aa r X i v : . [ g r- q c ] F e b Deconstructing Frame Dragging
L. Herrera ∗ Instituto Universitario de F´ısica Fundamental y Matem´aticas,Universidad de Salamanca, Salamanca 37007, Spain (Dated: February 9, 2021)The vorticity of world lines of observers associated to the rotation of a massive body was reportedby Lense and Thirring more than a century ago. In their example the frame dragging effect inducedby the vorticity, is directly (explicitly) related to the rotation of the source. However in many othercases it is not so, and the origin of vorticity remains obscure and difficult to identify. Accordingly, inorder to unravel this issue, and looking for the ultimate origin of vorticity associated to frame drag-ging, 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 phys-ical quantities present in all these cases, which determine the vorticity and may legitimately beconsidered as responsible for the frame dragging. Doing so we provide a comprehensive physicalpicture of frame dragging. Some observational consequences of our results are discussed.
PACS numbers: 04.20.-q; 04.20.Cv; 04.30.NkKeywords: Frame dragging; Super–energy; Gravitational radiation
I. INTRODUCTION
The dragging of inertial frames produced by self–gravitating sources, whose existence has been recentlyestablished by observations [1], is one of the most remark-able effect predicted by the general theory of relativity(GR) (see [2, 3]).The term frame dragging usually refers to the influenceof a rotating massive body on a gyroscope by producingvorticity in the congruence of world lines of observersoutside the rotating object. Although the appropriate-ness of the term “frame dragging” has been questionedby Rindler [4], it nevertheless has been used regularlyin the literature till nowadays, and accordingly we shalladopt such term here (see also [5, 6]).The basic concept for the understanding of this effectis that of vorticity of a congruence, which describes therotation of a gyroscope attached to the congruence, withrespect to reference particles.Two different effects may be detected by means of gy-roscopes. One of this (Fokker–de Sitter effect) refersto the precession of a gyroscope following a closed or-bit around a spherically symmetric mass distribution. Ithas been verified with a great degree of accuracy by ob-serving the rotation of the earth–moon system aroundthe sun [7], but this is not the frame dragging effect weare interested in here. The other effect, the one we areconcerned with in this work, is the Lense–Thirring–Schiffprecession, which refers to the appearance of vorticity inthe congruence of world lines of observers in the gravita-tional field of a massive rotating ball. It was reported bythe first time by Lense and Thirring [8], and is usuallyreferred to as Lense-Thirring effect ( some authors sug-gest that it should be named instead, Einstein-Thirring- ∗ [email protected] Lense effect, see [9–11]). This result led Schiff [12] topropose the use of gyroscopes to measure such an effect.Since then this idea has been developed extensively (see[4, 13–21] and references cited therein).However, although the origin of vorticity may be eas-ily identified in the Lense–Thirring metric, as due to therotation of a massive object, it is not always explicitlyrelated to rotation of massive objects. In fact, in any vac-uum stationary space time (besides the Lense–Thirringmetric) we can detect a frame dragging effect, withoutspecifying an explicit link to the rotation of a massivebody [22].The situation is still more striking for the electro–vacuum space times. The point is that the quantity re-sponsible for the rotational (relativistic) multipole mo-ments in these spacetimes, is affected by the mass ro-tations (angular momentum), as well as by the electro-magnetic field, i.e. it contains contributions from both(angular momentum and electromagnetic field). This ex-plains why such a quantity does not necessarily vanish inthe case when the angular momentum of the source iszero but electromagnetic fields are present.The first known example of this kind of situation wasbrought out by Bonnor [23]. Thus, analyzing the gravi-tational field of a magnetic dipole plus an electric charge,he showed that the corresponding spacetime is station-ary and a frame dragging effect appears. As a matter offact all stationary electro–vacuum solutions exhibit framedragging [24], even though in some cases the angular mo-mentum of the source is zero. In this latter case the ro-tational relativistic multipole moments and thereby thevorticity, are generated by the electromagnetic field. Fur-thermore, as we shall see, electrodynamic radiation alsoproduces vorticity.Finally, it is worth recalling that vorticity is present ingravitationally radiating space–times. The influence ofgravitational radiation on a gyroscope through the vor-ticity associated with the emission of gravitational radi-ation was put forward for the first time in [25], and hasbeen discussed in detail since then in [26–33]. In this casetoo, the explicit relationship between the vorticity andthe emission of gravitational radiation was establishedwithout resorting to the rotation of the source itself.Although in many of the scenarios described above arotating object is not explicitly identified as the source ofvorticity, the fact remains that at purely intuitive level,one always associates the vorticity of a congruence ofworld lines, under any circumstance, to the rotation of“something”.The purpose of this work is twofold: on the onehand we shall identify the physical concept (the “some-thing”) behind all cases where frame dragging is ob-served, whether or not the angular momentum of thesource vanishes. On the other hand we would like to em-phasize the possible observational consequences of ourresults.As we shall see below, in all possible cases, the appear-ing vorticity is accounted for by the existence of a flowof superenergy on the plane orthogonal to the vorticityvector, plus (in the case of electro–vacuum spacetimes) aflow of electromagnetic energy on the same plane.Since superenergy plays a fundamental role in our ap-proach, we shall start by providing a brief introductionof this concept in the next section.
