Frame-dragging: meaning, myths, and misconceptions

Originally introduced in connection with general relativistic Coriolis forces, the term $\textit{frame-dragging}$ is associated today with a plethora of effects related to the off-diagonal element of the metric tensor. It is also frequently the subject of misconceptions leading to incorrect predictions, even of nonexistent effects. We show that there are three different levels of frame-dragging corresponding to three distinct gravitomagnetic objects: gravitomagnetic potential 1-form, field, and tidal tensor, whose effects are independent, and sometimes opposing. It is seen that, from the two analogies commonly employed, the analogy with magnetism holds strong where it applies, whereas the fluid-dragging analogy (albeit of some use, qualitatively, in the first level) is, in general, misleading. Common misconceptions (such as viscous-type"body-dragging") are debunked. Applications considered include rotating cylinders (Lewis-Weyl metrics), Kerr, Kerr-Newman and Kerr-dS spacetimes, black holes surrounded by disks/rings, and binary systems.


Introduction
The term "dragging" in the context of relativistic effects generated by the motion of matter was first coined by Einstein in his 1913 letter to Mach [1], in connection with the general relativistic Coriolis force generated in the interior of a spinning mass shell, causing the plane of a Foucault pendulum to be "dragged around". The term appeared again in the papers by Lense and Thirring [2], namely, a "dragging coefficient" was defined as the (inverse) ratio between the shell's angular velocity (Ω), and the angular velocity (Ω ) of the reference frame for which the Coriolis forces vanish in its interior (corresponding to the local inertial frame). Although the notion was already implicit in these works, it was not, however, until Cohen's 1965 paper [3] that the designation "dragging of inertial frames" first appeared; therein Ω was dubbed "angular velocity of the inertial frames" inside the shell. The underlying principle is the same as for the Coriolis forces that arise near a spinning body, in a reference frame fixed to the distant stars. It is also the same for the precession of gyroscopes placed therein [4,5], in which case one also talks about dragging of the "compass of inertia" (the compass of inertia being defined as a system of axes undergoing Fermi-Walker transport, and physically realized precisely by the spin vectors of a set of guiding gyroscopes [6][7][8]). All these effects can be assigned [9] to the action of a Coriolis or "gravitomagnetic" field generated by the source's motion, and they are all commonly refereed to as "frame-dragging" effects (e.g. [7,[10][11][12][13][14][15]).
Later, another class of effects [10,[16][17][18] started being dubbed frame-dragging as well. They pertain to axistationary metrics, and include: the fact that the observers of zero angular momentum (ZAMOs) have non-vanishing angular velocity in a coordinate system fixed to the distant stars (dragging of the ZAMOs); and, conversely, objects with zero angular velocity have non-zero angular momentum. These are facets of a principle which, as we shall see, is different from the frame-dragging class mentioned above.
More recently [19], yet another type of effect -a local (i.e., tidal) one, stemming from the curvature tensor -has been rightfully dubbed frame-dragging: the precession of a gyroscope with respect to a system of axes attached to guiding gyroscopes at an infinitesimally close point ("differential precession").
In order to gain intuition into the "frame-dragging" effects, two analogies have been put forth: the electromagnetic analogy, in particular, between the magnetic field and the general relativistic Coriolis field (thus dubbed "gravitomagnetic field"), and the fluid-dragging analogy. The former is based on solid equations, best known in weak field slow motion approximations [7,[20][21][22][23][24][25], but with exact versions [13,[26][27][28][29][30][31][32][33][34][35] holding in arbitrarily strong fields, and is known for providing a familiar and reliable formalism. The fluid analogy, initially proposed in [5,36], and then supported by other authors (e.g. [7,12,13]), consists of an analogy drawn between the effects created by the rotation of a body immersed in a fluid, and the frame-dragging effects. Albeit providing (to a limited extent) a certain qualitative intuition for the second class of effects mentioned above (dragging of the ZAMOs), it is generically misleading and the source of most misconceptions and incorrect predictions concerning frame-dragging. Some of these were noticed already in the literature, most notably in a paper by Rindler [37], where, in the framework of a linearized theory approximation, several inconsistencies of the fluid-dragging model are pointed out, and the gravito-electromagnetic analogy is recommended instead.
In this paper we start (Sec. 2) by observing that the three classes (or levels) of "frame-dragging" effects are governed by distinct mathematical objects (the gravitomagnetic potential 1-form A, the gravitomagnetic field, and the gravitomagnetic tidal tensor) corresponding to different orders of differentiation of A, and underlying physical principles (dragging of the ZAMOs, and dragging or differential dragging of the compass of inertia); moreover, these levels are largely independent, there existing solutions displaying only the first or second levels, as well as phenomena where different levels of frame-dragging act oppositely (Secs. 2.3 and 2.4). Section 3 is devoted to debunking the common wrong notion that when a source (e.g. a black hole) spins, it forces test bodies around into rotation ("body-dragging"). We start by generalizing Rindler's paper to the exact theory, using the exact 1+3 "gravitoelectromagnetic" (GEM) formalism, exemplifying with an imaginary space station around a spinning black hole. We consider also the reciprocal problem (a rotating ring around a non-spinning black hole), and, to clear any doubt that no such effect takes place, the static equilibrium positions for test particles in the equatorial plane of spinning black hole spacetimes (Kerr-de Sitter and Kerr-Newman). Finally, we consider a notable phenomenon driven by framedragging -bobbings in binary systems -where the body-dragging picture predicts the opposite of the true effect.

Distinct effects under the same denomination
We will derive here the exact equations describing the different types of frame-dragging (equations of "gravitoelectromagnetism", or GEM), considering stationary spacetimes, where their formulation (the so called 1+3 "quasi-Maxwell" formalism [26-28, 31-34, 38]) is particularly simple and intuitive. A generalization for arbitrary time-dependent fields is given in Appendix A. The line element ds 2 = g αβ dx α dx β of a stationary spacetime can generically be written in the form Observers of 4-velocity (i.e. whose worldlines are tangent to the timelike Killing vector field ∂ t ) are at rest in the coordinate system of (1); they shall be called "laboratory" observers. The quotient of the spacetime by the worldlines of the laboratory observers yields a 3-D manifold Σ in which h ij is a Riemannian metric, called the spatial or "orthogonal" metric. It can be identified in spacetime with the projector orthogonal to u α (space projector with respect to u α ), and yields the spatial distances between neighboring laboratory observers, as measured through Einstein's light signaling procedure [26].

