Mon. Not. R. Astron. Soc. 354, L55–L59 (2004) doi:10.1111/j.1365-2966.2004.08375.x

Non-linear electrodynamics and the gravitational redshift of highlymagnetized neutron stars

Herman J. Mosquera Cuesta1,2,3� and Jose M. Salim1

1Instituto de Cosmologia, Relatividade e Astrofısica (ICRA-BR), Centro Brasileiro de Pesquisas Fısicas, Rua Dr. Xavier Sigaud 150, Cep 22290-180, Urca,Rio de Janeiro, RJ, Brazil2Abdus Salam International Centre for Theoretical Physics Strada Costiera 11, Miramare 34014, Trieste, Italy3Centro Latino-Americano de Fısica, Avenida Wenceslau Braz 71, CEP 22290-140 Fundos, Botafogo, Rio de Janeiro, RJ, Brazil

Accepted 2004 September 3. Received 2004 August 7; in original form 2004 June 16

ABSTRACTWe show that non-linear electrodynamics (NLED) modifies in a fundamental basis the conceptof gravitational redshift (GRS) as introduced by Einstein’s theory of general relativity (GR).The effect becomes apparent when light propagation from super-strongly magnetized compactobjects, such as pulsars, is under focus. The analysis, here based on the (exact) non-linearLagrangian of Born & Infeld (1934), proves that unlike GR, where the GRS is independent ofany background magnetic field (B-field), when NLED is incorporated into the photon dynamics,an effective GRS appears, which happens to depend decidedly on the B-field pervading thepulsar. The resulting GRS tends to infinity as the B-field grows larger, as opposed to the Einsteinprediction. As in astrophysics, the GRS is used to infer the mass–radius relation, and thus theequation of state (EOS) of a compact star (for example, a neutron star; Cottam et al. 2002).This unexpected GRS critical change may mislead observers into considering that fundamentalproperty: the EOS. Hence, a correct procedure to estimate those crucial physical propertiesdemands a neat separation of the NLED effects from the pure gravitational ones in the lightemitted by ultra-magnetized pulsars.

Key words: gravitation – methods: analytical – stars: emission-line, Be – stars: magnetic fields– stars: neutron – pulsars: general.

1 I N T RO D U C T I O N

The idea that non-linear electromagnetic interaction, i.e. light prop-agation in vacuum, can be geometrized was developed by Novelloet al. (2000) and Novello & Salim (2001). Since then, a numberof physical consequences for the dynamics of a variety of systemshave been explored. In recent papers, Mosquera Cuesta & Salim(2003, 2004) presented the first astrophysical context where suchnon-linear electrodynamic effects were accounted for: the case of ahighly magnetized neutron star (NS) or pulsar. In that paper, non-linear electrodynamics (NLED) was invoked a la Heisenberg–Euler,which is an infinite series expansion of which only the first non-linear term was used for the analysis. An immediate consequenceof that study was an overall modification of the space–time geom-etry around the pulsar as ‘perceived’ by light propagating out of it.This translates into a fundamental change of the star surface redshift,the GRS, which might have been inferred from the absorption (oremission) lines observed in a super-magnetized pulsar by Ibrahimet al. (2002, 2003). The result proved to be even more dramatic forthe so-called magnetars: pulsars said to be endowed with magnetic

�E-mail: hermanjc@cbpf.br

fields (B-fields) higher than the Schafroth quantum electrodynamics(QED) critical B-field. In this Letter, we demonstrate that the sameeffect still appears if one calls for NLED in the form rigorouslyderived by Born & Infeld (1934), which is based on the special rel-ativistic limit to the approaching velocity of an elementary chargedparticle to a point-like electron. As compared to our previous re-sults, here we stress that from the mathematical point of view theBorn & Infeld (1934) NLED is described by an exact Lagrangian,the dynamics of which has been successfully studied in a wide set ofphysical systems (Delphenich 2003). The analysis presented nextproves that this physics affects not only the magnetar electrody-namics (our focus here) but also that one of newly born, highlymagnetized protoneutron stars and the dynamics of their progenitorsupernovae.