II. SUPERENERGY AND SUPER–POYNTINGVECTOR
The concept of energy is a fundamental tool in allbranches of physics, allowing to approach and solve a vastnumber of problems under a variety of circumstances.This explains the fact that since the early times of GRmany researchers have tried by means of very differentapproaches to present a convincing definition of gravita-tional energy, in terms of an invariant local quantity Allthese attempts, as is well known, have failed. The reasonfor this failure is easy to understand.Indeed, as we know, in classical field theory energy isa quantity defined in terms of potentials and their firstderivatives. On the other hand however, we also knowthat in GR it is impossible to construct a tensor expressedonly through the metric tensor (the potentials) and theirfirst derivatives (in accordance with the equivalence prin-ciple). Therefore, a local description of gravitational en-ergy in terms of true invariants (tensors of any rank) isnot 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 issuperenergy, which may be defined either from the Bel or from the Bel–Robinson tensor [34–36] (they both coincidein vacuum), and has been shown to be very useful whenit comes to explaining a number of phenomena in thecontext of GR.Both, the Bel and the Bel–Robinson tensors, are ob-tained by invoking the “structural” analogy between GRand Maxwell theory of electromagnetism. More specifi-cally, exploiting the analogy between the Riemann tensor( R αβγδ ) and the Maxwell tensor ( F µν ), Bel introduced afour–index tensor defined in terms of the Riemann tensorin a way which is a reminiscence of the definition of theenergy–momentum tensor of electromagnetism in termsof 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 morerecent references on this issue).Let us now introduce the electric and magnetic partsof the Riemann and the Weyl tensors as, E αβ = C ( R ) αγβδ u γ u δ , (1) H αβ = C ( R ) ∗ αγβδ u γ u δ , (2)where C ( R ) αγβδ is the Weyl (Riemann) tensor, thefour–vector u γ in vacuum is the tangent vector to theworld–lines of observers, and C ( R ) ∗ αγβδ is the dual of theWeyl(Riemann) tensor.A third tensor may be defined from the double dual ofthe Riemann tensor as X αβ = ∗ R ∗ αγβδ u γ u δ , (3)which in the case of the Weyl tensor X αβ coincide withthe electric part of the Weyl tensor (up to a sign).Next, from the analogy with electromagnetism thesuper–energy and the super–Poynting vector are definedby U ( R ) = 12 ( X αβ X αβ + E αβ E αβ ) + H αβ H αβ ,U ( C ) = E αβ E αβ + H αβ H αβ , (4) P ( R ) α = η αβγδ ( E βǫ H γǫ − X βǫ H ǫγ ) u δ ,P ( C ) α = 2 η αβγδ E βǫ H γǫ u δ . (5)where R ( C ) denotes whether the quantity is defined withRiemann (Weyl) tensor, and η αβγδ is the Levi–Civita ten-sor.In the next sections we shall bring out the role playedby the above introduced variables in the study of theframe dragging effect. III. FRAME DRAGGING IN VACUUMSTATIONARY SPACETIMES
As we mentioned in the Introduction, the first case offrame dragging analyzed in the literature was the Lense–Thirring effect. For didactical reasons we shall start byconsidering first this case and from there on, we shallconsider examples of increasing complexity. Thus, after-ward we shall consider the Kerr metric, an approximationof which is the Lense–Thirring spacetime, and finally weshall consider the general vacuum stationary spacetime case.
A. The Lense–Thirring precession
The Lense–Thirring effect is based on an approximatesolution to the Einstein equations which reads [8] ds = − (cid:18) − mr (cid:19) dt + (cid:18) mr (cid:19) (cid:0) dr + r dθ + r sin θdφ (cid:1) + 4 J sin θr dφdt. (6)It describes the gravitational field outside a spinningsphere of constant density, up to first order in m/r and J/r , with m and J denoting the mass and the angularmomentum respectively.Up to that order, it coincides with the Kerr metric, byidentifying ma = − J (7)where a is the Kerr parameter [38].Next, the congruence of the world–lines of observersat rest in the frame of (6) is described by the timelikevector u α whose components are u α = q − mr , , , , (8)from the above expression the vorticity vector, definedas usual by ω α = 12 η αηιλ u η u ι,λ , (9)has, up to order a/r and m/r , the following non–nullcomponents ω r = 2 ma cos θr , (10) ω θ = ma sin θr , (11)or Ω = ( ω α ω α ) / = mar p θ, (12)which at θ = π readsΩ = mar . (13) The above expression embodies the essence of theLense–Thirring effect. It describe the vorticity of theworld lines of observers, produced by the rotation ( J ) ofthe source. Such vorticity, as correctly guessed by Shiff[12], could be detected by a gyroscope attached to theworld lines of our observer.Even though in this case the vorticity is explicitly re-lated to the rotation of the spinning object which sourcesthe gravitational field, the fact that this link in manyother cases is not so explicitly established leads us to thequestion: what is (are) the physical mechanism(s) whichexplains the appearance of vorticity in the world linesof the observer? As we shall see in the next sectionsthe answer to this question may be given in terms of aflow of superenergy plus (in the case of electro–vacuumspacetimes) a flow of electromagnetic energy.So, in order to approach to this conclusion, let us cal-culate the leading term of the super–Poynting gravita-tional vector at the equator. Using (5) and (6) we obtainfor the only non–vanishing component (remember that invacuum both expressions for the super–Poynting vectorcoincide) P φ ≈ m r ar r , (14)It describes a flux of super–energy on the plane orthogo-nal to the vorticity vector. On the other hand it followsat once from (14) that P φ = 0 ⇔ a = 0 ⇔ ω α = 0.From the comments above, a hint about the link be-tween superenergy and vorticity (frame dragging) beginsto appear. In order to delve deeper on this issue let usnext consider the Kerr metric. B. Frame dragging in the Kerr metric