Sagnac effect and dragging of the ZAMOs
Unlike translational motion, which is inherently relative, rotation is not, and is physically detectable. A way of detecting the absolute rotation of an apparatus (i.e., its rotation relative to the "spacetime geometry" [10], whose meaning shall be clear below) is the Sagnac effect [34,35,[39][40][41][42][43]. It consists of the difference in arrival times of light-beams propagating around a closed path in opposite directions. In flat spacetime, where the concept was first introduced (see e.g. [35,39,40,43] and references therein), the time difference is originated by the rotation of the apparatus with respect to global inertial frames (thus to the "distant stars"), see e.g. Fig. 1 in [34]. In a gravitational field, however, it arises also in apparatuses which are fixed relative to the distant stars (i.e., to asymptotically inertial frames) [34,[40][41][42][43]; in this case one talks about "frame-dragging". Both effects can be read from the spacetime metric (1), encompassing the flat Minkowski metric expressed in a rotating coordinate system, as well as arbitrary stationary gravitational fields. Along a photon worldline, ds 2 = 0; by (1), this yields two solutions dt = A i dx i ± e −Φ h ij dx i dx j , the + sign corresponding to the future-oriented worldline; 1 therefore where dl ≡ h ij dx i dx j is the spatial distance element. Consider photons constrained to move within a closed loop C in the space manifold Σ; for instance, within an optical fiber loop, as depicted in Fig.  1(a). Using the + (−) sign to denote the anti-clockwise (clockwise) directions, the coordinate time it takes for a full loop is, respectively, t ± =¸± C dt =¸C e −Φ dl ±¸C A i dx i ; therefore, the Sagnac coordinate time delay ∆t is (e.g. [34]) where we identified A i dx i with the 1-form A ≡ A i dx i on the space manifold Σ. Consider now an axistationary spacetime, whose line element (1) simplifies to (in spherical-type coordinates) In spite of being at rest, the laboratory observers (2) have, in general, non-zero angular momentum. The component of their angular momentum along the symmetry axis is, per unit mass [10,34,44], which is zero iff A φ = 0. This manifests physically as follows. Take an observer at r = r 0 , and consider a circular optical loop (e.g., an optical fiber) of the same radius around the axis θ = 0, see Fig. 1(a). Such an observer will measure a Sagnac effect; i.e., it will see light beams emitted in opposite directions along the loop not completing it at the same time, the difference in (coordinate) arrival times being, according to Eq. (4), Only observers with zero angular momentum (ZAMOs) measure no Sagnac effect. These observers are such that (u ZAMO ) φ = 0, i.e., have angular velocity That the Sagnac effect vanishes for them can easily be seen by performing a local coordinate transformationφ = φ − Ω ZAMO (r 0 )t, leading to a coordinate system where the ZAMO at r 0 is at rest. The metric form thereby obtained is diagonal at r 0 :ḡ 0φ (r 0 ) = 0, hence ∆t S = 0. This singles out the ZAMOs as those who regard the ±φ directions as geometrically equivalent; for this reason they are said to be those that do not rotate with respect to "the local spacetime geometry" [10]. If 2 g αβ r→∞ → η αβ , the coordinate system in (5) corresponds to a rigid frame anchored to the asymptotic inertial frame at infinity (i.e., to the distant stars). Hence, • the "laboratory" observers, at rest in a frame fixed to the distant stars, have non-zero angular momentum (6) (measuring a Sagnac effect); • the zero angular momentum observers have non-zero angular velocity (8) in a coordinate system fixed to the distant stars (or as "viewed" from an observer at infinity).
These features are usually assigned to "frame-dragging"; we point out that it in fact consists of the dragging of the ZAMOs, and the gravitomagnetic object governing it is the potential 1-form A (or equivalently, the gravitomagnetic vector potential A). These are the frame-dragging effects involved in arranging the bodies' angular momentum/angular velocities in e.g. the black hole-ring and black hole-disk systems in [18,45], as well as in black saturn systems [17], as discussed in Sec. 3 and Fig.  2 below.

Dragging of the compass of inertia: gravitomagnetic field and Lense-Thirring effects
Consider a (point-like) test particle of worldline x α (τ ), 4-velocity dx α /dτ ≡ U α and mass m. The space components of the geodesic equation DU α /dτ = 0 yield 3 , for the line element (1) [26,27,[31][32][33][34], where γ = −U α u α = e Φ (U 0 − U i A i ) is the Lorentz factor between U α and u α , is the Levi-Civita covariant derivative with respect to the spatial metric h ij , with Γ(h) i jk the corresponding Christoffel symbols, and are vector fields living on the space manifold Σ with metric h ij , dubbed, respectively, "gravitoelectric" and "gravitomagnetic" fields. These play in Eq. (9) roles analogous to those of the electric ( E) and magnetic ( B) fields in the Lorentz force equation, The analogy motivates also dubbing A gravitomagnetic vector potential. Here,∇ denotes covariant differentiation with respect to the spatial metric h ij [i.e., the Levi-Civita connection of (Σ, h)], Equation (10) is the standard 3-D covariant acceleration, and Eq. (9) describes the acceleration of the curve obtained by projecting the time-like geodesic onto the space manifold (Σ, h), with U as its tangent vector [identified in spacetime with the projection of U α onto (Σ, h): The physical interpretation of Eq. (9) is that, from the point of view of the laboratory observers, the 2 Actually, the weaker condition that the congruence of observers at rest (2) is inertial at infinity suffices. 3 Computing the Christoffel symbols Γ i spatial trajectory will appear accelerated, as if acted upon by the fictitious force F GEM (standing here for "gravitoelectromagnetic" force 4 ). In other words, the laboratory observers measure inertial forces, which arise from the fact that the laboratory frame is not inertial ; in fact, G and H are identified in spacetime, respectively, with minus the acceleration and twice the vorticity of the laboratory observers: They may be regarded as the relativistic generalization of, respectively, the Newtonian gravitational field and the classical Coriolis field, encompassing them as limiting cases [9]. It is H that governs, via Eq. (9), the Coriolis (i.e., gravitomagnetic) forces generated inside a spinning hollow sphere, noted by Einstein [1,46] and Thirring [2]; or those acting on test particles in the exterior field of a spinning body, causing the Lense-Thirring orbital precession [2,7]. It governs also the "precession" of gyroscopes with respect to the reference frame associated to the coordinate system in (1): according to the Mathisson-Papapetrou equations [24,[47][48][49][50], under the Mathisson-Pirani spin condition [47,51], the spin vector S α of a gyroscope (i.e., a spinning pole-dipole particle) of 4-velocity U α = dx α /dτ is Fermi-Walker transported along its center of mass worldline x α (τ ), where a α ≡ DU α /dτ . The spin vector is spatial with respect to U α , S α U α = 0, and so, for a gyroscope whose center of mass is at rest in the coordinates of (1), U α = u α [see Eq.
(2)], the space part of Eq. (13) reads (using the Christoffel symbols in footnote 3, and noting that resembling the precession of a magnetic dipole µ in a magnetic field, D S/dτ = µ × B. Likewise, the Sagnac time delay in an optical gyroscope (i.e., a small optical loop C) is also governed by the gravitomagnetic field H, as can be seen by applying the Stokes theorem to (4), considering the surface S with boundary ∂S = C, is the Levi-Civita tensor of the space manifold (Σ, h), and Area S the "area vector" of the loop (see [34] and footnote on p. 7 therein).
The gravito-electromagnetic analogy { G, H} ↔ { E, B} also extends to the field equations: where the equations for∇ · G and∇ × H are, respectively, the time-time and time-space projections of the Einstein field equations R αβ = 8π T αβ − 1 2 g αβ T γ γ , with ρ ≡ T αβ u α u β and J α ≡ −T αβ u β , and the equations for∇· H and∇× G follow directly from (11). They strongly resemble the Maxwell equations in a rotating frame, see Table 2 of [32]. Equation (13) tells us that the gyroscope's axis is fixed with respect to a Fermi-Walker transported frame, which mathematically defines a locally non-rotating frame (e.g. [6,10]); it is said to follow the "compass of inertia" [6,7,32]. This agrees with the notion that gyroscopes are objects that oppose changes in the direction of their rotation axes. Hence, the gyroscope "precession" in (14) is thus in fact minus the angular velocity of rotation H/2 of the coordinate basis vectors relative to a locally non-rotating frame. Consider now the case that g αβ r→∞ → η αβ , in which the coordinate system in (1) corresponds to a rigid frame anchored to the asymptotic inertial frame at infinity. So, at infinity, the reference frame is inertial; however, at finite distance from the mass-energy currents that [by Eq. (17)] source H, one has, in general, H = 0, and so that same rigid frame is rotating (besides being accelerated, as the observers at rest therein are not freely falling, ∇ u u α = −G α = 0). One can thus say that the motion of the sources (or mass-energy currents, in general) drags the local inertial frames, or the local compass of inertia. In some literature this is cast as the appearance of vorticity [52,53], the two notions being equivalent 5 via Eq. (12).

