It has been suggested in recent literature that nonlinear andor gravitomagnetic general relativistic effects can play a leading role in galactic dynamics partially or totally replacing dark matter. Using the inlineformulammlmath displayinlinemmlmrowmmlmn1mmlmnmmlmommlmommlmn3mmlmnmmlmrowmmlmathinlineformula quasiMaxwell formalism we show on general grounds such hypothesis to be impossible.
['General relativity', 'alternative theories of gravity', 'General Relativity and Quantum Cosmology', 'Astrophysics - Astrophysics of Galaxies']
Relativistic effects cannot explain galactic dynamics
{'Relativistic effects cannot explain galactic dynamics': "L. Filipe O. Costa ∗ and José Natário †  \nCAMGSD, Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal (Dated: September 20, 2024)  \nIt has been suggested in recent literature that nonlinear and/or gravitomagnetic general relativistic effects can play a leading role in galactic dynamics, partially or totally replacing dark matter. Using the 1+3 'quasiMaxwell' formalism, we show, on general grounds, such hypothesis to be impossible.", 'I. Introduction': "1  \n- II. Equations of motion for massive particles and light in a stationary field  \n1  \n- A. Geodesic equation in the 'quasi-Maxwell' formalism  \n2  \n- 1. Timelike geodesics  \n2  \n- 2. Null geodesics  \n2  \n- 3. Timelike circular geodesics  \n3  \n- B. Field equations  \n3  \n- III. Gravitomagnetism cannot be the  \nculprit-gravitational lensing  \n3  \n- IV. Nonlinear GR effects work against attraction  \n5  \n- V. Relevant examples  \n5  \n- A. Post-Newtonian theory  \n5  \n- 1. System of N point bodies  \n5  \n- 2. Self-gravitating disks  \n6  \n- 3. Lensing around a spinning body  \n6  \n- B. The Balasin-Grumiller (BG) 'galactic' model  \n7  \n- C. Gravitomagnetic dipole model  \n8  \nVI. Conclusion  \n10  \nAcknowledgments  \n10  \nA. Lensing in the Schwarzschild and Kerr spacetimes  \n10  \nReferences  \n12", 'I. INTRODUCTION': "In a recent number of works, based both on linearized theory [1, 2] and exact models [3-7], it has been asserted that general relativistic effects (namely, gravitomagnetic ones) can have a significant impact [1, 2, 5, 7], or even totally account for the galactic flat rotation curves [3, 4, 6]. In the framework of a weak field slow motion approximation, this has been shown to be impossible, and such claims addressed, in Refs. [8-10]. It has however been argued [3-5, 7] that it is still  \npossible in the exact theory, due to nonlinear effects not captured in linearized theory (and not manifest 'locally' [3-5]). In support of these claims, the Balasin-Grumiller (BG) model [5-7], or variants of it [3, 4, 7], have been used. Such model has been recently debunked in our Ref. [11], where it was shown to actually consist of a static dust, held in place by unphysical singularities along the symmetry axis, and that what actually rotates in the model are the ill chosen reference observers. In the present paper we address the general problem: assuming only stationarity of the solution, we show that (i) gravitomagnetism cannot be a relevant driving force of galactic dynamics; (ii) nonlinear general relativistic effects cannot account for any amount of the needed dark matter.  \nNotation and conventions.Greek letters α, β, γ, ... denote 4D spacetime indices, running 0-3; Roman letters i, j, k, ... are spatial indices, running 1-3; ϵ αβγδ ≡ √ -g [ αβγδ ] is the 4D Levi-Civita tensor, with the orientation [1230] = 1 ; ϵ ijk ≡ √ h [ ijk ] is the Levi-Civita tensor in a 3D Riemannian manifold Σ of metric h ij . Arrow notation ⃗ X denotes spatial 3-vectors on Σ , with components X i . Indices of spatial tensors on Σ are raised and lowered with h ij : X i = h ij X j . The basis vector corresponding to a coordinate ϕ is denoted by ∂ ϕ ≡ ∂/∂ϕ , and its α -component by ∂ α ϕ ≡ δ α ϕ .", 'II. EQUATIONS OF MOTION FOR MASSIVE PARTICLES AND LIGHT IN A STATIONARY FIELD': "The line element ds 2 = g αβ dx α dx β of a stationary spacetime can generically be written as  \nds 2 = -e 2Φ ( dt -A i dx i ) 2 + h ij dx i dx j , (1)  \nwhere e 2Φ = -g 00 , Φ ≡ Φ( x j ) , A i ≡ A i ( x j ) = -g 0 i /g 00 , and h ij ( x k ) = g ij + e 2Φ A i A j . We are interested in the equations of motion with respect to the coordinate system in (1); it is thus useful to refer to quantities as measured by the observers at rest in such coordinates ('static' observers). These observers have 4-velocity u α = e -Φ ∂ α t ≡ e -Φ δ α 0 , and their worldlines are tangent to the timelike Killing vector field ∂ t . The quotient of the spacetime by their worldlines yields a 3D Riemannian manifold Σ with metric h ij , which these observers regard as the 'spatial metric', since dl = √ h ij dx i dx j yields the infinitesimal spatial distance between them as measured by Einstein's light signaling procedure (e.g., by radar) [12]. The 3D metric h ij is identified in spacetime with the space projector with respect to u α , h αβ ≡ g αβ + u α u β (observe that h 0 α = 0 ).  \nAs discussed in detail in [11], when the congruence of static observers is asymptotically inertial [which is the case when their acceleration and vorticity, Eq. (7) below, asymptotically vanish], the coordinate system in (1) has axes anchored to inertial frames at infinity, being a generalization of the International Astronomical Union reference system to the exact metric (1). Such reference frame is in practice realized setting it up fixed with respect to remote celestial objects (namely quasars), and it is with respect to such a frame that, for instance, galactic rotation curves are defined.", "A. Geodesic equation in the 'quasi-Maxwell' formalism": "Let x α ( λ ) be the geodesic worldline of a test particle (that can have zero mass) and λ an affine parameter along it. The space components of the geodesic equation d 2 x α /dλ 2 = -Γ α µν ( dx µ /dλ )( dx ν /dλ ) can be written as 1  \nd 2 x i dλ 2 +Γ( h ) i jk dx j dλ dx k dλ = ν [ ν ⃗ G + d⃗x dλ × ⃗ H ] i (2)  \nwhere ν ≡ -u α dx α /dλ ,  \nΓ( h ) i jk ≡ 1 2 h il ( h lj,k + h lk,j -h jk,l ) (3)  \nare the Christoffel symbols of the space metric h ij , and  \nG i = -Φ ,i ; H i = e Φ ϵ ijk A k,j ( ϵ ijk ≡ √ h [ ijk ]) (4)  \nare fields living on the space manifold Σ , dubbed, respectively, 'gravitoelectric' and 'gravitomagnetic' fields, for playing in Eq. (2) roles analogous to the electric and magnetic fields in the Lorentz force equation. For a 3-vector ⃗ X on Σ , one can define the 3D covariant derivative with respect to h ij  \n˜ DX i dλ ≡ dX i dλ +Γ( h ) i jk X j dx k dλ . (5)", '1. Timelike geodesics': 'For timelike worldlines, one can set λ = τ , so dx α /dτ ≡ U α is the 4-velocity, ν = -u α U α ≡ γ becomes the Lorentz factor between U α and u α , and Eq. (2) can therefore be written as [12-17] (cf. also [18, 19]),  \n˜ D ⃗ U dτ = γ [ γ ⃗ G + ⃗ U × ⃗ H ] (6)  \nwhere ⃗ U is the 3-vector of components ( ⃗ U ) i = U i , tangent to the spatial curve x i ( τ ) obtained by projecting the geodesic  \nonto Σ . Equation (6) describes the acceleration of such 3D projected curve [since the left-hand member of (2) is the standard 3D covariant acceleration]. Its physical interpretation is that, from the point of view of the static observers, the spatial trajectory of the test particle will appear accelerated, as if acted upon by fictitious forces (inertial forces), arising from the fact that the reference frame is not inertial . In fact, ⃗ G and ⃗ H are identified in spacetime, respectively, with minus the acceleration and twice the vorticity of the static observers:  \nG α = -∇ u u α ≡ -u α ; β u β ; H α = 2 ω α = ϵ αβγδ u γ ; β u δ . (7) √  \nHere ϵ αβγδ = [ αβγδ ] / -g is the 4D contravariant LeviCivita tensor, and in the correspondence with (4) it is useful to note that -g = he 2Φ . Decomposing U α in terms of its projections parallel and orthogonal to u α ,  \nU α = γ ( u α + v α ) ; γ ≡ -u α U α = 1 √ 1 -v α v α , (8)  \nwhere v α is the spatial velocity of the particle relative to the static observers [11, 15, 18], Eq. (6) can be rewritten as  \n˜ D ⃗ U dτ = γ 2 [ ⃗ G + ⃗v × ⃗ H ] . (9)', '2. Null geodesics': "In the case of light rays (null geodesics), of tangent k α = dx α /dλ , Eqs. (2) and (5) analogously yield  \n˜ D ⃗ k dλ = ν [ ν ⃗ G + ⃗ k × ⃗ H ] ; ν ≡ -k α u α . (10)  \nAn equivalent formulation in terms of the photon's null 4momentum is given in [19], Eqs. (10.2)-(10.3), and a Lagrangian formulation in [20]. The null vector k α decomposes, in terms of its projections parallel and orthogonal to u α , as [21]  \nk α = ν ( u α + v α ) (11)  \nwhere v α is a unit vector orthogonal to u α , yielding the photon's spatial velocity relative to u α . Therefore, ⃗ k = ν⃗v , and one can re-write (10) as  \n˜ D ⃗ k dλ = ν 2 [ ⃗ G + ⃗v × ⃗ H ] . (12)  \nEquations (10)-(12) tell us that, just like massive particles, light rays in a stationary spacetime behave analogously to charged particles under the action of an electric and magnetic fields [20]. A difference, however, is that the second term in the left-hand member of (2), which reads here Γ( h ) i jm k j k m , is typically of the same order magnitude as the first term in (10) [contrary to the case of Γ( h ) i jm U j U m = O ( v 2 ) for slowly moving massive particles].  \nIn an axistationary spacetime we have g 0 i dx i = g 0 ϕ dϕ . If there is reflection symmetry about the equatorial plane [22], then g αβ,z = 0 (or g αβ,θ = 0 ) along that plane; noticing that the geodesic equation can be written in the form (see e.g. [23]) dU α /dτ = g µν,α U µ U ν / 2 , then dU z = 0 therein, allowing equatorial geodesic motion. For circular equatorial geodesics, U α = U 0 ( δ α 0 +Ω geo δ α ϕ ) , where the angular velocity Ω geo ≡ dϕ/dt = U ϕ /U 0 is obtained from the r -component of the geodesic equation, dU r /dτ = g µν,r U µ U ν / 2 = 0 :  \nΩ geo ± = -g 0 ϕ,r ± √ g 2 0 ϕ,r -2 g ϕϕ,r e 2Φ G r g ϕϕ,r . (13)  \n̸  \nIn the equatorial plane, given an arbitrary radial coordinate r , through the coordinate transformation r w ( r ) = e Φ( r ) √ h ϕϕ ( r ) , the metric can be written in the form (1) with h ϕϕ = e -2Φ r 2 w . This is the case of the Weyl canonical coordinates (cf. e.g. Eq. (19.21) of [24]); it is also the case (to the accuracy at hand) of the radial coordinate r = √ x 2 + y 2 associated to the post-Newtonian (PN) coordinate system in Eq. (20) below. It follows from (8) that the magnitude ∥ v α geo ± ∥ = (1 -γ -2 geo ± ) 1 / 2 of the velocity of the circular geodesics relative to the static observers u α = e -Φ δ α 0 reads, in such coordinates, for G r w = 0 ,  \n∥ v α geo ± ∥ = 2 r w | G r w | | √ -g ( H z ) 2 -4 r w G r w (1 + r w G r w ) ± √ -gH z | (14) where we have noted, from (4), that A ϕ,r = e -Φ ϵ rϕi H i = e -2Φ √ -gH z .", 'B. Field equations': "The fields ⃗ G and ⃗ H that govern the geodesic equation (2) obey [15]  \n˜ ∇· ⃗ G = -4 π (2 ρ + T α α ) + ⃗ G 2 + 1 2 ⃗ H 2 ; ˜ ∇× ⃗ G = 0 ; (15)  \n˜ ∇· ⃗ H = -⃗ G · ⃗ H ; ˜ ∇× ⃗ H = -16 π ⃗ J +2 ⃗ G × ⃗ H , (16)  \nwhere ρ ≡ T αβ u α u β and J α ≡ -T αβ u β are, respectively, the mass-energy density and current 4-vector as measured by the static observers of 4-velocity u α = e -Φ δ α 0 . Here ˜ ∇ denotes covariant differentiation with respect to the spatial metric h ij , with Christoffel symbols (3). The equations for ˜ ∇· ⃗ G and ˜ ∇ × ⃗ H are, respectively, the time-time and time-space projections, with respect to u α , of the Einstein field equations R αβ = 8 π ( T αβ -1 2 g αβ T γ γ ) ; the equations for ˜ ∇ · ⃗ H and ˜ ∇× ⃗ G follow from (4).  \nFigure 1. Gravitational lensing by galactic dark matter halos. When the observer, foreground, and emitting galaxies are aligned, an Einstein ring forms. Nearly perfect Einstein rings have been detected, such as in the system B1938+666, or the 'Cosmic Horseshoe' J1148+1930. The majority of the lensing effect is estimated to come from dark matter. The gravitomagnetic field cannot mimic such effects, since the inertial force ⃗v × ⃗ H does not contribute to make light rays converge in such setting; e.g., in the equatorial plane, it deflects photons on both sides of the body in the same direction (the effect being anyway negligible for any realistic galactic model).  \n<!