Rapidly Rotating Pulsar Radiation in Vacuum Nonlinear Electrodynamics

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provided by Springer - Publisher Connector Eur. Phys. J. C (2016) 76:612 DOI 10.1140/epjc/s10052-016-4464-3

Regular Article - Theoretical Physics

Rapidly rotating pulsar radiation in vacuum nonlinear electrodynamics

V. I. Denisov1,I.P.Denisova2,A.B.Pimenov1,V.A.Sokolov1,a 1 Physics Department, Moscow State University, Moscow 119991, Russia 2 Moscow Aviation Institute (National Research University), Volokolamskoe Highway 4, Moscow 125993, Russia

Received: 5 September 2016 / Accepted: 23 October 2016 / Published online: 8 November 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com

13 Abstract In this paper we investigate the corrections of high-field (surface fields more than Bp > 10 G) radio loud vacuum nonlinear electrodynamics on rapidly rotating pulsar pulsars can be explained by pair-production suppression due radiation and spin-down in the perturbative QED approach to photon splitting [19]. (post-Maxwellian approximation). An analytical expression In this paper we calculate vacuum nonlinear electrody- for the pulsar’s radiation intensity has been obtained and ana- namics corrections to electromagnetic radiation of rapidly lyzed. rotating pulsar and analyze pulsar spin-down under these corrections. This paper is organized as follows. In Sect. 2, we present 1 Introduction vacuum nonlinear electrodynamics models and discuss their main physical properties and predictions. In Sect. 3 pulsar Vacuum nonlinear electrodynamics effects is an object that radiation in post-Maxwellian approximation is calculated. piques a great interest in contemporary physics [1–4]. First of Section 4 is devoted to an analysis of the pulsar’s spin-down all it is related to the emerging opportunities of experimental vacuum nonlinear electrodynamics influence. In the last sec- research in terrestrial conditions using extreme laser facili- tion we summarize our results. ties like extreme light infrastructure (ELI) [5–7], Helmholtz International Beamline for extreme fields (HIBEF) [8]. It opens up new possibilities in fundamental physics tests [9– 2 Vacuum nonlinear electrodynamics theoretical models 11] with an extremal electromagnetic field intensities and particle accelerations that have never been obtained before. Modern theoretical models of nonlinear vacuum electrody- At the same time the investigation of the effects of vac- namics suppose that the electromagnetic field Lagrange func- uum nonlinear electrodynamics in astrophysics gives us an tion density L = L(I(2), I(4)) depends on both independent ki kl mi additional opportunity to carry out versatile research using invariants I(2) = FikF and I(4) = FikF Flm F of the the natural extreme regimes of strong electromagnetic and electromagnetic field tensor Fik. The specific relationship gravitational fields with intensities unavailable yet in lab- between the Lagrange function and the invariants depends oratory conditions. Compact astrophysical objects with a on the choice of the theoretical model. Nowadays the most strong field, such as pulsars and magnetars, are best suited promising models are Born–Infeld and Heisenberg–Euler for vacuum nonlinear electrodynamics research. Nowadays, electrodynamics. there are many effects of vacuum nonlinear electrodynam- Born–Infeld electrodynamics is a phenomenological the- ics predicted in the pulsar’s neighborhood. For example vac- ory originating from the requirement of self-energy finite- uum electron–positron pair production [12] and photon split- ness for a point-like electrical charge [20]. In subsequent ting [13], photon frequency doubling [14], light by light scat- studies, the attempts of quantization were performed [21,22] tering, and vacuum birefringence [15], transient radiation ray and also it was revealed that Born–Infeld theory describes bending [16,17] and normal waves delay [18]. Some of the the dynamics of electromagnetic fields on D-branes in string predicted effects are indirectly confirmed by astrophysical theory [23–25]. As the main features