Competing effects -circular geodesics
Let U α be the 4-velocity of a test particle moving along an equatorial circular geodesic in an axistationary spacetime (with reflection symmetry about the equatorial plane [54]), and L = g µν U µ U ν /2 the corresponding Lagrangian. The angular velocity Ω geo ≡ dφ/dt = U φ /U 0 of the circular geodesics is readily obtained from the Euler-Lagrange equations, g φφ,r Ω 2 geo + 2g 0φ,r Ω geo + g 00,r = 0 .
Using g 0φ = −g 00 A φ , noticing that reflection symmetry implies, in the equatorial plane, A φ,θ = 0, we have, by (11), Hence, the gravitomagnetic clock effect consists of the sum of two contributions originating from the two types of frame-dragging in Secs. 2.1 and 2.2: the Sagnac time delay (7) around the circular loop, due to the dragging of the ZAMOs, governed by A φ , plus a term due to the gravitomagnetic (or Coriolis) forces generated by the dragging of the compass of inertia, governed by the gravitomagnetic Cylinder H H = 0 Figure 1: (a) Sagnac effect in an optical loop around a spinning body (dragging of the ZAMOs): a "laboratory" observer, at rest with respect to the distant stars, sends light beams propagating in opposite directions along the loop; they take different times to complete the loop, the co-rotating one arriving first, by a time difference ∆tS < 0, Eq. (7). (b) For particles in circular geodesics around a Kerr black hole it is the other way around: the co-rotating one has the longer period. This is down to the combination of two oppositely competing effects: the dragging of the ZAMOs, tending to decrease the period of the co-rotating orbit vs. the gravitomagnetic force mγ U × H in Eq. (9) (dragging of the compass of inertia), which is repulsive/attractive for co/counter-rotating orbits, thereby slowing/speeding up the orbit, respectively. The latter effect prevails, so that ∆tgeo = ∆tS + ∆tH > 0. (c) Around an infinite spinning cylinder of the Weyl class, H = 0; hence, only the dragging of the ZAMOs subsists, and the situation for circular geodesics is opposite to (b) [thus similar to (a)]: the co-rotating geodesic has the shortest period, the difference reducing exactly to the Sagnac time delay, ∆tgeo = ∆tS < 0.
field H. The physical interpretation of the latter is as follows: for circular orbits, the gravitomagnetic , being centrifugal or centripetal depending on the ±φ direction of the orbit, thus respectively decreasing or increasing the overall attraction, and, consequently, the velocity of the orbits. See Figure 1 It is important to notice that the two contributions in (20) are independent. In fact, there are solutions for which H vanishes whilst A is non-zero, as is the case of the Lewis metric of the Weyl class, describing the exterior gravitational field produced by infinitely long rotating cylinders. The metric is given, in star-fixed ("canonical") coordinates, by Eq. (61) of [34], yielding, in the form (1), Here 0 ≤ λ m < 1/4 and j are, respectively, the Komar mass and angular momentum per unit length, and α the parameter governing the angle deficit (j > 0, for a cylinder spinning in the positive φ direction). Trivially dA = 0 ⇒ H = 0, hence ∆t H = 0 and so i.e., the difference in period between co-and counter-rotating circular geodesics equals precisely the Sagnac time delay for photons, the co-rotating one having a shorter period. The two contributions can even be opposing, as is the case for the Kerr metric. We have in this case, in the equatorial plane, (21) and so (observe that always 6 r > 2M , in order for the counter-rotating orbit to exist) i.e., the two contributions have opposite signs. Namely, the dragging of the ZAMOs tends to decrease the period of the orbit co-rotating with the black hole by ∆t S , as compared to the counter-rotating orbit; however, the gravitomagnetic force mγ U × H (dragging the compass of inertia) does the opposite, tending to increase the period of the co-rotating orbit (case in which U × H is repulsive), as compared to the counter-rotating orbit (case in which U × H is attractive), by ∆t H , see Fig.  1(b). Since ∆t H > −∆t S , it is the gravitomagnetic force that prevails, making the co-rotating orbits slower overall than the counter-rotating ones.
The gravitomagnetic force and the corresponding ∆t H have a close electromagnetic analogue in the magnetic force q U × B exerted on charged test particles orbiting a spinning charged body, see Eq. (30) of [34] -only with a different sign, which manifests that (anti-) parallel mass/energy currents have a repulsive (attractive) gravitomagnetic interaction, as opposed to magnetism, where (anti-) parallel charge currents attract (repel) (see [59] and Sec. 13.6 of [60]).

