Using Dominant and Weak Energy Conditions for Build New Classe Of

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arXiv:1705.05744v2 [physics.gen-ph] 15 Feb 2018 Abstract. Rodrigues E. Manuel Holes of Black Classe Regular New build for Energy Conditions Weak and Dominant Using JCAP to submission for Prepared frglrbakhlso hc w ae r smttclyReissner asymptotically are cases two which of holes black regular of erglradta h ekadDmnn nryCniin r sim are Conditions symmetry Energy the Dominant and with Weak els the conditio that energy and the regular of be requirement the through Electrodynamics, ercbakhlssltoso n4 in of solutions holes black metric Silva a c b audd eFıia PF nvriaeFdrld Pará, 66075 do Federal Universidade F´ısica, PPGF, de Faculdade apsUiestai eAattb,CP64000 Abaetetu 68440-000, CEP Abaetetuba, Federal Universitário de Universidade Campus Tecnologia, e Ciências Exatas de Faculdade E-mail: Tucuru´ı, Computa¸cão,Tucuru´ı, Pará, 68464-000, Federal da Brazil Universidade Engenharia de Faculdade a [email protected] esuytegnrlzto ftems ucino lse fregu of classes of function mass the of generalization the study We T 0 0 = , [email protected] T a,b 1 1 ftetno nrymmnu,w osrc e eea cla general new a construct we energy-momentum, tensor the of D dad .B Junior B. L. Ednaldo oigfo h opigo eea eaiiywt Non-linear with Relativity General of coupling the from coming , [email protected] s moigta h ouinmust solution the that Imposing ns. a Pará, Brazil ba, oPra apsUniversitário Pará, de Campus do a,c 10 Belém, Pará,-110, Brazil oParádo -Nordstrom. laeul aifid o mod- for satisfied, ultaneously acsV eS. de V. Marcos a peial sym- spherically lar ss Contents

1 Introduction 1

2 The energy conditions and the regular black hole solutions 2

3 The new class of regular black hole solutions 3

4 Conclusion 8

1 Introduction

Since the beginning of General Relativity solutions of black holes has gained a lot of attention. A black hole has the following basic characteristics [1]: It is a solution of Einstein’s equations and there is a surface that separates the causal structure of space-time, called the event horizon. These solutions initially always presented a singularity covered by this horizon, thus representing a great difficulty of physical interpretation. In 1935, Einstein and Rosen suggested a small modification [2], changing the signal of the contribution of the electromagnetic field, to formulate a geometry that could escape the singularity within the event horizon, thus creating the wormholes. This persistence in obtaining a solution in which geodesics were not to be interrupted by a singularity continued until a suggestion of Bardeen [3], known as Bardeen’s regular black hole. This new structure was always regular in all µν space-time, not presenting singularity in the curvature scalars and their contractions, as R, Rµν R µναβ and = RµναβR , besides having an event horizon. KMany solutions of regular black holes have arisen since then, which always present some characteristics in common, as it is the case of Bardeen. Most solutions have spherical symmetry [4]-[20] and there are some with axial symmetry [21]-[28]. We can also have regular solutions in modified gravity theory [29]-[31]. There are some key features in getting new solutions for regular black holes. One is that the metric and the curvature scalars do not have divergence in all space-time. This in spherical symmetry is sufficient to ensure regularity [1]. Another feature that arises when trying to avoid the appearance of a singularity, the energy distribution is similar to that of a de Sitter-like solution, with pr = pt = ρ (Sakarov criterion [32]) for p being the radial or tangential pressure, and ρ the energy density, clearly− violating the Strong Energy Condition (SEC), ρ +3p 0. Thus, a good possibility to obtain new solutions appears in the coupling of the gravitation with≥ the call Non-Linear Electrodynamics (NED) [33], where this equation of state is common. The Bardeen’s regular black hole can be interpreted as a solution coming from an NED [34]. There are also regular solutions that behave asymptotically as Reissner-Nordstrom or others that do not have this characteristic, as is the case of Bardeen. Another important characteristic is that coming from the so-called energy conditions. For solutions with spherical symmetry, it can be shown that every regular solution of black hole violates in some region the SEC [35], but the Dominant Energy Condition (DEC) and Weak Energy Condition (WEC) can always be met, as is the case of Bardeen which always satisfies WEC. We have new possibilities for attempting to construct solutions of regular black holes with spherical symmetry that always satisfy a WEC, this has been started in the work to Balart et al [36]. The idea is that one can generalize quite to obtain regular solutions through the requirement to always satisfy WEC. Following this direction started in [36], we can show that there are still several possibilities for new solutions of regular black holes, which can be asymptotically Reissner-Nordstrom or not. From the requirement to satisfy WEC, regularity and to be asymptotically Reissner-Nordstrom, one can construct metrics much more general than those already presented in the literature. This is the central idea in this work. The structure of the work is divided as follows. In section 2 we present the structure of the Einstein’s equations related to the models of regular solutions and the WEC, DEC and SEC. In