of Born–Infeld electro- observations. For instance the evidence of the absence of the dynamics one can note the absence of birefringence (however, there are modifications of the Born–Infeld theory [26] a e-mail: [email protected] with the vacuum birefringence predictions) and dichroism for 123 612 Page 2 of 8 Eur. Phys. J. C (2016) 76 :612 electromagnetic waves propagating in external electromag- predictions were experimentally proved in Delbrück light netic field [27]. Furthermore, this theory has a distinctive by light scattering [30], nonlinear Compton scattering [31], feature – the value of electric field depends on the direction Schwinger pair production in multiphoton scattering [1]. At of approach to the point-like charge. This property was noted the same time the recent astrophysical observations [32,33] by the authors of the theory and also eliminated by them in point on the absence of the vacuum birefringence effect the subsequent model development [28]. which favors the Born–Infeld theory prediction. The mea- Lagrangian function in Born–Infeld electrodynamics has surements performed for the speed of light in vacuum show the following form: that it does not depend on wave polarization with the accuracy δc/c < 10−28. So clarification of the vacuum nonlinear elec- ⎧ ⎫ ⎨ ⎬ trodynamics status requires the expansion of the experimen- 1 a2 a4 a4 L =− 1 − I( ) − I( ) + I 2 − 1 , tal test list both in terrestrial and astrophysical conditions. π 2 ⎩ 2 4 (2) ⎭ 4 a 2 4 8 The main hopes as regards this way to proceed are assigned (1) to the experiments with ultra-high intensity laser facilities [4] and astrophysical experiments with X-ray polarimetry [34] where a is a characteristic constant of theory, the inverse in pulsars and magnetars neighborhood. value of which has a meaning of maximum electric field for As follows from the Lagrangians (1)–(2) vacuum non- the point-like charge. For this constant only the following linear electrodynamics’ influence becomes valuable only − − , ∼ / estimation is known: a2 < 1.2 × 10 32 G 2. in strong electromagnetic fields, comparable to E B 1 a , ∼ The other nonlinear vacuum electrodynamics – the Hei- for Born–Infeld theory and E B Bc for Heisenberg– senberg–Euler model [15,29] – was derived in quantum field Euler electrodynamics [35]. In the case of relatively weak , << theory and describes one-loop radiative corrections caused by fields (E B Bc) the exact expressions (1) and (2) can be vacuum polarization in a strong electromagnetic field. Unlike decomposed and written [36] in the form of a unified para- the Born–Infeld electrodynamics, this theoretical model pos- metric post-Maxwellian Lagrangian: sesses vacuum birefringent properties in a strong field. 1 2 The effective Lagrangian function for Heisenberg–Euler L = I( ) + ξ (η − η )I + η I( ) , π 2 2 1 2 2 (2) 4 2 4 (5) theory has the following form: 32 ∞ ξ = / 2 = . × −27 −2 α 2 −σ σ where 1 Bc 0 5 10 G , and the post- I(2) Bc e d 2 L = − xyσ ctg(xσ)cth(yσ) Maxwellian parameters η1 and η2 depend on the choice of 16π 8π 2 σ 3 0 the theoretical model. In the case of Heisenberg–Euler elec- η η σ 2 trodynamics the post-Maxwellian parameters 1 and 2 are + (x2 − y2) − 1 dσ, (2) coupled to the fine structure constant α [37]: 3 2 3 13 where B = m c /eh¯ = 4.41 × 10 G is the value of the α − 7α − c η = = 5.1 × 10 5,η= = 9.0 × 10 5. (6) characteristic field in quantum electrodynamics, e and m are 1 45π 2 180π the electron charge and mass, α = e2/hc¯ is the fine structure constant, and for brevity we use the notations For Born–Infeld electrodynamics these parameters are equal to each other and they can be expressed through the i 1 field induction 1/a typical of this theory [37]: x =−√ (B2 − E2) + i(BE) 2Bc 2 2 2 η = η = a Bc < . × −6. 1 1 2 4 9 10 (7) − (B2 − E2) − i(BE) , (3) 4 2 The electromagnetic field equations for the post-Maxwellian vacuum electrodynamics with the Lagrangian (5) are equiv- 1 1 y = √ (B2 − E2) + i(BE) alent [35] to the equations of Maxwell electrodynamics of 2Bc 2 continuous media, 1 + (B2 − E2) − i(BE) . (4) ∂ F + ∂ F + ∂ F = 0, (8) 2 m ik i km k mi