2 N S M A S S – R A D I U S R E L AT I O N

NSs, formed during the death throes of massive stars, are among themost exotic objects in the Universe. They are supposed to be essen-tially composed of neutrons, although some protons and electronsare also required in order to guarantee stability against Pauli’s exclu-sion principle for fermions. As remnants of supernova explosionsor accretion-induced collapse of white dwarfs, they are (canonical)

C© 2004 RAS

L56 H. J. Mosquera Cuesta and J. M. Salim

objects with extremely high density ρ ∼ 1014 g cm−3 for mass ∼1 M� and radius R ∼ 10 km, and are supposed to be endowedwith typical magnetic fields of about B � 1012 G, as expected frommagnetic flux conservation during the supernova collapse.

In view of its density, a NS is also believed to trap in its core asubstantial part of even more exotic states of matter. It is almost aconsensus that these new states might exist inside and may dominatethe structural properties of the star. Pion plus kaon Bose–Einsteincondensates could appear, as well as ‘bags’ of strange quark mat-ter (Miller 2002). This last one is believed to be the most stablestate of nuclear matter (Glendenning 1997), which implies an ex-tremely dense medium, the physics of which is currently under se-vere scrutiny. The major effect of these exotic constituents is man-ifested through the NS mass–radius ratio (M/R). Most researchersin the field consider that the presence of such exotic components notonly makes the star more compact (i.e. smaller in radius) but alsolowers the maximum mass it can retain.

The fundamental properties of NSs, their mass (M) and radius(R), provide a direct test of the equation of state (EOS) of coldnuclear matter, a relationship between pressure and density that isdetermined by the physics of the strong interactions between theparticles that constitute the star. It is admitted that the most directprocedure of estimating these properties is by measuring the grav-itational redshift of spectral lines produced in the NS photosphere.The EOS relates M and R directly, and hence a measurement ofthe GRS at the star surface leads to a strong constraint on the M/Rratio. In this connection, observations of the low-mass X-ray binaryEXO0748-676 by Cottam et al. (2002) lead to the discovery of ab-sorption lines in the spectra of a handful of X-ray bursts, with mostof the features associated with Fe26 and Fe25 n = 2–3 and O8 n = 1–2transitions, all at a redshift z = 0.35 (see its definition in equation 2below). The conclusions regarding the nature of the nuclear mattermaking up that star thus seem to exclude some models in whichthe NS material is composed of more exotic matter than the coldnuclear one, such as the strange quark matter or kaon condensatesfor M = 1.4–1.8 M� and R = 9–12 km.

But the identification of absorption lines was also achieved bySanwal et al. (2002) using Chandra observations of the isolated NS1E1207.4-5209. The lines observed were found to correspond toenergies of 700 and 1400 eV, which they interpreted as the signatureof singly ionized helium in a strong magnetic field.1 The inferredredshift is 0.12–0.23. Because, as well as gravity (see equation 2),a magnetic field has nontrivial effects on the line energy (for anexample, see equation 1), Sanwal et al. (2002) could not make theircase for a correct and accurate identification of the lines. Neithercould they decidedly rule out alternative interpretations, such as thecyclotron feature, which is expected from interacting X-ray binaries,as prescribed by the relation

EHe = 3.2 (1 + z)−1

(Bsc

1015G

)keV. (1)

To gain insight into the most elusive properties of a NS (its massand radius), astronomers have several techniques at their disposal. Itsmass can be estimated, in some cases, from the orbital dynamics ofX-ray binary systems, while attempts to measure its radius proceedvia high-resolution spectroscopy, as done very recently by Sanwalet al. (2002) upon studying the star 1E1207.4-5209, and by Cottamet al. (2002) by analysing type I X-ray bursts from the star EXO0748-

1 Note that the B-field strength in this source is unknown because no spin-down was measured in it.

10 12.5 15 17.5 20log10(B−Field)

0

0.2

0.4

0.6

0.8

1

(g00

eff /g

00)

Figure 1. The ratio between the effective time–time metric componentsgeff

00 and the background g00 as a function of the magnetic field strengthB-field. Notice that the ratio tends to zero as the B-field grows larger, whichmeans that the effective (gravitational + NLED) redshift tends to infinitywhereas the standard redshift (GRS) remains constant for a given M/R ratio,as it is independent of B.