–1– section 3 we obtain a new class of solutions of regular black holes through the exigences of the previous section. We present our conclusions and perspectives on section 4.

2 The energy conditions and the regular black hole solutions

We will brieﬂy present how the possible models of regular black holes are related to agreement with WEC and DEC. Let us ﬁrst consider a metric that is spherically symmetrical and static

ds2 = ea(r)dt2 eb(r)dr2 r2 dθ2 + sin2 θdφ2 (2.1) − − where a(r) and b(r) are arbitrary functions of the radial coordinate. The action of the theory is given by the coupling between gravitation and NED

S = d4x√ g [R + 16πG ] , (2.2) − LNED Z where R is the curvature scalar and NED NED(F ) is an arbitrary function of the scalar F = µν L ≡ L (1/4)Fµν F , with Fµν = ∂µAν ∂ν Aµ being the Maxwell tensor. This theory falls to Maxwell for the speciﬁc choice = F . One− known form of formulation for regular black holes is to introduce LNED a mass function M(r) through component g11 of the metric by

2M(r) e−b(r) =1 . (2.3) − r Making the functional variation of the action (2.2) in relation to the metric we have the following Einstein’s equations

1 dM 1 d2M T 0 = T 1 = ,T 2 = T 3 = , (2.4) 0 1 4πGr2 dr 2 3 8πGr dr2 where ∂ (F ) T = g LNED F βF . (2.5) µν µν LNED − ∂F µ νβ

0 1 We have used a(r)= b(r) which arises from the spherical symmetry and the equation G0 G1 =0 − 0 1 0 1 − 2 (symmetry of the energy-momentum tensor T0 = T1 ). We can identify T0 = ρ and T1 = pr,T2 = 3 − T3 = pt, with ρ being the energy density and pr and pt are the radial and tangential pressure respectively.− Now we can establish the energy conditions [2]

SEC(r)= ρ + p +2p 0 , (2.6) r t ≥ W EC (r)= ρ + p 0 , (2.7) 1,2 r,t ≥ W EC (r)= ρ 0, (2.8) 3 ≥ DEC (r)= ρ p 0 , (2.9) 2,3 − r,t ≥ where, in view of the identities DEC1(r) W EC3(r) and NEC1,2(r) W EC1,2(r), this conditions are not written. Performing coordinate transformation≡ r =1/x, from (≡2.4) we have

x4 dM x4 dM d2M ρ(x)= = p ,p = 2 + x . (2.10) −4πG dx − r t −8πG dx dx2

–2– Then we see directly W EC1(x) = 0 which is always satisﬁed. We also have DEC2(x)=2W EC3(x). So let’s focus on the energy conditions