Many attempts to ﬁnd the experimental status for each ∂ Qki 4π of these theories were made a long time ago, but nowadays =− jk, (9) it still remains ambiguous. There is experimental evidence ∂xi c in favor of each of them. Heisenberg–Euler electrodynamics with the speciﬁc nonlinear constitutive relations [18] 123 Eur. Phys. J. C (2016) 76 :612 Page 3 of 8 612

ki = ki + ξ (η − η ) ki + η ki , where Fki is the electromagnetic field tensor of the rotating Q F 1 2 2 I(2) F 4 2 F(3) (10) (0) magnetic dipole m in Maxwell electrodynamics and f ik is ki kn mi the vacuum nonlinear correction. Substituting (13)into(10) where F( ) = F Fnm F is the third power of the elec- 3 and retaining only the terms linear in a small value f ik it can tromagnetic field tensor. The tensor Qik can be separated be found that into two terms Qki = Fki + Mki, one of which, Mki, will have a meaning similar to the matter polarization tensor in Qik f ik + Fik + Mik , (14) electrodynamics of continuous media. (0) (0) Also it should be noted that in a post-Maxwellian approx- ik ik = ik( nj ) imation the stress-energy tensor T and Poynting vector S where M(0) M F(0) is the polarization tensor calcu- have the form lated in the approximation of the Maxwell electrodynamics nj ik field F(0). The electromagnetic field equations (8)–(9) with ik = 1 ( + ξη ) ik − g T 1 1 I(2) F(2) 2I(2) account of (13)–(14) then will take the form 4π 8 (0) (0) (0) + ξ(η + η ) 2 − η ξ , ∂m F + ∂i F + ∂k F + ∂m fik + ∂i fkm + ∂k fmi = 0, 1 2 2 I(2) 4 2 I(4) (11) ik km mi ki ki ∂ f ki ∂ F( ) ∂ M( ) 4π + 0 + 0 =− jk. (15) ∂xi ∂xi ∂xi c μ 0μ c 0μ S = cT = 1 + ξη I( ) F , (12) 4π 1 2 (2) The solution of these equations may be obtained by the suc- ik = ni mk cessive approximation method. In the initial approximation where F(2) g Fnm F is the second power of the electro- (0) ik we assume that Fmi is the solution of the Maxwell electromagnetic field tensor, g is the metric tensor; and the Greek dynamics equations index takes the value μ = 1, 2, 3. As was shown in [38], post-Maxwellian approximation ( ) ( ) ( ) ∂ F 0 + ∂ F 0 + ∂ F 0 = 0, turns out to be very convenient for vacuum nonlinear electro- m ik i km k mi ∂ ki dynamics analysis, so we will use this representation (8)–(12) F( ) 4π 0 =− jk, (16) to calculate the radiation of the rapidly rotating pulsar. ∂xi c

