Electromagnetically-Induced Frame Dragging Around Astrophysical

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Electromagnetically-Induced Frame-Dragging around Astrophysical Objects

AndrésF. Gutiérrez-Ruiz∗ and Leonardo A. Pachón† Grupo de F´ısica Atómica y Molecular, Instituto de F´ısica, Facultad de Ciencias Exactas y Naturales, Universidad de Antioquia UdeA; Calle 70 No. 52-21, Medell´ın, Colombia. (Dated: June 23, 2015) Frame dragging (Lense-Thirring effect) is generally associated with rotating astrophysical objects. However, it can also be generated by electromagnetic fields if electric and magnetic fields are simultaneously present. In most models of astrophysical objects, macroscopic charge neutrality is assumed and the entire electromagnetic field is characterized in terms of a magnetic dipole component. Hence, the purely electromagnetic contribution to the frame dragging vanishes. However, strange stars may possess independent electric dipole and neutron stars independent electric quadrupole moments that may lead to the presence of purely electromagnetic contributions to the frame dragging. Moreover, recent observations have shown that in stars with strong electromagnetic fields, the magnetic quadrupole may have a significant contribution to the dynamics of stellar processes. As an attempt to characterize and quantify the effect of electromagnetic frame-dragging in these kind of astrophysical objects, an analytic solution to the Einstein-Maxwell equations is constructed here on the basis that the electromagnetic field is generated by the combination of arbitrary magnetic and electric dipoles plus arbitrary magnetic and electric quadrupole moments. The effect of each multipole contribution on the vorticity scalar and the Poynting vector is described in detail. Corrections on important quantities such the innermost stable circular orbit (ISCO) and the epicyclic frequencies are also considered.