2.4
Gravitomagnetic "tidal" effects: "differential" dragging and force on gyroscopes A third class of frame-dragging effects that has been (more recently) discussed in the literature [19,61,62], distinct from those in Secs. 2.1 and 2.2, is the "differential precession" of gyroscopes. It consists of the precession of a gyroscope relative to a frame attached to the spin axes of guiding gyroscopes at a neighboring point. The effect was originally derived 7 in [19]; we briefly re-derive it below in a straightforward manner, using Fermi coordinates. The spin vector of a gyroscope in a gravitational field is Fermi-Walker transported along its worldline, according to the Mathisson-Papapetrou-Pirani Equation (13). Consider an orthonormal tetrad eα Fermi-Walker transported along the worldline L(τ ) of the set of gyroscopes 1. There is a coordinate system {X α }, rectangular at L, and adapted to such tetrad (∂/∂X α | L = eα), the so-called [64] "Fermi coordinates", where the metric takes, to order O(X 2 ), locally the form [65], setting therein ω i = 0). Here a α = DU α /dτ is the acceleration of the fiducial worldline L(τ ) = L(T ), and R αβγδ ≡ R αβγδ (T ) are components of the curvature tensor evaluated along L. Let e α ≡ ∂/∂X α denote the coordinate basis vectors and Γ α βγ its Christoffel symbols, Γ α βγ e α = ∇ e β e γ . Along L, the vectors e α are Fermi-Walker transported, so ∇ e0 e i , e j | X k =0 = Γ j 0i (L) = 0. Hence gyroscopes moving along L (by definition), or momentarily at rest (U i = 0) at some event P 1 : (13). However, at some location X i 2 outside L (X i 2 = 0), we have Γ j 0i (X 2 ) = R j ik0 X k 2 = 0, and so the basis vectors e α are no longer Fermi-Walker transported. That means that gyroscope 2, at the location X i 2 , will precess with respect to this 6 The innermost circular geodesics, in each direction, are the photon orbits whose radius is r ph± = 2M {1 + cos[2 arccos(∓a/M )/3]} [16], and so r ph− ≥ 3M . 7 In a perhaps less straightforward manner though, and with unnecessary restrictions. Namely, in [19] it is assumed that (besides being momentary comoving) the gyroscopes at L and X i 2 have the same acceleration. This is not necessary, as shown here; in order for (25) to hold, one needs only U i 2 = 0, i.e., that gyroscope 2 has momentarily zero "Fermi relative velocity" [63] with respect to gyroscopes at L. Moreover, the results therein hold only for vacuum, as the magnetic part of the Weyl tensor is used instead of H αβ . coordinate system. If the gyroscope is therein at rest (U i 2 = 0), we have where in the first equality we noticed that S 0 2 = O(X 2 ) and, in the second, that U 0 2 = (−g 00 ) −1/2 = 1 + O(X 2 ), while only terms to first order in X are to be kept in (24) to the accuracy at hand. In the last equality we noticed that U α = δ α 0 and that, by the definition of the coordinate system {X α } (see e.g. Fig. 13.4 of [10]), X k 2 = δx k are components of the vector δx α = (0, δx i ) at L tangent to the (unique) spatial geodesic emanating orthogonally from L and passing through the event P 2 : (T 2 , X i 2 ), whose length equals that of the geodesic segment. It can be interpreted as the "separation vector" between P 2 and the simultaneous (T 2 = T 1 ) event P 1 ∈ L. Using the space projection relation (cf. Eq. (5) in [32] where H αβ ≡ R αµβν U µ U ν = σλ αµ R σλβν U µ U ν /2 is the "gravitomagnetic tidal tensor" (or "magnetic" part of the Riemann tensor) as measured along L, and in the second equation we noted that . Thus, δ Ω G is the angular velocity of precession of gyroscopes at location X i 2 with respect to the Fermi frame locked to the guiding gyroscopes at the neighboring worldline L (of course, this is just minus the angular velocity of rotation of the basis vectors e i | X2 relative to Fermi-Walker transport). We can cast this effect as a differential dragging of the compass of inertia.
Another manifestation is in the geodesic deviation equation [7,10]). In vacuum, one can decompose the Riemann tensor in terms of the gravitoelectric (E u ) αβ ≡ R αµβν u µ u ν and gravitomagnetic (H u ) αβ = R αµβν u µ u ν tidal tensors as measured by some laboratory observer u α , see decomposition (44) of [50]; hence, with respect to such an observer, the relative acceleration of nearby test particles (of 4-velocity U α ) comes in part from (H u ) αβ , which could thus be measured through gravity gradiometers [68].

