 × 10 Hz, −4 −3 result: a body at rest in the coordinates of (106) is well known corresponding to r˜ ∼ 47, ρ = 6 × 10 kg m , the Magnus to be comoving with the background fluid, so relative to it precession Ω ∼ 10−10Hz becomes the dominant spin-induced 2 the spatial mass/energy current J is zero, implying that the precession: Ω ∼ 10ΩSO ∼ 7 × 10 ΩSS, amounting to an or- o Magnus force (34) vanishes. bital inclination of ∼ 0.6 per cycle, reaching a peak angle α 0 i o Consider now an observer of 4-velocity U = U (1, v ), αpeak ∼ 9 in a time interval tpeak = π/Ωs ≈ 50yr (≈ 15.5 moving with respect to the coordinate system of (106); i.e, cycles). with a “peculiar” velocity v 6= 0. Such an observer measures Moreover, LISA’s band is just above the frequency for a nonzero antisymmetric gravitomagnetic tidal tensor (108); which Ω meets the magnitude of ΩSS (according to the den- this means that a spinning body moving with velocity v suf- sity profile in Eq. (89), for the conservative choice of pa- α βα fers a spin-curvature force F = −H Sβ, whose compo- rameters made above); Ω being already within the same or- nents read, in terms of the metric parameters, der of magnitude as ΩSS in the lower part of LISA’s band −5 0 (fGW ∼ 10 Hz). F = 0; F = A(t, r, θ)S × v , (109)

i i j k where (v×S) ≡  jk0v S , and the coordinate system is that VI. MAGNUS EFFECT IN COSMOLOGY: THE FLRW of (106). On the other hand, the energy-momentum tensor METRIC corresponding to the metric (106) is that of a perfect fluid, T αβ = (ρ+p)uαuβ +pgαβ, where uα is the fluid’s 4-velocity ; thus J α = −T αβU = γ(ρ + p)uα − pU α, where γ ≡ A setup of especial interest to consider is the gravitational β −u U α. From Eq. (107), we have that Magnus effect for a spinning body moving through an ho- α mogeneous medium (or “cloud”) representing the large scale α α β σ γ F = 4πγ(ρ + p) βσγ u S U ; (110) matter distribution of our universe. As discussed in Sec. III B 3, this is problematic in the framework of a PN approxi- since the fluid is at rest in the coordinate system of (106), we α α 0 16 mation, and of an analogy with electromagnetism, in partic- have that u = δ0 and γ = U , and therefore ular when the cloud is assumed infinite (in all directions), 0 2 α F = −4π(ρ + p)(U ) v × S . (111) due to the indeterminacy of FWeyl. General relativity how- ever admits a well known exact solution corresponding to an This equation leads to an important conclusion: in the gen- homogeneous isotropic universe (finite or not) — the FLRW eral case that ρ + p 6= 0, a spinning body arbitrarily moving spacetime, believed to represent the large scale structure of in the FLRW metric suffers a net force in the direction of the our universe. The metric is Magnus effect; such force is the only force that acts on the  dr2  body, deviating it from geodesic motion. If the weak energy ds2 = −dt2 + a2(t) + r2dθ2 + r2 sin2 θdφ2 . condition holds, ρ + p ≥ 0 (cf. e.g. Eq. (9.2.19) of Ref. 1 − kr2 F = F S × v similarly (106) [89]), we have that ( Mag) is parallel to , to the Magnus effect of fluid dynamics 17 It is well known that this is a conformally flat metric, that is, . This is the case for its Weyl tensor vanishes: Cαβγδ = 0. This makes this met- ric remarkable in this context: the Weyl force vanishes, and therefore, the total spin-curvature force exerted on a spinning 16 One may check that Eq. (109) indeed equals (111) using the Friedman body (if any) reduces to the Magnus force, Eq. (32), equations (e.g. Eqs. (5.11)-(5.12) of [88]) a˙ (t)2 + k 8πρ + Λ a¨(t) 4π Λ α α α = ; − = (ρ + 3p) − FWeyl = 0 ⇒ F = FMag . (107) a(t)2 3 a(t) 3 3

