Deconstructing Frame Dragging

The vorticity of world lines of observers associated to the rotation of a massive body was reported by Lense and Thirring more than a century ago. In their example the frame dragging effect induced by the vorticity, is directly (explicitly) related to the rotation of the source. However in many other cases it is not so, and the origin of vorticity remains obscure and difficult to identify. Accordingly, in order to unravel this issue, and looking for the ultimate origin of vorticity associated to frame dragging, we analyze in this manuscript very different scenarios where frame dragging effect is present. Specifically we consider general vacuum stationary spacetimes, general electro-vacuum spacetimes, radiating electro-vacuum spacetimes and Bondi--Sachs radiating spacetimes. We identify the physical quantities present in all these cases, which determine the vorticity and may legitimately be considered as responsible for the frame dragging. Doing so we provide a comprehensive physical picture of frame dragging. Some observational consequences of our results are discussed.


I. INTRODUCTION
The dragging of inertial frames produced by selfgravitating sources, whose existence has been recently established by observations [1], is one of the most remarkable effect predicted by the general theory of relativity (GR) (see [2,3]).
The term frame dragging usually refers to the influence of a rotating massive body on a gyroscope by producing vorticity in the congruence of world lines of observers outside the rotating object. Although the appropriateness of the term "frame dragging" has been questioned by Rindler [4], it nevertheless has been used regularly in the literature till nowadays, and accordingly we shall adopt such term here (see also [5,6]).
The basic concept for the understanding of this effect is that of vorticity of a congruence, which describes the rotation of a gyroscope attached to the congruence, with respect to reference particles.
Two different effects may be detected by means of gyroscopes. One of this (Fokker-de Sitter effect) refers to the precession of a gyroscope following a closed orbit around a spherically symmetric mass distribution. It has been verified with a great degree of accuracy by observing the rotation of the earth-moon system around the sun [7], but this is not the frame dragging effect we are interested in here. The other effect, the one we are concerned with in this work, is the Lense-Thirring-Schiff precession, which refers to the appearance of vorticity in the congruence of world lines of observers in the gravitational field of a massive rotating ball. It was reported by the first time by Lense and Thirring [8], and is usually referred to as Lense-Thirring effect ( some authors suggest that it should be named instead, Einstein-Thirring- * lherrera@usal.es Lense effect, see [9][10][11]). This result led Schiff [12] to propose the use of gyroscopes to measure such an effect. Since then this idea has been developed extensively (see [4,[13][14][15][16][17][18][19][20][21] and references cited therein).
However, although the origin of vorticity may be easily identified in the Lense-Thirring metric, as due to the rotation of a massive object, it is not always explicitly related to rotation of massive objects. In fact, in any vacuum stationary space time (besides the Lense-Thirring metric) we can detect a frame dragging effect, without specifying an explicit link to the rotation of a massive body [22].
The situation is still more striking for the electrovacuum space times. The point is that the quantity responsible for the rotational (relativistic) multipole moments in these spacetimes, is affected by the mass rotations (angular momentum), as well as by the electromagnetic field, i.e. it contains contributions from both (angular momentum and electromagnetic field). This explains why such a quantity does not necessarily vanish in the case when the angular momentum of the source is zero but electromagnetic fields are present.
The first known example of this kind of situation was brought out by Bonnor [23]. Thus, analyzing the gravitational field of a magnetic dipole plus an electric charge, he showed that the corresponding spacetime is stationary and a frame dragging effect appears. As a matter of fact all stationary electro-vacuum solutions exhibit frame dragging [24], even though in some cases the angular momentum of the source is zero. In this latter case the rotational relativistic multipole moments and thereby the vorticity, are generated by the electromagnetic field. Furthermore, as we shall see, electrodynamic radiation also produces vorticity.
Finally, it is worth recalling that vorticity is present in gravitationally radiating space-times. The influence of gravitational radiation on a gyroscope through the vor-ticity associated with the emission of gravitational radiation was put forward for the first time in [25], and has been discussed in detail since then in [26][27][28][29][30][31][32][33]. In this case too, the explicit relationship between the vorticity and the emission of gravitational radiation was established without resorting to the rotation of the source itself.
Although in many of the scenarios described above a rotating object is not explicitly identified as the source of vorticity, the fact remains that at purely intuitive level, one always associates the vorticity of a congruence of world lines, under any circumstance, to the rotation of "something".
The purpose of this work is twofold: on the one hand we shall identify the physical concept (the "something") behind all cases where frame dragging is observed, whether or not the angular momentum of the source vanishes. On the other hand we would like to emphasize the possible observational consequences of our results.
As we shall see below, in all possible cases, the appearing vorticity is accounted for by the existence of a flow of superenergy on the plane orthogonal to the vorticity vector, plus (in the case of electro-vacuum spacetimes) a flow of electromagnetic energy on the same plane.
Since superenergy plays a fundamental role in our approach, we shall start by providing a brief introduction of this concept in the next section.