dM d2M SEC(x)= x4 2 + x 0 , (2.11) dx dx2 ≤ dM d2M W EC (x)= x4 4 + x 0 , (2.12) 2 dx dx2 ≤ dM W EC (x)= x4 0, (2.13) 3 dx ≤ d2M DEC (x)= x5 0 . (2.14) 3 dx2 ≥ Now we will focus only on solutions that always satisfy the WEC and DEC. If we require the WEC and the regularity of the metric we must have [36] dM x + , x4 c , (2.15) → ∞ − dx → 1 with c1 being a positive constant. We still have if the solution behaves asymptotically as Reissner- Nordstrom [36] dM x 0, =0 . (2.16) → − dx 6 In the next section we’ll show how to build new classes of regular black hole solutions.

3 The new class of regular black hole solutions

The work of Balart et al suggests a mass function M(x), which behaves asymptotically as Reissner- Nordstrom and is general as

dM c1 = α 4/α , (3.1) − dx (1 + c2x ) with c1 and c2 being positive constants, and α positive integer. Actually α need not be positive integer if we do not require the solution to be asymptotically Reissner-Nordstrom. We can verify that the W EC1,2,3(x) and the DEC1,2,3(x) are always satisﬁed for the model (3.1) and SEC(x) is violated to some region, as expected [35]. We can now show that there are several other possibilities of generalization of this mass function. Taking the requirements (2.15), (2.16) and the WEC We can construct the following mass function

c4 c6 dM c1(c8 + c3x + c5x ) = α (4+c )/α , (3.2) − dx (1 + c2x ) 7 that integrating from inﬁnity to x, for Re[α] 0, Re[c c ] < 3 and Re[c c ] < 3, results in ≥ 4 − 7 6 − 7 −α c −α − c x c x 4 x M(x)= c c (4+c7)/αx−3−c7 8 F d , d , d , + 3 F d , d , d , 1 2 3+ c 2 1 1 2 3 − c 3 c + c 2 1 2 4 6 − c 7 2 − 4 7 2 c xc6 n x−α h i h i + 5 F d , d , d , , (3.3) 3 c + c 2 1 2 5 7 − c − 6 7 2 3+ c 4+ c 3+ c +oα 3 c + c 3 c + c + α d = 7 , d = 7 , d 7 , d = − 4,6 7 , d = − 4,6 7 . (3.4) 1 α 2 α 3 α 4,5 α 6,7 α with 2F1[i,k,l,z] being the Gauss’s Hypergeometric function. The mass function (3.3) presents the following characteristics: does not introduces divergences in the metric; may compose a metric that is asymptotically Reissner-Nordstrom; and generates solutions that satisfy the energy conditions NEC,

–3– DEC e WEC. The mass function also can presents a important restriction in the asymptotic limit. We will explain this better. Astrophysical black holes are usually modeled through axially symmetrical and charged solutions, like Kerr-Newman. These black holes present a physical restriction q2 << m2 [37], where m = lim M(x) is the ADM mass for asymptotically ﬂat solutions. So that, we may x→0 consider the mass function (3.3) as a toy model for an astrophysical solution with spherical symmetry −17 (without rotation), where the ADM mass must have the following restrictions 10 M⊙

c7+α c1r c3 c2 NED = c +4+α (c4 + c7 + 2) c4 2 + 7 c4 α L κ2 (c + rα) α r r − − 2 c2 c5 c5 + α (c7 c6 + 2) c + c7 +2 (c6 + 2) c 2 , (3.5) r − r 6 − r 6 − ! c7+4+α 2 2 1 c7+4+α α α κ q (c2 + r ) = r , (3.6) F c3 c2 c5 c2 c2c7 L c c α (c c )+ c +4 + c α (c c )+ c +4 α +4 1 r 4 r 4 − 7 4 r 6 r 6 − 7 6 − r c7+2+α 10 c1r c3 c2 F = c +4+α (c4 c7) + c4 +4 + 7 c4 α κ2q (c + rα) α r − r 2 c5 c2 c2c7 + c (c6 c7) α + c6 +4 α +4 , (3.7) r 6 − r − r ! with κ2 =8πG. Taking the mass function (3.3) the WEC, DEC and SEC on (2.12)-(2.14) are