Levels of Gravitomagnetism/"Frame-Dragging"
Governing gravitomagnetic object Physical effect A (gravitomagnetic vector potential) Dragging of the ZAMOs: • Sagnac effect • part of gravitomagnetic "clock effect" Dragging of the compass of inertia: • local Sagnac effect in light gyroscope • part of gravitomagnetic "clock effect" • Differential precession of gyroscopes (for a H αβ -H relation valid for orthonormal frames in arbitrary spacetimes, see Eq. (110) of [32]). We can thus say that H αβ is essentially [to leading order, in the case of (27)] a derivative of the gravitomagnetic field H. So, we can cast the different frame-dragging effects in the literature into the three levels of gravitomagnetism in Table 1, corresponding to three different orders of differentiation of the gravitomagnetic 1-form A.
We emphasize that these levels are independent: there exist solutions where only the first level (A) is intrinsically non-zero, as we have seen in Sec. 2.3; and others possessing the first two levels, but where the third vanishes. Examples of the latter are the Gödel universe and the uniform Som-Raychaudhuri metrics which, as discussed in detail in Sec. VII.B.3 of [69], are stationary solutions possessing zero gravitoelectric field ( G = 0) and non-zero uniform gravitomagnetic field H, leading, by virtue of Eq. (27), H αβ to vanish for the rest observers therein, whilst both H = 0 and A = 0. In such metrics, by Eq. (26), no force acts on gyroscopes at rest (whose worldlines are geodesic); and, by Eq. (25), they moreover do not precess with respect to neighboring guiding gyroscopes.
3 Frame-dragging is never "draggy" 8 -no body-dragging It is a widespread myth that when a massive body (e.g. a black hole) rotates, it drags everything around it. It seemingly originates from the fluid dragging analogy proposed originally in [5,36] and reinforced in [12,13], disseminating in the subsequent literature that the dragging of inertial frames can be explained in analogy with the dragging of a viscous fluid by an immersed rotating body. It is sometimes also portrayed as the whirlpool analogy [13,70]. Such mental pictures can be extremely misleading [22,37,70]; frame-dragging, in none of its levels, produces that type of effects. What is dragged are the ZAMOs, and the compass of inertia. The former leads to Sagnac and other akin global effects (having no local effect on the motion of test particles); the latter originates Coriolis (or gravitomagnetic) inertial forces on test particles. Such forces, however, affect only bodies moving with respect to the chosen reference frame, and are orthogonal to the body's velocity; hence, never of the viscous/dragging type. This later aspect was stressed in an illuminating paper by Rindler [37], based on linearized theory, in the framework of a weak field and slow motion approximation. Rindler's conclusion can be readily generalized to the exact theory using Eq. (9) [for stationary spacetimes, or with Eq. (38) of Appendix A for arbitrary fields]. Imagine that an advanced civilization builds a circular space station around a Kerr black hole, as illustrated in Fig. 2 (a), and imagine that it is initially set fixed with respect to the distant stars (which can be set up by pointing a telescope to a distant star). Regardless of the black hole's rotation, the station will not be dragged into any rotational motion: since, initially, U = 0, by Eq. (9) the only inertial force (per unit mass) acting on any of its mass elements is the radial force produced by the gravitoelectric field (i.e., by the relativistic generalization of the Newtonian field): F GEM /m = γ 2 G = γ 2 G r ∂ r ; this force is counteracted by the stresses in the station's structure (which prevent it from collapsing), and so U = 0 for all mass elements at all times. That is, the station remains static, with e.g. its hatches pointing to the same fixed stars. It is, in particular, unaffected by the dragging of the compass of inertia. The situation is only distinct from that around a static black hole in that here the station's angular momentum per unit mass in non-zero [cf. Eq. (6)], implying, e.g., that a Sagnac effect will be measured in an optical fiber loop along the station, see Sec. 2.1 and Fig. 1(a) [the difference in arrival times for beams propagating in opposite directions along the station being given by ∆t S in (22), making therein r = R]. It implies also that it is impossible for the crew members along the station to all have their clocks synchronized (see e.g. [34], Secs. 5.3.2-5.3.3). Imagine now that a crew member starts throwing objects (test particles); by having a non-zero velocity ( U = 0) with respect to the star-fixed reference frame, gravitomagnetic (i.e., Coriolis) forces mγ U × H will now act on them, cf. Eq. (9). The gravitomagnetic field H in the equatorial plane of the Kerr spacetime is given by Eq. (21); it is orthogonal to the plane, pointing up. Consider, in particular, as depicted in Fig. 2 (a), one test particle dropped from rest from the station, and another one launched with an initial outwards radial velocity. The in-falling particle is indeed deflected in the direction of the black hole's rotation; however the outgoing one is deflected in the opposite direction, contradicting the naive fluid dragging picture. This exemplifies how different Coriolis forces are from viscous dragging-type forces. In other words, the dragging of the compass of inertia is a phenomenon very different from "body-dragging." Another example of this difference is provided by the circular geodesics around the black hole, depicted in Fig. 1 (b): the orbits are stationary, their motion is not accelerated/decelerated by the black hole's rotation (as would be the case if there were some viscous coupling). The Coriolis forces mγ U × H are in this case radial, consequently just changing the overall gravitational attraction. Since they are repulsive (attractive) for co-(counter-) rotating orbits, their effect is to actually (for a fixed radius r) make the co-rotating geodesic slower than the counter-rotating one, somewhat at odds with the naive dragging picture, as pointed out in [22,70,71].
It is also instructive to consider the reciprocal of the setting in Fig. 2 (a), i.e., a non-spinning black-hole perturbed by a rotating ring or disk around it, as depicted in Fig. 2 (b). The gravitational field produced by such setting is given by the perturbative treatments in [45] (for a ring), and in [18] (for a disk perturbing a Schwarzschild black hole). Again, regardless of the ring/disk's rotating motion, the black hole is not dragged around, in the sense of acquiring any "angular acceleration". In fact, these solutions are stationary. The impact of ring/disk's rotation is (via the dragging of the ZAMOs) causing a black hole with zero angular momentum to have a non-zero horizon angular velocity [18,45,72] and, conversely, a non-rotating black hole to have a non-zero angular momentum [45,72] -thereby even introducing a certain ambiguity in a black hole's spinning/nonspinning character. Let us see this in detail. First observe that (u ZAMO ) α ∝ δ 0 α = ∇ α t , and so the ZAMOs are orthogonal to the hypersurfaces of constant time t, i.e., such hypersurfaces are their rest spaces. We can thus say that, at each point, these hypersurfaces are rotating, with respect to Boyer-Lindquist coordinates (anchored to inertial observers at infinity) with an angular velocity Ω ZAMO (r, θ) = −g 0φ /g φφ , cf. Eq. (8) (also called the "dragging angular velocity" e.g. [18]). The horizon is a 2-surface embedded in a hypersurface of constant t, having the special property that therein Ω ZAMO (r + ) ≡ Ω BH becomes independent 9 of θ; so, the horizon rotates rigidly with angular velocity Ω BH , sometimes dubbed the "black hole's angular velocity" [10,74,75]. The black hole's angular momentum is given [18,45,76] by the Komar integral associated with the axisymmetry Killing vector field ξ α = ∂ α φ : J = −(1/16π)´S BH dξ, for S BH some 2-surface enclosing the black hole but not the ring (in the case of the disk, S is the BH horizon). The ring's angular momentum is J R = −(1/16π)´S dξ − J, for S enclosing the whole system. For a BH-ring system, when the BH's angular momentum is zero (J = 0), the horizon angular velocity is non-zero, and given by Eqs. (75) and (68) of [45], where R is the ring's circumferential radius. Conversely, when the horizon's angular velocity is zero (Ω BH = 0), the BH's angular momentum is non-zero (negative): where A BH is the horizon's area, cf. Eqs. (76), (68), and (26) of [45]. It is also worth mentioning that entirely analogous conclusions to the above can be drawn using "black saturns" [17], which are exact (4+1)-dimensional solutions describing a black hole surrounded by a black ring.

Test particles in static equilibrium around spinning black holes
Perhaps the sharpest counter-examples to the notion of body-dragging, and how drastically actual frame-dragging (in any of its levels) differs from it, is the existence of static equilibrium positions for test particles around some spinning black hole spacetimes, such as the Kerr-de Sitter and (for charged particles) the Kerr-Newman spacetimes. The metrics of both these spacetimes are encompassed in the line element (Kerr-Newman-dS metric [77]) where Λ is the cosmological constant, and Q the black hole's charge.

Kerr-de Sitter spacetime
This metric is obtained by setting Q = 0 in (28)- (30). The gravitoelectric field [i.e., minus the acceleration of the laboratory observers, Eq. (12)] is, in the equatorial plane (θ = π/2), It vanishes at r = 3 √ 3M Λ −1 ≡ r static ("static radius" [78]), where the cosmological repulsion exactly balances the gravitational attraction. By virtue of Eq. (9), this means that no gravitational inertial force is exerted on a particle at rest at r = r static , F GEM = mD U /dτ = 0; i.e. particles placed there remain static (dx α /dτ = 0), while following a geodesic worldline (DU α /dτ = d 2 x α /dτ 2 = 0). They are not dragged in any way by the black hole's rotation, no tangential force being needed to hold them in place, unlike one might think based on the viscous fluid dragging analogy. Frame-dragging has no detectable effect on such particles, only causing them (via the dragging of the ZAMOs) to have non-zero angular momentum,

