Black Holes and Solitons in an Extended Proca Theory JHEP05(2017)114 5 10 19 = 0 1 ]

Home , Proca action

JHEP05(2017)114 Springer May 5, 2017 April 4, 2017 May 19, 2017 : : : Accepted Received Published b 10.1007/JHEP05(2017)114 doi: Published for SISSA by , and Mokhtar Hassaine a [email protected] , . 3 Christos Charmousis a 1703.07676 The Authors. c Black Holes, Classical Theories of Gravity

We study a massive vector tensor theory that acquires mass via a standard , [email protected] Instituto de y Universidad Matemática F´ısica, deCasilla Talca, 747, Talca, Chile E-mail: [email protected] Laboratoire de Physique Théorique,CNRS, UniversitéParis-Sud, UniversitéParis-Saclay, Orsay, 91405 France b a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: flat black holes. For aand particular non coupling trivial we geometry findrole everywhere. particle of like In solitons an all that adS effectiveconstant. have solutions cosmological a the regular constant Proca distinctly mass different term from plays the the bare cosmological Abstract: Proca term but alsoWe via find a static higher solutions order thathigher term are order containing asymptotically term an flat, adS explicit regularizesasymptotically or coupling the regular Lifshitz to effect for black curvature. arbitrary of holes. couplings. the Since This Proca the is mass true in term, particular generically for solutions asymptotically are Eugeny Babichev, Black holes and solitons in an extended Proca theory JHEP05(2017)114 5 10 19 = 0 1 ]. If instead of a scalar C 4 14 6 – 2 = 0 = 1, 1 κ C 12 Λ β = 2 β 2 = 0 case with – 1 – µ 9 κ β = 0 κ ], their stability, their cosmological implications, the is that of the Maxwell field coupled with GR, Einstein- 1 3 1 = Λ = 0 or µ in 4 dimensions. As a result EM theory cannot drive slow roll 2 = 0 and arbitrary 1 = 1, for C κ 1 17 Vector tensor theories have been approached in many different ways and since a long The list references are indicative and not exhaustive-the interested readerBy should conformal consult we mean within that the the energy momentum tensor of the electromagnetic field is traceless in 4 4.2 Lifshitz black holes: topological 3.1 The spherically3.2 symmetric adS black Planar holes horizon and black solitons holes 4.1 The case 1 2 inflation. Furthermore, it cannot seed observed astrophysical magnetic fields (of thereferences size provided here of for akey more references. complete bibliography. We are however including here some ofdimensions. the older time. A convenient starting point Maxwell theory (EM). Thiswith theory gravity is describing massless, theand has standard two has model propagating U(1) photon’s vectorduality Abelian degrees interactions and of gauge is freedom symmetry. conformal (for the Additionally, photon) the theory has electro-magnetic field one considers a vectorIn degree of this freedom, paper this provides weis another will instructive way study to to modify black overview gravity. facets hole vector since and tensor the soliton theories advent solutions as of they of General Relativity have vector (GR). appeared tensor in theories. many It different a lot of attention recently anddifferent an important facets effort of is made theseexistence by and the theories nature community to of [ understand compacttensor objects etc. theories Most as of they thelot attention are of has the been characteristics simplest given one to of scalar finds modified in theories, more and complicated additionally, theories present [ a 1 Introduction The unknown natureRelativity of at very dark large distance energy scales. and As a dark result, modified matter gravity theories question have attracted the validity of General 5 Conclusions A Integration of the metric function for the case 4 Solutions for Contents 1 Introduction 2 Theory and field3 equations The general solution for 1 = 4 JHEP05(2017)114 φ µ ∇ ]. This ]. This ]. 23 10 – 32 , 21 31 ] and references 30 ]. Furthermore, Proca 29 [ 3 ]). 28 , 27 ]). Moreover, Horndeski showed that there 16 , – 2 – 15 ]. One of course loses U(1) gauge symmetry and 35 – ]. 33 6 ] for which he demonstrated that there is a unique additional ] for such a theory. The additional term, akin to a “Paul” term 18 , 18 , ]. One then deals with an extension of Horndeski theory or vector- 17 , 17 32 14 , ] (for black hole solutions see [ – 31 26 one can start with a shift symmetric Horndeski theory and replace [ 11 – 4 ] theory, introduces a certain strain for far away asymptotics in a similar µ 24 A 20 ]. In such theories gauge symmetry is absent and the vector has generically 4 ]); one has to go beyond EM theory in order to pursue such and other effects, , 9 5 – 19 7 Another course of action is that of adding a mass term to the EM theory; thus losing Another interesting approach has been to try to keep U(1) gauge symmetry while Introducing more complicated kinetic terms one gets vector tensor theories studied If the Proca mass is associated to dark energyCertain and, intrinsic therefore vector to interactions the are size of not the obtainable universe, from it this is simple still substitution within [ 3 4 introduces in some formstringent or constraints another if a mass our term field for is the still vector. thethe Adding photon allowed a photon (coupled mass mass constraint to term bounds. the has standard model (!)). and time scaling.we We will will see see tothe explicitly remedy same this the manner arising effects aswithin). of here Horndeski In Proca considered in part scalar mass theby tensor one later the theories vector can sections. (see consider [ tensor higher But galileons order as as terms constructed in in [ U(1) gauge invariance and dealingtudinal with an degree Einstein-Proca of theory