Let U α = U 0(1, vi), where U 0 = dt/dτ and vi = U i/U 0 = 17 We note that this result is contrary to that estimated in [8]. Equation (111) dxi/dt, be the 4-velocity of some arbitrary observer. The is, supposedly, the exact relativistic solution for the problem addressed in [8]: the force exerted on a spinning body moving in a medium (“field of gravitomagnetic tidal tensor it measures has, as only nonva- stars”) of uniform density ρ, representing the large scale stellar distribution nishing components, of the universe. For the accuracy at hand in [8], Eq. (111) yields F = −4πρv × S. The force estimated in [8] however does not agree with this k result, having even the opposite direction (i.e., it is anti-Magnus). Hij = H[ij] = ijk0v A(t, r, θ) (108) 21 ordinary matter, radiation, or DM. In the case ρ = −p, corre- to its spin axis, and has the same direction as the Magnus ef- sponding to cosmological constant/dark energy, the Magnus fect of fluid dynamics. The effect was seen moreover to have a force vanishes, F = 0. This can also be equivalently seen close analog in electromagnetism; namely in the force exerted from the fact that the Ricci tensor for a cosmological constant in a magnetic dipole inside a current slab. Such setting, and is Rαβ = Λgαβ, which, via Eqs. (27)-(28) (with Cαβγδ = 0) its gravitational counterpart, provided useful toy models for implies Hαβ = 0 for all observers. Or from the fact that the ef- the understanding of the contrast between the two parts of the fective energy-momentum tensor of a cosmological constant spin-curvature force: the Magnus force, which depends only is T αβ = pgαβ, which does not lead to any spatial mass- on the body’s angular momentum and on the mass-density energy currents with respect to any observer: J α = −pU α, current of the medium relative to it, and the dependence of α β α so (see Eq. (10)) h βJ = 0 for all U . Other dark energy the Weyl force on the details and boundary conditions of the models have been proposed however, for which ρ 6= −p (e.g. system. This dependence shows clearly in the astrophysical [90–93]), and that, as such, would generate a Magnus force. systems studied, and means also that some problems tried to Candidates even exist for which ρ < −p [93] (violating the be addressed in earlier literature were not well posed. weak energy condition), leading to an anti-Magnus force. Gravitational Magnus effects could have interesting signa- We thus come to another interesting conclusion: the grav- tures in cosmology or in astrophysics. They are shown to lead itational Magnus force on spinning celestial bodies acts as a to secular orbital precessions that might be detectable by fu- probe for the matter/energy content of the universe, in partic- ture astrometric or gravitational-wave observations. These ef- ular, for the ratio ρ/p, and for the different dark energy can- fects are considered here in three astrophysical settings: DM didates. The bodies that should be more affected are rotating halos, BH accretion disks, and the FLRW metric. In DM ha- galaxies (which one can treat as extended bodies) with large los, due to their low density, the effects are typically small, peculiar velocities v. The effect is any case very small, given being more noticeable for bodies with large “Møller radius” the constrains in place: WMAP results [94] show our universe S/m, yielding a further possible test for the existence of DM to be nearly flat, implying an average density close to the crit- and its density profile (in addition e.g. to the dynamical ical value ρ ≈ 10−26kg m−3; assuming an equation of state friction effect proposed in [47]). In accretion disks, due to of the form p = −wρ, the parameter w is constrained from their high density, the orbital precession caused by the Mag- observations to be within −1.2 . w < −1/3 (e.g. [25, 93]). nus force is more important; it can be comparable, or larger, Taking w ∼ −0.8, considering a galaxy with the same diam- than the spin-spin and spin-orbit precessions, and, in the fu- eter 2R ≈ 55kpc and Møller radius S/m as the Milky way, ture, possibly detectable in the gravitational radiation emitted and moving with a peculiar velocity v ∼ 103km s−1 (a rea- by binary systems with a disk. In the FLRW spacetime (de- sonably high value [95]), it would take about 104× age of scribing the standard cosmological model), is is shown that a the universe in order for the deflection caused by the Magnus Magnus force acts on any spinning body moving with respect force, ∆x ∼ (1/2)∆t2F/m ∼ 2π∆t2ρvS/m, be of order the to the background fluid, it is the only covariant force acting galaxy’s size. on the body (deviating it from geodesic motion), and has the Finally, we note that a reciprocal force Fbody,cloud, in the same direction of its fluid dynamics counterpart. It should af- likes of that computed in Secs. III B-IV, cannot be computed fect primarily galaxies with large peculiar velocities v. All here; such integrals assume that everywhere the metric can be forms of matter/energy give rise to such Magnus force except taken as nearly flat, so that vectors at different points can be for dark energy if modeled with a cosmological constant, so added. The metric (106), however, is not asymptotically flat it acts as a probe for the nature of the energy content of the (grr diverging at infinity), so any such integrals are not valid universe. mathematical operations here. This is the general situation in the exact theory: Fcloud,body ≡ F (the spin-curvature force) is a physical force always well defined, whereas its reciprocal, ACKNOWLEDGMENTS Fbody,cloud, is not. We thank J. Natário for enlightening discussions, and C. Moore and D. Gerosa for correspondence and very useful sug- VII. CONCLUSIONS gestions. We also thank the anonymous Referees for valu- able remarks and suggestions that helped us improve this In the wake of earlier works [6,7] where a gravitational work. V.C. acknowledges financial support provided under analogue to the Magnus effect of fluid dynamics has been sug- the European Union’s H2020 ERC Consolidator Grant “Mat- gested, we investigated its existence in the rigorous equations ter and strong-field gravity: New frontiers in Einstein’s the- of motion for spinning bodies in general relativity (Mathisson- ory” Grant Agreement No. MaGRaTh–646597. Research at Papapetrou equations). We have seen that not only the effect Perimeter Institute is supported by the Government of Canada takes place, as it is a fundamental part of the spin-curvature through Industry Canada and by the Province of Ontario force. Indeed, as made manifest by writing it in tidal tensor through the Ministry of Economic Development & Innova- form, such force can be exactly split into two parts: one due to tion. L.F.C. is funded by FCT/Portugal through Grant No. α the magnetic part of the Weyl tensor (the Weyl force FWeyl), SFRH/BDP/85664/2012. This project has received funding α plus the Magnus force (FMag), which arises whenever, relative from the European Union’s Horizon 2020 research and inno- to the body, there is a spatial mass/energy current nonparallel vation programme under the Marie Sklodowska-Curie Grant 22

