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Non-linear effects on radiation propagation around a charged compact object R. R. Cuzinatto1, C. A. M. de Melo1,2, K. C. de Vasconcelos3, L. G. Medeiros3, and P. J. Pompeia4,5 ∗ † ‡ § ¶ 1 Instituto de Ciência e Tecnologia, Universidade Federal de Alfenas. Rod. José Aurélio Vilela (BR 267), Km 533, n◦11999, CEP 37701-970, Poços de Caldas, MG, Brazil. 2 Instituto de F´ısica Teórica, Universidade Estadual Paulista. Rua Bento Teobaldo Ferraz 271 Bloco II, P.O. Box 70532-2, CEP 01156-970, São Paulo, SP, Brazil. 3 Departamento de F´ısica Teórica e Experimental and Escola de Ciência e Tecnologia, Universidade Federal do Rio Grande do Norte. Campus Universita´rio, s/n - Lagoa Nova, CEP 59078-970, Natal, RN, Brazil. 4Instituto de Fomento e Coordenação Industrial, Departamento de Ciência e Tecnologia Aeroespacial. Pra¸ca Mal. Eduardo Gomes 50, CEP 12228-901, São José dos Campos, SP, Brazil. 6 5 Instituto Tecnolo´gico de Aeronau´tica, Departamento de Ciência e Tecnologia Aeroespacial. Pra¸ca Mal. Eduardo Gomes 50, 1 CEP 12228-900, São José dos Campos, SP, Brazil. 0 2 Thepropagation ofnon-linearelectromagnetic wavesiscarefully analyzed onacurvedspacetime created by static spherically symmetric mass and charge distribution. We compute how non-linear c electrodynamicsaffectsthegeodesicdeviationandtheredshiftofphotonspropagatingnearthismas- e D sive charged object. In thefirst order approximation, the effects of electromagnetic self-interaction can be distinguished from the usual Reissner-Nordström terms. In the particular case of Euler- 2 HeisenbergeffectiveLagrangian, we findthat theseself-interaction effects might beimportant near 2 extremal compact charged objects. Key-words: Non-linear electrodynamics; Black holes; Redshift; Light bending; Geodesic path. ] c PACS numbers: 04.20.Jb,04.70.Bw,41.20.Jb q - r g I. INTRODUCTION in the literature the direct influence of the background [ electric field on radiation propagationis ignored. 3 GeneralizationsofMaxwellelectrodynamicshavebeen In the early 1970’s it was realized that the self- v proposedsinceitwasestablishedandtheyaremotivated interaction of NLED modifies the dispersion relation of 4 by several reasons such as experimental constraints on a photon propagating in a region with a background 3 the eventual photon mass [1–3], classical aspects of vac- electric field [47]. From the geometrical point of view, 2 uum polarization [4, 5], electrodynamics in the context this modification can be mapped to an effective metric 6 of strings and superstrings [6–10], etc. Among the sev- in the flat spacetime [48–52]. This effective metric can 0 eralgeneralizations,thereisagroupknownasNonlinear be promptly generalized to a curved spacetime metric . 1 Electrodynamics (NLED) which is characterized by pre- through the minimal coupling prescription. As a conse- 0 sentingnonlinearfieldequations. ExamplesofNLEDare quence, there are two simultaneous effects on radiation 5 Born-Infeldtheory[11–14]andEuler-Heisenbergelectro- propagation: first, the path of light rays may be altered 1 : dynamics[4]. Theformerwasproposedtolimitthemax- duetoself-interactionoftheelectricfieldviatheeffective v imumvalueoftheelectricfieldofapointcharge[15]and metric; second, the photon dynamics is affected by the Xi thelastarisesasaneffectiveactionofone-loopQED[16]. spacetime curvature. In fact, some preliminary results concerning photon propagation in non-linear interaction Sincethedecadeof1980,severalapplicationsofNLED r a in the context of gravitation were proposed [17–22], in- with