I. INTRODUCTION tions is not available because of the lack of an analytic exact solution with such a complex electromagnetic field Frame dragging (Lense-Thirring effect) is the configuration. quintessential hallmark of general relativity and is the Moreover, recent observations have shown that in result of the spacetime vorticity. It was detected by the stars with strong electromagnetic fields, the magnetic Gravity Probe B [1] and traditionally, it has been asso- quadrupole may have significant contributions on the dy- ciated with rotating stellar astrophysical objects and in namics of stellar processes. Therefore, it is well known other astrophysical contexts, such as Galactic models, the by now that the actual configuration of the magnetic frame dragging has been associated to the presence of field of strongly magnetized stars may depart from the magnetogravitational monopoles [2, 3]. Surprisingly, if dipole configuration [20]. As an attempt to describe the the astrophysical object does not rotate but possesses spacetime geometry surrounding these kind of astrophysi- both electric and magnetic fields, the spacetime vorticity cal objects, an analytic solution to the Einstein-Maxwell does not vanish [4–6]. In this case, frame dragging is of field equations is constructed here on the basis that the purely electromagnetic nature and it is associated with electromagnetic field is generated by the combination the existence of a nonvanishing electromagnetic Poynting of arbitrary magnetic and electric dipole plus arbitrary vector around the source [5–9]. magnetic and electric quadrupole moments. This ana- In most of the early models of astrophysical objects, lytic exact solution allows for analyzing, e.g., the effect macroscopic charge neutrality is assumed [10] and the of each multipole contribution on the vorticity scalar and magnetic field is characterized in terms of a pure dipole the Poynting vector (see Sec.V). Moreover, it is possi- component [11–14]. This idea was endorsed by the con- ble to predict corrections to important quantities such firmation of many predicted features of a star with a as the innermost stable circular orbit and the epicyclic arXiv:1504.01763v2 [gr-qc] 19 Jun 2015 dipole field using two- and three-dimensional magneto- frequencies. hydrodynamic numerical simulations of magnetospheric To motivate further the derivation of the model consid- accretion [15–18]. Thus, under this configuration the ered here, the observational evidence for the existence of purely electromagnetic contribution to the vorticity ten- nondipolar fields in a variety of astrophysical objects is sor vanishes. However, astrophysical objects such as discussed next. strange stars may possess independent electric dipole (see Sec. 10.4 in Ref. [19]) and neutron stars independent electric quadrupole moments (see below), which may lead II. OBSERVATIONAL EVIDENCE OF to the presence of purely electromagnetic contributions NON-DIPOLAR FIELDS to frame dragging. A precise account of those contribu- Measurements of magnetic fields of strongly magnetized stars, based on the Zeeman–Doppler imaging technique [21], have shown that for these kind of astrophysical ob- ∗ [email protected] jects the magnetic field has a complicated multipolar † [email protected] topology in the vicinity of the star [22–26]. This feature 2 certainly is of prime relevance in, e.g., the accretion-disk Maxwell field equation have been developed, e.g. [46–48], dynamics in binary systems because if the quadrupole and (iii) systematic studies on the construction of exact component dominates, then the flow of matter into the solution from its physical content have been performed star will certainly differ from the well-known dynamics [49, 50], a new analytic exact solution to the Einstein- induced by a pure dipole field [25]. Maxwell field equations is introduced below. This so- Complex configurations of magnetic fields are also lution provides physical insight, e.g., into the influence present in stars such as the T-Tauri stars. They are of high order electromagnetic multipole moments in the young stellar objects of low mass that present variations frame dragging and in studying quasi-periodic oscillations in their luminosity. An important subclass of this kind (QPOs), which become a useful tool to identify the char- of stars are the so-called classical T-Tauri stars (cTTS) acteristics of the compact objects present in Low Mass because they present accretion from the circumstellar disk X-Ray Binaries (LXRB) [51–53]. The model presented [27]. Recent evidence points out that in classical T-Tauri here is a member of the N-solitonic solution derived in stars, the magnetic field near the star is strongly non- Ref. [54]. In AppendixA the relevant equations of the dipolar [26]. In the particular case of V2129 Oph, there derived metric are summarized. exits a dominant octupole, 0.12 T, and a weak dipole In terms of the quasicylindrical Weyl-Lewis-Papapetrou component, 0.035 T, of the magnetic field [28]. Under- coordinates xµ = (t, ρ, z, φ), the simplest form of the line standing the circumstellar disk dynamics, under complex element for the stationary axisymmetric case was given field topologies, could provide insight into the formation by Papapetrou [55], of planets and the evolution of the star itself. 2 µ ν The discussion above also applies to white dwarfs. The ds = gµν dx dx , (1) first observations indicated that only a small fraction of white dwarfs appear to exhibit magnetic fields. However, with gtt = −f(ρ, z), gtφ = f(ρ, z)ω(ρ, z), gφφ = 2 −1 2 the observational situation changed significantly by the ρ f (ρ, z) − f(ρ, z)ω (ρ, z) and gzz = gρρ = discovery of strong-field magnetic white dwarfs [29–31], e2γ(ρ,z)f −1(ρ, z). The metric functions f, ω and γ can which are known to cover a wide range of field strengths be obtained from the Ernst complex potentials E and Φ ∼ 1 − 100T and deviates from the simple dipolar configu- (see details in Ref. [46]). The Ernst potentials obey the ration [32, 33]. Recent spectropolarimetric observations relations [46] in white dwarf have shown that, in addition to the dipole term, the quadrupole and octupole terms make significant (Re E + |Φ|2)∇2E = (∇E + 2Φ∗∇Φ) · ∇E, (2) contributions to the field when it is represented as an (Re E + |Φ|2)∇2Φ = (∇E + 2Φ∗∇Φ) · ∇Φ . axisymmetric multipolar expansion [34]. Many studied cases indicate that higher multipole components or non From a physical viewpoint, the Ernst potentials are rele- axisymmetric components may be required in a realistic vant because they lead to the definition of the analogues of model of white dwarfs (see, e.g., Ref [35]). the Newtonian gravitational potential, ξ = (1−E)/(1+E), Magnetars constitute an additional source of motiva- and the Coulomb potential, q = 2Φ/(1 + E). The real tion. They are characterized by their extremely powerful part of ξ accounts for the matter distribution and its magnetic fields, covering strengths from ∼ 108 to ∼ 1011 T imaginary part for the mass currents. Besides, the real [36, 37], so that they can have occasional violent bursts. part of the q potential denotes the electric field and its However, certain magnestars, such as the SGR 0418+5729, imaginary part the magnetic field. undergo this bursting phenomenon even with a weak mag- The Ernst equations (2) can be solved by means of the 8 netic fields (∼ 10 T) [38]. For SGR 0418+5729, the Sibgatullin’s integral method [47, 48], according to which observed X-ray spectra cannot be fit with this low field the complex potentials E and Φ can be calculated from strength; hence, it has been suggested that a hidden non- specified axis data E(z, ρ = 0) and Φ(z, ρ = 0) [47, 48]. dipole field component must be present to explain the Motivated by the accuracy [41, 42] and the level of gen- bursting episodes [39]. erality of the analytic solution derived in Ref. [42], the In summary, there is sufficient observational evidence Ernst potential E(z, ρ = 0) is chosen as in Ref. [42]. To to develop a consistent analytic closed representation construct the exact solution that represents the electro- of the exterior spacetime around stars with non-dipolar magnetic field configuration described above, the Ernst magnetic fields. potential Φ(z, ρ = 0) is chosen following the prescription in Ref. [49]. The Ernst potentials on the symmetry axis read III. ANALYTIC FORMULAE OF THE MODEL z3 − z2(m + ia) − kz + is E(z, ρ = 0) = , z3 + z2(m − ia) − kz + is By combining the facts that (i) almost all the analytic (3) closed form models for relevant astrophysical objects have z2ς + z(υ + iµ) + iζ + χ Φ(z, ρ = 0) = . been conceived in the frame of stationary axisymmetry z3 + z2(m − ia) − kz + is geometry (see [40–45] for the case of neutron stars), (ii) powerful tools to construct exact solution to the Einstein- The physical meaning of the parameters in Eq. (3) is 3 derived from the multipole moments calculated using the Fodor-Hoenselaers-Perjésprocedure [56] (see also Ref. [57]). For the present case,

2 3 P0 = m, P1 = iam, P2 = (k − a )m, P3 = −im(a − 2ak + s), 1 P = m 70a4 − 210a2k + 13aς(µ − iυ) + 140as + 10k(7k − m2 + ς2) + 3(µ2 + υ2) , 4 70 1 P = − im −21a5 + 84a3k + a2(−6µς − 63s + 6iυς) (4) 5 21 + a −63k2 + 6k(m − ς)(m + ς) + µ2 + 5iζς + υ2 + 5χς + k (3µς + 42s − iυς) + 2i(µζ + υχ) + 7s(ς2 − m2) ,