II. SUPERENERGY AND SUPER-POYNTING VECTOR
The concept of energy is a fundamental tool in all branches of physics, allowing to approach and solve a vast number of problems under a variety of circumstances. This explains the fact that since the early times of GR many researchers have tried by means of very different approaches to present a convincing definition of gravitational energy, in terms of an invariant local quantity All these attempts, as is well known, have failed. The reason for this failure is easy to understand.
Indeed, as we know, in classical field theory energy is a quantity defined in terms of potentials and their first derivatives. On the other hand however, we also know that in GR it is impossible to construct a tensor expressed only through the metric tensor (the potentials) and their first derivatives (in accordance with the equivalence principle). Therefore, a local description of gravitational energy in terms of true invariants (tensors of any rank) is not possible within the context of the theory.
Thus, the following alternatives remain: • To define energy in terms of a non-local quantity.
• To resort to pseudo-tensors.
• To introduce a succedaneous definition of energy.
One example of the last of the above alternatives is superenergy, which may be defined either from the Bel or from the Bel-Robinson tensor [34][35][36] (they both coincide in vacuum), and has been shown to be very useful when it comes to explaining a number of phenomena in the context of GR.
Both, the Bel and the Bel-Robinson tensors, are obtained by invoking the "structural" analogy between GR and Maxwell theory of electromagnetism. More specifically, exploiting the analogy between the Riemann tensor (R αβγδ ) and the Maxwell tensor (F µν ), Bel introduced a four-index tensor defined in terms of the Riemann tensor in a way which is a reminiscence of the definition of the energy-momentum tensor of electromagnetism in terms of the Maxwell tensor. This is the Bel tensor.
The Bel-Robinson tensor is defined as the Bel tensor, but with the Riemann tensor replaced by the Weyl tensor (C αβγδ ) (see [37] for a comprehensive account and more recent references on this issue).
Let us now introduce the electric and magnetic parts of the Riemann and the Weyl tensors as, where C(R) αγβδ is the Weyl (Riemann) tensor, the four-vector u γ in vacuum is the tangent vector to the world-lines of observers, and C(R) * αγβδ is the dual of the Weyl(Riemann) tensor.
A third tensor may be defined from the double dual of the Riemann tensor as which in the case of the Weyl tensor X αβ coincide with the electric part of the Weyl tensor (up to a sign). Next, from the analogy with electromagnetism the super-energy and the super-Poynting vector are defined by where R(C) denotes whether the quantity is defined with Riemann (Weyl) tensor, and η αβγδ is the Levi-Civita tensor.
In the next sections we shall bring out the role played by the above introduced variables in the study of the frame dragging effect.

III. FRAME DRAGGING IN VACUUM STATIONARY SPACETIMES
As we mentioned in the Introduction, the first case of frame dragging analyzed in the literature was the Lense-Thirring effect. For didactical reasons we shall start by considering first this case and from there on, we shall consider examples of increasing complexity. Thus, afterward we shall consider the Kerr metric, an approximation of which is the Lense-Thirring spacetime, and finally we shall consider the general vacuum stationary spacetime case.