The calculations performed in the previous subsectioncan be very easily repeated for the Kerr metric.In Boyer–Linquist coordinates the Kerr metric takesthe form ds = (cid:18) − mrr + a cos θ (cid:19) dt − (cid:18) mar sin θr + a cos θ (cid:19) dtdφ + (cid:18) r + a cos θr − mr + a (cid:19) dr + ( r + a cos θ ) dθ + (cid:18) r sin θ + a sin θ + 2 mra sin θr + a cos θ (cid:19) dφ , (15)and the congruence of world–lines of observers at rest in(15) are defined by the time–like vector u α with compo- nents u α = q − mrr + a cos θ , , , . (16)This congruence is endowed with vorticity, described bya vorticity vector ω α whose non–vanishing componentsare ω r = 2 mra cos θ ( r − mr + a )( r + a cos θ ) − ( r − mr + a cos θ ) − , (17)and ω θ = ma sin θ ( r − a cos θ )( r + a cos θ ) − ( r − mr + a cos θ ) − . (18)The above expressions coincide with (10) and (11) up tofirst order in m/r and a/r .Finally, using the package GR- Tensor running on Maple we obtain for the super–Poynting vector (5) P µ = ( P t , , , P φ ) , (19)with P t = − m ra sin θ ( r − mr + a sin θ + a )( r + a cos θ ) − ( r − mr + a cos θ ) − × (cid:18) r − mr + a cos θr + a cos θ (cid:19) − / , (20) P φ = 9 m a ( r − mr − a cos θ + 2 a )( r + a cos θ ) − ( r − mr + a cos θ ) − × (cid:18) r − mr + a cos θr + a cos θ (cid:19) − / . (21)From the above expressions it follows, as in the precedentcase, that there is an azimuthal flow of superenergy aslong as a = 0, inversely the vanishing of such a flowimplies a = 0. Once again frame dragging appears to betightly related to a circular flow of superenergy on theplane 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 vortic-ity, since a specific interior for the Kerr metric is avail-able [39]. The remarkable fact is the presence of a non–vanishing T φt component of the energy–momentum ten-sor of the source, which, defining as usual an energy–momentum flux vector as: F ν = − V µ T νµ (where V µ denotes the four velocity of the fluid), implies that inthe 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 rem-iniscence of an effect appearing in stationary Einstein–Maxwell systems. Indeed, in all stationary Einstein–Maxwell systems, there is a non vanishing componentof 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 bea distinct physical property of rotating fluids, which hasbeen overlooked in previous studies of these sources, andthat is directly related to the vorticity (see eqs.(8) and(18) in [39]). C. Frame dragging in a general stationary vacuumspacetime
Let us now consider the general stationary and axisym-metric vacuum case.The line element for a general stationary and axisym-metric vacuum spacetime may be written as [40, 41] ds = − f dt +2 f ωdtdφ + f − e γ ( dρ + dz )+( f − ρ − f ω ) dφ , (22)where x = t ; x = ρ ; x = z and x = φ and metricfunctions depend only on ρ and z , which must satisfy thevacuum field equations: γ ρ = 14 ρf (cid:2) ρ (cid:0) f ρ − f z (cid:1) − f (cid:0) ω ρ − ω z (cid:1)(cid:3) , (23) γ z = 12 ρf (cid:0) ρ f ρ f z − f ω ρ ω z (cid:1) , (24) f ρρ = − f zz − f ρ ρ − f ρ (cid:0) ω ρ + ω z (cid:1) + 1 f (cid:0) f ρ + f z (cid:1) , (25) ω ρρ = − ω zz + ω ρ ρ − f ( f ρ ω ρ + f z ω z ) , (26)where subscripts denote partial derivatives.Then following the same protocol as in the previouscases, we define the four velocity vector for an observerat rest in the frame of (22), which reads u α = ( f − / , , , . (27)The super-Poynting vector can now be calculated forthe general class of spacetimes represented by the abovemetric (22), (i.e.: without making any assumption aboutthe matter content of the source), and one gets (usingagain GR–Tensor) P µ = ( P t , , , P φ ) with P t = ωP φ , (28) where P φ is given by (again in the general case, i.e.:without taking into account the field equations): P φ = f / e − γ ρ − { A } , or using the field equations (23-26) in the above ex-pression P φ = − f − / e − γ ρ − { A } , where A
11 and A
12 are given in the Appendix A.Now, in [24] it has been shown that for the generalmetric (22) the following relations hold H αβ = 0 ⇔ ω α = 0 ⇔ ω = 0 . (29)and of course as it follows from (5) H αβ = 0 ⇒ P µ = 0 . (30)In order to establish a link between vorticity and thesuper–Poynting vector of the kind already found for theKerr (and Lense–Thirring) metric we still need to provethat the vanishing of the super–Poynting vector impliesthe vanishing of the vorticity, i.e. we have to prove that P µ = 0 ⇔ H αβ = 0 ⇔ ω α = 0 ⇔ ω = 0 . (31)Such a proof has been carried out in [22], but is quitecumbersome and therefore we shall omit the details here.Thus based on (31) we conclude that for any stationaryspacetime, irrespectively of its source, there is a framedragging effect associated to a flux of superenergy on theplane orthogonal to the vorticity vector.We shall next analyze the electro–vacuum stationarycase. IV. FRAME DRAGGING INELECTRO–VACUUM STATIONARYSPACETIMES