676. In those systems, success was achieved in determining thoseparameters, or the relation between them, by looking at the generalrelativistic effect known as the GRS of excited ions near the NSsurface. Gravity effects cause the observed energies of the spectrallines of excited atoms to be shifted to lower values by a factor

1

(1 + z)≡

[1 − 2G

c2

(M

R

)]1/2

, (2)

with z being the GRS. Because this redshift depends on the M/Rratio, measuring the displacement of the spectral lines leads to anindirect, but highly accurate, estimate of the star radius.

The above analysis continues to be valid whenever the effectsof the NS B-field are negligible. However, if the NS is pervadedby a super-strong B-field (Bsc), as for magnetars, there is then thepossibility, for a given field strength, for the gravity effects to beemulated by the electromagnetic ones. In what follows, we provethat this is the case if NLED (a la Born & Infeld 1934) is taken intoaccount to describe the general physics taking place on the pulsarsurface. Our major result proves that for very high magnetic fields(B � 1014 G) the redshift induced by NLED can be as high as thatproduced by gravity alone, whereas in the extreme limit (B � 1015

G) it largely overtakes it (see Figs 1 and 2). In this way, the NLEDemulates the gravitation. In such stars, then, care should be exercisedwhen putting forward claims regarding the M/R ratio or the EOS ofthe observed pulsar.

3 N L E D A N D T H E E F F E C T I V E M E T R I C

It is well known that extremely strong magnetic fields induce thephenomenon of vacuum polarization, which manifests, dependingon the magnetic field strength, as either real or virtual electron–positron pair creation in a vacuum. Although this is a quantum ef-fect, we stress that it can also be described classically by includingcorrections to the standard linear Maxwell Lagrangian. Because oftheir interaction with this excited background of pairs, the modi-fication of the dispersion relation for photons, as compared to theone from Maxwell dynamics, is one of the major consequences ofthe non-linearities introduced by the NLED Lagrangian. Among theprincipal alterations associated to this new dispersion relation is the

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NLED and GRS of highly magnetized neutron stars L57

10 12.5 15 17.5 20log

10(B−Field)

0

2

4

6

8

10

z Effe

ctiv

e

Figure 2. The plot shows the variation, for a given constant M/R ratio,of the effective redshift (which in turns depends on the effective time–timemetric component geff

00 ) as a function of the B-field strength. Notice that ittends to infinity as the B-field grows larger, whereas the standard redshift(GRS) takes on a fixed value z � 1.

modification of the photon trajectory, which is the main focus in thisLetter. [A discussion on light-lensing in compact stars is given byMosquera Cuesta, de Freitas Pacheco & Salim (2004).] We arguehere that, for extremely supercritical B-fields, NLED effects forcephotons to propagate along accelerated curves.

In case the non-linear Lagrangian density is a function only of thescalar F ≡ F µν Fµν , say L(F), the force accelerating the photons isgiven as

kα||νkν =(

4L F F

L FFµ

β Fβνkµkν

)|α

, (3)

where kν is the wavevector, and LF and LFF stand, correspondingly,for first and second partial derivatives with respect to the invariantF. Here, and also in equation (9) below, the symbols | and || stand,respectively, for partial and covariant derivatives. This feature allowsfor this force, acting along the path of the photons, to be geometrized(Novello et al. 2000; Novello & Salim 2001; De Lorenci et al. 2002)in such a way that in an effective metric

geffµν = gµν + gNLED

µν (4)

the photons follow geodesic paths. In such a situation, the standardgeometric procedure used in general relativity to describe the pho-tons can now be used upon replacing the metric of the backgroundgeometry, whichever it is, by that of the effective metric. In thiscase, the modified redshift is now proven to have a couple of com-ponents: one due to the gravitational field and another stemmingfrom the magnetic field. As the shift in energy, and width, producedby the effective metric ‘pull’ of the star on laboratory-known spectrallines increases directly with the strength of the effective potentialassociated to the effective metric, this shift has two contributions:one coming from gravitational effects and another from NLED ef-fects. In the case of hypermagnetized stars (e.g. magnetars), bothcontributions may be of the same order of magnitude. This clearlyhinders the imposition of constraints on the M/R ratio. This diffi-culty can be overcome by taking into account that the contributionof the B-field, which depends on both the angular coordinates θ andφ, differs from that of the gravitational field which is fully isotropic.