Kerr-Newman spacetime
An analogous situation occurs in the Kerr-Newman spacetime, in this case for charged particles. This metric is obtained by setting Λ = 0 in (28)- (30); the corresponding electromagnetic 4-potential 1-form is A = Qr(a sin 2 θdφ − dt)/Σ. Consider now a (point-like) test particle of 4-velocity U α , mass m, and charge q. Its equation of motion is DU α dτ = (q/m)F αβ U β , whose space part, in the "laboratory frame", reads, cf. Eqs. (9) and (43), where E and B denote, respectively, the space components of the electric field (E u ) α ≡ F αβ u β and magnetic field (B u ) α ≡ F αβ u β as measured by the laboratory observers (2). In the equatorial plane, E and G read The equilibrium positions are given by the conditionD U /dτ = 0 = U ; noticing that U = 0 ⇒ U α = u α ⇒ γ = 1 [cf. Eq. (2), and recall that γ ≡ −U α u α ], this condition becomes (in the equatorial plane) which, in the non-naked scenario M 2 > Q 2 + a 2 , and outside the ergosphere, r > M + M 2 − Q 2 , yields the single solution [79] with Q and q having the same sign (as expected), and |q| > m. Hence, a particle with such a charge, placed at r = r static , remains at rest, with no gravitomagnetic nor magnetic force acting on it since

"Bobbings" in binary systems
Frame-dragging plays a crucial role in the dynamics of binary systems. Among other effects, it is at the origin of the bobbings observed in numerical simulations [80,81] of nearly identical black holes with anti-parallel spins along the orbital plane ("extreme kick configurations", see Figs. [3][4], in which the whole binary (i.e., its center of mass) oscillates "up and down." They are an example of a setting where the naive viscous fluid analogy, and its associated "body-dragging" misconception, lead to predictions opposite to the real effects. The setting is illustrated in Fig. 3; according to the fluid dragging analogy, in phases A and C there would be no dragging forces, which would be maximum in B and D, each black hole dragging the other into the plane of the illustration in phase B, and outwards in phase D. None of this is true, however.
The gravitomagnetic field of a body (or black hole) with spin vector J is given, at leading order, and in its post-Newtonian (PN) rest frame, by (e.g. [7,9]) similar to that of a magnetic dipole, with −2 J in the place of the magnetic moment µ. Its field lines are depicted in Fig. 4(a). The gravitomagnetic force exerted by black hole 1 (BH 1) on black hole 2 (BH 2) is, in the former's PN instantaneous rest frame,  Figure 4: (a) The field lines of the gravitomagnetic field H produced by a spinning black hole (of spin vector J); it is similar (far from the BH) to that of a magnetic dipole oriented oppositely to J. (b) The actual picture of the corresponding spin-orbit gravitomagnetic forces M v × H involved in the bobbing motion, with phase D suppressed since the result is similar to B. The situation is opposite to that in Fig. 3: the forces vanish when the spins are orthogonal to the axis connecting the black holes (phases B and D, as therein v1 H2, v2 H1), and have maximum magnitude when they lie along such an axis, pushing the pair out of the plane of the illustration in phase A, and inwards in phase C. v 21 = v 2 − v 1 is the velocity of BH 2 relative to BH 1, and in the last equality we used U α = U 0 (1, v), and approximated U ≈ v ≡ d x/dt. Using the vector identity (5.2a) of [83], it reads

H J
where r 21 ≡ x 2 − x 1 is the position vector of BH 2 relative to BH 1, and in the second equality we noticed that ( v 21 × r 21 )· J 1 = 0 since the black hole's spins lie in the orbital plane, and v 21 · r 21 ≈ 0 for quasi-circular motion. The analogous expression for F GM2,1 = M 1 v 12 × H 2 follows by interchanging 1 ↔ 2 in (32). It is clear from Fig. 4 that these forces vanish in phases B and D, as therein the BHs' velocities are aligned with their gravitomagnetic fields: v 1 H 2 , v 2 H 1 . They have maximum magnitude in phases A and C, when each black hole's velocity is orthogonal to the other's spinvector, pointing out of the plane of the illustration in phase A, and inwards in phase C. That is, the situation is opposite in phasing to the "body-dragging" picture of Fig. 3. (It is rather surprising that in [84], where Fig. 3 is reproduced, and at the same time equations equivalent to the above are presented, this disagreement has seemingly gone unnoticed).

The origin of the bobbing
The gravitomagnetic forces ( F GM1,2 and F GM2,1 ) each black hole exerts on the other, which are spin-orbit interactions, point in the same direction, seemingly causing the whole pair to accelerate. Albeit true, it is not the whole story, as these are not the only spin-orbit forces in this system. Each black hole also exerts a spin-curvature force given by Eq. (26), which is of the same order of magnitude [albeit of different nature, not an inertial force but a "real", covariant force, causing the body to deviate from geodesic motion, as discussed in Sec. 2.4], as we shall now see. Compact bodies such as black holes have angular momentum J ∼ < M 2 , hence the gravitomagnetic acceleration in Eq. (32) is of the order vJ/r 3 ∼ < vM 2 /r 3 ∼ vU (∇U ) ∼ O(5), i.e., it arises at 1.5 order in the post-Newtonian (PN) expansion, see Appendix A.1. Thus, we need equations of motion for spinning (pole-dipole) bodies accurate to 1.5 PN order. These follow by taking the 1.5 PN limit of the Mathisson-Papapetrou Equation (26), which reduces (see Appendix A.3) to (44), with F the 1.5PN limit of the spin-curvature force F i = −H βi J β . The force exerted by BH 2 on BH 1 is is the gravitomagnetic tidal tensor produced by BH 2 as "measured" by BH 1 (Eqs. (94) and (88)- (89) in [50]); it reads [50,[85][86][87] where, again, in the second equality we noticed that ( v 12 × r 12 ) · J 1 = 0 for spins lying in the orbital plane, that v 12 · r 12 ≈ 0 for quasi-circular motion, and that v 12 = v 1 − v 2 = − v 21 . The analogous expression for F 1,2 follows by interchanging 1 ↔ 2 in (33). Observe that it is F 2,1 (not F GM2,1 ) which should be regarded (in the context of a PN approximation) as the "reaction" to the gravitomagnetic force F GM1,2 in Eq. (32), as these are the ones that depend on J 1 ; and note that they do not cancel out: That is, the spin-orbit interactions do not obey an action-reaction law [87]. It is this mismatch that causes the whole binary to bob. Consider the PN frame momentarily comoving with BH 1; the 1.5PN gravitational field generated by BH 1 is, in such frame, described by the metric (39) in Appendix A.1, with w given by Eq. (47) with v 1 = 0, and U = M 1 /r 1 , A = 2 r 1 × J 1 /r 3 1 , cf. Eqs. (48). The equation of motion for BH 2 is then, by Eq. (44), with G =∝ r 21 [cf. Eq. (51) with v 1 = 0]. The extreme kick configuration corresponds to the case that Since v 21 and G lie both in the orbital plane xOy, by (35) the z-coordinate acceleration of BH 2 reduces to where J = | J 1 | = | J 2 |, v is the magnitude of the BH's velocities v 2 = − v 1 with respect to the binary's center of mass frame, Ω = v/r, and in the last equality we assumed (without loss of generality) the system to be at phase A of Fig. 4 at t = 0. By analogous steps one finds the same z-coordinate acceleration for BH 1, d 2 z 1 /dt 2 = d 2 z 2 /dt 2 ; that is, the bobbing is synchronous, the whole binary (thus its center of mass) oscillating up and down. It should be stressed that such bobbing, and the mismatch between action and reaction in the spin-orbit forces, do not imply a violation of any conservation principle; on the contrary, as shown in [84], it is a necessary consequence of the interchange between mechanical momentum of the bodies and field momentum (in the sense of the Landau-Lifshitz pseudotensor [88]).
It should also be remarked that in Ref. [82], where the scheme in Fig. 3 was originally presented, still the author was remarkably not misled into a qualitatively wrong phasing. Therein a notion of "space dragging", seemingly different from a viscous fluid analogy, and seemingly imparting a maximum bobbing velocity at stages B and D is used (hence apparently closer to the phenomena involved in the ZAMOs dragging, as if each black hole was "trying" to maintain constant its orbital angular momentum about the other). Such a notion agrees in phasing with Eq. (36), which, integrating, yields dz 2 /dt = 2vJ sin(Ωt)/(r 3 12 Ω): it is indeed in phases B and D (φ = Ωt = π/2 and φ = 3π/2, respectively) that the bobbing velocities have maximum magnitude. However we notice that, as discussed in Sec. 2, the gravitomagnetic forces (dragging of the compass of inertia) are actually of different origin from the ZAMOs dragging; and that such a notion is, at best, complicated and much less intuitive than the gravitomagnetic picture.
Finally, we made use above of a PN frame momentarily comoving with BH 1, which greatly simplified computations, since, in this case: the gravitomagnetic field H 1 produced by BH 1 reduces to that generated by its spin, Eq. (31); the gravitomagnetic force F GM1,2 encodes the whole spin-orbit inertial force acting on BH 2, and G lies along r 12 . In other PN frames (such as one momentarily comoving with the binary's center of mass, as depicted in Figs. 3-4), the description is more complicated, as H 1 then includes a contribution due to BH 1's translational motion, Eq. (52), and part of the spin-orbit inertial force is then encoded in the gravitoelectric field G [having then a non-vanishing z-component, see Eq. (51)]; however, the total spin-orbit inertial force is the same, and equal to (32). Likewise, the spin-curvature force does not depend on the chosen PN frame, so neither does the z-coordinate acceleration, which holds for a generic PN frame, cf. Eq. (56). The derivation for a generic PN frame is given in Appendix A.3.