-- image -->", 'III. GRAVITOMAGNETISM CANNOT BE THE CULPRIT-GRAVITATIONAL LENSING': "Alongside the galactic rotation curves, there is a set of other effects consistently indicating hypothetical dark matter (DM) halos around galaxies. Any viable alternative to DM must explain these effects. One of the most compelling of such indications is the observed galactic gravitational lensing, which cannot be accounted for by visible baryonic matter alone. It is well known that, when the emitting object (light source), the foreground galaxy (lens), and the observer are nearly aligned (see Fig 1), roughly circular rings (Einstein rings [25-31]) are formed, as in the case of the system B1938+666 [27, 32, 33] (the first detected complete Einstein ring), or the 'Cosmic Horseshoe' J1148+1930 [34-38]. The majority of the mass causing such lensing effect is estimated to consist of dark matter [33, 35-38], and is consistent with the halos' shape being roughly spherical or moderately deformed (namely prolate [37], as seems to be more common [39]). Anomalies in such rings have actually been recently proposed [40] as a mean of determining the nature of dark matter, namely distinguishing between weakly interacting massive particles and axions.  \nThese effects could not take place if the gravitomagnetic field ⃗ H were the main driver of galactic dynamics. For a rotating body, under the assumptions of stationarity and reflection symmetry about the equatorial plane, ⃗ H is axisymmetric, orthogonal to the equatorial plane, and pointing in the same direction along the whole plane. It follows from Eq. (12) that the gravitomagnetic 'force' ⃗v × ⃗ H on light rays passing through opposite sides of the body points in the same direction, thus not contributing to produce symmetric convergence of light rays along the line of sight of observers aligned with the light source and the foreground galaxy (setting illustrated in Fig. 1). This can be seen also using the Gauss-Bonnet theorem [41-47]: the light ray trajectories C ± illustrated in Fig. 1 are the projections of the photon's null geodesics C (4) ± onto  \nthe space manifold Σ , π Σ C (4) ± = C ± . Let S be an oriented 2-surface on Σ bounded by C ± : ∂ S = C + ∪ ( -C -) , where -C -is the curve with the opposite orientation of C -. Then  \nθ R = S Kd S + ˆ C + κ g dλ -ˆ C -κ g dλ -θ S -2 π [ χ ( S ) -1] , (17)  \nwhere K is the Gaussian curvature of S and χ ( S ) its Euler characteristic [if simply connected, namely in the absence of singularities, χ ( S ) = 1 ], and κ g is the geodesic curvature [41, 48, 49] of a curve C of tangent vector ˙ C ( λ ) . Considering a positively oriented orthonormal frame { e ˆ 1 , e ˆ 2 } on S along C such that e ˆ 1 = ˙ C/ ∥ ˙ C ∥ (see Fig. 1), the geodesic curvature of C is given by [48]  \nκ g = 1 ∥ ˙ C ∥ ⟨ ˜ ∇ ˙ C e ˆ 1 , e ˆ 2 ⟩ = ⟨ ˜ ∇ ˙ C ˙ C,e ˆ 2 ⟩ ∥ ˙ C ∥ 2 ,  \nwhere ˜ ∇ is the Levi-Civita covariant derivative of 2 (Σ , h ) , with Christoffel symbols (3), as defined above. For light rays, ˙ C = ⃗ k , ∥ ˙ C ∥ 2 = h αβ k α k β = ν 2 [cf. Eq. (11)], and e ˆ 1 = ⃗v . Recalling that the left-hand member of (2) is the 3D covariant acceleration ˜ ∇ ˙ C ˙ C , by Eq. (12) we have ˜ ∇ ˙ C ˙ C = ν 2 [ ⃗ G + ⃗v × ⃗ H ] ; hence,  \nκ g = ⟨ ⃗ G,e ˆ 2 ⟩ + ⟨ ⃗v × ⃗ H,e ˆ 2 ⟩ ≡ G ˆ 2 +( ⃗v × ⃗ H ) ˆ 2 . (18)  \nFor the setting in Fig. 1, in the equatorial plane, ( ⃗v ± × ⃗ H ) ˆ 2 = -∥ ⃗ H ∥ ; hence, the gravitomagnetic contributions to the convergence angle θ R in Eq. (17) have opposite signs. In particular, for the case where the rays are approximately symmetric, as required for a nearly perfect Einstein ring, we have, due to the reflection symmetry about the source-lens-observer axis, ' C + ∥ ⃗ H ∥ dλ ≈ ' C -∥ ⃗ H ∥ dλ and ' C + G ˆ 2 dλ ≈ -' C -G ˆ 2 dλ ; therefore,  \nˆ C + κ g dλ -ˆ C -κ g dλ ≈ 2 ˆ C + G ˆ 2 dλ ,  \nso the gravitomagnetic force does not contribute to θ R in Eq. (17). Indeed, the effect of the gravitomagnetic force would be to cause the light rays that reach an observer aligned with the source and foreground galaxy to arrive at different angles from each side, as exemplified in Appendix A for the case of a Kerr black hole. Moreover, in the far field regime, the gravitomagnetic field of a galaxy (as for any spinning body) is dipole-like,  \nFigure 2. Gravitomagnetic field ⃗ H produced by a galactic disk. At large distances, it is dipole-like, deflecting photons with impact parameter ⃗ b orthogonal to the equatorial plane in a direction opposite to those for which ⃗ b lies in the equatorial plane (in both cases, photons through opposite sides of the body being deflected in the same direction); this produces no convergence of light rays along the line of sight of observers aligned with the light source and the foreground galaxy.  \n<!-- image -->  \nas illustrated in Fig. 2; hence, light rays with impact parameter ⃗ b orthogonal to the equatorial plane are deflected in a direction parallel to that plane (thus orthogonal to ⃗ b ), which again produces no convergence, cf. Secs. V A 3 and Appendix A. As a consequence, not only ⃗ H does not contribute, as it acts against the formation of Einstein rings (actually precluding them for small light sources). This is well known in the case of the Kerr spacetime [20, 50-52], discussed in Appendix A, and for generic rotating sources in the weak field limit [53], discussed in Sec. V A 3 below. This is in spite of, for astrophysical lenses, ⃗ H (thus ⃗v × ⃗ H ) being smaller (typically much smaller) than ⃗ G along the light ray's path.  \nObserve now the crucial difference in magnitude (for a given ⃗ H ) between the gravitomagnetic forces ⃗v × ⃗ H on light and on massive astrophysical bodies,  \nphotons : v = 1 ⇒ | ⃗v × ⃗ H | ∼ | ⃗ H | stars in galaxy : v > 10 -3 ⇒ | ⃗v × ⃗ H | > 10 -3 | ⃗ H | .  \nBy Eq. (9), in order for ⃗v × ⃗ H to have a relevant impact on the galactic rotation curves, one would need | ⃗ H | ≳ 10 3 | ⃗ G | . Such a gravitomagnetic field (impossible for a physical source, cf. Sec. V A 3) would yield a gravitational lens with a gravitomagnetic force three orders of magnitude larger than the gravitoelectric term: | ⃗v × ⃗ H | ∼ | ⃗ H | ≳ 10 3 | ⃗ G | (thus than the Newtonian contribution). This, besides precluding Einstein rings, would produce bending angles much larger than observed, as can be checked using the Gauss-Bonnet theorem (employing it, with κ g as given in Eq. (18) above, to the quadrilateral in e.g. Fig. 2 of [41]; cf. also Eq. (30) therein), or, in linearized theory, using Eq. (28) below, leading to extreme lensing effects also very different from those observed. This is a trait  \ncommon to the 'galactic' models based on gravitomagnetism proposed in the literature, as exemplified in Secs. V A 3 and VBbelow.", 'IV. NONLINEAR GR EFFECTS WORK AGAINST ATTRACTION': "The analysis in the previous section shows that not only the gravitomagnetic field ⃗ H cannot replace the role of dark matter in gravitational lensing, as the observed deflection angles constrain ⃗ H in galaxies to be, at most, within the order of magnitude of ⃗ G (actually having to be considerably smaller, in order to allow for the Einstein rings). For the motion of stars, since v > 10 -3 , this effectively renders the gravitomagnetic term ⃗v × ⃗ H in Eq. (9) irrelevant compared to the gravitoelectric term. The measured star velocities constrain likewise the magnitude of ⃗ G to | ⃗ G | r ∼ v 2 > 10 -6 ; from (14), we have thus, for the velocity of the circular equatorial geodesics,  \n∥ v α geo ± ∥ = √ r w G r W + O (10 -9 ; 10 -6 | ⃗ H | / | ⃗ G | ) , (19)  \nwhich is ruled by G α , as expected (and formally similar to the Keplerian law). We stress that the approximate expression (19) follows merely from observational constrains, not linearized theory; indeed ⃗ G , so far, remains a nonlinear field. The same conclusion 3 can be reached from the angular velocity (13), which yields Ω geo ± = e Φ ( -2 G r /g ϕϕ,r ) 1 / 2 (1 + [terms < 10 -3 ]) (where one can take g ϕϕ = r 2 , since it is also well known from experiment that the galactic space metric is very weakly curved).  \nThus it remains only to clarify whether nonlinear GR effects can amplify ⃗ G in order to produce an attractive effect able to sustain the rotation curves. It is clear however, from the first of Eqs. (15), that the nonlinear terms ⃗ G 2 and ⃗ H 2 / 2 act as effective negative 'energy' sources for ⃗ G [13], countering the attractive effect of the source term 2 ρ + T α α ( = ρ for dust). Hence, they only aggravate the need for dark matter. This is manifest already in post-Newtonian (PN) theory, see Sec. V A below. It is also clear, based on PN theory, that the effect is anyway negligible in any realistic galactic model, given the weakness of the galactic gravitational field (cf. Secs. V A 1 and V A 2). It is huge, however, in some galactic models proposed in the literature. Namely, in the Balasin-Grumiller dust model (Sec. V B below), the effect is such that it completely kills off the attractive contribution from the dust's mass density ρ , yielding ⃗ G = 0 [11].", 'A. Post-Newtonian theory': "In the post-Newtonian (PN) expansion of general relativity, at first (1PN) order, the metric can be written, in geometrized units, as [54-56]  \ng 00 = -1 + 2 w -2 w 2 + O (6) ; g i 0 = A i + O (5) ; g ij = δ ij (1 + 2 w ) + O (4) , (20)  \nwhere O ( n ) ≡ O ( ϵ n ) , ϵ is a small dimensionless parameter such that U ∼ ϵ 2 , U is minus the Newtonian potential, and w = U + O (4) consists of the sum of the Newtonian potential U plus nonlinear terms of order ϵ 4 . The bodies' velocities are assumed such that v ≲ ϵ (since, for bounded orbits, v ∼ √ U ), and time derivatives increase the degree of smallness of a quantity by a factor ϵ ; for example, ∂U/∂t ∼ Uv ∼ ϵU . The geodesic equation for a test particle can be written as [54]  \nd 2 ⃗x dt 2 = (1+ v 2 -2 U ) ⃗ G + ⃗v × ⃗ H -3 ∂U ∂t ⃗v -4( ⃗ G · ⃗v ) ⃗v + O (6) (21)  \nwith  \n⃗ G = ∇ w -∂ ⃗ A ∂t + O (6) ; ⃗ H = ∇× ⃗ A + O (5) . (22)  \nIn the stationary case (i.e., neglecting all time derivatives), Eq. (21) yields the post-Newtonian limit of Eq. (6), as can be seen noting that ˜ D ⃗ U/dτ = ( U 0 ) 2 d 2 ⃗x/dt 2 -v 2 ⃗ G +4( ⃗ G · ⃗v ) ⃗v + O (6) and ( U 0 ) -2 = 1 -v 2 -2 U + O (4) .", '1. System of N point bodies': "For a system of N gravitationally interacting point particles, the metric potentials read, in the harmonic gauge (e.g. [54, 56-58]),  \n̸  \nw = ∑ a M a r a   1 + 2 v 2 a -∑ b =a M b r ab -1 2 ⃗r a · ⃗a a -( ⃗r a · ⃗v a ) 2 2 r 2 a   ⃗ A = -4 ∑ a M a r a ⃗v a ; U = ∑ a M a r a , (23)  \nwhere M a is the mass of particle 'a', ⃗r a ≡ ⃗x -⃗x a , ⃗x is the point of observation, ⃗x a is the instantaneous position of particle ' a ', ⃗v a = ∂⃗x a /∂t its velocity, ⃗a a = ∂⃗v a /∂t its coordinate acceleration, and ⃗r ab ≡ ⃗x a -⃗x b . The gravitoelectric and  \ngravitomagnetic fields (22) read  \n̸  \n⃗ G = -∑ a M a r 3 a ⃗r a   1 -2 ∑ b M b r b -∑ b =a M b r ab +2 v 2 a -3( ⃗r a · ⃗v a ) 2 2 r 2 a -1 2 ( ⃗r a · ⃗a a ) ] +3 ∑ a M a r 3 a ( ⃗r a · ⃗v a ) ⃗v a + 7 2 ∑ a M a ⃗a a r a ; (24)  \n⃗ H = -4 M a r 3 a ∑ a ⃗v a × ⃗r a . (25)  \nIn the case of a single body at rest, the potentials (23) reduce to w = M/r , ⃗ A = 0 , and (20) yields the 1PN limit of the Schwarzschild metric in isotropic or harmonic coordinates (obtained, to the accuracy at hand, from the usual Schwarzschild coordinates { t, ϱ, θ, ϕ } through the substitution ϱ = r + M ; cf., e.g., [58], pp. 268-270). Equations (24)-(25) yield in this case ⃗ H = 0 and  \n⃗ G = -M r 3 ( 1 -2 M r ) ⃗r .  \nThis is smaller than the Newtonian (0PN) gravitoelectric field ⃗ G N = -⃗rM/r 3 , showing that the nonlinear contribution decreases the gravitational attraction. The angular velocity of circular geodesics, Eq. (13), is  \nΩ geo ± = ± [ √ M r 3 -3 2 √ M 3 r 5 ] ,  \nwhich, accordingly, is slower than the Newtonian angular velocity Ω N = ± M 1 / 2 /r 3 / 2 .  \nStars in a galaxy can be considered, to a good approximation, point masses. In the Milky Way their velocities are of the order v a ∼ 10 -3 , and their coordinate acceleration a a of the order v 2 a /r ∼ 10 -6 /r ; hence, the velocity and acceleration dependent terms in (24) are smaller by a 10 6 factor compared to the Newtonian term ⃗ G N = -∑ a M a ⃗r a /r 3 a . The same applies to the velocity dependent terms in (21), by Eqs. (23)(25). The remaining relativistic corrections are the nonlinear terms in the first line of Eq. (24) (which are negative, thus decreasing the magnitude of ⃗ G ), plus the negative term -2 U ⃗ G in (21); all of them decrease the attraction, thus working against the sought effect for explaining the galactic rotation curves (the corrections being anyway negligible, comparing to the Newtonian terms).", '2. Self-gravitating disks': 'The gravitational field of self-gravitating stationary fluids is described, to first post-Newtonian order, by the metric (20) with w = U [59, 60]. The relativistic Euler equations are integrable provided that the angular momentum per unit mass j depends only on the angular velocity Ω [59]. A function j (Ω) ,  \nappropriate for toroids (including thin hollow disks, case that could represent a disk galaxy), is given in Eq. (8) of [59], leading to the rotation curve  \nΩ = Ω N [ 1 -2 1 -δ Ω 2 N r 2 -4 h N 1 -δ ] -A ϕ r 2 (1 -δ ) (26)  \n(cf. Eq. (18) of [59]), where Ω N is the Newtonian result, δ ∈ [ -∞ , 0] \\ {-1 } is a parameter, and h N the specific enthalpy at Newtonian (0PN) accuracy. The nonlinear contribution in the second term, again, slows down the rotation. This contribution, as well as the third and fourth terms of Eq. (26), are anyway negligible for the Milky Way: Ω 2 N r 2 ∼ 10 -6 ≳ h N ; and A ϕ , from its definition in e.g. Eqs. (8.4) of [58], is of order A ϕ /r 2 ∼ 10 -3 U/r ∼ 10 -6 Ω N (in agreement with the conclusion in [8]).', '3. Lensing around a spinning body': "The gravitational field of any isolated stationary matter distribution is described, to linear order, by the metric  \nds 2 = ( -1+2 U ) dt 2 +2 A i dx i + δ ij (1 + 2 U ) dx i dx j , (27)  \nwhich follows from linearizing (20) (cf. e.g. Eq. (27.32) of [61]). Consider light rays from a distant light source, at impact parameter ⃗ b , being scattered by the distribution (see e.g. Fig. 1 of [62]). Let ⃗v be a unit vector parallel to the ray's direction [i.e., the photon's spatial velocity, as defined in Eq. (11)]. The change in the ray's direction, ∆ ⃗v = ⃗v f -⃗v in , is given by [28, 63] (cf. also [62])  \n∆ ⃗v = 2 ˆ ∞ -∞ ⃗ Gdt +∆ ⃗v H ; ∆ ⃗v H = ˆ ∞ -∞ ⃗v × ⃗ Hdt , (28)  \nwhere ⃗ G = ∇ U and ∆ ⃗v H is the gravitomagnetic deflection. The gravitomagnetic vector potential of the distribution is ⃗ A = 2 ⃗r × ⃗ S/r 3 , where ⃗ S is the angular momentum. Notice that this is a dipole-type potential, of dipole moment -2 ⃗ S . The corresponding gravitomagnetic field is [cf. Eqs. (22)] ⃗ H = 2 ⃗ S/r 3 -6( ⃗ S · ⃗r ) ⃗r/r 5 , formally identical to the magnetic field of a magnetic dipole of moment ⃗ µ = -2 ⃗ S , as is well known (e.g. [64]). Equations (28) yield in this case [62, 65]  \n∆ ⃗v H = 4 b 2 [ 2 b 2 ( ⃗ b · ( ⃗v in × ⃗ S ) ⃗ b -( ⃗v in × ⃗ S ) ] .  \nFor pairs of light rays at opposite impact parameters ± ⃗ b (see Fig. 2), ∆ ⃗v H is the same, thus not contributing to produce convergence. For ⃗v in and ⃗ b mutually orthogonal and lying in the equatorial plane (e.g. ⃗v in = ∥ ⃗v in ∥ ⃗e x , ⃗ b = ±∥ ⃗ b ∥ ⃗e y , ⃗ S = ∥ ⃗ S ∥ ⃗e z , as depicted in Fig. 2), the photons are deflected in the direction ˆ v in × ˆ S ( = -⃗e y , in Fig. 2) along the equatorial plane; for the same ⃗v in , but now ⃗ b orthogonal to the equatorial plane ( ⃗ b ∥ ± ⃗ S ), the photons are again deflected parallel to the equatorial plane, but now in the opposite direction -ˆ v in × ˆ S .  \nConsidering the Newtonian potential of a spherical source, U = M/r , and 0 ≤ S/M 2 < 1 , Eq. (27) yields the linearized Kerr metric. It produces weak lensing images very similar to those displayed for the exact metric in Figs. (8)(c)(d). Namely, for large enough light sources (see Appendix A), the lens acts like a shifted non-spinning lens [66, 67]. For extended objects (like fast-spinning stars) larger values of S/M 2 are possible; but still the gravitomagnetic field ⃗ H along the ray's trajectory is smaller than (or, at best, when grazing the object, within the order of magnitude of) ⃗ G , since S ≲ Mv rot R (where R is the body's radius and v rot its rotational velocity), and so in the body's exterior | ⃗ H | / | ⃗ G | ∼ v rot R/r < 1 , with R/r < 1 and v rot < 1 .  \nTo entertain the possibility of ⃗ H having an impact on the galactic rotation curves, however, a much larger field would need to be considered, as discussed in Sec. III. According to Eq. (9), one would need | ⃗ H | ≳ 10 3 | ⃗ G | since, for stars in a galaxy, v > 10 -3 . Besides impossible for a rotating body, as shown above, such ⃗ H would, by Eq. (28) (since, for light, v = 1 ) lead to extreme deflection angles, orders of magnitude larger than observed, and of a very different type. This can be exemplified with the gravitomagnetic equatorial potential A ϕ in Eq. (23) of [9], claimed in some literature to flatten the rotation curves (but shown in [9] to be actually sourced by unphysical singularities). We have, using the values in [9] for the galaxy NGC 1560, and the Newtonian potential U in Eq. (15) therein, | ⃗ H | / | ⃗ G | > Kr -7 . 3 / 30 , with K = 6400kpc 7 . 3 / 30 = 1 /C , thus | ⃗ H | / | ⃗ G | > 10 3 throughout the whole galaxy ( R ≲ 10 kpc ). Numerical computation of the ray trajectories yields an equatorial plot resembling that in Fig. 3, with rays hugely deflected in the same direction on both sides of the lens, and not even crossing along the lenssource axis.", "B. The Balasin-Grumiller (BG) 'galactic' model": "The BG solution is described by the metric [5]  \nds 2 = -( dt -Ndϕ ) 2 + r 2 dϕ 2 + e µ ( dr 2 + dz 2 ) ; (29) N ( r, z ) = V 0 2 ∑ ± [ √ ( z ± r 0 ) 2 + r 2 -√ ( z ± R ) 2 + r 2 ] + V 0 ( R -r 0 ) , (30)  \nclaimed to describe a galactic dust model in comoving coordinates, with r 0 the radius of the bulge region, and R roughly the radius [6] of the galactic disk in the equatorial plane. It has, however, been recently shown in [11] to actually consist of a dust static with respect to the asymptotic inertial frame (hence totally unsuitable as a galactic model), held in place by unphysical singularities along the symmetry axis, and the claimed flat rotation curves to be but an artifact of an unsuitable choice of reference observers (the zero angular momentum observers-ZAMOs, which undergo circular motions with respect to inertial frames at infinity, due to the artificially large frame-dragging effects created by the singu-  \nlarities). It turns out to be also an archetypal example for Secs. III and IV.  \nSince ⃗ G = 0 and ⃗ J = 0 [11], Eqs. (15)-(16) yield ˜ ∇· ⃗ H = 0 , ˜ ∇× ⃗ H = 0 , and  \n˜ ∇· ⃗ G = -4 πρ + 1 2 ⃗ H 2 = 0 . (31)  \nLinearization of this equation leads to the empty space equation ρ = 0 ; hence, the solution has no linear or Newtonian limit, being thus a purely nonlinear solution. In fact, it is an extreme example of the phenomenon discussed in Sec. IV: the repulsive action of the nonlinear contribution ⃗ H 2 / 2 is such that it cancels out exactly the attractive gravitational effect of the dust's energy density (allowing the gravitoelectric field ⃗ G to vanish [11]).  \nThe equation for null geodesics, from Eqs. (2), (10), and (12), becomes here  \nd 2 x i dλ 2 +Γ( h ) i jl k j k l = ν ⃗ k × ⃗ H = ν 2 ⃗v × ⃗ H .  \nIf one approximates, as done in [5], µ ≈ const . , the space metric h ij becomes flat, Γ( h ) i jl become the Christoffel symbols of a cylindrical coordinate system in Euclidean space, and the spacetime an example of a solution where light deflection is solely driven by the gravitomagnetic 'force' ν 2 ⃗v × ⃗ H . The numerical results, for the parameters suggested in [5] for the Milky Way ( r 0 = 1kpc , R = 100kpc , V 0 = 220km / s ) and a point light source in the equatorial plane at a distance 1 Gpc from the origin (within the typical order of magnitude of the sources' distance of observed Einstein rings, e.g. [33, 37]), are plotted in Fig. 3. For a finite light source of radius 20 kpc , the resulting images, obtained with the GYOTO [68] ray-tracing code, are shown in Fig. 4 for different source positions.  \nThey are consistent with the expectation for the gravitomagnetic field as plotted in Fig. 5 of [11], generated by a pair of rods with opposite Newman-Unti-Tamburino (NUT) charges located along the z -axis at r 0 < z < R and -R < z < -r 0 . In the equatorial plane, ⃗ H | z =0 = | H | ∂ z points always in the same direction orthogonal to the plane; hence, light rays passing through opposite sides of the galaxy are deflected in the same direction (the ∂ y direction). The rays do not cross along the line of sight of an observer aligned with the light source and foreground galaxy, resulting in a single image, shown in Fig. 4(b) for an observer at x O = -0 . 1 Gpc ; such image is moreover deformed and shifted in the negative y -direction (by a huge angle ≈ 150 arcsec ). Observe that, for a spherical lens with M = 10 12 M ⊙ (approximately the Milky Way's mass), this aligned setting would yield instead an Einstein ring with angular radius [26, 27, 69] θ E = √ 4 Md ls / ( d s d l ) ≈ 9 arcsec , where d ls = x s = 1Gpc is the distance between the lens and the source (located at x l = 0 ), and d s = ( x s -x O ) = 1 . 1 Gpc and d l = -x O = 0 . 1 Gpc are the distances between the observer and, respectively, the source and lens.  \nLight sources shifted from the optical axis in the negative y -direction yield also a single image, consistent with the fact  \nFigure 3. Gravitational light deflection (numerical results) in the Balasin-Grumiller solution for r 0 = 1kpc , R = 100kpc , V 0 = 220km / s , µ ≈ 1 , and a light source at ( x, y, z ) = (10 3 Mpc , 0 , 0) . Left panel: in the equatorial plane, light rays through opposite sides of the body are deflected in the same direction. Right panel: light rays starting out in the z O x plane. Each color corresponds to a pair of rays with symmetric angles about the equator; these are made to diverge along the z -direction, and are moreover deflected along the positive y -direction if they approach the z -axis at | z | < R , and in the opposite direction at | z | > R . This is consistent with the presence of a pair of axial (oppositely charged) NUT rods located at r 0 < | z | < R , whose gravitomagnetic field (Fig. 5 of [11]) drives the deflection. Multiple images of the source occur at some points [see Fig. 4(d)], namely on the y > 0 side of the equatorial plane, where light rays cross; but no Einstein rings are possible.  \n<!-- image -->  \nθ  \nFigure 4. Images corresponding to the setting in Fig. 3, for a spherical light source of of radius R s = 20kpc , at a distance 1 . 1 × 10 3 Mpc from the observer, located along the x -axis at ( x O , y O , z O ) = ( -10 2 Mpc , 0 , 0 ). Angular coordinates represent altitude (zero in the equatorial plane) and azimuth (zero along the negative x -semi-axis) in the observer's sky. Panel (a) is the unlensed image of the light source, located at ( x s , y s , z s ) = (10 3 Mpc , 0 , 0 ), in Minkowski spacetime. Panels (b)-(c) are images in the Balasin-Grumiller spacetime. In (b) the source is again located at ( x s , y s , z s ) = (10 3 Mpc , 0 , 0 ) (i.e., observer, lens, and source all aligned along the x -axis); the result is a single image, oblately deformed, and shifted in the negative y -direction. Panel (c) is the single image of a source located, in the x O z plane, at ( x s , y s , z s ) =(10 3 Mpc , 0 , 1 Mpc) . In panel (d) the source is located in the equatorial plane at ( x s , y s , z s ) =(10 3 Mpc , 1 . 64 Mpc , 0) [i.e., ( r s , ϕ s , z s ) =(10 3 Mpc , 339 arcsec , 0) ]. The setting is equivalent to an observer at ( x O , y O , z O ) = ( -10 5 kpc , 164 kpc , 0 ) in the left panel of Fig. 3; the result are multiple distorted images of the source. All the images correspond to numerical simulations performed with the GYOTO ray-tracing code [68].  \n<!