Our warning (see Fig. 2!) then holds. The identification and anal-ysis of spectral lines from highly magnetized NSs must take intoaccount the two possible different polarizations of the received pho-

tons, in order to be able to discriminate between redshifts producedeither gravitationally or electromagnetically. Putting this claim inperspective, we stress that if the characteristic redshift (or the M/Rratio) were to be inferred from this source, care should be takenbecause for this super-strong B-field, estimated upon the identifiedline energy and width or via the pulsar spin-down, such redshiftbecomes of the order of the gravitational one (GRS) expected froma canonical NS. It is, therefore, not clear whether one can conclu-sively assert something about the M/R ratio of any magnetar undersuch dynamical conditions.

3.1 The method of effective geometry

In this Letter, we want to investigate the effects of non-linearitiesof very strong magnetic fields in the evolution of electromagneticwaves, described here as the surface of discontinuity of the electro-magnetic field (represented hereafter by F µν). Because in the pulsarbackground there is only a magnetic field,2 the invariant G = Bµ

Eµ is not a functional in the Lagrangian. For this reason, we will re-strict our analysis to the simple class of gauge-invariant Lagrangiansdefined by

L = L(F). (5)

The surface of discontinuity3 for the electromagnetic field willbe represented by �. We also assume that the field F µν is contin-uous when crossing � and that its first derivative presents a finitediscontinuity (Hadamard 1903):

[Fµν]� = 0 (6)

and

[Fµν|λ]� = fµνkλ, (7)

respectively. The notation

[Fµν]� ≡ limδ→0+

(J|�+δ − J|�−δ) (8)

represents the discontinuity of the field through the surface �. Thetensor f µν is called the discontinuity of the field, whilst

kλ ≡ �|λ (9)

is called the propagation vector. From the least action principle, weobtain the following field equation:

(L F Fµν)||µ = 0. (10)

Applying the Hadamard conditions (6) and (7) to the discontinuityof the field in equation (10), we obtain

L F f µνkν + 2L F Fξ Fµνkν = 0, (11)

where ξ is defined by

ξ.= Fαδ fαδ. (12)

Both the discontinuity conditions and the electromagnetic fieldtensor cyclic identity lead to the following dynamical relation:

fµνkλ + fνλkµ + fλµkν = 0. (13)

2 For the present analysis, we assume a slowly rotating NS for which theelectric field induced by the rotating dipole is negligible.3 Of course, the entire discussion onwards could alternatively be rephrasedin terms of concepts more familiar to the astronomy community as that oflight rays used for describing the propagation of electromagnetic waves inthe optical geometry approximation.

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L58 H. J. Mosquera Cuesta and J. M. Salim

In the particular case of a polarization such that ξ = 0, it followsfrom equation (10) that

f µνkν = 0. (14)

Thus, by multiplying equation (13) by kλ and using the result ofequation (14), we obtain

fµνkλkλ = 0. (15)

This equation expresses that for this particular polarization thediscontinuity propagates with the metric f µν of the backgroundspace–time. For the general case, when ξ �= 0, we multiply equa-tion (13) by kαgαλ Fµν to obtain

ξkνkµgµν + 2Fµν f λν kλkµ = 0. (16)

From this relation and equation (11), we obtain the propagationlaw for the field discontinuities, in this case given as(

L F gµν − 4L F F Fµα Fαν

)kµkν = 0, (17)

where

Fµα Fαν = −B2hµν − Bµ Bν . (18)

Equation (17) allows us to interpret the term inside the parenthesis(being multiplied by kµ kν) as an effective geometry

gµν

eff = L F gµν − 4L F F Fµα Fαν . (19)

Hence one concludes that the discontinuities will followgeodesics in this effective metric.

3.2 Born–Infeld NLED

One can start the study of the NLED effects on the light propagationfrom hypermagnetized NSs, with the Born–Infeld (BI) Lagrangian(recall that a typical pulsar has no relevant electric field, i.e. E isnull)

L = L(F), (20)

with

L = −b2

2

[(1 + F

b2

)1/2

− 1

], (21)

where

b = e

R20

= e

e4/(

m20c8

) ⇒ b = m20c8

e3= 9.8 × 1015 e.s.u. (22)

(electrostatic units).In order to obtain the effective metric that stems from the BI

Lagrangian, one has therefore to compute the derivatives of theLagrangian with respect to F. The first of them reads

L F = −1

4(1 + F/b2)1/2, (23)

with its second derivative being

L F F = 1

8b2(1 + F/b2)3/2. (24)

The L(F) BI Lagrangian produces, according to equation (19), aneffective contravariant metric given as

gµν

eff = −1

4(1 + F/b2)1/2gµν + B2

2b2(1 + F/b2)3/2(hµν + lµlν) ,

(25)

where we define the tensor hµν as the metric induced in the referenceframe perpendicular to the observers, which are determined by the

vector field Vµ, and lµ ≡ Bµ/|Bγ B γ |1/2 as the unit 4-vector alongthe B-field direction.