W EC (x)= c x4(1 + c xα)−d2 (1 + c xc4 + c xx6 ) (3.8) 2 − 1 2 3 5 W EC (x)= c x−c7−α 4 c c xα + c xc4 (4 + c + c xα(c c )) 3 − 1 − 2 7 3 4 2 4 − 7 h +c xc6 (4 + c + c xα(c c )) , (3.9) 5 6 2 6 − 7 i SEC(x)= c x4(1 + c xα)−(d2+1) 2 c xα(2 + c )+ c xc4 (2 + c + c xα(c 2 c )) + − 1 2 − 2 7 3 4 2 4 − − 7 h +c xc6 (2 + c + c xα(c 2 c )) , (3.10) 5 6 2 6 − − 7 i DEC (x)= c x4(1 + c xα)−(d2+1) c xα(4 + c ) c xc4 (c + c xα(c 4 c )) + 3 1 2 2 7 − 3 4 2 4 − − 7 h c xc6 (c + c xα(c 4 c )) . (3.11) − 5 6 2 6 − − 7 i We see that the SEC can never be satisﬁed in all space-time, always in some region it will be violated, being consistent with the result of [35].

1 To find LNED and LF we solve the Einstein’s equations in terms of these functions. To determine the electric field 10 µν F , we solve the modified Maxwell equations ∇µ (F LF ) = 0, with the electric charge q as an integration constant.

–4– If we look for solutions that only satisfy the WEC and DEC we will ﬁnd solutions that can be singular, since the general mass function (3.3) is not restricted to only regular solutions. Therefore, we must ﬁrst ensure regularity in all space-time. For this, we will calculate the curvature scalar and Kretschmann for the following metric

− ds2 = (1 2xM(x)) dt2 x−4 (1 2xM(x)) 1 dx2 x−2 dθ2 + sin2 θdφ2 , (3.12) − − − − which results in d2M R(x)= 2x5 , (3.13) − dx2 d2M 2 = R Rijkl =4x10 + K ijkl dx2 dM d2M d2M dM 2 dM +32x9 + 16x8M + 80x8 + 96x7M + 48x6M 2. (3.14) dx dx2 dx2 dx dx If there is a divergence in the curvature scalar then there will also be a divergence in the Kretschmann scalar. For spherically symmetrical solutions it is then convenient to ﬁrst check the regularity of the curvature scalar. Thus, by taking the mass function (3.3), the curvature scalar becomes