Agreement No. 690904, and from FCT/Portugal through only surviving component is along ez, and that, by symmetry, the project UID/MAT/04459/2013. The authors would like one needs only to integrate over the octant x > R, y > R, to acknowledge networking support by the COST Action z > R, multiplying then the result by a factor of 8. The slab CA16104. orthogonal to the y axis corresponds to setting zm = ∞, ym = h/2; taking afterwards the limit h → ∞ amounts to

Appendix A: Infinite clouds and Fubini’s theorem 3 y=∞ n z=∞ h y Bdipd x = 8|y=R |z=R − arctan ˆr>R z #) In Sec.IIB we considered two different ways of obtaining Ry 4 an infinite cloud in all directions (a slab orthogonal to the y + arctan µez = πµez . (A1) p 2 2 2 3 axis delimited by −h/2 < y < h/2, and a slab orthogonal z y + z + R to the z axis delimited by −h/2 < z < h/2, in the limit The slab orthogonal to the z axis corresponds to ym = ∞, h → ∞), and seen that the force Fdip,cloud exerted on them by the magnetic dipole is different in each case. We consider zm = h/2; taking afterwards the limit h → ∞ amounts to here yet another route: an infinite cloud obtained by taking n h y the limit R → ∞ of a sphere of radius R. From Eq. (20), B d3x = 8|z=∞ |y=∞ − arctan ˆ dip z=R y=R which holds for any sphere containing the dipole (finite or in- r>R z #) finite), we have then Fdip,cloud = −8πµjey/3; which is yet Ry 8π + arctan µez = − µez . (A2) another different result, comparing to (23), and to the force zpy2 + z2 + R2 3 exerted on the slab orthogonal to z (Fdip,cloud = 0). These inconsistencies stem from a fundamental a mathematical prin- The integrals (A1) and (A2) differ only in the order of the ciple, embodied in Fubini’s theorem [20, 21] (in turn related integrations (or of the infinite limits) over the y and z coor- with Riemann’s series theorem, e.g. [36]); namely, that the dinates; yet the outcome is very different. This is a conse- multiple integral of a function which is not absolutely conver- quence of Fubini’s theorem [20, 21]: the double integral of a gent (i.e., the integral of the absolute value of the integrand function which is not absolutely convergent is not, in general, does not converge) depends on the way the integration is per- well defined; when written as a iterative integral, the result formed. This is the case of the integrals mentioned above. may depend on the order of integration. Since the problem of Take e.g. the “spherical” infinite cloud; we have considering an infinite cloud by taking initially a slab of width h y z π 2π R |1 − 3 cos2 θ| either along or along (and taking afterwards the limit |Bz |d3x =µ h → ∞), boils down to the order of the iterations in the mul- ˆ dip ˆ ˆ ˆ r r0R R R R bodies, and then naively applying a Newtonian-like action-  Ry  y=ym z=zm x=∞ reaction principle. This is problematic however. Although for = −8|y=R |z=R |x=R arctan µez zr the toy model, stationary settings of Sec. III B, we have shown " # that the force Fbody,cloud exerted by the body on the cloud in- y=ym z=zm Ry y = 8| | arctan − arctan µez deed equals minus its reciprocal F ≡ F (the force y=R z=R p 2 2 2 z cloud,body z y + z + R exerted by the cloud on the body), this is not true in general dynamics where y and z denote the upper integration bounds for the : the gravitomagnetic interactions, similarly to their m m do not obey respective coordinates. In the first equality we noticed that the magnetic counterparts, an action-reaction law of the type of FA,B = −FB,A. For instance, the force exerted by the spinning body on an individual particle of the cloud does not (contrary to the belief in some literature) equal mi- 18 The lower bound R in the integrals actually amounts to leave a cube of side nus the force exerted by the particle on the spinning body. 2R outside the integral, not a sphere of radius R; that does not however This is a leading order effect, which is a consequence of the have any effect on the outcome, in the limit h → ∞. interchange between field momentum and the “mechanical” 23