-- image -->  \nthat in the y < 0 side of the setting in Fig. 3 light rays do not cross. The same applies to sources shifted along the x O z plane, as shown in Fig. 4(c). Some light rays do cross on the y > 0 side of the setting in Fig. 3, leading to multiple images of the source, shown in Fig. 4(d) (for the equivalent situation of an observer along the x -axis and a source displaced in the positive y -direction by the same angle relative to the axis). These images display moreover a huge angular separation [about 290 arcsec in Fig. 4(d)]. Therefore, it is clear that the BG model cannot produce Einstein rings such as those observed in [32, 33]. The lensing effects it generates, besides of a very different type, are moreover of a magnitude much larger than observed.", 'C. Gravitomagnetic dipole model': "In [70] a galactic model based on a so-called 'gravitomagnetic dipole' is proposed. It consists of the solution obtained in [71, 72] from a (nonlinear) superposition of two NUT solutions, interpreted as a pair of NUT 'objects' of equal masses and equal in magnitude, but opposite, NUT charges. The lengthy expression for the metric is given in Eqs. (1)-(10) of [70]. It depends only on three parameters: the mass M and NUT charge ± Q NUT ( ± ν , in the notation of [70, 71]) of each object, and their separation 2 k . For sufficiently large k , the objects are interpreted as a pair of NUT black holes connected by a spinning Misner string [70]. The gravitomagnetic  \n<!-- image -->  \nFigure 5. Gravitational light deflection (numerical results) in the gravitomagnetic dipole model, for Q NUT = ± M , separation 2 k = 3 . 6 × 10 6 M , and a light source at a distance r = 10 10 M . No convergence of light rays is produced, which immediately rules out this setting as a viable galactic model. Left panel: rays in the equatorial plane are made to diverge. Right panel: light rays starting out in the z O x plane (each color corresponds to a pair of rays with symmetric angles about the x -axis). They are deflected orthogonally to that plane; along the positive y -direction if they approach the z -axis in the region between the two NUT black holes, | z | < k , and in the opposite direction otherwise, as expected from the gravitomagnetic field plotted in Fig. 6.  \n<!-- image -->  \nFigure 6. Left panel: x -z plot of the gravitomagnetic field ⃗ H for the gravitomagnetic dipole model with Q NUT = ± M , and separation 2 k = 3 . 6 × 10 6 M ; color represents field strength (blue=weak, yellow=strong). Right panel: radial component of the equatorial gravitoelectric field ⃗ G , as a function of r . Albeit attractive, | G r | does not monotonically decrease with r .  \n<!-- image -->  \nfield (4) plotted in Fig. 6 is indeed consistent with a pair of opposite gravitomagnetic monopoles (NUT singularities). We choose the values ν ≃ M and k = 1 . 8 × 10 6 M as proposed in Eqs. (82) and (85) of [70], claimed therein 4 to yield, for M = 2 . 6 × 10 11 M ⊙ ( M ⊙ ≡ solar mass), a nearly flat velocity profile for circular equatorial geodesics approximately  \nmatching the observed flat region of the Milky Way rotation curve.  \nThe trajectories of light rays emitted from a light source in the equatorial plane at r = 10 10 M , corresponding (again, for M = 2 . 6 × 10 11 M ⊙ ) to 10 2 Mpc , i.e., roughly 100 times the average distance between nearby galaxies, is plotted in Fig. 5. No convergence of light rays is produced (thus no Einstein rings, nor multiple image of the source); the rays diverge. This completely rules out this model as a viable galactic model.  \nLight rays starting in the equatorial plane diverge along that plane; the effect, however, is actually not caused by the gravitomagnetic field. It occurs as well when Q NUT = 0 ⇒ ⃗ A = ⃗ H = 0 (in which case the setting is held static by the tension of the string). In fact, the plot corresponding to Q NUT = 0 is only slightly changed comparing to the left  \npanel of Fig. 5, being then symmetric with respect to the x -axis. The effect (which, in appearance, reminds a repulsive scattering) occurs as well for massive particles (i.e., timelike worldlines). Inspection of the radial equation d 2 r/dτ 2 = -Γ r αβ U α U β shows that it stems from the weakness of the attractive term -Γ r 00 E 2 ( g 00 ) 2 = e -2Φ G r E 2 (see Footnote 1; here E ≡ -U α ∂ α t = -U 0 is the particle's energy per unit mass), due to the unusual behavior of the gravitoelectric field ⃗ G , plotted in Fig. 6: although always attractive ( G r < 0 ), its magnitude does not monotonically increase with decreasing r ; in fact, within the region r/M < 1 . 5 × 10 6 (which includes the flat profile range), | ⃗ G | decreases approaching r = 0 , with lim r → 0 ⃗ G = 0 . This contrasts with e.g. the situation in the Schwarzschild geometry (where | G r | monotonically increases until the horizon, where lim r → r + G r = -∞ ), or in Levi-Civita or Lewis-Weyl cylinders [16], and causes the centrifugal term -Γ r ϕϕ L 2 ( g ϕϕ ) 2 = (1 / 2) g ϕϕ,r L 2 ( g ϕϕ ) 2 /g rr to sharply dominate the scattering (hence the apparent repulsion). Here L ≡ U α ∂ α ϕ = U ϕ is the test particle's angular momentum per unit mass. The same conclusion can be reached by inspecting the effective potential V eff ( E,L,r ) that follows from the equation U α U α = -1 ⇔ ( dr/dτ ) 2 = -2 V eff , so that d 2 r/dτ 2 = -dV eff /dr .  \nLight rays starting out in the z O x plane are deflected orthogonally to that plane, via the action of the gravitomagnetic field ⃗ H .  \nIt should moreover be noted that (as correctly noticed in [70]) the z -distance 2 k = 3 . 6 × 10 6 M between the NUT black holes is comparable or larger than the galaxy size, which adds to the unrealistic character of the model.", 'VI. CONCLUSION': "We have shown that, in light of the experimentally measured galactic rotation curves and gravitational lensing, general relativity cannot resolve the missing mass problem usually attributed to dark matter, and considering nonlinear corrections only aggravates the problem. In the process we wrote the equations for light propagation in the quasi-Maxwell formalism, and by gravitational lensing ruled out any galactic model (linear or nonlinear) based on gravitomagnetism, as well as the recent 'dipole' models (gravitomagnetic or not, linear or not).", 'Acknowledgments': 'We thank A. Pollo, E. Malec, P. Mach, and D. Grumiller for correspondence and very useful discussions. We are also indebted to T. Paumard and F. Vincent for generous and crucial help in implementing the GYOTO raytracing code, and to the anonymous referee for valuable remarks and suggestions. L.F.C. and J.N. were supported by FCT/Portugal through Projects UIDB/MAT/04459/2020 and UIDP/MAT/04459/2020.', 'Appendix A: Lensing in the Schwarzschild and Kerr spacetimes': "For comparison with the lensing effects of the models in Secs. V B and V C, in order to evince their unphysical character, we present in Figs. 7 and 8 the corresponding ray trajectories and lensed images produced by a Schwarzschild black hole (displaying some essential features common to other compact and spherically symmetric lenses) and, to display the role of the gravitomagnetic field, of a fast spinning Kerr black hole.  \nIn the Schwarzschild case, the deflection is the same for rays starting at equal (in magnitude) angles relative to the axis connecting the source and the black hole; this leads to perfect Einstein rings as viewed by observers along this axis [20, 29, 69, 73, 74] ('optical axis'; here, the x -axis). Namely, the image of an idealized point source is a (infinitely bright) circle; for a real, finite source, the image is a ring of finite width [27, 28], exemplified in the simulation in Fig. 8(b).  \nIn the Kerr spacetime, the lens is not spherically symmetric; consequently, for an idealized point source, the Einstein ring does not form [20, 50-52]. The primary caustic surface [75] behind the black hole no longer degenerates into the optical axis, having instead a finite astroid section (see e.g. Figs. 2-3 in [50]), and the image of point sources (inside or outside it) are a finite set of points (not some 1D contour) [50, 52, 75]. For finite sources, however, rings can still form, when the source is large enough to cover to whole caustic section [27, 28]. In this case, the impact of the black hole's spin is the following. The gravitomagnetic field (which is approximately dipole-like, as depicted in Fig. 2) causes the deflection angles of equatorial light rays through opposite sides of the black hole to differ (left panel of Fig. 7). Hence, rays starting at equal but opposite angles relative to the x -axis will not cross along that axis; conversely, those that do, arrive at different angles. On light rays with initial direction along the z O x plane, the gravitomagnetic field causes a deflection along the y -direction, causing them not to cross along the x -axis (besides not contributing to their convergence which, as in the Schwarzschild case, is along the z -direction). Consequently, the caustic behind the black hole is transversely displaced (in the y -direction) from the optical axis, cf. e.g. Fig. 2 in [50]. As a result, the image of a finite source along the optical axis will in general be two arcs, as exemplified in the simulation in Fig. 8(c); if the source extends until the caustic location (and covering it) the arcs will close into a deformed ring, weakened at the poles, and slightly shifted in the negative y -direction. This is similar to the situation in a Schwarzschild lens of the same mass, but for a source displaced from the optical axis. For a source centered at the center of the caustic section, and fully covering it, the image will be an almost perfect Einstein ring similar to that produced by a Schwarzschild lens of the same mass for a source along the optical axis, only shifted along the y -direction, as shown in Fig. 8(d). The ring's angular diameter (hence the lens power), in particular, is the same. (Such a ring is also slightly deformed [51]; the effect is however almost unnoticeable in the simulations performed.) It is thus clear that the gravitomagnetic field does not enhance the optical effect generating Einstein rings. Indeed, for sources  \nFigure 7. Gravitational light deflection (numerical results) in the Schwarzschild and in a Kerr spacetime with rotation parameter a = S/M = 0 . 9 . The source is at a distance r = 100 M . First row: equatorial plane; second row: light rays starting out in the z O x plane. Each color corresponds to a pair of rays with symmetric angles about the x -axis. In the Schwarzschild case, the deflection is the same for rays starting at equal (in magnitude) angles relative to the x -axis connecting the source and the black hole; this leads to perfectly circular Einstein rings as viewed by observers along the x -axis [Fig. 8 (b)]. In the Kerr spacetime, the deflection of equatorial light rays through opposite sides of the black hole differs; rays starting at equal (in magnitude) angles will not cross along the x -axis; and those that do cross along the x -axis arrive at different angles. Light rays with initial direction along the z O x plane suffer (in addition to the convergence along the z -axis), a deflection along the y -direction. These gravitomagnetic effects do not enhance the convergence of light rays for observers along the x -axis, instead mimicking a non-aligned Schwarzschild lens, as shown in Figs. 8 (c)-(d).  \n<!-- image -->  \nFigure 8. Images produced by the settings in Fig. 7, for a spherical light source of radius R s = 3 M . Panel (a) is the unlensed image of the light source in Minkowski spacetime. Panel (b) is the image of the source when placed behind a Schwarzschild black hole (of mass M ) along the optical axis (i.e., light source, black hole, and observer all aligned along the x -axis). The source is located at ( x s , y s , z s ) = (100 M, 0 , 0) , and the observer at ( x O , y O , z O ) = ( -20 M, 0 , 0) ; the black hole is at the origin. The resulting image is a perfect Einstein ring. Panel (c): image produced by a similarly aligned setting for a Kerr black hole with a = 0 . 9 ; it results in a pair of arcs. Panel (d): for the same Kerr black hole and observer position, the light source is now placed at the center of the primary caustic ( x s , y s , z s ) ≈ (100 M, -6 M, 0) , which is transversely displaced [52] from the optical axis by ∆ y ≈ -6 M . (The caustic section is therein about 0 . 1 M wide, cf. Eq. (19) of [52], much smaller than the source's diameter). The image is an Einstein ring almost identical to panel (b), only transversely shifted in the y -direction. 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