Because the geodesic equation of the discontinuity (that definesthe effective metric) is conformal invariant, one can multiply thislast equation by the conformal factor

−4

(1 + F

b2

)3/2

(26)

to obtain

gµν

eff =(

1 + F

b2

)gµν − 2B2

b2(hµν + lµlν) . (27)

By noting that

F = Fµν Fµν = −2(E2 − B2), (28)

and recalling that E = 0 in the case of a canonical pulsar, we findthat F = 2 B2. Therefore, the effective metric reads

gµν

eff =(

1 + 2B2

b2

)gµν − 2B2

b2(hµν + lµlν) , (29)

or, equivalently,

gµν

eff = gµν + 2B2

b2V µV ν − 2B2

b2lµlν . (30)

As one can check, this effective metric is a functional of thebackground metric gµν , the 4-vector velocity field of the inertialobservers Vν , and the spatial configuration (orientation lµ) of theB-field.

Because the concept of gravitational redshift (GRS) is associatedwith the covariant form of the background metric, one needs tofind the inverse of the effective metric gµν

eff given above. With thedefinition of the inverse metric

gµν

eff geffνα = δµ

α, (31)

one obtains the covariant form of the effective metric as

geffµν = gµν − 2B2/b2

(2B2/b2 + 1)VµVν + 2B2/b2

(2B2/b2 + 1)lµlν . (32)

In order to write the covariant time–time effective metric compo-nent explicitly, one can start by figuring out that (a) both the emitterand observer are in inertial frames, so V µ = δ

µ

0 /(g00)1/2, and that(b) the magnetic field is a pure radial field. In this case, one canwrite Bµ = Bµ(r ), where Bµ ≡ B lµ = B lr, which implies that thelµ time, polar and azimuthal vector components become l t = 0, lθ

= 0 and lφ = 0. By using all these assumptions in equation (19),one arrives to the time–time effective metric component (in the Ap-pendix we derive the corresponding grr metric component, which isalso interesting to notice)

gefft t = gtt − 2B2/b2

(2B2/b2 + 1)gtt , (33)

or, similarly,

gefft t =

(1

2B2/b2 + 1

)gtt . (34)

This effective metric corresponds to the result already derived inour previous papers (Mosquera Cuesta & Salim 2003, 2004). Westress, meanwhile, that our previous result was obtained by usingthe approximate Lagrangian

L(F) = −1

4F + µ

4

(F2 + 7

4G2

), (35)

C© 2004 RAS, MNRAS 354, L55–L59

NLED and GRS of highly magnetized neutron stars L59

where

µ = 2α2

45

(h- /mc)3

mc2,

with α = e2/(4 πh- c), G ≡ F µν F∗ µν , F∗ µν ≡ (1/2) ηµναβ F αβ , and Fas defined above. This Lagrangian is built up on the first two terms ofthe infinite series expansion associated with the Heisenberg–Euler(1936) Lagrangian, which proved to be valid for B-field strengthsnear the QED critical field B ∼ 1013.5 G. In this Letter, we overrunthat limit. In fact, from equation (34) it is straightforward to verifythat the ratio geff

00/g00 � 1, and that it decreases all the way down tozero as the B-field attains higher strength values. This means thatthe effective surface redshift grows unbounded as B becomes largerand larger. Both results are displayed in Figs 1 and 2. Fig. 2 alsoconfirms that for canonical NSs the gravitational (GRS) redshift re-mains constant even for fields larger then the Schafroth QED limit.