2 Q0 = ς, Q1 = υ + i(aς + µ),Q2 = −a ς − aµ + kς + χ + i(aυ + ζ), 2 3 2 Q3 = −a υ − aζ + kυ + i(−a ς − a µ + a(2kς + χ) + kµ − sς), 1 Q = a4ς + a3(µ − iυ) + a2(−3kς − iζ − χ) + a 140sς − (µ − iυ)(140k − 3m2 − 10ς2) 4 70 1 + ς ς(kς + iζ + χ) + (µ − iυ)2 + (7k − m2)(kς + iζ + χ) + 7s(µ − iυ) , 7 (5) 1 Q = 21ia5ς + 21a4(υ + iµ) − 21ia3(4kς + iζ + χ) + a2 63isς − i(63k + 8ς2)(µ − iυ) 5 21 + ia −ς ς(8kς + iζ + χ) + 9µ2 − 16iµυ − 7υ2 + (21k − 2m2)(3kς + 2iζ + 2χ) + 42s(µ − iυ)} − i(µ − iυ)(−21k2 + 2km2 + µ2 + υ2) + kς2(υ − iµ) + ς(−42iks − 6µζ + 6iµχ + 7im2s + 8iζυ + 8υχ) + 21s(ζ − iχ) + 7isς3 .

Symbol Associated Multipole Moment is the usual rotation-induced deformation and a second ς Electric monopole contribution km that accounts for a possible intrinsic de- υ Electric dipole formation of the star [59, 60]. An analogous argument can χ Electric quadrupole be formulated in the case of the electric moments. The µ Magnetic dipole real part of Q2 accounts for the electric quadrupole contri- ζ Magnetic quadrupole bution to the total electromagnetic quadrupole moment. The terms −a2ς and −aµ account for the rotation-induced Table I. Summary of the electromagnetic parameters and the redistribution of the electric charge and deformation of multipole moments they are related to. the magnetic dipole; whereas, the term kς accounts for the contribution from the charge distributed over the intrinsic deformed mass. The additional parameter χ is Specifically, the interpretation of the parameters based on added to account for any additional possible contribution the multipole expansion in Eqs. (4) and (5) is as follows. to the total quadrupole moment. The real parameter m corresponds to the total mass, a The multipole expansion in Eq. (5) shows that even if to the total angular moment per unit mass while k and the magnetic dipole parameter is zero (µ = 0), a magnetic s are related to the mass-quadrupole moment and the dipole component (imaginary part of Q1) is present pro- differential rotations, respectively. For later convenience, vided by the rotation of the electric charge Q0. Similarly, an electric monopole contribution Q0, characterized by even if the magnetic quadropole parameter ζ is set to the parameter ς, was introduced above. Parameters υ zero, a rotating electric dipole can induce a magnetic and µ are associated with the electric and magnetic dipole quadrupole (imaginary part of Q2). For the electric part moments, respectively, whereas χ and ζ with the electric of the multipole expansion in Eq. (5), an analogous be- and magnetic quadrupole moments, respectively. The havior is observed, namely, a rotating magnetic dipole existence of an electric dipole is theorized for strange stars can induce an electric quadropole moment and a rotating (See Ref. [58]). A summary of the arbitrary parameters magnetic quadrupole can generate an electric octupole and the multipole moments they are related to can be moment. Based on these processes, the astrophysical found in TableI. source can afford a non-vanishing induced Poynting vec- The mass moment P2 governs the deformation of the tor and, correspondingly, an induced non-vanishing flux star and it is composed of two parts: the term a2m that of electromagnetic energy around the source that will 4

contribute to the frame dragging induced solely by the z[km] mass currents. 50

IV. CHARACTERIZATION OF THE 0 ELECTROMAGNETIC FIELDS

50 To describe the electromagnetic properties of the so- − a b lution, the electric and magnetic fields, in the spacetime 100 surrounding the star, are calculated by means of the − expressions z[km] 1 50 E = F uβ ,B = − γδF uβ , (6) α αβ α 2 αβ γδ 0 where Fαβ is the electromagnetic field tensor Fαβ = 2 A[β;α], Aµ = (0, 0,Aφ, −At) is the electromagnetic four- 50 − potential, uα is a time–like vector and αβγδ is the totally c d antisymmetric tensor of positive orientation with norm 100 αβγδ − 100 50 0 50 ρ[km] 100 50 0 50 ρ[km] αβγδ = −24 [61]. For a congruence of observers at − − − − rest in the frame of (1), the four–velocity√ is defined by the time–like vector uα = (1/ f, 0, 0, 0). The vectorial Figure 1. Magnetic field force lines for m = 2.071 km, fields have components in the ρ and z directions only. j = 0.194, Q = −2.76 km3, s = −2.28 km4, ς = 0 km, The components of the electric field are given by υ = 0 km2, χ = 0, µ = 1 km2 for (a) ζ = 0 km3, (b) 3 3 3 √ √ ζ = 15 km , (c) ζ = 30 km and (d) ζ = 50 km . The non- f f electromagnetic parameters correspond to the model 2 for the E = − A ,E = − A , (7) ρ e2γ t,ρ z e2γ t,z equation of state L in Ref. [62] (see also TableII). and for the magnetic field by f 3/2 B = (−ωA + A ) , (8) Model m [km] j Q [km3] s [km4] ρ ρe2γ t,z φ,z M2 2.071 0.194 -2.76 -2.28 f 3/2 B = − (−ωA + A ) . (9) M3 2.075 0.324 -7.55 -10.5 z ρe2γ t,ρ φ,ρ M4 2.080 0.417 -12.2 -22.0 M5 2.083 0.483 -16.2 -33.9 The explicit form of the fields can be found in AppendixB.