Electro–vacuum solutions to the Einstein equationspose 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 mag-netic dipole with an electric charge in the center. Thesurprising result is that, for this spacetime, the worldlines of observers at rest with respect to the electromag-netic source are endowed of vorticity (i.e. the resultingspacetime is not static but stationary).In order to explain the appearance of vorticity in thespacetime generated by a charged magnetic dipole Bon-nor resorts to a result pointed out by Feynmann in hisLectures on Physics [42], showing that for such a sys-tem (in the context of classical electrodynamics) thereexists a non–vanishing component of the Poynting vec-tor describing a flow of electromagnetic energy round incircles. This strange result leads Feynmann to write that“it shows the theory of the Poynting vector is obviouslynuts”. However, some pages ahead in the same book,when discussing the “paradox” of the rotating disk withcharges and a solenoid, Feynmann shows that this “cir-cular” flow of electromagnetic energy is absolutely nec-essary in order to preserve the conservation of angularmomentum. In other words the theory of the Poyntingvector not only is not “nuts”, but is necessary to rec-oncile the electrodynamics with the conservation law ofangular momentum.Based on the above comments Bonnor then suggeststhat, in the context of GR, such a circular flow of energyaffects inertial frames by producing vorticity of congru-ences of particles, relative to the compass of inertia. Inother words Bonnor suggests that the “something” whichrotates thereby generating the vorticity, is electromag-netic energy.The interesting point is that this conjecture was shownto be valid for a general axially symmetric stationaryelectro–vacuum metric [24].Indeed, assuming the line element (22) for the space-time admitting an electromagnetic field, it can be shownthat the variable responsible for the rotational multipolemoments, which in its turn determine the vorticity ofthe congruence of world lines of observers, is affected by,both, the electromagnetic field and by the mass rota-tions (angular momentum) [24]. This explains why thevorticity does not necessarily vanish in the case when theangular momentum of the source is zero but electromag-netic fields are present. At any rate, it is important tostress that in such cases, the super–Poynting vector doesnot vanish either.We shall next consider the presence of vorticity due togravitational and electromagnetic radiation.
V. VORTICITY AND RADIATION
We shall now analyze the generation of vorticity relatedto the emission of gravitational and/or electromagneticradiation. As we shall see, the emission of radiation is al-ways accompanied by the appearance of vorticity of worldlines of observers. Furthermore, the calculations suggestthat once the radiation process has stopped, there is stilla remaining vorticity associated to the tail of the wave,which allows in principle to prove (or disprove) the vio-lation of the Huyghens principle in a Riemannian space-time (see [43–49] and references therein for a discussionon this issue), by means of observations.
A. Gravitational radiation and vorticity
Since the early days of GR, starting with the worksof Einstein and Weyl on the linear approximation of theEinstein equations, a great deal of work has been doneso far in order to provide a consistent framework for thestudy of gravitational radiation. Also, important collab- oration efforts have been carried on, and are now underconsideration, to put in evidence gravitational waves bymeans of laser interferometers [50].However it was necessary to wait for more than half acentury, until Bondi and coworkers [51] provided a firmtheoretical evidence of the existence of gravitational ra-diation without resorting to the linear approximation.The essential “philosophy” behind the Bondi formal-ism, consists in interpreting gravitational radiation as thephysical process by means of which the source of the field“informs” about any changes in its structure. Thus theinformation required to forecast the evolution of the sys-tem (besides the “initial” data) is thereby identified withradiation itself, and this information is represented bythe so called “news function”. In other words, whateverhappens at the source, leading to changes in the field, itcan only do so by affecting the news function and viceversa. Therefore if the news function is zero over a timeinterval, there is no gravitational radiation over that in-terval. Inversely, non vanishing news on an interval im-plies the emission of gravitational radiation during thatinterval. Thus the main virtue of this approach resides inproviding a clear and precise criterion for the existenceof gravitational radiation.The above described picture is reinforced by the factthat the Bondi mass of a system is constant if and onlyif there are no news.In order to facilitate discussion let us briefly intro-duce the main aspects of the Bondi approach. Bondi andcoworkers start with the general form of an axially (andreflection) symmetric asymptotically flat metric given by ds = (cid:18) Vr e β − U r e γ (cid:19) du + 2 e β dudr + 2 U r e γ dudθ − r (cid:0) e γ dθ + e − γ sin θdφ (cid:1) , (32)where V, β, U and γ are functions of u, r and θ .The coordinates are numbered x , , , = u, r, θ, φ re-spectively. u is a timelike coordinate such that u = constant defines a null surface. In flat spacetime thissurface coincides with the null light cone open to the fu-ture. r is a null coordinate ( g rr = 0) and θ and φ are twoangle coordinates.Regularity conditions in the neighborhood of the polaraxis (sin θ = 0), implies that as sin θ − > V, β, U/ sin θ, γ/ sin θ, (33)each equals a function of cos θ regular on the polar axis.Then the four metric functions are assumed to be ex-panded in series of 1 /r , which using field equations pro-duces γ = c ( u, θ ) r − + (cid:20) C ( u, θ ) − c (cid:21) r − + ..., (34) U = − ( c θ + 2 c cot θ ) r − + ..., (35) V = r − M ( u, θ ) − (cid:20) N θ + N cot θ − c θ − cc θ cot θ − c (1 + 8 cot θ ) (cid:21) r − + ..., (36) β = − c r − + ... (37)where letters as subscripts denote derivatives, and4 C u = 2 c c u + 2 cM + N cot θ − N θ . (38)The three functions c, M and N depend on u and θ ,and are further related by the supplementary conditions M u = − c u + 12 ( c θθ + 3 c θ cot θ − c ) u , (39) − N u = M θ + 3 cc uθ + 4 cc u cot θ + c u c θ . (40)In the static case M equals the mass of the systemwhereas N and C are closely related to the dipole andquadrupole