4 D I S C U S S I O N A N D C O N C L U S I O N

In a very interesting couple of Letters by Ibrahim et al. (2002, 2003)was reported the discovery of cyclotron line resonance featuresin the source SGR 1806-20, said to be a magnetar candidate byKouveliotou et al. (1998). The 5.0-keV feature discovered withRossi X-ray Transient Experiment (XTE) is strong, with an equiv-alent width of ∼500 eV and a narrow width of less than 0.4 eV(Ibrahim et al. 2002, 2003). When these features are interpreted inthe context of accretion models, one arrives to M/R > 0.3 M�km−1, which is either inconsistent with NSs or it requires a low(5–7) × 1011 G magnetic field, which is said not to correspond toany soft gamma-ray repeaters (SGRs) (Ibrahim et al. 2003). In themagnetar scenario, meanwhile, the features are plausibly explainedas being ion–cyclotron resonances in an ultra-strong B-field B sc ∼1015 G (see equation 1). The spectral line is said to be consistentwith a proton–cyclotron fundamental state whose energy and widthare close to model predictions (Ibrahim et al. 2003). According toIbrahim et al. (2003), the confirmation of this findings would allowto estimate the gravitational redshift (the GRS), mass and radius ofthe quoted magnetar SGR 1806-20.

Although the quoted spectral line in Ibrahim et al. (2002, 2003)is found to be a cyclotron resonance produced by protons in thathigh field, we raise the possibility that it could also be due to NLEDeffects in the same super-strong magnetic field of SGR 1806-20,as suggested by equation (34). If this were the case, no conclusiveassertion on the M/R ratio of the compact star glowing in SGR1806-20 could be consistently made, because a NLED effect mightwell be emulating the standard gravitational effect associated withthe pulsar surface redshift.

In summary, because we started with a more general and exactLagrangian than that of Born & Infeld (1934), we can assert thatthe result here derived is inherently generic to any kind of non-linear theory describing the electromagnetic interaction, and thus isuniversal in nature. Consequentially, absorption or emission linesfrom magnetars, if these stellar objects do exist in nature [see PerezMartınez, Perez Rojas & Mosquera Cuesta (2003) for argumentscontending their formation], they cannot be safely used as an un-biased source of information regarding the fundamental parametersof a NS pulsar such as its mass, radius or EOS.

AC K N OW L E D G M E N T S

We are grateful to the anonymous referee for all the suggestionsand criticisms that helped us to improve this manuscript. JMS ac-knowledges Conselho Nacional de Desenvolvimento Cientıfico e

Tecnologico (CNPq/Brazil) for Grant No. 302334/2002-5. HJMCthanks Fundacao de Amparo a Pesquisa do Estado de Rio de Janeiro(FAPERJ/Brazil) for Grant-in-Aid 151.684/2002.

R E F E R E N C E S

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482, 134Delphenich D. H., 2003, preprint (hep-th/0309108)Glendenning N. K., 1997, Compact stars: Nuclear physics, particle physics

and general relativity. Springer-Verlag, New York, p. 390Hadamard J., 1903, Lecons sur la propagation des ondes et les equations de

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ApJ, 574, L51Ibrahim A. I., Swank J. H., Parke W., 2003, ApJ, 584, L17Kouveliotou C. et al., 1998, Nat, 393, 235Miller C., 2002, Nat, 420, 31Mosquera Cuesta H. J., Salim J. M., 2003, preprint (astro-ph/0307513)Mosquera Cuesta H. J., Salim J. M., 2004, ApJ, 608, 925Mosquera Cuesta H. J., de Freitas Pacheco J. S., Salim J. M., 2004, Phys.

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A P P E N D I X

An additional outcome of the above procedure is related to the mod-ifications to the radial component of the background metric. For apure radial B-field, the 4-vector unit lµ can be written as

lµ ≡ δrµ√−grr

= √−grrδrµ, (36)

which renders

lr = δµr√−grr

, (37)

and consequently

lµlµ = −1 ⇒ lr lr grr = −1. (38)

Therefore, the third term in equation (32) reduces to

Bµ Bνgµν = B2lr lr grr . (39)

In this way, one can verify that such a radial–radial effectivemetric component is given by the relation

geffrr = grr − 2B2/b2

(2B2/b2 + 1)grr =

[1 − 2B2/b2

(2B2/b2 + 1)

]grr , (40)

or, equivalently,

geffrr =

(1

1 + 2B2/b2

)grr . (41)

Some astrophysical consequences of this fundamental change inthe radial component of the background metric will be explored ina forthcoming paper.

This paper has been typeset from a TEX/LATEX file prepared by the author.

C© 2004 RAS, MNRAS 354, L55–L59