corresponding to rotating magnetic dipole m and a current k 3 Rapidly rotating pulsar radiation in post-Maxwellian density which by j is represented in the right hand side of nonlinear electrodynamics these equations. In this case, from (15) it follows that the vacuum nonlinear electrodynamics’ corrections fik may be Pulsars are the compact objects best suited for vacuum non- obtained as a solution of linearized equations: linear electrodynamics tests in astrophysics. They possess ∂m fik + ∂i fkm + ∂k fmi = 0, (17) sufficiently strong magnetic fields with the strength vary- ∂ ki ∂ ki ∼ 9 ∼ 14 f M(0) ing from Bp 10 G up to Bp 10 G; as these values are + = 0. (18) ∂xi ∂xi close to Bc the vacuum nonlinear electrodynamics’ influence can be manifested. At the same time, the pulsar’s fast rotation To satisfy the homogeneous equation (17) electromagnetic k may enhance the nonlinear influence on its radiation. potential A should be introduced fki = ∂k Ai −∂i Ak.Using Let us consider a pulsar of radius Rs, rotating around an this potential the inhomogeneous equation (18) under the axis passing through its center with the angular velocity ω. Lorentz gauge will take the form We shall suppose that the rotation is fast enough, so the linear ki velocity for the points on the pulsar’s surface is comparable ∂ M( ) ω / ∼ ∂ ∂n Ak = 0 . (19) to the speed of light Rs c 1. We assume that the pulsar’s n ∂xi magnetic dipole moment m is inclined to the rotation axis at the angle θ0, therefore the cartesian coordinates of this vector It is more convenient to rewrite the last equation in terms of ki vary under rotation as m ={mx = m sin θ0 cos ωt, m y = the antisymmetric Hertz tensor defined by m sin θ0 sin ωt, mz = m cos θ0}. As the vacuum nonlinear electrodynamics’ influence in ∂ ki Ak =− . (20) post-Maxwellian approximation has the character of a small ∂xi correction to Maxwell theory one can represent the total electromagnetic field tensor Fki in the form In this case Eq. (19) will take the simple form

ki = ki + ki, −∂ ∂n ki = ki = ki , F F(0) f (13) n M(0) (21) 123 612 Page 4 of 8 Eur. Phys. J. C (2016) 76 :612

n where =−∂n∂ is the D’Alembert operator. Six indepen- find the components of the electromagnetic field tensor fik dent equations in (21) may be expressed in vector form by and radiation properties such as the Poynting vector S and introducing the Hertz electric and magnetic Z potentials the tonal intensity I . The Poynting vector components rep- [39]: resented by (12) in post-Maxwellian electrodynamics can be simplified by the radiative asymptotic condition Sμ ∼ 1/r 2, α α0 α 1 αμν = , Z = μν, (22) which actually means that for the radiation description we 2 can use the Maxwellian expression for this vector: where αμν is the Levi-Civita symbol and all of the indices take values α, μ, ν = 1, 2, 3. In terms of these potentials μ 0μ c 0μ S = cT ∼ F( ) . (29) Eq. (21) can be rewritten 4π 2

= P0, Z = M0, (23) Finally, the total intensity can be obtained by integrating of the Poynting vector by the surface with the normal n directed where the source vectors P0 and M0 are expressed from polar- to the observer located at the large distance r >> Rs from (0) ization tensor Mik by the equalities the pulsar:

α α0 α 1 αμν (0) P = M( ), M = Mμν . (24) 0 0 0 2 I = (Sn)r 2d, (30) The explicit components of these vectors may easily be obtained in Minkowski space-time with the use of (10) and where d is the solid angle. (24): Solutions of Eq. (23) with the right hand side (25), (26) lead to the following expression for the pulsar radiation inten- = ξ{η ( 2 − 2) + η ( ) }, P0 2 1 E0 B0 E0 2 2 B0 E0 B0 (25) sity:

ω4 B2 R6 B2 2 2 2 p s 2 2 p 9 1 M0 = 2ξ{η1(E − B )B0 − 2η2(B0 E0)E0}, (26) I = sin θ 1 + 24Y η − η 0 0 3c3 0 35Y 3 B2 15 1 2 c where E0 and B0 are the electromagnetic field components 311 ×Ci(2Y ) sin2 θ + 4Y 9 η − η Ci(2Y ) of the rotating magnetic dipole in Maxwell electrodynamics, 0 2 45 1 the expressions for which are well described in the literature η − η + 3 1 15 2 ( 4 − 2) [40] and the field vectors themselves have the form Y 2Y 3Y 5 3(m(τ) r)r − r 2m(τ) m˙ (τ) − η − η ( ) 2 θ B (r, t) = − 18 1 10 2 cos 2Y sin 0 0 5 2 r cr η − η 2 Y 45 2 311 1 6 4 2 3(m˙ (τ) r)r (m¨ (τ) r)r − r m¨ (τ) + (2Y − 3Y ) + (172η1 − 60η2)Y + + , (27) 6 15 cr 4 c2r 3 η − η 2 30 2 2 1 6 4 −336η1 cos(2Y ) + Y (2Y − Y ) [r, m˙ (τ)] [r, m¨ (τ)] 5 E0(r, t) = + , (28) 41η + 85η cr 3 c2r 2 − 1 2 Y 2 + 9η + 5η sin(2Y ) sin2 θ 5 1 2 0 where τ = t−r/c is the retarded time and the dot corresponds 1 311η1 − 45η2 to the derivative of the magnetic dipole moment m(τ) with + (2Y 8 − Y 6 + 3Y 4) 3 15 respect to the retarded time τ. Therefore, the right hand side +( η − η ) 2 + η ( ) , of Eq. (23) can be obtained by using of (25)–(28). Equations 15 2 141 1 Y 84 1 sin 2Y (31) (23) themselves are the inhomogeneous hyperbolic equations the exact solution methods of which are well developed and where the following notations are used for brevity: k = ω/c described in the literature [41–43]. Since we are interested and Y = kRs,also Bp is the surface magnetic field inductance ( ) = x cos u only in the radiative solutions for the pulsar’s field, when and Ci x ∞ u du is an integral cosine. solving Eq. (23) one should retain only the terms decreas- It is obvious that the intensity obtained can be represented ing not faster than ∼ 1/r with the distance to the pulsar. in a form which distinguishes the Maxwell radiation intensity At the same time there are no restrictions on the rotational and the vacuum nonlinear electrodynamics correction. In this velocity so ωRs/c ∼ 1. Due to the excessive unwieldiness representation it is convenient to introduce the “correction here we will not represent the whole solutions for the Hertz function” (θ0, Y ), which is a multiplier before the scaling 2/ 2 potentials and Z, but we will use the results for them to factor Bp Bc determining how strong the vacuum nonlinear 123 Eur. Phys. J. C (2016) 76 :612 Page 5 of 8 612 electrodynamics’ influence on the pulsar radiation is, 1.6 1.4 ω4 2 6 2 3.4 e−5 2 Bp Rs Bp 3.5 e−5 I = 2 θ + (θ , Y ) . 1.2 3 sin 0 1 2 0 (32) 3c Bc 1 3.8 e−5