a background electric field and in the presence of a BH were obtained in the context of Euler-Heisenberg cludingapplicationstocosmology[23–29]andspherically and Born-Infeld electrodynamics [53, 54]. In the present symmetric solutions of charged Black Holes (BH) [30– work, the authors intend to generalize these results for 36]. Moreover,generalizationsofReissner-Nordstro¨mso- a generic NLED and particularly show that, despite the lutionwithNLEDwerestudiedwherestabilityandther- fact that the background electric field does not generate modynamicspropertiesoftheBHwereanalyzed[37–45]. effective horizons (horizons that would be sensed only Inparticular,thegeodesicmotionoftestparticlesaround Born-InfeldBHwasstudiedinLinareset al.[46]andref- by radiation), this field directly influences the geometric redshift and geodesic deviation. erences therein. However, in most of the papers found The paper is organized as follows. In Sect. 2, the sphericallysymmetric solutionforagenericNLEDisde- termined. In Sect. 3, minimal coupling prescription is used to generalize the effective metric in flat spacetime ∗ [email protected] † [email protected] to a curved one. In this section, it is also shown that ‡ [email protected] the effective metric does not produce any new horizon. § [email protected] In Sect. 4, the influence of the background electric field ¶ pedropjp@ifi.cta.br in geometric redshift and geodesic deviation is analyzed. 2 Final remarks are presented in Sect. 5. III. EFFECTIVE METRIC In NLED, the electromagnetic field is self-interacting II. GENERAL SOLUTION FOR NLED and this is directly reflected on photon propagation. In particular,whenradiationpropagatesinaslowlyvarying The matter Lagrangian for a general NLED invariant electromagneticfieldasabackground,theself-interaction under parity is: between the radiation and the background field can be = (F,G2), (1) described by an effective metric: η¯µν [47–49]. In flat L L spacetime one has where F 1FµνF and G 1F˜µνF . The quan- tity F ≡is −th4e electµroνmagneti≡c fi−el4d tensµoνr and Fµν is η¯µνkµkν = ηµν +λ FµβFβν kµkν =0, (9) µν ± its dual: Fµν ≡ 12ηµνρσFρσ, where ηµνρσ = ε√µνρgσ is the where (cid:0) (cid:1) totally antisymmetric tensor constructed from−theeLevi- ( + )+2F ¯ √δ Civitasymebolεµνρσ andthe determinantg ofthemetric λ = −LF LGG LFF L± , (10) tensor gµν.1 ± 2 G2L¯+2LF (LGGF −LFGG)−L2F The energy-momentum tensor is given by: with (cid:2) (cid:3) T = F Fλ +G g g , (2) µν LF µλ ν LG µν −L µν δ = 2F 2 + ( ) 2 witThheLFsta≡tic∂∂FLspahnedricLaGlly≡sy∂∂mGLm. etric line element is: (cid:2) (cid:0)L+FF2LGGGL−FFLLFGGG(cid:1) −LLF2FGLG−G2−LFLLFFFG(cid:3)2(11) ds2 =eν(r)dt2 eλ(r)dr2 r2dθ2 r2sin2θdφ2 , (3) and (cid:2) (cid:0) (cid:1) (cid:3) − − − where ν(r) and λ(r) are functions determined by the L¯=LFFLGG−L2FG. (12) gravitationalfield equations. In order to preserve space- The signal in Eq. (10) indicates birefringence [47, 56], time symmetry, one supposes the matter content to be ± i.e. the possible dependence of the photon propagation constituted by a static and spherically symmetric elec- on its polarization, which is a typical phenomenon of tric charge. Hence, only the components F = F = 01 10 − NLED.Whenminimalcouplingprescription(η g ) E(r) of the field strength Fµν are non-null. Conse- µν → µν − isperformed,theeffectivemetriconthecurvedspacetime quently G = 0. In this case, the asymptotically flat exis found to be: terior Schwarzchild-de Sitter-NLED solution of Einstein equations is: g¯µν =gµν +λ FµβF ν, (13) β ± 2m 1 Λ ν(r)= λ(r)=ln 1 + S(r) r2 . (4) where gµν is the spacetime