Conclusions
Generically very feeble in the solar system, where they have been subject of different experimental tests [89][90][91][92][93][94][95][96] (including dedicated space missions [91,95], as well as some controversies [97][98][99][100]), gravitomagnetic effects become preponderant in the strong field regime, shaping the orbits of binary systems and the waveforms of the emitted gravitational radiation [101][102][103][104][105]. Yet they are still commonly misunderstood, in particular those dubbed "frame-dragging". Pertaining initially to the dragging of the compass of inertia, the usage of the term has been extended to other gravitomagnetic effects, as well as to persistent misconceptions fueled by the deceptive fluid-dragging analogy -possibly even by the very term "dragging". We aimed in this paper to deconstruct such misconceptions, explaining what the different types of frame-dragging effects consist of, and the relationship between them. We split them into three different levels (Table 1), governed by three distinct mathematical objects, corresponding to different orders of differentiation of the gravitomagnetic potential 1-form A. The first level (Sec. 2.1), governed by A itself, may be cast physically (in axistationary spacetimes) as the dragging of the ZAMOs, which comprises effects such as the Sagnac effect in optical loops around the source, or the arrangement of the bodies' angular velocities/angular momentum in black holes surrounded by disks or rings (Sec. 3). It contributes also to the gravitomagnetic clock effect (Sec. 2.3). The second level, governed by the gravitomagnetic field H, and physically interpreted as the dragging of the compass of inertia (Sec. 2.2), includes the gravitomagnetic (or Coriolis) forces on test bodies and the Lense-Thirring orbital precessions, gyroscope precession, and is also responsible for the remainder of the gravitomagnetic clock effect. The third level (Sec. 2.4), governed by the gravitomagnetic tidal tensor H αβ ("magnetic" part of the Riemann tensor), and interpreted as a differential dragging of the compass of inertia, physically manifests in the relative precession of nearby sets of gyroscopes, and in the spin-curvature force on a gyroscope. These levels are largely independent, in that there exist spacetimes possessing only the first (which we exemplified with spinning Lewis-Weyl cylinders, Sec. 2.3), and others possessing only the first two levels, but missing the third (e.g. the Gödel universe). In the case of the first two levels (dragging of the ZAMOs vs dragging of the compass of inertia), their effects can actually be opposite, as exemplified here by the case of circular geodesics in the Kerr spacetime (Sec. 2.3).
Two main analogies are commonly used to help to physically interpret the frame-dragging effects: the analogy with magnetism (which created the term "gravitomagnetism") and the fluid dragging analogy. The former is clearly useful for the second and third levels of frame-dragging, since not only the gravitational effects comprised therein have an electromagnetic analogue, as the exact equations describing them exhibit, in the appropriate formalism, exact analogies (up to constant factors) with the electromagnetic counterparts (Secs. 2.2-2.4). As for the first level, they have no analogue in classical electromagnetism (only in quantum electrodynamics 10 ). The fluid model, on the other hand, draws an analogy with the dragging of a viscous fluid by a moving/rotating body. It gives some qualitative intuition (in a loose sense) for the first level of frame-dragging, in that in all the systems considered herein (black hole spacetimes, and infinite spinning cylinders) the ZAMOs are dragged in the same direction of the source's rotation. The parallelism ends there however: other features of any fluid flow, such as the unavoidable dragging of immersed bodies along with the flow, have no parallel in any of the frame-dragging levels. The dragging of the compass of inertia, in particular (which was the original motivation for such analogy), is a very different (sometimes opposite, cf. Secs. 3 and 3.2) phenomenon from body-dragging. Some of these inconsistencies have been pointed out in the literature, namely in a paper by Rindler [37] -based, however, on a weak field slow motion linearized theory approach, the question remaining as to whether the conclusions would fully hold in the exact case. We generalized them here (Sec. 3) to the exact theory, using the exact 1+3 GEM formalism. We considered first (as an application akin, on the whole, to [37]) a space station around a spinning black hole (seen to remain stationary, without acquiring any rotation), as well as the situation for test particles launched from it, where those with initial outwards radial velocity are deflected in the direction opposite to the black hole's rotation. We considered also the reciprocal problem -a rotating ring around a 'non-spinning' black hole (either with zero angular momentum, or zero horizon angular velocity), pointing out that the black hole acquires no angular acceleration of any sort, the solutions being stationary. As a very physical, and perhaps sharpest example to convince the reader that the (all too common) notion of "body-dragging" is wrong, and its underlying viscous fluid-dragging analogy extremely misleading, we considered equilibrium positions for test particles in spinning black hole solutions (namely Kerr-Newman and Kerr-de Sitter). Finally, we considered a notable phenomenon which is driven by frame-dragging -the bobbings in "extreme kick" binary systems -and where the viscous-dragging picture predicts the exact opposite of the real effect. Acknowledgments L.F.C. and J.N. were supported by FCT/Portugal through projects UIDB/MAT/04459/2020 and UIDP/MAT/04459/2020.
Consider a congruence of observers of 4-velocity u α , and a (point-like) test particle of worldline x α (τ ) and 4-velocity dx α /dτ = U α . Let U α ≡ h α β U β be the spatial projection of the particle's velocity with respect to u α , cf. Eq. (3); it yields, up to a γ (≡ −u α U α ) factor, the relative velocity of the particle with respect to the observers (cf. e.g. [28,50,63]). It is the variation of U α along x α (τ ) that one casts as inertial forces (per unit mass); the precise definition of such variation involves some subtleties however. For that we need a connection∇ (i.e., a covariant derivative) for spatial vectors that (i) in the directions orthogonal to u α should equal the projected Levi-Civita spacetime connection:∇ X Z α = h α β ∇ X Z β , for any X α and Z α orthogonal to u α , so that it corrects for the trivial variation of the spatial axes in the directions orthogonal to u α (for instance, of a non-rectangular coordinate system in flat spacetime) which is not related to inertial forces and does not vanish in an inertial frame; (ii) along the congruence, becomes an ordinary time-derivative ∂ u , so that it yields the variation of U α with respect to a system of spatial axes undergoing a transport law specific to the chosen reference frame. The most natural of such choices is spatial axes corotating with the observers ("congruence adapted" frame [32]; arguably, the closest generalization of the Newtonian concept of reference frame [6,29]). For an orthonormal basis eα, whose general transport law along the observer congruence can be written as (e.g. [10,32]) that amounts to choosing Ω α (the angular velocity of rotation of the spatial axes relative to Fermi-Walker transport) equal to the observer's vorticity: Ω α = ω α , as defined in (12). If the congruence is rigid, this ensures that the axes eî point to fixed neighboring observers. The connection that yields the variation of a spatial vector X α with respect to such a frame is∇ α X β ≡ h β γ ∇ α X γ + u α β δγλ u γ X δ ω λ , cf. Eq. (51) of [32]; and the inertial or "gravitoelectromagnetic" force on a test particle as measured in such frame is the variation of U α along x α (τ ) with respect to∇, that is, ∇ U U α ≡DU α /dτ . Since, for geodesic motion, ∇ U U α = 0, it follows, using (3), that∇ U U α = −γ(∇ U u α + α δγλ u γ U δ ω λ ), where γ ≡ −U α u α . Finally, from the decomposition (e.g. Eq. (135) of [69]) where σ αβ = h µ α h ν β u (µ;ν) − θh αβ /3 and θ ≡ u α ;α are, respectively, the congruence's shear and expansion, we have (using ∇ U u α ≡ U β ∇ β u α ) [32] where the "gravitoelectric" G α and "gravitomagnetic" H α fields are again given by Eqs. (12) (being, respectively, minus the observers' acceleration, and twice their vorticity), and m is the particle's mass. For a congruence of observers (2) tangent to a time-like Killing vector field in a stationary spacetime (or for any rigid congruence of observers in general), σ αβ = 0 = θ, and so Eq. (38) reduces to Eq. (9).