Figure1 shows the force lines of the magnetic field for Table II. Realistic numerical solutions for rotating neutron stars derived by Pappas and Apostolatos [62] for the equation various values of the magnetic quadrupole parameter ζ of state L. Here, m is the total mass of the star, j is the and for realistic values of the mass and mass current dimensionless spin parameter: j = J/M 2 (being J the angular multipoles. Specifically, the vacuum multipole moments of momentum), Q is the quadrupole moment and s is the current the solution mass, angular moment and mass quadrupole octupole moment. See Table VI of [62]. and current octupole have been fixed to the numerical ones obtained in Ref. [62]. They are listed in TableII. In particular, for Fig.1.a, ζ = 0 km3;1.b, ζ = 10 km3; 1.c, ζ = 25 km3 and for Fig.1.d, ζ = 50 km3. The increasing of the separation between consecutive force in TableII. Specifically, for Fig.2.a, χ = 0 km3;2.b, lines indicates that the magnetic field decreases while χ = 10 km3;2.c, χ = 25 km3 and for Fig.2.d, χ = 50 km3. the distance increases. Figure1 not only shows how the As above, it is clear that reflection symmetry is broken reflection symmetry around the plane z = 0 is broken and that at large distances the electric dipole component because of the magnetic quadrupole [6], but also shows dominates the field configuration. that at large distances from the source, the magnetic field behaves like a magnetic dipole despite the presence of In general, the electromagnetic field of astrophysical strong non-dipolar contributions. objects is expected to be a combination of the results depicted in Figs.1 and2. Moreover, as shown below, the breaking of the reflection symmetry has an important Figure2 shows the force lines of the electric field for a role on the vorticity of the spacetime. After character- variety of values of the electric quadrupole for realistic ize the electromagnetic fields, the generation of purely values of the mass and mass current multipoles listed electromagnetic frame dragging is discussed below. 5

z[km] as discussed below, it could decrease the total vorticity of the spacetime. Figure (3) depicts the Poynting vector 50 circulation around the source, when µ is chosen parallel (l.h.s. panel) and antiparallel (r.h.s. panel) to the star’s

0 rotation axis.

50 − y[km] a b 100 − 20 z[km] 0 50 20 − 0 40 − 40 20 0 20 x[km] 40 20 0 20 x[km] 50 − − − − − c d 100 Figure 3. Vector ﬁeld of the Poynting vector for the star model 3 4 − 100 50 0 50 ρ[km] 100 50 0 50 ρ[km] m = 2.071 km, j = 0.194, Q = −2.76 km , s = −2.28 km . − − − − The electromagnetic multipoles were chosen zero except ς = 0.1 km, µ = 1 km2 in the left panel and µ = −1 km2 in the Figure 2. Electric ﬁeld force lines for m = 2.071 km, j = 0.194, right side. The direction of the Poynting vector is due the Q = −2.76 km3, s = −2.28 km4, ς = 0 km, υ = 1 km2, sign change in the multipole expansions in Eq. (5). Figures µ = 0 km2, ζ = 0 for (a) χ = 0 km3, (b) χ = 10 km3, (c) are presented in the quasi-Cartesian auxiliary coordinates χ = 25 km3 and (d) χ = 50 km3. r → px2 + y2 and φ = arctan(y/x).