moment respectively.Next, Bondi defines the mass m ( u ) of the system as m ( u ) = 12 Z π M ( u, θ ) sin θdθ, (41)which by virtue of (39) and (33) yields m u = − Z π c u sin θdθ. (42) The two main conclusions emerging from the Bondi’sapproach are • If γ, M and N are known for some u = a (constant),and c u (the news function) is known for all u inthe interval a ≤ u ≤ b , then the system is fullydetermined in that interval. • As it follows from (42), the mass of a system isconstant if and only if there are no news.In the light of these comments the relationship betweennews function and the occurrence of radiation becomesclear.Let us now calculate the vorticity of the world lines ofobservers at rest in the frame of (32). For such observersthe four-velocity vector has components u α = (cid:18) A, e β A , U r e γ A , (cid:19) (43)with A ≡ (cid:18) Vr e β − U r e γ (cid:19) / . (44)Using (9) , we easily obtain ω α = (cid:0) , , , ω φ (cid:1) (45)with ω φ = − e − β r sin θ { β θ e β − e β A θ A − (cid:0) U r e γ (cid:1) r + 2 U r e γ A A r + e β (cid:0) U r e γ (cid:1) u A − U r e γ A β u e β } (46)and for the absolute value of ω α we getΩ ≡ ( − ω α ω α ) / = e − β − γ r { β θ e β − e β A θ A − (cid:0) U r e γ (cid:1) r + 2 U r e γ A r A + e β A (cid:0) U r e γ (cid:1) u − β u e β A U r e γ } (47) Feeding back (34)–(37) into (47) and keeping onlyterms up to order r , we obtainΩ = − r ( c uθ + 2 c u cot θ ) + 1 r [ M θ − M ( c uθ + 2 c u cot θ ) − cc uθ + 6 cc u cot θ + 2 c u c θ ] . (48)Let us now analyze the expression above. First of allobserve that, up to order 1 /r , a gyroscope in the grav-itational field given by (32) will precess as long as the system radiates ( c u = 0). Indeed, if we assume c uθ + 2 c u cot θ = 0 (49)then c u = F ( u )sin θ (50)which implies, due to the regularity conditions (33) F ( u ) = 0 = ⇒ c u = 0 . (51)In other words the leading term in (48) will vanish if andonly if c u = 0.Let us now analyze the term of order 1 r . 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 =constant) the system is static, in which case c u = 0 (52)which implies , because of (40) M θ = 0 (53)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 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-radiativemotions [51].Thus, in the interval u ∈ ( u , u f ) the leading term ofvorticity is given by the term of the order 1 /r in (48). For u > u f there is a vorticity term of order r describingthe effect of the tail of the wave on the vorticity. Thisprovides an “observational” possibility to find evidencefor the violation of the Huygens’s principle.Following the line of arguments of the preceding sec-tions, we shall establish a link between vorticity and acircular flow of superenergy on the plane orthogonal tothe vorticity vector. For doing so, let us calculate thesuper–Poynting vector ( P µ ), defined by (5). We obtainthat the leading terms for each super–Poynting compo-nent are P r = − r c uu , (54) P θ = − r sin θ { [2 c uu c + c uu c u ] cos θ + (cid:2) c uu c θu + c uu c θ (cid:3) sin θ } , (55) P φ = P φ = 0 . (56)The vanishing (at all orders) of the azimuthal compo-nent ( P φ ), is expected from the reflection symmetry ofthe Bondi metric, which is incompatible with the pres-ence of a circular flow of superenergy in the φ direction.Since the vorticity vector, which is orthogonal to theplane of rotation, has in the Bondi spacetime only onenon–vanishing contravariant component ( φ ), then theplane 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 theBondi case.In order to strength further the case for the super-Poynting vector as the physical origin of the mentionedvorticity, we shall consider next the general radiativemetric without axial and reflection symmetry.The extension of the Bondi formalism to the case with-out any kind of symmetries was performed by Sachs [52].In this case the line element reads (we have found moreconvenient to follow the notation given in [53] which isslightly different from the original Sachs paper) ds = ( V r − e β − r e γ U cosh 2 δ − r e − γ W cosh 2 δ − r U W sinh 2 δ ) du + 2 e β dudr + 2 r ( e γ U cosh 2 δ + W sinh 2 δ ) dudθ + 2 r ( e − γ W cosh 2 δ + U sinh 2 δ ) sin θdudφ − r ( e γ cosh 2 δdθ + e − γ cosh 2 δ sin θdφ + 2 sinh 2 δ sin θdθdφ ) , (57)where β , γ , δ , U , W , V are functions of x = u , x = r , x = θ , x = φ .The general analysis of the field equations is similar tothe one in [51], but of course the expressions are far morecomplicated (see [52, 53] for details). In particular, thereare now two news functions. Let us first calculate the vorticity for the congruenceof observers at rest in (57), whose four–velocity vector isgiven by u α = A − δ αu , (58)where now A is given by A = ( V r − e β − r e γ U cosh 2 δ − r e − γ W cosh 2 δ − r U W sinh 2 δ ) / . (59)Thus, (9) lead us to ω α = ( ω u , ω r , ω θ , ω φ ) , (60)where ω u = − A sin θ { r e − β ( W U r − U W r ) + (cid:2) r sinh 2 δ cosh 2 δ ( U e γ + W e − γ ) + 4 U W r cosh δ (cid:3) e − β γ r + 2 r e − β ( W e − γ − U e γ ) δ r + e β [ e − β ( U sinh 2 δ + e − γ W cosh 2 δ )] θ − e β [ e − β ( W sinh 2 δ + e − γ U cosh 2 δ ] φ } , (61) ω r = 1 e β sin θ { r A − [(( U e γ + W e − γ ) sinh 2 δ cosh 2 δ + U W cosh δ ) γ u + ( W e − γ − U e γ ) δ u + 12 ( W U u − U W u )]+ A [ A − ( W e − γ cosh 2 δ + U sinh 2 δ )] θ − A [ A − ( W sinh 2 δ + U e γ cosh 2 δ )] φ } , (62) ω θ = 12 r sin θ { A e − β [ r A − ( U sinh 2 δ + W e − γ cosh 2 δ )] r − e β A − [ e − β r ( U sinh 2 δ + e − γ W cosh 2 δ )] u + e β A − ( e − β A ) φ } , (63)and ω φ = 12 r sin θ { A e − β [ r A − ( W sinh 2 δ + U e γ cosh 2 δ )] r − e β A − [ r e − β ( W sinh 2 δ + U e γ cosh 2 δ )] u + A − e β ( A e − β ) θ } . (64)Expanding the metric functions in series of 1 /r as in theprevious case, using the field equations and