metric given in Eq. (3) – see − (cid:18) − r r − 3 (cid:19) appendix A of Novello et al. [49] for more details. This Weconsiderc=GN =1. Theconstantmisthegeomet- way,photons(i.e. weakradiationpropagatingfieldsthat ric mass, Λ is the cosmologicalconstant and do not disturb the spacetime) will follow geodesics described by the following effective line element S(r) 2 r2 E2+ dr. (5) F ≡ −L L eν e ν The field equationsZfor F(cid:0)µν are: (cid:1) ds¯2 = [1+λ E2]dt2− [1+λ− E2]dr2−r2dΩ2. (14) ± ± ∂ √ gFµν =0 and ∂ F =0. (6) ν F [ρ µν] Thislineelementcontainsbothspacetime(gravitational) L − This implie(cid:0)s: (cid:1) effects–via thefactoreν –andtheinfluenceoftheback- q groundfield–throughthefunction 1+λ E2 . Thean- LFE = r2, (7) gularpartofthelineelementisdΩ2(cid:2)=dθ2±+sin(cid:3)2θdφ2,as usual. It is also worth mentioning that the NLED under where q is the total electric charge. consideration influences photon trajectory both gravita- Note that an explicit solution can be obtained when an specific (F,G2) is chosen. For instance, Maxwell tionally – through S(r) – and electromagnetically – by L means of λ . For instance, in Maxwell electrodynamics: Electrodynamics is recoveredwhen =1 and LF λ =0and±S(r)isgivenbyEq.(8). Ontheotherhand, q q2 in±Born-Infeld theory, where E = S(r)= , (8) r2 ⇒ r which is the well known Reissner-Nordstro¨msolution. =b2 1 1 2F G2 , (15) LBI " −r − b2 − b4 # 1 EI~naflnadtsB~paacreetitmhee,eFlec=tri(cE~a2n−dmB~2a)g/n2etaicndfieGld=srEe~s·pBe~c.tiTvehley.vFectaonrds o[5n4e].fiNndosteλ±tha=t,b2b−o1Eth2;inSM(r)axiswgeilvleanndbyBeolrlinp-tIincfefuldncctaisoenss, GaretheonlyLorentzandgaugeinvariantfunctionsofFµν [55]. no birefringence is present [48, 57]. 3 The first point to be analyzed here concerns the exis- which leads to: tence of effective event horizons. For this goal, an inves- q a q 2 tigation of the radial null geodesics is carried on using E 1 + . (22) Eq. (14). The result is: ≃ r2 − 2 r2 (cid:20) (cid:16) (cid:17) (cid:21) dt 2 = 1+λ±E2 e−ν =e−2ν, (16) Notice that the condition (18) implies: dr [1+λ E2]eν (cid:18) (cid:19) (cid:2) ± (cid:3) a q 2 1. (23) which is exactly the same expression obtained consider- + r2 ≪ (cid:16) (cid:17) ing propagationwith the spacetime metric Eq. (3). This Now Eq. (22) is substituted in Eq. (5) leading to means that the horizons “seen” by photons are gener- atedexclusivelybygravitationaleffectsofthespacetime. q2 k q4 Note that this result is formally independent of the par- S(r) , (24) ≃ r − 20r5 ticular type of NLED under consideration because the factors scaling with λ cancel out in Eq. (16). However, where k a . The change of notation k instead of a + + we emphasize that the±gravitationaleffects of the NLED isdone in≡ordertodistinguishgravitationaleffects ofthe are still present in ν. NLED from purely electromagnetic effects, i.e. k is re- Although the horizons do not depend on the effective lated to the spacetime metric and a to the effective metric,thelaterisimportantinotherphenomenasuchas metric. ± geometricredshiftandlightdeflection. Thesetwoeffects Finally, the components of the metric Eq. (3) in the are analyzed in the following section. first order approximation are rewritten as: eν q 2 k q4 g¯ = W (r) 1 a , IV. INFLUENCE OF NLED ON RADIATION 00 [1+λ E2] ≃ − ± r2 − 20r6 PROPAGATION ± (cid:20) (cid:16) (cid:17) (cid:21) (25) 1 1+λ E2 q 2 k q4 In this section, an investigation on how an specific − = ± W (r) 1+a , NatLioEnDwiinllflbueenccoensdtuhceteradd.iNatoioinnflreudesnhcieftoafnthdegceoosdmesoiclodgeicvai-l g¯11 (cid:2) e−ν (cid:3) ≃ (cid:20) ±(cid:16)r2(cid:17) (cid:21)− 