V. VORTICITY SCALAR AND POYNTING VECTOR The change of the circulation of the Poynting vector can be undesrtood from the potential q(ρ, z) that is analoguos The physics of the frame dragging states that the rota- to the Coloumb potential (see above). Under the condition tion of the source induces a twist in the neighbourhood that the reflection symmetry is not broken, υ = 0 and that drags any frame of reference near the source. In the ζ = 0, a sign change of the magnetic dipole parameter µ case of spacetimes endowed by complex electromagnetic is equivalent to change the sign of the coordiante z with fields and mass currents, frame dragging originates from a global minus sign, namely, q(ρ, z; υ = 0, ζ = 0, −µ) = ∗ a combination of the vorticity of the electromagnetic field −q(ρ, −z; υ = 0, ζ = 0, µ) = −q (ρ, z; υ = 0, ζ = 0, µ). and the vorticity associated with the mass currents. Therefore, under the conditions that reflection symmetry Poynting vector.–The Poynting vector S carries the imposses [49], all the electric field moments change sing information about the electromagnetic energy flux in the and this leads to a global sing of the Poynting vector. spacetime. Because of the axially symmetric character Vorticity Scalar.–For an observer at rest respect to (1), of the spacetime, only the component along the unitary the vorticity tensor is defined by ωαβ = u[α;β] + u˙ [αuβ]. The direct calculation of ω for (1) yields to [6] vector êφ survives [6]. In terms of the Ernst potentials αβ φ √ ∗ 2γ S = fIm Φ,ρΦ,z /(4πρe )[63] or more conveniently   0 0 0√ 0 √ 0 0 0 − 1 fω f  2 √ ,ρ  Sφ = ∇Φ∗ × ∇Φ, (10) ωαβ =  1  . (11) 2γ  0 0 0 − 2 fω,z  8πρe 1 √ 1 √ 0 2 fω,ρ 2 fω,z 0 where it is clear that Sφ vanishes if Φ is purely real, purely imaginary or when their real and imaginary parts are pro- To quantify the vorticity, it is convenient to introduce the portional to each other. Due to the complex combination vorticity scalar that is defined by the contraction of the of the electromagnetic moments with the mass currents, vorticity tensor and reads their effects on the Poynting vector (and subsequently to 1 q α β 2 2 2 p 2γ 2 the vorticity) are often subtle. However, the multipole ex- ωv = ωβ ωα = f f(ω,ρ + ω,z)/ 2e ρ , (12) pansion in Eq. (5) allows for a detailed description of each −2 ∗ −2 contribution. For instance, if υ = 0, χ = 0 and |aς| < |µ|, with ω,ρ = −ρf =(E,z + 2Φ Φ,z) and ω,z = ρf =(E,ρ + ∗ it is then clear that, to leading order, a sign change in the 2Φ Φ,ρ). More explicitly, it reads magnetic dipole moment parameter µ changes the sign of the magnetic field. This has the effect of changing the e−γ q ω = √ =[E + 2Φ∗Φ ]2 + =[E + 2Φ∗Φ ]2 (13) rotation direction of the Poynting vector (see Fig.3) and v 2f ,z ,z ,ρ ,ρ 6

ω [Hz] The vorticity scalar can be understood in terms of its v fluid mechanics analogue. It represents the rotation of M2 M3 M4 M5 the fluid. In the general relativistic case, it can be related 125 to the rotation velocity of a family of congruences. 100 Equation (13) is very useful to analyze the contributions to the vorticity because the electromagnetic and 75 mass currents terms can be easily identified there. More- over, this identification can be accompanied by further 50 expressing these contributions in terms of the parameters 25 of the Ernst potentials (3). In particular, in absence of 0 electromagnetic fields, the imaginary part of E is asso- 10 12 14 16 18 ρ[km] ciated with mass currents. Thus, for static sources one could attempt to assume E as real in Eq. (13) and focus only in the electromagnetic contributions encoded in Φ. Figure 4. Vorticity generated by the mass currents. All the However, because E and Φ are not independent objects electromagnetic moments are zero in this case. The vorticity [see, e.g., Eq. (2)], in the presence of electromagnetic fields, of the spacetime was calculated for the models 2 (j = 0.194), 3 E and Φ are complex even for non-spinning astrophysical (j = 0.324), 4 (j = 0.417) and 5 (j = 0.483) with the equation objects. Notwithstanding, if Φ is purely real or purely of state L in Ref. [62]. They are listed in the TableII imaginary, for non-rotating objects, the imaginary part of E vanishes [see, e.g., Eq. (2)]. Therefore, interest here star for a fixed value of the electric monopole and for a is in identifying when the electromagnetic contribution variety of magnetic dipole moments. The vorticity scalar does not vanish. decreases (increases) from its value in the vacuum situa- In doing so, it is convenient to express ωv = |=(∇E + ∗ √ γ ∗ tion (µ = 0) when the dipole is parallel (anti–parallel) to 2Φ ∇Φ)|/( 2fe ), so that the term =Φ ∇Φ vanishes if the star’s rotation–axis. The reason for this phenomenon Φ is purely real, purely imaginary or when their real and can be understood in terms of Eq. (13). To do so, note imaginary parts are proportional to each other. Based that E decreases monotonically as a function of z and ρ; on the discussion above, see also Eq. (5), the purely thus, its derivatives carries a negative sign. Although the real case corresponds to the absence of magnetic fields same argument applies to Φ, when the sign of µ changes, and the purely imaginary case to the absence of electric see above, the Poynting vector changes its circulation fields. In the cases when the Poynting vector is zero, the and that is enough to change the sign of its contribu- electromagnetic contribution to the vorticity vanishes; this tion to the vorticity scalar [see Eq. (13)]. Alternatively, generalizes the results in Ref. [6] where a particular space- assume that the reflection symmetry exits and set the time was considered. The particular case of proportional mass currents parameters a and s to zero, for this case, real and imaginary parts leads to the case of proportional Φ(ρ, z; a = 0, s = 0, −µ) = Φ∗(ρ, z; a = 0, s = 0, µ) and electric and magnetic fields and corresponds to the cases therefore =Φ∗∇Φ changes sign. studied in Refs. [4,6]. Note that the considerations above on the vorticity The lower panel of Fig.5 depicts the functional depen- scalar and the Poynting vector are completely general dence of the vorticity scalar on the distance from the star and valid for any stationary axially symmetric spacetime. for a fixed value of the electric dipole and for a variety of magnetic dipole moments. The contribution in this To study the contribution of the electromagnetic energy case is weaker than in the case of an electric monopole. flux to the vorticity, characterize first the contributions of However, results in the lower panel of Fig.5 are more the rotation of the source. In this case, the Ernst potential realistic than those in the upper panel. Interestingly, in Φ that encodes all the electromagnetic moments is zero the lower panel, there is no change in the sign of the and E is complex. For fast rotating stars, the vorticity electromagnetic contribution to the vorticity scalar when scalar (and the Lense-Thirring effect) is larger than for µ changes sign. The reason for this relies on the fact that slow rotating stars, this can be seen in the Fig. (4). for υ =6 0 or ζ 6= 0, the reflection symmetry around the When one includes an electromagnetic field to a partic- equatorial plane breaks down and the arguments provided ular star model, the Ernst potential Φ is no longer zero, above do not apply, i.e., Φ(ρ, z; a = 0, s = 0, −µ) does not and it has an important role in the vorticity scalar. The equate to Φ∗(ρ, z; a = 0, s = 0, µ). particular contribution depends on the structure gener- Based on the processes described above on the genera- ated by the multipole expansion and the Poynting vector. tion of a Poynting vector from the fields induced by the Before considering the purely electromagnetic contribu- mass currents (a 6= 0 and s 6= 0), it is clear that even tion to the frame–dragging in realistic situations (zero if the parameters associated with the electric field (ς, υ or negligible total electric charge), consider the case of a and χ) are set to zero, but the parameters associated source with a dipole magnetic field and electric charge. with the magnetic field (µ and ζ) are non-vanishing, then The upper panel of Fig. (5) depicts the functional de- a rotationally-induced electromagnetic contribution to pendence of the vorticity scalar on the distance from the the vorticity scalar is present. A similar scenario takes 7