feeding back the resulting expressions into (61, 62, 63, 64) we get forthe leading term of the absolute value of ω µ Ω = − r [( c θu + 2 c u cot θ + d φu csc θ ) + ( d θu + 2 d u cot θ − c φu csc θ ) ] / , (65)which of course reduces to (48) in the Bondi (axially andreflection symmetric) case ( d = c φ = 0). It is worthstressing the fact that now we have two news functions( c u , d u ).Next, the calculation of the super–Poynting vectorgives 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 justpresent the leading terms for each super–Poynting com-ponent, they read P r = − r ( d uu + c uu ) , (67)0 P θ = − r sin θ { [2( d uu + c uu ) c + c uu c u + d uu d u ] cos θ + (cid:2) c uu c θu + d uu d θu + ( c uu + d uu ) c θ (cid:3) sin θ ++ c uu d φu − d uu c φu + ( d uu + c uu ) d φ } , (68) P φ = 2 r { c uu d u − d uu c u − ( d uu + c uu ) d ] cos θ ++ (cid:2) c uu d θu − d uu c θu − ( c uu + d uu ) d θ (cid:3) sin θ + ( d uu + c uu ) c φ − ( c uu c φu + d uu d φu ) } (69)from which it follows P φ = − r sin θ { sin θ [ d uθ c uu − d uu c uθ ] + 2 cos θ [ c uu d u − d uu c u ] − [ c uu c uφ + d uu d uφ ] } , (70)this component of course vanishes in the Bondi case.From the expressions above we see that the main con-clusion established for the Bondi metric, is also valid inthe most general case, namely, there is always a non–vanishing component of P µ on the plane orthogonal to aunit vector along which there is a non–vanishing compo-nent of vorticity, and inversely, P µ vanishes on a planeorthogonal to a unit vector along which the componentof vorticity vector vanishes. The link between the super–Poynting vector and vorticity is thereby firmly estab-lished.So far we have shown the appearance of vorticity in sta-tionary vacuum spacetimes, stationary electro–vacuumspacetimes and in radiative vacuum spacetimes (Bondi–Sachs), and have succeed in exhibiting the link be-tween this vorticity and a circular flow of electromagneticand/or super–energy, on the plane orthogonal to the vor-ticity vector. It remains to analyze the possible role ofelectromagnetic radiation in the appearance of vorticity.The next section is devoted to this issue. B. Electromagnetic radiation and vorticity
The relationship between electromagnetic radiationand vorticity has been unambiguously established in [54].The corresponding calculations are quite cumbersomeand we shall not reproduce them here. Instead we shallhighlight the most important results emerging from suchcalculations.The formalism used to study the general electro–vacuum case (including electromagnetic radiation) wasdeveloped by van der Burg in [55]. It represents a gener-alization of the Bondi–Sachs formalism for the Einstein–Maxwell system. Thus, the starting point is the Einstein–Maxwell sys-tem of equations, which reads R µγ + T µγ = 0 , (71) F [ µν,δ ] = 0 , (72) F µν ; ν = 0 , (73)where R µγ is the Ricci tensor and the energy momentumtensor T µγ of the electromagnetic field is given as usualby T µν = 14 g µν F γδ F γδ − g γδ F µγ F νδ . (74)Then, following the script indicated in [51], i.e. ex-panding the physical and metric variables in power seriesof 1 /r and using the Einstein–Maxwell equations, one ar-rives at the conclusion that if a specific set of functions isprescribed on a given initial hypersurface u = constant ,the evolution of the system is fully determined providedthe four functions, c u , d u , X, Y are given for all u . Thesefour functions are the news functions of the system. Thefirst two ( c u , d u ) are the gravitational news functions al-ready mentioned before for the purely gravitational case,whereas X and Y are the two news functions correspond-ing to the electromagnetic field, these appear in the se-ries expansion of F , F . Thus, whatever happens atthe source leading to changes in the field, it can only doso by affecting the four news functions and viceversa.Following the same line of arguments, an equation forthe decreasing of the mass function due to the radiation(gravitational and electromagnetic) similar to (42) canbe obtained, it reads m u = − Z π Z π ( c ∗ u ¯ c ∗ u + 12 X ∗ ¯ X ∗ ) sin θdθdφ, (75)1where c ∗ = c + id, X ∗ = X + iY, (76)and bar denotes complex conjugate.Having arrived at this point we can now proceed to cal-culate the vorticity, the super–Poynting vector and theelectromagnetic Poynting vector. The resulting expres-sions are available in [54], since they are extremely long,here we shall focus on the main conclusions emergingfrom them.First, the vorticity vector (9) is calculated for the four–vector u α given by (58). The important point to stresshere is that the absolute value of ω µ can be written gener-ically as Ω = Ω G r − + · · · + Ω GEM r − + · · · , (77)where subscripts G , GEM and EM stand for gravi-tational, gravito–electromagnetic and electromagnetic.The “gravitational” subscript refers to those terms con-taining exclusively functions appearing in the purelygravitational case and their derivatives. “Electromag-netic” terms are those containing exclusively functionsappearing in F µν and their derivatives, whereas “gravito–electromagnetic” subscript refers to those terms contain-ing functions of either kind and/or combination of both.Finally, we calculate the electromagnetic Poynting vec-tor defined by S α = T αβ u β , (78)and the super–Poynting vector defined by (5). Since weare not operating in vacuum, P ( C ) α and P ( R ) α are dif-ferent, we shall use P ( C ) α for the discussion.The resulting expressions are deployed in [54]. Letus summarize the main information contained in suchexpressions.First, we notice that the leading terms for each super–Poynting (contravariant) component are P u = P u G r − + · · · ,P r = P r G r − + · · · ,P θ = P θ G r − + · · · + P θ GEM r − + · · · ,P φ = P φ G r − + · · · + P φ GEM r − + · · · , (79)whereas for the electromagnetic Poynting vector we canwrite S u = S u EM r − + S u GEM r − · · · , (80) S r = S r EM r − + S r GEM r − + · · · , (81) S θ = S θ EM r − + S θ GEM r − + · · · , (82) S φ = S φ EM r − + S φ GEM r − + · · · . (83) Next, there are explicit contributions from the electro-magnetic news functions in Ω GEM as well as in P φ GEM and P θ GEM . More so, the vanishing of these contribu-tions in P φ GEM and P θ GEM implies the vanishing of thecorresponding contribution in Ω