F other body, divided by the mass: a = (q2/M)r /r3 . The a) b) dip,1 a 12 12 v electric force exerted by particle 1 on particle 2 is then 2 1 v 2 r 2 q r12 v 21 1 FEL1,2 = qE1 = q (1 + ϕ1 − ) 3 1 2 2Mr12 r F 12 1,2 µ Magnetic F dipole and its reciprocal, the force FEL2,1 = qE2 exerted by particle 2,1 2 on particle 1, is r r 1 2 F 2 2 x 1,dip 2 q r21 2 q r12 FEL2,1 = q (1+ϕ2− ) 3 = −q (1+ϕ2− ) 3 2Mr21 r21 2Mr12 r12 FIG. 8. Two situations where action-reaction law is not obeyed in where we noted that r12 ≡ r1 −r2 = −r21. Thus, the electric electrodynamics: a) two orthogonality moving point charges (Feyn- forces are of opposite direction and of nearly equal magnitude: man paradox); b) interaction of a magnetic dipole with a particle of F ≈ −F (they slightly differ because ϕ 6= ϕ ). the cloud. In a), the electric forces each particle exerts on the other EL1,2 EL2,1 1 2 are nearly opposite, but particle 2 exerts a nonvanishing magnetic The same however does not apply to the magnetic forces: par- force qv × B on particle 1, whereas particle 1 does not exert any ticle 1 exerts no magnetic force on particle 2, FM1,2 = 0, 1 2 3 magnetic force on particle 2. In b), the force exerted by the dipole since, at the site of particle 2, B1 = qv1 × r12/r12 = 0, (particle 2) on the moving particle 1 is only minus half the force that whereas particle 2 exerts a nonvanishing magnetic force on the latter exerts on the former, F1,dip = −2Fdip,1. particle 1:

2 q v1v2 FM2,1 = qv1 × B2 = − 2 ey . momentum that the bodies/matter possess. Below we discuss r12 this issue in detail, starting by the electromagnetic case. Therefore

F1,2 = FEL1,2 6= F2,1 = FEL2,1 + FM2,1 1. Magnetism showing that an action-reaction law (in a naive Newtonian It is well known that electromagnetic forces do not obey the sense) does not apply here. This example, sometimes called action reaction law, and that the center of mass position of a the “Feynman paradox,” is due to Feynman, see Ref. [96] p. system of charged bodies is not a fixed point. Notice that this 26-5 and 27-11, and Fig. 26-6 therein. Further discussions on does not imply any violation of the conservation equations for this problem are given in e.g Sec. 8.2.1 of Ref. [18], and, in the total energy-momentum tensor; in fact, it is a necessary more detail, in Ref. [97]. consequence of the interchange between mechanical momen- tum of the bodies and electromagnetic field momentum . We analyze next some examples relevant to the problem at hand. b. Interaction of a magnetic dipole with individual particles of the cloud