ω [Hz] v on the dynamics of orbiting of particles. One of the most µ =−10km2 µ =−5km2 µ =0km2 distinctive points between both theories is the frame- 2 2 46 µ =5km µ =10km dragging, but is not an observable by itself. A way to measure it, is to appeal to the dynamics around the 36 source and characterizing the behavior of, e.g., neutral test particles [40, 53]. 26 Observationally, the Keplerian motion of matter could be useful to model the quasi-periodic oscillations (QPOs) 16 that are present in the low-mass X-ray binaries (LMXBs) containing a neutron star [67]. These oscillations occur 6 10 12 14 16 18 ρ[km] at frequencies in the range of kHz and come in pairs, the upper and the lower mode correspond to the frequencies

ωv[Hz] of Keplerian motion and periastron precession of the µ =−10km2 µ =−5km2 µ =0km2 accreted matter in the close vicinity of the star [52]. An 2 2 46 µ =5km µ =10km additional effect is that these equatorial orbits will exhibit a relativistic nodal precession due to frame dragging [53], 36 causing a detectable signal in the spectra of LMXBs [68]. The following subsections are devoted to quantifying the 26 magnetic quadrupole effect in the ISCO and the epicyclic frequencies for different models of neutron stars [62]. 16

6 Inﬂuence of the ﬁeld on the ISCO radii 10 12 14 16 18 ρ[km]

The dynamics of an orbiting particle can be analyzed Figure 5. Electromagnetically generated vorticity by the pres- by using the Lagrangian formalism. To do so, consider ence of a magnetic dipole and a fixed electric monopole of a particle of rest mass m0 = 1 moving in the space- ς = 0.1 km (upper panel with υ = 0, χ = 0 and ζ = 0) and an time described by the metric functions in Eq. (1), the 2 electric dipole of υ = 10 km (lower panel with ς = 0, χ = 0 Lagrangian of the particle is then given by and ζ = 0). The vorticity of the spacetime was calculated for the model 2 for the equation of state L in Ref. [62]. Parameters 1 L = g x˙ µx˙ ν , (14) are listed in TableII. 2 µν where the dot denotes differentiation with respect to the place if the parameters associated with the magnetic field proper time τ, xµ(τ) are the Weyl-Lewis-Papapetrou are set to zero and non-vanishing electric parameters are coordinates. For a stationary and axisymmetric spacetime, considered. In particular, even in the low rotation regime, there are two constants of motion related to the time a non-negligible electric field is generated [64] and may be coordinate t and azimuthal coordinate φ. Thus, the energy important to characterize the evolution of the electromag- and the canonical angular momentum, respectively, are netic structure in neutron stars. Moreover, in the case of conserved (see Ref. [69] for details). Assuming that the fast rotation, the frame-dragging caused by a Kerr black motion takes place in the equatorial plane of the star z = 0, hole significantly distorts the structure of an external and taking into account the four velocity normalization µ ν magnetic field and this scenario may be relevant for low for massive particles gµν u u = −1, one can identify an accreting black holes as the one present in the Milky Way effective potential that characterizes the motion in the center [65, 66]. plane (see, e.g., Ref. [70]) The most relevant contributions to the vorticity scalar from the electromagnetic field were considered above. E2g + 2ELg + L2g V (ρ) = 1 − φφ tφ tt . (15) However, in general, all the multipole moments contribute eff 2 gtφ − gφφgtt to the vorticity scalar and the details of the net result may deviate from those discussed above; albeit, the magnitude For a particle moving in a circular orbit, the energy E of the effect is not expected to differ from the predictions and the canonical angular momentum L are determined in Fig.5. by the conditions Veff (ρ) = 0 and dVeff /dρ = 0 (see, e.g., Ref. [71]). The marginal stability condition reads 2 2 d Veff /dρ = 0. Thus, the ISCO’s radius is determined VI. ORBITAL EQUATORIAL MOTION AND by solving numerically the previous equation for ρ. EPICYCLIC FREQUENCIES The importance of determining the ISCO is that accretion disks extend from the last stable orbit to exterior In general relativity, as in Newtonian gravitation, all zones, so the ISCO is an inner boundary for the accreted the physical characteristics of the source have an effect matter. As stated above, the inclusion of a magnetic 8