A. The Lense-Thirring precession
The Lense-Thirring effect is based on an approximate solution to the Einstein equations which reads [8] It describes the gravitational field outside a spinning sphere of constant density, up to first order in m/r and J/r 2 , with m and J denoting the mass and the angular momentum respectively. Up to that order, it coincides with the Kerr metric, by identifying where a is the Kerr parameter [38]. Next, the congruence of the world-lines of observers at rest in the frame of (6) is described by the timelike vector u α whose components are from the above expression the vorticity vector, defined as usual by has, up to order a/r and m/r, the following non-null components or Ω = (ω α ω α ) 1/2 = ma r 3 1 + 3 cos 2 θ, which at θ = π 2 reads Ω = ma r 3 .
The above expression embodies the essence of the Lense-Thirring effect. It describe the vorticity of the world lines of observers, produced by the rotation (J) of the source. Such vorticity, as correctly guessed by Shiff [12], could be detected by a gyroscope attached to the world lines of our observer.
Even though in this case the vorticity is explicitly related to the rotation of the spinning object which sources the gravitational field, the fact that this link in many other cases is not so explicitly established leads us to the question: what is (are) the physical mechanism(s) which explains the appearance of vorticity in the world lines of the observer? As we shall see in the next sections the answer to this question may be given in terms of a flow of superenergy plus (in the case of electro-vacuum spacetimes) a flow of electromagnetic energy.
So, in order to approach to this conclusion, let us calculate the leading term of the super-Poynting gravitational vector at the equator. Using (5) and (6) we obtain for the only non-vanishing component (remember that in vacuum both expressions for the super-Poynting vector coincide) It describes a flux of super-energy on the plane orthogonal to the vorticity vector. On the other hand it follows at once from (14) that From the comments above, a hint about the link between superenergy and vorticity (frame dragging) begins to appear. In order to delve deeper on this issue let us next consider the Kerr metric.

B. Frame dragging in the Kerr metric
The calculations performed in the previous subsection can be very easily repeated for the Kerr metric.

Finally, using the package GR-Tensor running on
Maple we obtain for the super-Poynting vector (5) with From the above expressions it follows, as in the precedent case, that there is an azimuthal flow of superenergy as long as a = 0, inversely the vanishing of such a flow implies a = 0. Once again frame dragging appears to be tightly related to a circular flow of superenergy on the plane orthogonal to the vorticity vector.
In the present case we can delve deeper in the rela-tionship between the source of the field and the vorticity, since a specific interior for the Kerr metric is available [39]. The remarkable fact is the presence of a nonvanishing T φ t component of the energy-momentum tensor of the source, which, defining as usual an energymomentum flux vector as: F ν = −V µ T νµ (where V µ denotes the four velocity of the fluid), implies that in the equatorial plane of our system (within the source) energy flows round in circles around the symmetry axis. This result, as we shall see in the next section, is a reminiscence of an effect appearing in stationary Einstein-Maxwell systems. Indeed, in all stationary Einstein-Maxwell systems, there is a non vanishing component of the Poynting vector describing a similar phenomenon [23,24] (of electromagnetic nature, in this latter case). Thus, the appearance of such a component seems to be a distinct physical property of rotating fluids, which has been overlooked in previous studies of these sources, and that is directly related to the vorticity (see eqs. (8) and (18) in [39]).

C. Frame dragging in a general stationary vacuum spacetime
Let us now consider the general stationary and axisymmetric vacuum case.
The line element for a general stationary and axisymmetric vacuum spacetime may be written as [40,41] where x 0 = t; x 1 = ρ; x 2 = z and x 3 = φ and metric functions depend only on ρ and z, which must satisfy the vacuum field equations: where subscripts denote partial derivatives. Then following the same protocol as in the previous cases, we define the four velocity vector for an observer at rest in the frame of (22), which reads The super-Poynting vector can now be calculated for the general class of spacetimes represented by the above metric (22), (i.e.: without making any assumption about the matter content of the source), and one gets (using again GR-Tensor) where P φ is given by (again in the general case, i.e.: without taking into account the field equations): or using the field equations (23)(24)(25)(26) in the above expression where A11 and A12 are given in the Appendix A. Now, in [24] it has been shown that for the general metric (22) the following relations hold and of course as it follows from (5) In order to establish a link between vorticity and the super-Poynting vector of the kind already found for the Kerr (and Lense-Thirring) metric we still need to prove that the vanishing of the super-Poynting vector implies the vanishing of the vorticity, i.e. we have to prove that Such a proof has been carried out in [22], but is quite cumbersome and therefore we shall omit the details here.
Thus based on (31) we conclude that for any stationary spacetime, irrespectively of its source, there is a frame dragging effect associated to a flux of superenergy on the plane orthogonal to the vorticity vector.
We shall next analyze the electro-vacuum stationary case.