a. Simplest example: Two moving point charges (Feynman Consider a system composed of a magnetic dipole µ = µez paradox) placed at the origin (call it particle 2), and a particle of the cloud (particle 1, of charge q) in the equatorial plane, and at Consider the setup in Fig.8a: a pair of point particles with the instantaneous position depicted in Fig.8b. The magnetic equal charge q and mass M, and with orthogonal velocities, field created by the magnetic dipole is given by Eq. (5) ; the one (particle 1) moving directly towards the other with v1 = force it exerts on particle 1 is thus v1ex, and the other moving orthogonally with v2 = v2ey. v × µ vµ To first “post-Coulombian” [13, 17, 30] order, the electric and Fdip,1 = qv × Bdip = −q = q ey . (B2) r3 r3 magnetic fields generated by a moving charge are [17] (cf. 21 21 also [19, 30]) Let us now compute the force that particle 1 exerts on the dipole. The magnetic field created by a generically moving ra 1 aa q 3 charge is, from Eq. (B1), Bcharge = qv × r/r ; the force Ea = q(1 + ϕa) 3 − q ; Ba = 3 va × ra (B1) ra 2 ra ra i j,i i it exerts on the dipole is Fcharge,dip = Bchargeµj ≡ ∇ (µ · where ra ≡ x − xa, x is the point of observation, xa is the Bcharge), cf. Eqs. (4)-(5); explicitly: instantaneous position of particle “a”, a its acceleration, and a µ × v (v × r) · µ Fcharge,dip = q − 3q r . (B3) 2 2 r3 r5 va 1 3 (ra · va) ϕa ≡ − (ra · aa) − 2 . 2 2 2 ra Hence, the force exerted by particle 1 on the dipole is

For a system of two bodies, to 1PC accuracy, aa in the equa- µ × v (v × r12) · µ 2vµq F1,dip = q 3 − 3q 5 r12 = − 3 ey . tion above is to be taken as the Coulomb force caused by the r21 r21 r21 24

Comparing with Eq. (B2), again we see that action does not 1 meet reaction: F1,dip = −2Fdip,1 (the sign is opposite as v expected, but the magnitudes do not meet). FSO2,1 r21 Let us now consider particle 3 lying at x3 = r3ez, and moving (again) with velocity v = vex. The force that the FSO1,2 dipole exerts on it, Fdip,3 = qv × Bdip(x3), is, from Eq. (21), MagneticBlack S dipoleHole v × µ 3(µ · r )v × r vµq F = −q + q 23 23 = −2 e . dip,3 r3 r5 r3 y 23 23 23 FIG. 9. A situation where the action reaction law is not obeyed in PN F Its reciprocal (the force that particle 3 exerts on the dipole) is, gravity (analogue of Fig.8b): the spin-orbit force SO2,1 exerted by the spinning body (e.g. a BH) on a moving particle is not the same from Eq. (B3), as the spin-orbit force FSO1,2 that the latter exerts on the former: FSO1,2 = −3FSO2,1/2. µ × v (v × r32) · µ vµq F3,dip = q 3 − 3q 5 r32 = 3 ey r23 r23 r23 2. Gravitomagnetism since the second term of the second expression vanishes. Thus, again, action does not meet reaction, only now it is the Analogous examples to the ones above can be given in grav- magnitude of the force on the particle that is twice that on the ity — two point masses momentarily in orthogonal motion, dipole: F = −2F . dip,3 3,dip the interaction of the spinning body with individual particles of the cloud (Sec. III B 2), and with the entire cloud — with entirely analogous conclusions. (As for the infinite wire of c. A magnetic dipole and an infinite straight wire Sec. B1c, it cannot be mirrored here since the metric of an infinitely long cylindrical mass is not asymptotically flat). Be- Consider again a magnetic dipole µ = µez placed at the low we discuss in detail the especially important second ex- origin, and an infinitely long wire placed along the straightline ample. (parallel to the x axis) defined by y = y0, z = 0, with a current I = Σj flowing through it in the positive x direction, j = jex. Σ is the cross sectional area of the wire. The force a. Interaction of a spinning body with individual particles of the exerted by the magnetic dipole (placed at the origin, and with cloud µ = µez) on the wire is, from Eq. (21), Consider a system composed of a spinning body momentar- j × µ 2µI ily at rest (body 2, of mass M2, and angular momentum S), Fdip,wire = j × Bdip = − 3 = 2 ey. ˆwire ˆwire r y0 and a point mass (body 1, of mass M1) moving with velocity v1 = v, as illustrated in Fig.9. Let us now compute its reciprocal, i.e. the force that the wire To 1.5PN order, the coordinate acceleration of a (spinning exerts on the dipole. The magnetic field generated by the wire or nonspinning) body in a gravitational field is generically is well known to be (e.g. Sec. 5.3 of [18], or Eqs. (14.22)- given by Eqs. (44), (40), (46). The coordinate acceleration (14.24) of [96]) of the spinning body (body 2), due to the gravitational field generated by the moving point mass 1, is then (since v2 = 0) 2I 0 Bwire = 2 02 [y ez − zey] 2 (z + y ) d x2 1 2 = (1 − 2U1)G1 + F (B4) dt M2 0 i,j where y = y − y0. Thus Bwire has the only nonvan- z,y y,z 2 02 2 where U = M /r is the Newtonian potential of body 1 and ishing components Bwire = Bwire = 2I(z − y )(z + 1 1 1 02 −2 y ) . Therefore, the force exerted on the dipole, Fwire,dip = F is the spin-curvature force on body 2. The latter amounts j,i to the whole spin-orbit force that acts on body 2, so we may Bwireµjei ≡ ∇(µ · Bwire), is write F = FSO1,2. From Eq. (40), 2µI F = − e = −F . 1 wire,dip y2 y dip,wire F j = F j = Hi,jS − (S × G˙ )j . (B5) 0 SO1,2 2 1 i 1