For most known rapidly rotating pulsars [44] with Y ∼ 1 , rad 0.8 0 the factor B2/B2 1 is small, which matches the require- θ 4.3 e−5 p c 0.6 ments of the post-Maxwellian approximation. At the same e−5 time this means that the vacuum nonlinear electrodynam- 0.4 4.9 ics corrections will be sufficiently suppressed in comparison 0.2 with the Maxwell electrodynamics radiation. However, this 5.5 e−5 assessment may be waived for special sources of so-called 0 fast radio bursts (FRB’s), six cases of which have recently 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Y been discovered [46]. One of the hypotheses explaining the nature of FRBs assumes that their source is a rapidly rotating Fig. 1 Correction function isolines and the best contrast line neutron star with the strong surface magnetic field Bp > Bc called blitzar [45]. In this case vacuum nonlinear electrody- ∝ ω4 2 θ namics corrections to the pulsar radiation became significant the total pulsar luminosity, which is I sin 0, but at the but at the same time this makes a strict solution (31) inappli- same time, as mentioned, this will decrease the vacuum non- cable because it was obtained in the low-field limit. So our linear electrodynamics correction on the Maxwell radiation θ ∼ π/ further evaluations will be applied to the case of the typi- background. For instance, if 0 2 the correction will ∼ . cal rapidly rotating pulsar, for instance PSR B1937+21 with most significantly stand out for the pulsars with Y 0 5. So 8 the correction function (θ0, Y ) plays the role of a contrast. Bp ∼ 4.2 × 10 G Bc, and maybe for blitzars but with the And the red line in Fig. 1 marks the relation between θ0 and restriction Bp < Bc. The main purpose of our analysis will be the identification of new qualitative features of the pul- Y for the best contrast. sar radiation and comparing vacuum nonlinear corrections Another distinctive feature of the pulsar radiation is to the electromagnetic radiation with the other weak energy manifested in a sophisticated, non-polynomial dependence loss mechanisms. between the radiation intensity (31) and the angular veloc- Let us investigate the properties of the correction func- ity, which greatly complicates the analysis. Performing a power-law approximation of (31) will allow us to describe tion (θ0, Y ). First of all, it should be noted that there is no radiation when the pulsar dipole moment is coaxial with the vacuum nonlinear electrodynamics’ influence on the pulsar spin-down, in traditional terms of braking-indices and the rotation axis i.e. when θ0 is zero. The correction func- torque-functions [47]. It also provides a possibility for com- tion depends both on the angle θ0 and the angular velocity parison of the pulsar spin-down caused by different non- through Y = ωRs/c,so(θ0, Y ) may be represented as a surface defined in the region where its coordinates take val- electromagnetic dissipative factors with the power-law relation between the radiation intensity and angular velocity, for ues 0 ≤ Y < 1 and 0 ≤ θ0 ≤ π/2. Some isolines – the instance with the quadrupole gravitational radiation. Let us relations θ0(Y ) at which this surface takes a constant value, investigate the features of the pulsar spin-down as a result (θ0, Y ) = const – are represented in the Fig. 1, the numer- of the radiation, with the amendments of vacuum nonlinear ical values for which were obtained with the η1 and η2 from the Heisenberg–Euler theory. electrodynamics. The obtained isolines differ from each other by the abso- lute value of the correction function but all of them have a pronounced extremum at some point which lies on the red 4 Pulsar spin-down line. This means that for each fixed angle θ0 between the pulsar dipole moment and the rotation axis there is an angular The observed spin-down rate [47] can be expressed by the velocity at which the vacuum nonlinear electrodynamics cor- derivative of the angular velocity as rections become the most pronounced. Increasing Y at con- 2 3 2 θ I 2Bp Rs Bp stant 0 up to the value marked by the red line increases the ω˙ =− =− sin2 θ Y 3 + Y 3 , (33) ω 0 2 correction of vacuum nonlinear electrodynamics. The sub- J 3J Bc sequent Y and angular velocity increase become ineffective because the vacuum corrections in this case will be reduced. where J is the pulsar’s inertia momentum and the dot means It should be noted that increasing Y → 1 also will enhance the time derivative. For a description in terms of torque func- 123 612 Page 6 of 8 Eur. Phys. J. C (2016) 76 :612