IV. FRAME DRAGGING IN ELECTRO-VACUUM STATIONARY SPACETIMES
Electro-vacuum solutions to the Einstein equations pose a challenge concerning the frame dragging effect. This was pointed out for the first time by Bonnor in [23] by analyzing the gravitational field produced by a magnetic dipole with an electric charge in the center. The surprising result is that, for this spacetime, the world lines of observers at rest with respect to the electromagnetic source are endowed of vorticity (i.e. the resulting spacetime is not static but stationary).
In order to explain the appearance of vorticity in the spacetime generated by a charged magnetic dipole Bonnor resorts to a result pointed out by Feynmann in his Lectures on Physics [42], showing that for such a system (in the context of classical electrodynamics) there exists a non-vanishing component of the Poynting vector describing a flow of electromagnetic energy round in circles. This strange result leads Feynmann to write that "it shows the theory of the Poynting vector is obviously nuts". However, some pages ahead in the same book, when discussing the "paradox" of the rotating disk with charges and a solenoid, Feynmann shows that this "circular" flow of electromagnetic energy is absolutely necessary in order to preserve the conservation of angular momentum. In other words the theory of the Poynting vector not only is not "nuts", but is necessary to reconcile the electrodynamics with the conservation law of angular momentum.
Based on the above comments Bonnor then suggests that, in the context of GR, such a circular flow of energy affects inertial frames by producing vorticity of congruences of particles, relative to the compass of inertia. In other words Bonnor suggests that the "something" which rotates thereby generating the vorticity, is electromagnetic energy.
The interesting point is that this conjecture was shown to be valid for a general axially symmetric stationary electro-vacuum metric [24].
Indeed, assuming the line element (22) for the spacetime admitting an electromagnetic field, it can be shown that the variable responsible for the rotational multipole moments, which in its turn determine the vorticity of the congruence of world lines of observers, is affected by, both, the electromagnetic field and by the mass rotations (angular momentum) [24]. This explains why the vorticity does not necessarily vanish in the case when the angular momentum of the source is zero but electromagnetic fields are present. At any rate, it is important to stress that in such cases, the super-Poynting vector does not vanish either.
We shall next consider the presence of vorticity due to gravitational and electromagnetic radiation.

V. VORTICITY AND RADIATION
We shall now analyze the generation of vorticity related to the emission of gravitational and/or electromagnetic radiation. As we shall see, the emission of radiation is always accompanied by the appearance of vorticity of world lines of observers. Furthermore, the calculations suggest that once the radiation process has stopped, there is still a remaining vorticity associated to the tail of the wave, which allows in principle to prove (or disprove) the violation of the Huyghens principle in a Riemannian spacetime (see [43][44][45][46][47][48][49] and references therein for a discussion on this issue), by means of observations.