GEM , and viceversa.From the above it is clear that electromagnetic ra-diation as described by electromagnetic news functionsdoes produce vorticity. Furthermore we have identifiedthe presence of electromagnetic news both in the Poynt-ing and the super–Poynting components orthogonal tothe vorticity vector. Doing so we have proved that aBonnor–like mechanism to generate vorticity is at work inthis case too, but with the important difference that nowvorticity is generated by the contributions of, both, thePoynting and the super–Poynting vectors, on the planesorthogonal to the vorticity vector.
VI. DISCUSSION
We started this manuscript with two goals in mind.On the one hand we wanted to identify the fundamentalphysical phenomenon which being present in all scenariosexhibiting frame dragging, could be considered as theresponsible for the frame dragging effect. In other wordswe wanted to identify the factor that mediates betweenthe source of the gravitational field and the appearanceof vorticity, in any scenario.On the other hand, we wanted to explore the observa-tional consequences that could be derived from our anal-ysis.Concerning our first goal, it has been clearly estab-lished that in vacuum, the appearance of vorticity is al-ways related to the existence of circular flow of super–energy in the plane orthogonal to the vorticity vector.This is true for all stationary vacuum spacetimes as wellas for general Bondi–Sachs radiative spacetimes.In the case of electro–vacuum spacetimes, we havecircular flows of super–energy as well as circular flowsof electromagnetic energy in the plane orthogonal tothe vorticity vector. This is true in stationary electro–vacuum spacetimes as well as in spacetimes admitting,both, gravitational and electromagnetic radiation. Par-ticularly remarkable is the fact that electromagnetic ra-diation does produce vorticity.All this having been said, a natural question arises con-cerning our second goal, namely, what observational con-sequences could be derived from the analysis presentedso far?First of all it should be clear that the established factthat the emission of gravitational radiation always en-tails the appearance of vorticity in the congruence of theworld lines of observers, provides a mechanism for de-tecting gravitational radiation. Thus, any experimentaldevice intended to measure rotations could be a potentialdetector of gravitational radiation as well. We are wellaware of the fact that extremely high sensitivities have tobe reached, for these detectors to be operational. Thus,2from the estimates displayed in [25], we see that for alarge class of possible events leading to the emission ofgravitational radiation, the expected values of Ω rangefrom Ω ≈ − s − to Ω ≈ − s − . Although theseestimates are twenty years old and deserves to be up-dated, we believe that probably the sensitivity of the ac-tual technology is still below the range of expected valuesof vorticity. Nevertheless, the intense activity deployedin recent years in this field, invoking ring lasers, atominterferometers, atom lasers, anomalous spin–precession,trapped atoms and quantum interference (see References[56–68] and references therein), besides the incrediblesensitivities obtained so far in gyroscope technology andexhibited in the Gravity Probe B experiment [1], makeus being optimist in that this kind of detectors may beoperating in the foreseeable future.In the same order of ideas the established link betweenvorticity and electromagnetic radiation, has potential ob-servational consequences which should not be overlooked.Indeed, intense electromagnetic outbursts are expectedfrom hyperenergetic phenomena such as collapsing hy-permassive neutron stars and Gamma Ray Bursts (see[69] and references therein). Therefore, although the con-tributions of the GEM terms in (77) are of order 1 /r , incontrast with the G terms which are of order 1 /r , the co-efficient of the former terms usually exceeds the latter bymany orders of magnitude, which opens the possibilityto detect them more easily.Finally, the association of the sources of electromag-netic fields (charges and currents) with vorticity, suggeststhe possibility to extract information about the former,by measuring the latter. Thus, in [23], using the datacorresponding to the earth, Bonnor estimates that thevorticity would be of the order of Ω ≈ × − s − .Although this figure is so small that we do not expectto be able to measure it in the near future, the strengthof electromagnetic sources in very compact objects couldproduce vorticity many orders of magnitude larger.To summarize. If we adopt the usual meaning of theverb “to explain” (a phenomenon), as referred to the ac-tion of expressing such a phenomenon in terms of fun-damental concepts, then we can say that we have suc-ceed in explaining the frame dragging effect as due tocircular flows of super–energy and electromagnetic en-ergy (whenever present) in planes orthogonal to vorticityvector. This result in turn, creates huge opportunitiesto obtain information from self–gravitating systems bymeasuring the vorticity of the congruence of world–linesof observers. ACKNOWLEDGMENTS
This research was funded by Ministerio de Ciencia, In-novacion y Universidades. Grant number: PGC2018–096038–B–I00.