Table 1 Expansion coefficients Let us compare the pulsar spin-down caused by nonlinear 4 4 4 4 4 vacuum electrodynamics and dissipation caused by gravi- θ0 α7 × 10 α6 × 10 α5 × 10 α4 × 10 α3 × 10 tational waves radiation. Among several possible ways of π/21.7 −4.02.4 −0.3 −0.2 gravitational radiation by an isolated pulsar we will choose π/31.5 −3.82.7 −0.5 −0.3 the two most relevant scenarios – quadrupole mass radiation π/61.1 −3.63.3 −0.9 −0.5 [50] and the radiation caused by Rossby waves [51], called r-modes. Quadrupole gravitational radiation may originate by the tions, the right hand side of Eq. (33) should be represented strain caused by the pulsar rotation, which is especially likely in a polynomial form of the angular velocity, for rapidly rotating pulsars. The spin-down under this kind of radiation can be represented by N N ω n 3 n Rs ε2 Y (θ0, Y ) = αn(θ0) Y = αn(θ0) , (34) 5 32 GJ 5 c ω˙ = K Q ω =− ω , (38) n n 5 c5 ε where αn are decomposition coefficients and the number where G is a gravitational constant and is the pulsar ellip- of the terms N should be selected sufficient to ensure the ticity, which is in accordance with modern representations − required accuracy of the decomposition. We will take the ε<10 4 [47]. number of terms in the expansion (34) equal to N = 8. This Another reason for gravitational wave emission by an iso- choice ensures the accuracy of a power-law approximation lated pulsar are the oscillations modes induced by the pulsar for the pulsars with Y > 0.6 better than 0.1%. It should rotation. Gravitational radiation is caused by the instability be noted that the series does not converge at Y ∼ 1 but its of such oscillations. As was shown by Owen et al. [52]for replacement by the partial sum with the specially selected young rapidly rotating pulsars, spin-down caused by r-modes number of terms allows one to accomplish the polynomial can be expressed in the form approximation, which provides a good match with the exact ∼ 217π F2GM2 R6β2 expression near Y 1 but leads to significant errors when ω˙ = K ω7 =− s sat ω7, R 7 2 7 (39) Y 1. In this case the expansion coefficients (with the η1 3 5 Jc and η2 from the Heisenberg–Euler theory) for the terms pro- where M and Rs are the pulsar mass and its radius, the r-mode viding the largest contribution are represented in Table 1. −7 −5 oscillations saturation amplitude 10 ≤ βsat ≤ 10 was The coefficients not listed in the table are small and can be defined by [53], and the dimensionless constant F as has been neglected in further consideration. shown in [54] is to be strictly bounded within 1/(20π) ≤ For quantitative analysis, we will take the inclination angle F ≤ 3/(28π). equal to θ = π/2. This choice is justified because it provides 0 So the torque function for the quadrupole gravitational the greatest total intensity of the pulsar radiation and in our radiation KQ can be compared with the nonlinear electrody- comparison of nonlinear electrodynamics spin-down with the namics torque K5 and the r-mode radiation torque KR can other non-electrodynamical dissipative factors, it gives the be compared with the torque K7. For this comparison we upper limit of the nonlinear electrodynamics’ influence. suppose the pulsar with the typical radius Rs = 30 km, mass After the expansion, the right hand side of the spin-down M = 2M and inertia momentum J = 1045 gcm2.Alsowe equation (33) will take the form assume that the dipole moment inclination is θ0 = π/2 and the post-Maxwellian parameters correspond to Heisenberg– ω˙ = + ωn, KM Kn (35) Euler theory (the choice of Born–Infeld parameters in first n estimation gives a similar order). For the pulsar with the surface magnetic field Bp ∼ where KM corresponds to the torque function of the dipole 1011G, for which ellipticity reaches the maximum value magnetic radiation in Maxwell electrodynamics [48,49]: −4 −6 ε ∼ 10 , the r-mode saturation amplitude βsat ∼ 10 2 6 and F = 1/(20π), and we have the following estimation: 2B R − − K =− p s 2 θ , K /K ∼ 1.3 × 10 11 and K /K ∼ 2.6 × 10 7.Sothe M 3 sin 0 (36) 5 Q 7 R 3Jc quadrupole and r-mode gravitational radiation torque will and K are the torques originating from the nonlinear vacuum significantly exceed the nonlinear electrodynamics torque n ∼ω5 ∼ω7 electrodynamics: coupled with the terms and in the spin-down equation. For another parameter set the opposite case takes place. − Bp 2 Rs n 3 If the pulsar distortion and ellipticity is two orders of mag- = α (θ ) . − Kn n 0 KM (37) nitude lower (ε ∼ 10 6), and the pulsar field is stronger Bc c 123 Eur. Phys. J. C (2016) 76 :612 Page 7 of 8 612

13 Bp ∼ 10 G than K5/KQ ∼ 12.6 and K7/KR ∼ 25.6. gravitational torque and play a more significant role in the However, it should be noted that rapidly rotating pulsars with spin-down equation under certain conditions. such a strong field have not been observed yet. Nevertheless, the theoretical models assuming the blitzars as the sources of fast radio bursts [45] do not eliminate the possibility of such Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm strong electromagnetic fields for the rapidly rotating pulsar. ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, Therefore the obtained ratio between the torques seems very and reproduction in any medium, provided you give appropriate credit exotic but still cannot be completely discarded. to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP3.

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