A. Gravitational radiation and vorticity
Since the early days of GR, starting with the works of Einstein and Weyl on the linear approximation of the Einstein equations, a great deal of work has been done so far in order to provide a consistent framework for the study of gravitational radiation. Also, important collab-oration efforts have been carried on, and are now under consideration, to put in evidence gravitational waves by means of laser interferometers [50].
However it was necessary to wait for more than half a century, until Bondi and coworkers [51] provided a firm theoretical evidence of the existence of gravitational radiation without resorting to the linear approximation.
The essential "philosophy" behind the Bondi formalism, consists in interpreting gravitational radiation as the physical process by means of which the source of the field "informs" about any changes in its structure. Thus the information required to forecast the evolution of the system (besides the "initial" data) is thereby identified with radiation itself, and this information is represented by the so called "news function". In other words, whatever happens at the source, leading to changes in the field, it can only do so by affecting the news function and vice versa. Therefore if the news function is zero over a time interval, there is no gravitational radiation over that interval. Inversely, non vanishing news on an interval implies the emission of gravitational radiation during that interval. Thus the main virtue of this approach resides in providing a clear and precise criterion for the existence of gravitational radiation.
The above described picture is reinforced by the fact that the Bondi mass of a system is constant if and only if there are no news.
In order to facilitate discussion let us briefly introduce the main aspects of the Bondi approach. Bondi and coworkers start with the general form of an axially (and reflection) symmetric asymptotically flat metric given by ds 2 = V r e 2β − U 2 r 2 e 2γ du 2 + 2e 2β dudr + 2U r 2 e 2γ dudθ − r 2 e 2γ dθ 2 + e −2γ sin 2 θdφ 2 , where V, β, U and γ are functions of u, r and θ. The coordinates are numbered x 0,1,2,3 = u, r, θ, φ respectively. u is a timelike coordinate such that u = constant defines a null surface. In flat spacetime this surface coincides with the null light cone open to the future. r is a null coordinate (g rr = 0) and θ and φ are two angle coordinates.
Regularity conditions in the neighborhood of the polar axis (sin θ = 0), implies that as sin θ− > 0 each equals a function of cos θ regular on the polar axis. Then the four metric functions are assumed to be expanded in series of 1/r, which using field equations produces γ = c(u, θ)r −1 + C(u, θ) − 1 6 c 3 r −3 + ..., where letters as subscripts denote derivatives, and The three functions c, M and N depend on u and θ, and are further related by the supplementary conditions In the static case M equals the mass of the system whereas N and C are closely related to the dipole and quadrupole moment respectively.
Next, Bondi defines the mass m(u) of the system as which by virtue of (39) and (33) yields The two main conclusions emerging from the Bondi's approach are • If γ, M and N are known for some u = a(constant), and c u (the news function) is known for all u in the interval a ≤ u ≤ b, then the system is fully determined in that interval.
• As it follows from (42), the mass of a system is constant if and only if there are no news.
In the light of these comments the relationship between news function and the occurrence of radiation becomes clear.
Let us now calculate the vorticity of the world lines of observers at rest in the frame of (32). For such observers the four-velocity vector has components Using (9) , we easily obtain with and for the absolute value of ω α we get Feeding back (34)-(37) into (47) and keeping only terms up to order 1 r 2 , we obtain Let us now analyze the expression above. First of all observe that, up to order 1/r, a gyroscope in the gravitational field given by (32) will precess as long as the system radiates (c u = 0). Indeed, if we assume which implies, due to the regularity conditions (33) In other words the leading term in (48) will vanish if and only if c u = 0.
Let us now analyze the term of order 1 r 2 . It contains, besides the terms involving c u , a term not involving news (M θ ). Let us now assume that initially (before some u = u 0 =constant) the system is static, in which case which implies , because of (40) and Ω = 0 (actually, in this case Ω = 0 at any order) as expected for a static field. Then let us suppose that at u = u 0 the system starts to radiate (c u = 0) until u = u f , when the news function vanishes again. For u > u f the system is not radiating although (in general) M θ = 0 implying time dependence of metric functions. This class of spacetimes is referred to as non-radiative motions [51].
Thus, in the interval u ∈ (u 0 ,u f ) the leading term of vorticity is given by the term of the order 1/r in (48). For u > u f there is a vorticity term of order 1 r 2 describing the effect of the tail of the wave on the vorticity. This provides an "observational" possibility to find evidence for the violation of the Huygens's principle.
Following the line of arguments of the preceding sections, we shall establish a link between vorticity and a circular flow of superenergy on the plane orthogonal to the vorticity vector. For doing so, let us calculate the super-Poynting vector (P µ ), defined by (5). We obtain that the leading terms for each super-Poynting component are The vanishing (at all orders) of the azimuthal component (P φ ), is expected from the reflection symmetry of the Bondi metric, which is incompatible with the presence of a circular flow of superenergy in the φ direction. Since the vorticity vector, which is orthogonal to the plane of rotation, has in the Bondi spacetime only one non-vanishing contravariant component ( φ ), then the plane of the associated rotation is orthogonal to the φ direction. Therefore, it is the θ component of P µ the physical factor to be associated to the vorticity, in the Bondi case.
In order to strength further the case for the super-Poynting vector as the physical origin of the mentioned vorticity, we shall consider next the general radiative metric without axial and reflection symmetry.
The extension of the Bondi formalism to the case without any kind of symmetries was performed by Sachs [52]. In this case the line element reads (we have found more convenient to follow the notation given in [53] which is slightly different from the original Sachs paper) ds 2 = (V r −1 e 2β − r 2 e 2γ U 2 cosh 2δ − r 2 e −2γ W 2 cosh 2δ − 2r 2 U W sinh 2δ)du 2 + 2e 2β dudr + 2r 2 (e 2γ U cosh 2δ + W sinh 2δ)dudθ + 2r 2 (e −2γ W cosh 2δ + U sinh 2δ) sin θdudφ − r 2 (e 2γ cosh 2δdθ 2 + e −2γ cosh 2δ sin 2 θdφ 2 + 2 sinh 2δ sin θdθdφ), where β, γ, δ, U , W , V are functions of x 0 = u, x 1 = r, The general analysis of the field equations is similar to the one in [51], but of course the expressions are far more complicated (see [52,53] for details). In particular, there are now two news functions.
Let us first calculate the vorticity for the congruence of observers at rest in (57), whose four-velocity vector is given by where now A is given by Thus, (9) lead us to where and Expanding the metric functions in series of 1/r as in the previous case, using the field equations and feeding back the resulting expressions into (61,62,63,64) we get for the leading term of the absolute value of ω µ which of course reduces to (48) in the Bondi (axially and reflection symmetric) case (d = c φ = 0). It is worth stressing the fact that now we have two news functions (c u , d u ). Next, the calculation of the super-Poynting vector gives the following result P µ = (0, P r , P θ , P φ ).
(66) The explicit terms are too long and the calculations are quite cumbersome (see [28] for details), so let us just present the leading terms for each super-Poynting component, they read from which it follows this component of course vanishes in the Bondi case. From the expressions above we see that the main conclusion established for the Bondi metric, is also valid in the most general case, namely, there is always a nonvanishing component of P µ on the plane orthogonal to a unit vector along which there is a non-vanishing component of vorticity, and inversely, P µ vanishes on a plane orthogonal to a unit vector along which the component of vorticity vector vanishes. The link between the super-Poynting vector and vorticity is thereby firmly established.
So far we have shown the appearance of vorticity in stationary vacuum spacetimes, stationary electro-vacuum spacetimes and in radiative vacuum spacetimes (Bondi-Sachs), and have succeed in exhibiting the link between this vorticity and a circular flow of electromagnetic and/or super-energy, on the plane orthogonal to the vorticity vector. It remains to analyze the possible role of electromagnetic radiation in the appearance of vorticity. The next section is devoted to this issue.