A.1 Post-Newtonian approximation
The post-Newtonian expansion is a weak field and slow motion approximation tailored to bound astrophysical systems. It can be cast in different, equivalent ways; here, following [10,69,87,88,107,108], we frame the expansion in terms of a small dimensionless parameter , such that U ∼ 2 , where U is minus the Newtonian potential [i.e., taking the Newtonian limit of Eq. (1), Φ = −U ], and the bodies' velocities are assumed such that v ∼ < (since, for bounded orbits, v ∼ √ U ). In terms of "forces," the Newtonian force m∇U is taken to be of zeroth PN order [0PN, i.e., O(2)], and each factor 2 amounts to a unit increase of the PN order. Time derivatives increase the degree of smallness of a quantity by a factor ; for example, ∂U/∂t ∼ U v ∼ U . The 1PN expansion consists of keeping terms up to O(4) in the equations of motion [107]. This amounts to retaining terms up to O(4) in g 00 , O(3) in g 0i , and O(2) in g ij , effectively considering a metric of the form [20,88] g 00 = −1 + 2w − 2w 2 + O(6); g i0 = A i + O(5); g ij = δ ij (1 + 2U ) + O(4) (actually accurate to 1.5PN), where w consists of the sum of U plus non-linear terms of order 4 , w = U + O(4). For observers (2) at rest in a given coordinate system, by (12) G i = Γ i 00 /g 00 , H i = − ij k0 Γ k 0j /g 00 ; hence 11 In terms of coordinate acceleration, noting thatD U /dτ = (U 0 ) 2 d 2 x/dt 2 + 2 v∂ t U − v 2 G + 4( G · v) v + O(6), (U 0 ) −2 = 1 − v 2 − 2U + O(4), we have (cf. Eq. (7.17) of [20]), where F IPN stands for "post-Newtonian inertial force" (in order to distinguish from F GEM ). The absence of O(5) terms means that (41)- (42) are actually accurate to 1.5PN order.
"Linear dragging".-The contribution −∂ A/∂t to G in (40) (and thus to F IPN ) means that a time-varying gravitomagnetic vector potential induces a velocity-independent inertial force on a test particle. For instance, in the interior of an accelerating massive shell, inertial forces are induced in the same direction of the shell's acceleration, as first noted by Einstein ( [109], pp. 100-102; see also [110,111]). The effect has been studied in the framework of the exact theory in some special solutions (e.g. [112,113]), and has been experimentally confirmed to high accuracy (albeit indirectly, one may argue) in the observations of binary pulsars [114]. It is sometimes dubbed "translational dragging" [111] or "linear dragging" [112,115] (of inertial frames). We note that, in spite of such denominations, it is a component of the gravitoelectric field G; as such, in the given reference frame, it does not directly fit into the frame-dragging types in Table 1. It mixes, however, with the gravitomagnetic inertial acceleration v × H (dragging of the compass of inertia) in changes of frame; for instance, as exemplified in Secs. 3.2 and A.3.1, an inertial force which, in a given PN frame, consists solely of the gravitomagnetic term m v × H, in another frame can be partially incorporated in the term −m∂ A/∂t (this stems from the transformation laws for G and H, which exhibit a certain analogy with their electromagnetic counterparts, see [108]).
Hence F ISO matches the gravitomagnetic force (32) exerted on BH 2 as measured in the PN frame momentarily comoving with BH 1, obtained in Sec. 3.2. That is: in a generic frame, the gravitomagnetic force does not encode the whole spin-orbit inertial force, part of it being encoded in the gravitoelectric force M 2 G; however, the total spin-orbit inertial force is the same in all PN frames.
Since F mono ∈ xOy, we have, for the z-coordinate, matching the result (36) obtained in Sec. 3.2.