So, in this case, the action-reaction law is obeyed, just like Here H1 is the gravitomagnetic field generated by the trans- for the current slabs in Sec.IIA. This is the expected result lational motion of body 1; it is given by H1 = ∇ × A1, with because one is dealing here with magnetostatics, where there A1 = −4M1v/r1 (cf. e.g. [17, 31]), or, explicitly cannot be an exchange between mechanical and field momen- tum, for the latter is constant and equal to zero (the Poynting M1 H1 = −4 3 v × r1 (B6) vector is zero, since the electric field is zero). r1 25

The gravitoelectric field G1, to the accuracy at hand, is to For the setup in Fig.9, where particle 1 lies in the equa- be taken, in this expression, as the leading term G1 ' torial plane at the instantaneous position x1 = r1ez, and has 3 ˙ −Mr1/r1. In order to compute G1, one notes that r˙1 = −v, velocity v1 = v = vex, we have 19 and that r˙1 = −r1 · v/r1. One gets 3M  2r [(v × r ) · S] 3M Sv F = 1 v × S + 12 12 = 1 e F = SO1,2 3 2 3 y SO1,2 r12 r12 r12   3M1 2r12[(v × r12) · S] (v · r12)S × r12 (B11) 3 v × S + 2 + 2 . and r12 r12 r12 (B7) 2 2M1Sv FSO2,1 = M1v × H2 = 3 M1v × S = − 3 ey . r12 r12 Notice that this amounts to the whole spin-orbit force that acts (B12) on body 2: FSO1,2 = F . Therefore, F = −3F /2. The coordinate acceleration of body 1 (the point mass) is, SO1,2 SO2,1 Let us now consider particle 3 lying at x = r e (i.e., from Eqs. (44)-(46), 3 3 z on top of the spinning body, above the equatorial plane), and, d2x again, with velocity v3 = v = vex, see Fig.1b. The spin- 1 = (1 + v2 − 2U )G − 4(G · v)v + v × H (B8) orbit force that it exerts on the spinning body, F = F , dt2 2 2 2 2 SO3,2 is obtained from Eq. (B7), replacing therein r12 → r32, with where U2 = M2/r2 is the Newtonian potential of the spinning r32 ≡ r3 − r2, body (F = 0 in this case, since body 1’s spin is zero). The   3M3 2r32[(v × r32) · S] 3M3Sv gravitomagnetic field H2 generated by the spinning body 2 is F = v × S + = − e , SO3,2 r3 r2 r3 y given by Eq. (55) (replacing therein r → r2); therefore the 32 32 23 (B13) gravitomagnetic “force” M1v×H2 it exerts on body 1, which where we noted that (v × r32) · S = 0. The spin-orbit force amounts to the whole spin-orbit force FSO2,1 acting on body 1, is given by exerted by the spinning body on particle 3, FSO2,3 = M3v × H2, is   1 6 FSO2,1 v×H2 = 2v × S − (r21 · S)v × r21 = . 2M  3  4M Sv r3 r2 M 3 3 12 12 1 FSO2,3 = 3 v × S − 2 (r23 · S)v × r23 = 3 ey. (B9) r23 r23 r23 Using the vector identity (5.2a) of [98], we can re-write this (B14) result as Thus, again, action does not equal minus reaction: FSO2,3 = −4FSO3,2/3. FSO2,1 = Analogously to the electromagnetic case, this mismatch be- tween action and reaction does not mean a violation of any 3M 4 2r [(v × r ) · S] 2(v · r )S × r  1 12 12 12 12 conservation principle; it can be cast as an interchange be- − 3 v × S + 2 + 2 , r12 3 r12 r12 tween mechanical momentum of the bodies and field momen- (B10) tum (in the sense of the Landau-Lifshitz pseudotensor, see [31]). It is the same principle that is behind the famous bob- cf. Eq. (5.3a) of [98]. Comparing with (B7), we see that bings in binary systems [99], where the center of mass of the the spin-orbit interactions do not obey an action-reaction law: whole binary bobs up and down. F 6= −F . In other words, the spin-curvature force SO1,2 SO2,1 The examples above illustrate an important aspect depicted (B7) exerted by body 1 on body 2, is different from the gravit- in Fig.1b: cloud particles in the equatorial plane deflect in omagnetic “force” (B9) exerted by body 2 on body 1. Notice the negative y direction (i.e., to the left in Fig.1b), cf. Eq. that F is the analogue of the electromagnetic force F SO1,2 1,dip (B12), and push the spinning body to the right, as Eq. (B11) of Sec.B1b, and F the analogue of F . Therefore, SO2,1 dip,1 shows; however, cloud particles in regions outside the equa- the overall coordinate accelerations of the two bodies do not torial plane do the opposite: they deflect to the right [cf. Eq. likewise obey an action-reaction law: (B14)], and push the spinning body to the left, with a force d2x d2x whose magnitude is twice that of the force exerted by the par- M 2 6= M 1 . ticles at the equatorial plane, as shown by Eq. (B13). It is 2 dt2 1 dt2 the effect of the latter that eventually prevails in the cloud slab In fact, comparing (B4) to (B8), we see that actually all the PN (orthogonal to y) of Fig.3, where the net force on the cloud terms (not only the spin-orbit ones) differ; only the Newtonian points in the positive y direction (and the force on the spin- parts of M1G2 and M2G1 match up to sign. ning body points in the negative y direction), whereas in the special case of a slab orthogonal to z (i.e., to the body’s spin axis) the two effects exactly cancel out. The above explains also why in the earlier work [7] the au- 19 Transforming this expression to body 1’s rest frame (by noticing that the thors were misled into concluding that the force on the spin- velocity v1 = v of body 1 in the rest frame of body 2 equals minus the ning body was opposite to the Magnus effect (“anti-Magnus”, velocity v2 of body 2 in body 1’s rest frame: v = −v2), yields Eq. (97). upwards in Fig.3): the argument therein is based on the 26 asymmetric accretion that occurs in a spinning BH, i.e., the crucial here to distinguish between the radial component of absorption cross section being larger for counterrotating par- FGM (which determines its attractive/repulsive nature), from ticles than for corotating ones. This is a gravitomagnetic the force itself, and the overall deflection it causes. If one phenomenon, due to the fact that the gravitomagnetic force looks only at the equatorial plane, the reasoning in [7] is qual- FGM = Mv × H is attractive for counterrotating particles, itatively correct, since, as depicted in Fig.1b, particles in and repulsive in the corotating case. That can easily be seen the equatorial plane suffer a deflection opposite to that corre- considering again particles with velocity v = vex, as in Fig. sponding to a Magnus effect. That however overlooks the key 1b; from Eq. (55), it follows that the radial component of the fact that, as exemplified by the particles along the axis in Fig. 4 gravitomagnetic force is FGM · r/r = −2MvSy/r (attrac- 1b, there are regions outside the equatorial plane where parti- tive for positive y, repulsive for negative y). However, it is cles are deflected in the opposite direction, i.e., in the direction expected from a Magnus effect.

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