−d2 −α −d2 2c1c2 −c7 x α c4 α R(x)= α x 1+ c2(4 + c7)x + c3x (c4 + c2(c4 (4 + c7))x ) (1 + c2x ) c2 − − h +c xc6 (c + c (c (4 + c ))xα) . (3.15) 5 6 2 6 − 7 i 4 We can see that in the asymptotic limit x 0 we have R(x) x 0, for c4,6 0. Now for the solution to be regular in origin x + , we have→ to check the power∼ of→x that govern.≥ Then in this → ∞ limit we have the following powers that govern in the numerator of the curvature scalar α + c4 c7 and α + c c . Considering c c , then the highest power is α + c c , which subtracting from− 6 − 7 4 ≥ 6 4 − 7 the denominator, which is α, become c4 c7. For these considerations the dominant power of the − − curvature scalar, for this limit, becomes R(x) xc4 c7 . Then the curvature scalar goes to zero or to ∼ a constant for c4 c7. So let’s do c7 = c4 + k, with k 0, this ensures that the solution is regular at x + . ≤ ≥ → Let’s∞ now check the energy conditions for c c and c = c + k, with k 0 4 ≥ 6 7 4 ≥ − xα d2 W EC (x)= c c−d2 x−c4−k 1+ (1 + c xc4 + c xc6 ), (3.16) 2 − 1 2 c 3 5 2 − c cd2 xα d2 W EC (x)= 1 2 x−c4−k 1+ [4 c (c + k)xα + c xc4 (4 + c c kxα)+ 3 −(1 + c xα) c − 2 4 3 4 − 2 2 2 c6 α +c5x (4 + c6 + c2(c6 (c4 + k))x )] (3.17) − − c c−d2 xα d2 DEC (x)= 1 2 x−c4−k 1+ [c (4 + c + k)xα c xc4 (c c (4 + k)xα)+ 3 (1 + c xα) c 2 4 − 3 4 − 2 2 2 c xc6 (c + c (c (4 + c + k))xα)]. (3.18) − 5 6 2 6 − 4 We can see that the W EC (x) is always satisfied. However, in the W EC (x), the term [ c c kxα+c4 ] 2 3 − 2 3 can never be compensated, when k> 0, for it be satisfied. Then we take k = 0, which implies c7 = c4. α Now is the term [ c2c4x ] which must be compensated, which is solved for the choice c7 = c4 = α. So the WEC and− DEC can be fulfilled to (4 + α)c αc and c α, but previously we imposed 3 ≥ 2 6 ≥ c4 = α c6, which implies c6 = c4 = c7 = α, what is not interesting. Then we have the only possibility more appreciable≥ in that c = 0 the WEC and DEC are satisfied for [α/(4+α)]c c [(4+α)/α]c . 5 3 ≤ 2 ≤ 3 Then we take c2 = [(4 + α)/α]c3, and the WEC and DEC are always fulfilled. The mass function of

–5– this solution is given by

−(4+α)/α −α c3(4 + α) c3 α 3 4+ α 3+ α αx M(x)= c1 x 2F1 , , , + α 3 α α α −c3(4 + α) n 1 3+ α 4+ α 3 αx−α + 2F1 , , 2+ , . (3.19) 3+ α α α α −c3(4 + α) o Let’s check the properties for the particular case where α = 1. The curvature scalar, the Kretschmann scalar and the mass function are given by

c1(4+5c3x) M(x)= 4 , (3.20) 75c3(1+5c3x) 5 8c1c3x (6+5c3x) R(x)= 6 , (3.21) − (1+5c3x) 32c2x6 8 120c x + 2325c2x2 + 22500c4x4 + 37500c5x5 + 15625c6x6 (x)= 1 − 3 3 3 3 3 . (3.22) K 1875c2 (1+5c x)12 3 3 We then have the following limits 4c lim M(x), R(x), (x) = 1 , 0, 0 , (3.23) x→0{ K } 75c 3 8c 32c2 lim M(x), R(x), (x) = 0, 1 , 1 . (3.24) x→+∞{ K } −3125c4 29296875c8 3 3 We see here that this is a regular solution in all space-time and asymptotically ﬂat. We can expand the mass function on Taylor series to x << 1