Appendix A A
11 = (cid:2) − ρω ρ ( ω ρ + ω z ) γ ρ − ρω z ( ω ρ + ω z ) γ z − ω z ω ρρ ρ + ω z ω zz ρ + 4 ω z ω ρ ω ρz ρ − ω ρ ω zz ρ − ω ρ + ω ρ ω ρρ ρ (cid:3) f +3 ρ ( ω ρ + ω z )( ω z f z + ω ρ f ρ ) f − ρ ( − ργ z ω ρz + 2 γ z ω ρ ρ +2 ργ ρ ω ρ + γ z ω z + γ ρ ω ρ + ργ ρ ω zz − ργ ρ ω ρρ ) f (cid:2) ρ ( f z ω z + f ρ ω ρ ) γ ρ + 2 ρ ( ρf zz ω ρ − ρf ρz ω z − f z ω z +4 f ρ ω ρ − ρf z ω ρz − ρf ρρ ω ρ − ρf ρ ω ρρ + ρf ρ ω zz ) γ ρ +4 ρ ( f z ω z + f ρ ω ρ ) γ z + 2 ρ (4 f ρ ω z + ρf z ω ρρ − ρf ρz ω ρ + ρf ρρ ω z − ρf ρ ω ρz − ρf zz ω z − ρf z ω zz + 2 ω ρ f z ) γ z +4 ρ f ρz ω ρz − ρ f zz ω ρρ − ρ f ρρ ω zz + ρ f zz ω ρ − ρ f ρz ω z − ρ f ρρ ω ρ + ρ f zz ω zz + ρ f ρρ ω ρρ (cid:3) f − ρ ( f ρ + f z ) ω ρ γ ρ − ρ ( f ρ + f z ) ω z γ z + 3 ρ ( f ρρ f ρ ω ρ + f zz f z ω z + 2 f ρz f z ω ρ − f ρρ f z ω z + 2 f ρz f ρ ω z − f zz f ρ ω ρ ) ,A
12 = ω ρ ( − ω z − ω ρ ω z + ω ρ ) f + (cid:2) − ρω ρ f ρ ( ω z + ω ρ + 2 ω ρ ω z ) − ρf z ω z ( ω ρ + 2 ω z ω ρ + ω z ) (cid:3) f + (cid:2) − ρω zz ( ω ρ + 3 ω z ) + 4 ω ρ ( − ρω z ω ρz + ω z ) (cid:3) f + (cid:2) ρω z ( − ω z f z + ρω z f ρz − ρω ρ ω z f zz − ρω ρ f ρz − ρf ρ ω z ω zz − ω ρ f z − ρf ρ ω ρ ω ρz − ω z ω ρ f ρ − ρf z ω ρ ω zz + ρf z ω z ω ρz )+4 ρω ρ ( − ρω ρ f z ω ρz + ρω ρ f ρ ω zz + ω ρ f ρ + ρω ρ f zz ) (cid:3) f + (cid:2) − ρ ω ρ ( f z + f ρ ) − ρ f z ω ρ ω z (2 ω ρ f ρ + 5 ω z f z )+2 ρ f ρ ω z (2 ω z f z − ω ρ f ρ ) (cid:3) f + (cid:2) ρ ( f ρz ω z − f ρ ω zz ) − ρ ( f ρz ω ρz + f zz ω zz ) + 10 ρ f z ω z f ρ ω ρ ( f z ω z + f ρ ω ρ ) − ρ f z ( f ρ f z ω ρ + f ρ ω z ) + 2 ρ f ρ ( ω ρ − ω ρ ω z )+2 ρ f z ( ω z − ω z ω ρ ) (cid:3) f + (cid:2) − ρ f ρz ( ω ρ f z + ω z f ρ )+4 ρ f ρ ( f ρ ω ρ − f z ω z ) + 4 ρ (3 f ρ ω zz + 2 f zz f ρ ω ρ − ω ρz f ρ f z + f z ω zz − f zz ω z f z ) (cid:3) f + (cid:2) ρ f ρz ( f ρ ω z − f z ω z + 2 ω ρ f ρ f z )+4 ρ f zz ( f z ω ρ − f ρ ω ρ + 2 ω z f ρ f z ) + 4 ρ ω ρz ( − f z + 3 f z f ρ )+4 ρ ω zz ( − f ρ + 3 f ρ f z ) + 4 ρ (4 f ρ f z ω z − f z f ρ ω ρ − f ρ ω ρ + 3 f z ω z ) (cid:3) f + (cid:2) ρ ω ρ (14 f z f ρ − f z + 5 f ρ )+4 ρ ω z ( f ρ f z + 5 f z f ρ ) (cid:3) f − ρ ω z (2 f z f ρ + f z + f z f ρ ) − ρ ω ρ (2 f ρ f z + f ρ + f r hof z ) . 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