B. Electromagnetic radiation and vorticity
The relationship between electromagnetic radiation and vorticity has been unambiguously established in [54]. The corresponding calculations are quite cumbersome and we shall not reproduce them here. Instead we shall highlight the most important results emerging from such calculations.
The formalism used to study the general electrovacuum case (including electromagnetic radiation) was developed by van der Burg in [55]. It represents a generalization of the Bondi-Sachs formalism for the Einstein-Maxwell system. Thus, the starting point is the Einstein-Maxwell system of equations, which reads where R µγ is the Ricci tensor and the energy momentum tensor T µγ of the electromagnetic field is given as usual by Then, following the script indicated in [51], i.e. expanding the physical and metric variables in power series of 1/r and using the Einstein-Maxwell equations, one arrives at the conclusion that if a specific set of functions is prescribed on a given initial hypersurface u = constant, the evolution of the system is fully determined provided the four functions, c u , d u , X, Y are given for all u. These four functions are the news functions of the system. The first two (c u , d u ) are the gravitational news functions already mentioned before for the purely gravitational case, whereas X and Y are the two news functions corresponding to the electromagnetic field, these appear in the series expansion of F 12 , F 13 . Thus, whatever happens at the source leading to changes in the field, it can only do so by affecting the four news functions and viceversa.
Following the same line of arguments, an equation for the decreasing of the mass function due to the radiation (gravitational and electromagnetic) similar to (42) can be obtained, it reads where c * = c + id, and bar denotes complex conjugate.
Having arrived at this point we can now proceed to calculate the vorticity, the super-Poynting vector and the electromagnetic Poynting vector. The resulting expressions are available in [54], since they are extremely long, here we shall focus on the main conclusions emerging from them.
First, the vorticity vector (9) is calculated for the fourvector u α given by (58). The important point to stress here is that the absolute value of ω µ can be written generically as where subscripts G, GEM and EM stand for gravitational, gravito-electromagnetic and electromagnetic. The "gravitational" subscript refers to those terms containing exclusively functions appearing in the purely gravitational case and their derivatives. "Electromagnetic" terms are those containing exclusively functions appearing in F µν and their derivatives, whereas "gravitoelectromagnetic" subscript refers to those terms containing functions of either kind and/or combination of both. Finally, we calculate the electromagnetic Poynting vector defined by and the super-Poynting vector defined by (5). Since we are not operating in vacuum, P (C) α and P (R) α are different, we shall use P (C) α for the discussion. The resulting expressions are deployed in [54]. Let us summarize the main information contained in such expressions.
First, we notice that the leading terms for each super-Poynting (contravariant) component are P u = P u G r −4 + · · · , P r = P r G r −4 + · · · , P θ = P θ G r −4 + · · · + P θ GEM r −6 + · · · , whereas for the electromagnetic Poynting vector we can write Next, there are explicit contributions from the electromagnetic news functions in Ω GEM as well as in P φ GEM and P θ GEM . More so, the vanishing of these contributions in P φ GEM and P θ GEM implies the vanishing of the corresponding contribution in Ω GEM , and viceversa.
From the above it is clear that electromagnetic radiation as described by electromagnetic news functions does produce vorticity. Furthermore we have identified the presence of electromagnetic news both in the Poynting and the super-Poynting components orthogonal to the vorticity vector. Doing so we have proved that a Bonnor-like mechanism to generate vorticity is at work in this case too, but with the important difference that now vorticity is generated by the contributions of, both, the Poynting and the super-Poynting vectors, on the planes orthogonal to the vorticity vector.