8c1 2 3 2 2 3 1 2xM(x) = 1 x +2c1x + O(x )=1 2mx + q x + O(x ) , (3.25) − ∼ − 75c3 − 2 2 with m being the ADM mass and q the electric charge. When c1 = q /2 and c3 = [2q /(75m)] the solution becomes asymptotically Reissner-Nordstrom. We can verify the same behavior for all cases of integer α. We still have the possibility of fractional α. The functions , and F 10, in terms of the radial coordinate, are written as LNED LF 10125m4q2 4q4 + 180mq2r 1125m2r2 NED(r) = − , (3.26) 2 2 6 L κ (15mr +2q ) 6 κ2 15mr +2q2 F (r) = 6 6 , (3.27) L 22781250 m r 22781250qm6r4 F 10(r) = . (3.28) κ2 (15mr +2q2)6 The Lagrangian and the electric field are regular in all space-time. In the appendix B we will use this solution to show a way to construct regular solutions in the f(R) theory. Now we can turn our attention to another direction to construct solutions that satisfy WEC and DEC. For the condition of regular solution in which c7 = c4 and c4 c6 in (3.16)-(3.18), with α ≥ k = 0, we can see that the term [ c2c4x ] can be compensated through c6 = α. This generates a 2α − new term, [c2c4c5x ], which must be compensated for WEC and DEC to be met. So we can choose c4 = c7 =2α and thus the WEC and DEC are satisfied to [α/(2(2+α))]c5 c2 [(4+α)/(2α)]c5 and [α/(2(2 + α))]c c c [(4 + α)/(2α)]c c . In short, when c = c =2α,≤ c =≤α,[α/(2(2 + α))]c 2 5 ≤ 3 ≤ 2 5 4 7 6 5 ≤ c2 [(4 + α)/(2α)]c5 and [α/(2(2 + α))]c2c5 c3 [(4 + α)/(2α)]c2c5, the solution satisfies WEC and≤ DEC. The mass function becomes ≤ ≤ −α − 1 3 4 3 x M(x)= c c 2(2+α)/αx−3−2α F 2+ , 2+ , 3+ , + 1 2 3+2α 2 1 α α α − c 2 n −α α −α c3 2α 4 3 3+ α x c5x 4 3+ α 3 x + x 2F1 2+ , , , + 2F1 2+ , , 2+ , . (3.29) 3 α α α − c2 3+ α α α α − c2 o

–6– 2 Let’s look at a particular case for α =1,c2 =2c5 and c3 =5c5. In this case the mass function, curvature scalar and Kretschmann are 2 2 c1(4+10c5x + 25c5x ) M(x)= 5 , (3.30) 30c5(1+2c5x) 5 2 2 2c1c5x (11 + 40c5x ) R(x)= 7 , (3.31) − (1+2c5x) 2 6 8c1x 2 2 3 3 (x)= 2 14 (8 16c5x + 492c2x 314c5x + K 75c5(1+2c5x) − − +6313c4x4 2170c5x5 + 22650c6x6 + 10000c8x8). (3.32) 5 − 5 5 5 We then have the following limits 2c 5c 25c2 lim M(x), R(x), (x) = 1 , 0, 0 , lim M(x), R(x), (x) = 0, 1 , 1 .(3.33) x→0{ K } 15c x→+∞{ K } −8c4 384c8 5 5 3 We see here again that this is a regular solution in all space-time and asymptotically ﬂat. We can expand the mass function on Taylor series to x << 1

4c1 2 3 2 2 3 1 2xM(x) = 1 x +2c1x + O(x )=1 2mx + q x + O(x ) . (3.34) − ∼ − 15c5 −

2 2 When c1 = q /2 and c5 = [q /(15m)] the solution is asymptotically Reissner-Nordstrom. We can verify the same behavior for all cases of integer α. We still have the possibility of fractional α. The functions , and F 10, for the radial coordinate, becomes LNED LF 253125m4q2 4q6 42mq4r + 225m2q2r2 1350m3r3 NED(r) = − − , (3.35) 2 7 2 L 2 (2q + 15mr) κ 7 2 2q2 + 15mr κ2 F (r) = 5 5 2 2 2 2 , (3.36) L 759375m r (28q + 15mq r + 900m r ) 759375m5qr3 28q2 + 15mq2r + 900m2r2 F 10(r) = . (3.37) 2 7 2 2 (2 q + 15mr) κ The Lagrangian and the electric ﬁeld are well behaved in all space-time. Turning our attention to the WEC and DEC (3.16)-(3.18), we see that another possibility arises when c7 = c4 = α and c6 = α. In this case the WEC and DEC are fulﬁlled to [α/(4 + α)]c3 c2 [(4 + α)/α]c , 4 α and 4 −2c c α. The mass function becomes ≤ ≤ 3 ≥ ≥ 2 5 −α −α − c x 3 4+ α 3 x M(x)= c c (4+α)/αx−3−α 5 F 2+ , , 3+ , + 1 2 3+2α 2 1 α α α − c 2 n −α −α c3 α 3 4+ α 3+ α x 1 3+ α 4+ α 3 x + x 2F1 , , , + 2F1 , , 2+ , . (3.38) 3 α α α − c2 3+ α α α α − c2 o Taking the example c2 = [(4 + α)/α]c3, α = 2 we have