20r(626) constantwillbe consideredhere,i.e. Λ=0. Aperturba- where tive approachwill be adopted, in which nonlinear effects aresmallcorrectionsto the Maxwelltheory. L F,G2 is 2m q2 written as W(r)=1 + (27) (cid:0) (cid:1) − r r2 1 1 F,G2 F + a+F2+ a G2+ (3) (17) is the usual Reissner-Nordstro¨m term. L ≃ 2 2 − O In this approximation,the manner by whichthe event (cid:0) (cid:1) where (3) represents higher order terms and horizon is affected by the NLED can be studied. The O horizons r are obtained when the condition g11 = 0 is a+F a G 1. (18) considered±, which in our particular case gives: ∼ − ≪ The condition a >0 ensures that the energy density is 2m q2 k q4 positive definite±[34]. 1− r + r2 − 20r6 ≃0. (28) In the first order regime of perturbation and considering a radial electric field (F = E2 e G = 0), Eq. (10) The approximated solution to this equation is: 2 reduces to: r2 q4 (0) λ+ ≃ LFFF ≃a+ 1− a2+E2 , r± ≃r±(0)± 40 m±2−q2kr±6(0), (29) λ L LGG (cid:16) a 1(cid:17) a+E2+a E2 . (19) where p − ≃ F 2F GG ≃ − − 2 − L − L (cid:16) (cid:17) r =m m2 q2 (30) (0) Hence, ± ± − p is the usual Reissner-Nordstro¨m horizon. Notice that 1+λ E2 1+a E2. (20) ± ≃ ± r >r e r <r . (31) The next step is to obtain S(r). By substituting + +(0) − −(0) Eq. (17) into Eq. (7), one obtains: Hence, the effect of the nonlinearity of NLED is to dis- a q placetheexternalhorizonoutwardsandtheinternalhori- 1+ +E2 E , (21) zon inwards. 2 ≃ r2 (cid:16) (cid:17) 4 A. Geometric redshift whereαisthefine-structureconstantandm istheelec- e tron mass. This way, The geometric redshift of a light ray emitted radially at r and detected at r with r <r <r , is given by 16α2~3 28α2~3 1 2 + 1 2 a = , a = , (38) + 45m4 − 45m4 e e g¯ (r ) 00 2 1+z = . (32) sg¯00(r1) where a+ ∼a− ∼ 10−28cemrg3. In order to have an idea of the magnitude of the non- By using Eq. (25) and considering r = r and r , 1 2 → ∞ linearterminthecontextofEuler-Heisenbergelectrody- one gets: namics,ahypotheticalsituationisconsidered: astarlike 2m q2 k q4 q 2 −1/2 the Sun (with the same radius R and mass M ) with 1+z 1 + a W(r) . an electric charge producing an e⊙xtremal black⊙hole, i. ≃ − r r2 − 20r6 − ± r2 (cid:20) (cid:16) (cid:17) (cid:21) (33) e. m = q. This is the maximal charge permitted for a static black hole if the cosmic censorship hypothesis is The first three terms of the right hand side of this equa- considered. In this case, tion correspond to the usual Reissner-Nordstro¨m terms where the redshift is increased by the mass while the m M G m2 q2 cThhaerglaest(Mtwaoxwteerlml tsedrmes)criisberetshpeoenffsiebcltes ofofrthdeecNreLaEsDin:gthite. R → c2R⊙ ≃2×10−6 ⇒ R2 ∼ R2 ∼10−12, (39) ⊙ ⊙ ⊙ ⊙ fourth one is a consequence of the curvature of space- 2 2 a q a M time generated by non-linear electromagnetic terms and 2± R2 → 2±G R2⊙ ≃4×10−15. (40) the fifth one is due to the effective metric of the NLED, (cid:18) (cid:19) (cid:18) (cid:19) ⊙ ⊙ i.e., a redshift created by photons interacting with the background electric field. Note that both terms lead to Forthis hypotheticalsituation, the nonlinear term(scal- an increasing of z. It is also worth mentioning that the ingwitha )isnineordersofmagnitudesmallerthanthe influenceoftheeffectivemetricmakesredshiftdependent dominant±term mr . Although being negligible this term on the polarization since in general a =a . becomes rapidly important as r decreases. In particular, + Ifaweakfieldapproximationistaken6into−account,i.e. forRWD =10−3R (a white dwarfradius)the nonlinear term becomes of t⊙he same magnitude order of m, i.e. r 2m q2 1 e ≪1, (34) r ≪ r2 m a q 2 then Eq. (33) is simplified to R ∼ 2± R2 ∼10−3, (41) WD (cid:18) WD(cid:19) m 3m2 q2 a q 2 m2 q2 10 6. (42) z ≃ r + 2 r2 − 2r2 + 2± r2 , (35) RW2 D ∼ RW2 D ∼ − (cid:16) (cid:17) where the mass dependence is taken up to second order Thismightindicateanastrophysicalobjectforwhichthe and the term k rq2 2 rq22 is neglected, meaning that the NLED effects could be important indeed. curvature-born term disappears. (cid:0) (cid:1) Next, the geodesic deviation will be analyzed. The magnitude of the last term in Eq. (35) depends on the coupling constant of the NLED, which is a free parameter (up to constraints coming from experimental data). For instance, for Born-Infeld Lagrangian, B. Geodesic deviation of radiation Eq. (15), 1 Thegeodesicdeviationofalightraypropagatingfrom a =a = . (36) + − b2 infinity to a point at a distance r from the origin of our coordinatesysteminaregionwithstaticsphericallysym- The exception is Euler-Heisenberg electrodynamics metric gravitationaland electric fields is given by [58]: whichisobtainedasaneffectivetheoryatthelow-energy regimeofQED.2AccordingtoBialynickaBirulaandBia- lynicki Birula [47], Euler-HeisenbergLagrangianis given ∞ g¯11(r) dr φ(r) φ = , (43) by LEH ≃F + 24α52m~43 4F2+7G2 , (37) − ∞ Zr vuuut(cid:20)(cid:16)rr0(cid:17)2(cid:16)g¯g¯0000((rr0))(cid:17)−1(cid:21) r e where r is the maximum approximationdistance of the (cid:0) (cid:1) 0 light ray from the gravitational/electric source. In the approximationofsmallcorrectionstoMaxwelltheoryg¯ 00 2 Energiesmuchlessthanthatoftheelectronrestenergy. and g¯ are given by Eqs. (25) and (26), respectively. In 11 5 first order, the above expression is rewritten as This equation shows clearly the contributions of mass, electric chargeand the nonlinear term for the evaluation (φ(r) φ ) ofgeodesicdeviationintheweak-fieldapproximation. In − ∞ ± ≃ the case of the Euler-Heisenberg Lagrangian, Eq. (37), ∞ q 2 q2 q 2 f (r) 1 a f (r)+k f (r) dr, the contribution of each term can be evaluated by con- 1 − ± r2 2 r2 r2 3 siderations analogous to those of the previous section. Zr (cid:26) (cid:16) (cid:17) (cid:16) (cid:17) (cid:27) Preliminaryresults ofthe influence ofthe vacuumpolar- (44) ization effects described by Euler-Heisenberg electrody- where namics were obtained in De Lorenci et al. [53]. Particular cases are obtained if one takes one or more 1 r 2 −1/2 terms of Eq. (53) to be null. For instance, for an object f (r)= W(r ) W (r) , (45) of null charge the known result ∆φ = 4m is obtained. 1 r "(cid:18)r0(cid:19) 0 − # An interesting case is obtained when±onlyr0the effect due 4 toNLEDself-interaction(theinteractionoftheradiation W (r ) r W (r ) 1 1 0 − r0 0 with the background electric field) is considered, i.e. no f (r)= + , (46) 2 2 2 W (r ) (cid:16) r0(cid:17)2W (r) gravitational effects are taken into account. Essentially, 0 − r the metric considered in this calculation is the effective 6 W(cid:0)(r(cid:1)) r W (r) metric in a flat spacetime. Mathematically, this corre- 1 0 − r0 f (r)= 1 . (47) sponds to take W(r) = 1 and k = 0 in Eq. (44). The 3 40W(r) − W (r0)−(cid:16)rr0(cid:17)2W(r) integration of this expression leads to a purely electric   (PE) geodesic deviation:  (cid:0) (cid:1)  In Eq. (44) the first term is the standard effect of classi- 9π q 2 cal electrostatics in the context of general relativity, the ∆φPE a . (54) secondtermisthecontributionfromNLEDeffectivemet- ± ≃ ±16 (cid:18)r02(cid:19) ric, and the third one is the correction due to spacetime Hypothetically, this result can be used to verify experi- curvaturecomingfromthenonlinearbackgroundelectric