VI. DISCUSSION
We started this manuscript with two goals in mind. On the one hand we wanted to identify the fundamental physical phenomenon which being present in all scenarios exhibiting frame dragging, could be considered as the responsible for the frame dragging effect. In other words we wanted to identify the factor that mediates between the source of the gravitational field and the appearance of vorticity, in any scenario.
On the other hand, we wanted to explore the observational consequences that could be derived from our analysis.
Concerning our first goal, it has been clearly established that in vacuum, the appearance of vorticity is always related to the existence of circular flow of superenergy in the plane orthogonal to the vorticity vector. This is true for all stationary vacuum spacetimes as well as for general Bondi-Sachs radiative spacetimes.
In the case of electro-vacuum spacetimes, we have circular flows of super-energy as well as circular flows of electromagnetic energy in the plane orthogonal to the vorticity vector. This is true in stationary electrovacuum spacetimes as well as in spacetimes admitting, both, gravitational and electromagnetic radiation. Particularly remarkable is the fact that electromagnetic radiation does produce vorticity.
All this having been said, a natural question arises concerning our second goal, namely, what observational consequences could be derived from the analysis presented so far?
First of all it should be clear that the established fact that the emission of gravitational radiation always entails the appearance of vorticity in the congruence of the world lines of observers, provides a mechanism for detecting gravitational radiation. Thus, any experimental device intended to measure rotations could be a potential detector of gravitational radiation as well. We are well aware of the fact that extremely high sensitivities have to be reached, for these detectors to be operational. Thus, from the estimates displayed in [25], we see that for a large class of possible events leading to the emission of gravitational radiation, the expected values of Ω range from Ω ≈ 10 −15 s −1 to Ω ≈ 10 −19 s −1 . Although these estimates are twenty years old and deserves to be updated, we believe that probably the sensitivity of the actual technology is still below the range of expected values of vorticity. Nevertheless, the intense activity deployed in recent years in this field, invoking ring lasers, atom interferometers, atom lasers, anomalous spin-precession, trapped atoms and quantum interference (see References [56][57][58][59][60][61][62][63][64][65][66][67][68] and references therein), besides the incredible sensitivities obtained so far in gyroscope technology and exhibited in the Gravity Probe B experiment [1], make us being optimist in that this kind of detectors may be operating in the foreseeable future.
In the same order of ideas the established link between vorticity and electromagnetic radiation, has potential observational consequences which should not be overlooked. Indeed, intense electromagnetic outbursts are expected from hyperenergetic phenomena such as collapsing hypermassive neutron stars and Gamma Ray Bursts (see [69] and references therein). Therefore, although the contributions of the GEM terms in (77) are of order 1/r 3 , in contrast with the G terms which are of order 1/r, the coefficient of the former terms usually exceeds the latter by many orders of magnitude, which opens the possibility to detect them more easily.
Finally, the association of the sources of electromagnetic fields (charges and currents) with vorticity, suggests the possibility to extract information about the former, by measuring the latter. Thus, in [23], using the data corresponding to the earth, Bonnor estimates that the vorticity would be of the order of Ω ≈ 4 × 10 −33 s −1 . Although this figure is so small that we do not expect to be able to measure it in the near future, the strength of electromagnetic sources in very compact objects could produce vorticity many orders of magnitude larger.
To summarize. If we adopt the usual meaning of the verb "to explain" (a phenomenon), as referred to the action of expressing such a phenomenon in terms of fundamental concepts, then we can say that we have succeed in explaining the frame dragging effect as due to circular flows of super-energy and electromagnetic energy (whenever present) in planes orthogonal to vorticity vector. This result in turn, creates huge opportunities to obtain information from self-gravitating systems by measuring the vorticity of the congruence of world-lines of observers.