4 2 −1 c 45c x (27c c 2)+3x (225c c 14)+ 72c 5√3(2 27c3c5)cot √3c3x M(x)= 1 3 3 5 − 3 5 − 5 + − , 2 2 72 x (3c3x + 1) √c3 ! (3.39) and the limits

c1 3 lim M(x), R(x), (x) = sing[c3] sign[c5] , 0, 0 , (3.40) x→0{ K } c3 ∞ 3 8c 32c2 lim M(x), R(x), (x) = 0, 1 , 1 . (3.41) x→+∞{ K } −27c2 2187c4 3 3

–7– We see then that the solution is asymptotically non-flat, but regular in all space-time. There are still solutions with fractional α which are asymptotically flat. The functions , and F 10, for the coordinate r, are given by LNED LF 2 4 2 2c1 3c3 + r (9c3c5 1)+4c3r NED = − , (3.42) 2 2 4 L κ (3c3 + r ) 2 2 2 4 κ q 3c3 + r F = 8 2 , (3.43) L 2c1r ( 6mc 3c5 + c5r + 2) 6 − 2 2c1r 6c3c5 + c5r +2 F 10 = − . (3.44) 2 2 4 κ q (3c3 + r ) The Lagrangian and the electric field are regular in all space-time.

4 Conclusion

We assumed that the derivative of the mass function derived from the coupling of General Relativity 0 1 with Non-linear Electrodynamics, with symmetry T 0 = T 1, it should be finite for the asymptotic limit of radial infinity and be proportional to a positive power (negative for x = 1/r) in the limit of the origin of the radial coordinate. This led us to construct a more general mass function than that of the Balart case, which for some limit of the parameters may fall in this particular case. Imposing regularity in all space-time we arrive at a relation between the constants of the mass function, which imposing the WEC and DEC, results in three solutions that have two free constants, which were related to the mass ADM and the electric charge, for asymptotically flat cases. Two classes of solutions are asymptotically flat and asymptotically Reissner-Nordstrom, in the case c6 = α the solution is asymptotically non-flat only for α integer. − Solutions logically violate SEC in some region. But we believe that there are still other classes of regular solutions that simultaneously satisfy WEC and DEC, coming from the mass function (3.3).

Acknowledgements: M. E. R. thanks Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnológico - CNPq, Brazil, Edital MCTI/CNPQ/Universal 14/2014 for partial financial support.

Appendix A: More general mass function

We may consider a more general model of mass function where

N ni dM(x) c8 +Σi=1kix = c1 , (4.1) α (4+c7)/α − dx (1 + c2x ) that integrating results in

α − c 3+ c 4+ c 3+ c + α x M(x)= c c (4+c7)/αx−3−c7 8 F 7 , 7 , 7 , + 1 2 3+ c 2 1 α α α − c ( 7 2 k xni 4+ c 3+ c n 3+ c + α n xα +ΣN i F 7 , 7 − i , 7 − i , . (4.2) i=1 3+ c n 2 1 α α α − c 7 − i 2 ) The constants in the mass function must be adjusted to satisfy the energy conditions and the restriction 3M⊙

–8– Appendix B: Generalization to f(R) Gravity

In this appendix, we will show a way to ﬁnd regular solutions in f(R) theory coupled to the Non-linear Electrodynamics. The action that describes this theory is given by:

S = d4x√ g [f(R)+16πG ] . (4.3) f(R) − LNED Z The equations of motion may be written as 1 1 R g R = f −1 8πGT + g (f Rf ) (g ) f 8πG (eff), (4.4) µν − 2 µν R µν 2 µν − R − µν − ∇µ∇ν R ≡ Tµν where fR df(R)/dR. To construct the solutions we consider the line element (2.1) with the ansatz (2.3). From≡ the components (4.4) and assume a = b, we can write [30] −

fR(r)= c1r + c0, (4.5) that recover General Relativity for c0 = 1 and c1 = 0. The function f(R) is obtained, in terms of the radial coordinate, through dR(r) f(r)= f (r) dr, (4.6) R dr Z where the curvature scalar in given in terms of the mass function as

2rM ′′ +4M ′ R(r) = . (4.7) − r2 From (4.4) and the modified Maxwell equations, the functions , and F 10 are LNED LF 1 (r)= r2f(r)+4c M ′ +2c r , (4.8) LNED −16πGr2 0 1 2 8q πG − (r)= [(c r + c0) rM ′′ (2c + c r) M ′ 3c M + c r] 1 , (4.9) LF − r2 1 − 0 1 − 1 1 1 F 10(r)= 3c M + (2c + c r) M ′ r [c + (c + c r) M ′′] . (4.10) 8qπG { 1 0 1 − 1 0 1 } In this sense, all the relevant expression are written in terms of the mass function. So that, for which different mass function we have a different f(R) theory and a different Non-linear Electrody- namics. Is also important verify if the solutions satisfy the energy conditions. However, from the theorem proved in [31], the energy conditions are identical from the General Relativity. As an example, we will use the mass function (3.20) that we construct in the section 3 for α =1 2 2 with c1 = q /2 and c3 = [2q /(75m)]. Due the symmetry a = b, the curvature scalar is given by − 162000m4 45mq4r + q6 R(r)= . (4.11) 2 6 − (15mr +2q ) That is the same from General Relativity, so that does not present divergences. From the equation (4.6), we obtain

3 4 2 2 2 2 4 2700m q 60c0m 45mr + q + c1 3375m r + 180mq r +4q f(R)= . (4.12) 2 6 − (15mr +2q ) In ﬁg. 1 we show a parametric plot for f(R) R. In this we can see the non-linear behavior of this theory. ×

–9– 0

−20

−40

f(R) −60 2 m

−80

−100

−120 −70 −60 −50 −40 −30 −20 −10 0 m2R

−1 Figure 1. Parametric plot for f(R) as a function of R, with q = 0.1m, c1 = 2m e c0 = 1.

The functions , and F 10 becomes LNED LF

10 r 6 2 3 4 5 2 F (r)= 22781250c0m q r c1 r m (11390625mr+9112500q (r 3m) 2 2 6 − − κ q (15mr +2q ) +3037500m4q4r4 + 540000m3q6r3 +54000m2q8r2 + 2880mq10r + 64q12 , (4.13) 1 4 2 2 2 2 4 NED(r)= 10125c0m q r 1125m r 180mq r 4q L − 2 2 6 − − κ r (15mr +2q ) +c 64q12 + 2880mq10r 5400m2q8r(m 10r) 27000m3q6r2(9m 20r) 1 − − − − 1518750 m4q4r3(3m 2r) + 9112500m5q2r5 + 11390625m6r6 , (4.14) − − 6 κ2q2 15mr +2q2 (r)= c 64q12 + 2880mq10r + 54000q6r2m2(10mr + q2) LF − r3 1 +3037500m4q4r4 (11390625mr + 9112500q2)(3m r)m5r4 22781250c m6q2r3 . (4.15) − − − 0 If c0 = 1 and c1 = 0 we recover the result of General Relativity. The method present in this section can be used to construct much more regular solutions. Some examples are described in [30, 31].

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