field causes a dependence of all physical quantities of the on this in the next section. intensity on the field. From the International System of Units (SI), the conversion to geometrized units is given by Influence of the electromagnetic field in the Keplerian and the epicyclic frequencies −6√ 10 Gµ0 µgeom = µSI, (16) c2 Imposing the conditions of constant orbital radius, dρ/dτ = 0 and taking into account that dφ/dτ = where G is the gravitational constant, µ is the vacuum 0 Ω dt/dτ, the Keplerian frequency reads (see, e.g., permittivity, c is the speed of light and µ is given in units K Ref. [69]) of Am2. The units of ζ are Am3 and the conversion for the quadrupole reads p 2 dφ −gtφ,ρ ± (gtφ,ρ) − gφφ,ρgtt,ρ √ ΩK = = , (21) 10−9 Gµ dt gφφ,ρ ζ = 0 ζ . (17) geom c2 SI where “+” and “−” denotes the Keplerian frequency for In the same way, the electric multipole moments can be corotating and counter-rotating orbits, respectively. In written in geometrized units as: Ref. [40] the functional dependence of the Keplerian frequency on the magnetic dipole parameter µ was discussed, r Gµ here interest is in the functional dependence on the mag- ς = 0 ς , (18) geom 4π SI netic quadupole parameter ζ. The upper panel of Fig.7 r depicts the functional dependence on ζ, as in the case −3 Gµ0 υgeom = 10 υSI, (19) analyzed in Ref. [40], the Keplerian frequency increases 4π with increasing ζ because the ISCO’s radius decreases r Gµ (see Fig.6) as a consequence of the additional deforma- χ = 10−6 0 ς . (20) geom 4π SI tion of the space-time by the electromagnetic field. In particular, the energy of the electromagnetic field shifts Consider a superposition of a fixed dipolar and the marginally unstable region toward the star. quadrupole component that will be varied from 1 km3 to Analytical expressions for the radial and vertical fre- 50 km3. Figure6 depicts the ISCO radius as a function of quencies follow from allowing slightly radial and vertical the parameter µ and ζ for three realistic numerical solu- perturbations of the orbit. According to Ref. [69], the tions for rotating neutron stars models derived in Ref. [62]. radial and vertical epicyclic frequencies are given by The parameter µ is set to 1 km2 that corresponds to a magnetic dipole field of 1012 T. The quadrupole parame- 1 gαα g ν = − (g + g Ω )2 φφ ter ζ is chosen between 0 and 50 km3 that corresponds α tt tφ k 2 2π 2 ρ ,αα to magnetic quadrupoles from 0 to 5 × 1035 Am3, respec- g tively. In all these cases, the ISCO radius decreases for tφ −2(gtt + gtφΩk)(gtφ + gφφΩk) 2 (22) ρ ,αα 12.3 2 gtt +(gtφ + gφφΩk) 2 , ρ ,αα 12 with α = {ρ, z}. In the relativistic precession model 11.7 (RPM) [51], the periastron νp and the nodal νp frequencies [km] ρ z are the observational relevant ones, they are defined by ISCO 11.4 R Ω M2 p K 11.1 να = − να. (23) M3 2π M4 10.8 At the ISCO, the radial oscillation frequency equals the 0 10 20 30 40 ζ[km3 ] Keplerian frequency and only the vertical precession α = z is considered here. The influence of the magnetic quadrupole on the vertical precession frequency is Figure 6. ISCO radius as a function of magnetic quadrupole depicted in the lower panel of Fig.7. The behaviour and parameter ζ, with µ = 0 km2. The physical parameters for the underlying physical mechanism are analogous to those the star correspond to the models 2-4 listed in TableII. We for the Keplerian frequency. can se that the ISCO radius decreases for increasing ζ. The frequency νLT that characterizes the Lense– Thirring effect is given by (see, e.g., Ref. [69]) increasing ζ. This can be explained as a result of the deformation of the space-time by the energy stored in the 1 gtφ νLT = − , (24) electromagnetic (see, e.g., Ref. [40]). We elaborate more 2π gφφ 9

1905 VII. CONCLUDING REMARKS M2 1848 M3 M4 We present a new stationary axisymmetric nine- parameter closed-form analytic solution that generalizes 1792 the Kerr solution with arbitrary mass-quadrupole mo- [Hz]

K ment, octupole current moment, electric and magnetic 1736 Ω dipole and electric and magnetic quadrupole moments. 1680 The analytic form of its multipolar structure and their electric and magnetic ﬁelds are also presented. Accord- 1626 ing to the arguments presented through the paper, this 113 solution could be used to model the exterior gravitational and electromagnetic ﬁelds around strongly magnetized 100.3 stars, in particular white dwarfs. Besides, this model also could be used even for the description of exotic stars such 87.6 as the τ Sco recently reported by Donati et al. [72].