mentally if vacuum polarization is able to produce light field. Notethatthepresenceofthesecondtermindicates deviation, as predicted by Eq. (54). thatthe geodesicdeviationisdependentonthepolariza- tion of the radiation. The integral in Eq. (44) can be evaluated analytically V. CONCLUSION if the weak field-approximation(34) is considered: In this paper we have constructed the solution of Ein- r m 0 (φ(r) φ ) arcsin + h (r) stein equations for the exterior of spherically symmetric 1 − ∞ ± ≃ r r0 massandchargedistributionsinthecontextofnon-linear (cid:16) (cid:17) q2 1 q 2 electrodynamics. Theresultinglineelementisisgivenin h (r)+ a h (r), (48) − 2r2 2 2 ± r2 3 Eqs. (3), (4) and (5). It is asymptotically de Sitter and 0 (cid:18) 0(cid:19) depends onthe mass m, onthe cosmologicalconstantΛ, where and on the functional S(r). This object is an integral given in terms of the electrostatic field E~ and the La- h (r)=2 1 r0 2 1− rr0, (49) grangiandensity (F,G2) of the particular NLED to be 1 −r − r −s1+ rr0 taken into accounLt. (cid:16) (cid:17) First,wenoticedthattheradialnullgeodesicsfollowed 3 r 1 r r 2 h (r)= arcsin 0 0 1 0 (50) by photons in the curved effective metric can be calcu- 2 2 r − 2 r − r r lated as usual, being the NLED influence encapsulated (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) 9 r 1 r r 2 infunctionν(r) viaS(r). Thenon-linearityofelectrody- 0 0 0 h (r)= arcsin 1 3 8 r − 8 r − r namics appears explicitly in Eq. (22) for the magnitude r (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) of the electric field, which is calculated when small per- 1 r 2 r 3 + 1 0 0 . (51) turbations of Maxwell theory are considered. This leads 4 − r r r to a modification in the horizons of the charged BH as (cid:16) (cid:17) (cid:16) (cid:17) The deviationobservedina regionfar fromthe source perceived by the photons: displacing the internal and is given by: external horizons inwards and outwards respectively. The redshift of spectral lines is increased by the pres- ∆φ=2 φ(r ) φ π, (52) ence of the non-linear electrodynamic terms. Moreover, 0 | − ∞|− z is sensible to the polarization of the radiation due to resulting: self-interaction. In order to estimate the magnitude of the effects on z coming fromNLED, we decide to choose m 3πq2 9π q 2 a particular non-linear theory and study it in the weak- ∆φ 4 +a . (53) ± ≃ r0 − 4 r02 ±16 (cid:18)r02(cid:19) field regime. Using Euler-Heisenberg Lagrangian, one 6 concludes that extremal compact charged objects could Eq. (54). This could be used for comparison with ob- produceredshiftsforwhichthecontributionfromtheor- servationaldata in order to determine if the vacuum po- dinary (m/r) term is as important as the NLED-born larization actually can bend light rays. term. Finally, we calculated the geodesic deviation experi- enced by radiation nearby massive charged bodies. The Reissner-Nordstro¨m result is recovered along with small ACKNOWLEDGMENTS corrections to Maxwell electrodynamics due to NLED. The background electric field affects the geodesic path KCV acknowledges CAPES-Brazil for financial sup- of light rays, increasing the deviation that would be ex- port. LGM is grateful to FAPERN-Brazil for financial pected from the linear theory. We also obtained an support. expression for the deviation when only the effect due to NLED self-interaction is taken into account – see [1] L.-C.Tu,J.Luo, andG.T.Gillies,ReportsonProgress [24] V. A. De Lorenci, R. Klippert, M. Novello, and J. M. in Physics 68, 77 (2005). 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