[Hz] This solution allowed for a comprehensive analysis LT

ν 74.9 of the contribution of complex and intense electromag-

M2 netic fields to quintessential hallmark of general relativity, 62.2 M3 namely, the Lense–Thirring effect. It was shown that if M4 the value of the parameters is such that the reflection 49.9 symmetry is preserved [49], then a sign change of the 39 M2 magnetic dipole parameter µ is enough to change the direction of the circulation of the Poynting vector and the 25 M3 M4 sign of the electromagnetic contribution to the vorticity 12.5 scalar. The influence of complex electromagnetic fields on the ISCO’s radius and the epicyclic frequencies were [Hz]

p 0 z also considered and it was shown that for strong magnetic ν fields, the influence is not negligible. −12.5 Additional interest is in studying the effect of the magnetic field in the quadrupole moment of the star (see, e.g., −29.7 Refs. [73, 74]). This topic has gained recently theoretical 0 10 20 30 40 ζ[km3 ] and observational interest and is currently discussed under the title of I − Love − Q relations [74]. Due to the Figure 7. Influence of the magnetic quadrupole moment ζ in great generality of the present analytic exact solution, this the epicyclic frequencies for different models of neutron stars with ς = 0, µ = 1 km2, υ = 0 and χ = 0. Vacuum parameters subject will be explored in a forthcoming contribution of the star are listed in TableII. [75].

ACKNOWLEDGMENTS and could be relevant to model the horizontal branch oscillations observed in the Low Mass X-Rax Binaries(LMXBs) Fruitful discussions with Prof. CésarA. Valenzuela- [51]. The strength of the frame–dragging effect increases Toledo are acknowledged with pleasure. This work was for fast rotating objects and, as shown in the central panel partially supported by Fundaciónpara la Promociónde of Fig.7, for stronger magnetic fields. Figure7 also shows la Investigacióny la Tecnolog´ıadel Banco la República that the electromagnetically-induced frame dragging has Grant No. 2879. and by Comitépara el Desarrollo de la a direct effect on the orbiting particles, even if they are Investigación(CODI) of Universidad de Antioquia under neutral. the Estrategia de Sostenibilidad 2015-2016.

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Appendix A: Metric Functions of the Developed Solution

This appendix summarizes the relevant equations of the developed metric. The potentials in the symmetry axis can be written as [54]:

3 3 X ei X fi e(z) = 1 + , f(z) = , (A1) z − β z − β i=3 i i=3 i with 2 j 2mβj iζ + (ς + iµ)βj ej = (−1) , fj = , i, k 6= j . (A2) (βj − βk)(βj − βi) (βj − βk)(βj − βi)

Then, the Ernst potentials and the metric functions in whole spacetime are derived with the aid of the Sibgatullin’s integral method [47, 48]. By using the representation proposed in Ref. [76] and used also Ref. [42],

A + B C E = , Φ = , (A3) A − B A − B

AA¯ − BB¯ + CC¯ AA¯ − BB¯ + CC¯ Im[(A + B)H¯ − (A¯ + B¯)G − CI¯] f = , e2γ = , ω = , ¯ ¯ 6 ¯ ¯ ¯ (A − B)(A − B) Y AA − BB + CC KK¯ rn n=1 12 where X X X X A = aij kri rj rk ,B = bijri rj,C = cijri rj ,K = aij k , 1≤ipi ri 1≤i

Ψlm|nps = f(αn) ∆lm|ps + f(αp) ∆lm|sn + f(αs) ∆lm|np ,

e(z) +e ˜(z) + 2f˜(z)f(z) = 0. (A4)

Appendix B: Electromagnetic Field: Analytic Form

The At potential is the real part of the electromagnetic Ernst potential Φ, and the potential Aφ can be calculated as 0 the real part of the Kinnersley potential K = Aφ + iAt [77], which can be obtained using the Sibgatullin’s method and can be written as I (f + f ) K = −i 1 2 . (B1) A − B 13

Thus, the closed–form expressions for the electric and magnetic ﬁelds are

( ) Λ C,ρ − C ln(A − B),ρ Eρ = − Re , p|A|2 − |B|2 + |C|2 A − B ( ) Λ C,z − C ln(A − B),z Ez = − Re , p|A|2 − |B|2 + |C|2 A − B

( ) Im(A − B)H¯ + (A¯ − B¯)G − CI¯] Λp|A|2 − |B|2 + |C|2 (f¯ + f¯ ) I¯ − I¯ln(A¯ − B¯) B = E + Im 1 2 ,z ,z , ρ ρ|A − B|2 z ρ|A − B|2 A − B ( ) Im(A − B)H¯ + (A¯ − B¯)G − CI¯] Λp|A|2 − |B|2 + |C|2 (f¯ + f¯ ) I¯ − I¯ln(A¯ − B¯) B = − E + Im 1 2 ,ρ ,ρ . z ρ|A − B|2 ρ ρ|A − B|2 A − B when

|K|2 Q6 r Λ = n=1 n . |A − B|