freedom with (toto an the be additional two longi- a vectorial photon onestheories with of introduce mass EM). unusual then If asymptotics, the there similarwell Proca are to known field that a stringent is such Horndeski constraints theories assumed Maxwell are term, more it akin is to in Lifshitz fact spacetimes with anisotropic space way to a Proca massimplications term, of which Horndeski-Maxwell we theory willtheory turn have was to been shown in studied to a be morevery moment. a recently early Stability direct [ on and consequence [ of cosmological a Kaluza-Klein cascade of Lovelock theory EM theory in the presencea of curvature. theory describing In photons other coupled wordsvariation to one of gravity, can the albeit still fine nonminimally consider structure (for constant this seeis constraints theory [ on a as the Birkhoff theorem [ in Fab 4 [ extension of EM [ term which can beflat added spacetime, to U(1) gauge EM symmetry enjoying andelectromagnetic second the duality order following field and properties: equations conformal (atthe the coupling Maxwell same Maxwell equations is time theory). absent in the in This this term extension, involves the in double contrast dual to curvature tensor which modifies problem Jacobson and collaborators considered spontaneousby breaking fixing of the Lorentz symmetry normtheory of is the a vector nice andinvariance. prototype thus theory introducing where Einstein-Aether one theory can [ study inasking detail if the the EM breaking action of can be Lorentz generalised. Horndeski in a series of nice papers studied an the review [ for example, related to the CMB [ early on [ polarisations, of which, the time component of the vector is always a ghost. To remedy this the micro-Gauss) as the magnetic field is strongly diluted during inflation (see for example JHEP05(2017)114 ]. ]) = 0. 44 42 (2.1) (1.1) , , µ 41 43 µν , 41 βZ − ]. There it was . 41 2 ν A A µν µ g A 1 2 µν − ). A set of such solutions while one gets asymptotic ν βG β 1.1 A + µ we solve for arbitrary coupling 4 was studied [ 2 A / 4 A ]. We will see that the additional 2 2 2 µ 2 = 1 40 µ ]) and also, less surprisingly, Lifshitz – β − 45 − ], that there existed a simple procedure 2 38 F 43 ) reads, αβ 1 4 F 1.1 − – 3 – αβ F 2Λ µν − we solve for a particular coupling in all generality g 1 4 R 3 = 0 while it is EM with a higher order term if . Using previous work on black holes for scalar tensor = 0 and generic coupling β − β x µ 4 σ ν F g d ]), it was realized [ µσ ]). To construct these, as we will see, it is important to allow − 50 F √ – 52 , 1 2 46 Z 51 − ] = µν g g, A = 0 the case of a particular coupling [ + Λ S µ ] (see also [ µν 36 G := In the next section we set up the theory in question and then show how the field When one is seeking black hole solutions with additional fields one is confronted with µν E 2 Theory and field equations The variation with respect to the metric of ( respect to two variables.and find In black section hole butbut also by soliton setting solutions. tosolutions zero In are an section found. integration We constant. round Asymptotically up flat, our results adS and but perspectives also in Lifshitz the final section. effective negative cosmological constantblack (see holes also (see [ alsofor [ both Proca and galileon terms. equations boil down to two coupled field equations, one algebraic and one ODE with we will discard someothers. which are We non willasymptotically relevant most mathematically flat importantly and solutions classify will for theadS discover possible solutions numerous solutions in the andterm presence show we of that will a one find Proca gets everywhere term. regular Furthermore, solitons allowing with for the the Proca Proca mass term playing the role of an for generic values oftheories the coupling [ to relate certain scalarwas tensor obtained to for vector certain tensoraction couplings. solutions for of static We ( and will spherical undertake symmetry. here We will a find full previous analysis solutions of in the our above analysis, This action is Einstein-ProcaRecently for there was some activityFor the finding case hairy solutions forpostulated the that only above for theory this [ We particular will coupling see there explicitly existed that asymptotically this flat is solutions. not the case. Asymptotically flat black holes can be found curvature-vector term modifies this conclusion providingfor hairy black asymptotically holes flat (see also spacetimes. [ the Putting following these action considerations together we will consider the presence of curvature.spacetime. In This may other have words interesting one cosmological could implications. still haveno Maxwell hair equations theorems. inhairy flat solutions It as is was known discussed that by Einstein-Proca Bekenstein, [ theories for example do not admit Nevertheless one can consider theories where an effective mass is created but only due to JHEP05(2017)114 (2.9) (2.5) (2.6) (2.7) (2.8) (2.2) (2.3) (2.4) . i 0 α Λ # A ) β ) ) field ν r + dr. βκ A ( ) ) 2 r 2 r χ µ ( µ ( ( + 2 k χ 0 respectively. It is 2 0 ∇ , ) ) r p 1 α + β 2 2 1. Spherical symmetry h − ∇ µ A dt 2 ( α ) r , = + " r ( A . ( ) 2 2 1 ( κ a r β A = 0 ( − ν = 0 a ∇ = dr κ ) α ∇ µ 2 β µν µ ) . 2 4 0 βκ G dx ∇ a ) µ − ∇ ( µ − 1 2 r . + 2 2 βκ ( βκ rh k 2 − βA ) is trivial or alternatively the metric A 0 1 4 r ) r = 0 equation gives the + ) + 2 2 ( = 0 = (1 ν 2 − 0 µ 2 + 2 χ rr r r h A 0 + 0 ,A r # ( E 2 ν µ ) 2 0 h 2 ( β ,κ µ 0 r Z ) β A 2 2 µ α Λ 2 A 2 2 r 1 a Ω – 4 – β R α 2 µ ) r a d βaa α )( h r A + 2 − 2 )( ( ) which is no longer a gauge term in the presence − β A ) r r h 2 r 4 2 ) αβ ( − 2 = 0 equation is then immediately verified and hence ( r βκ M µ + 0 χ G k 2 µν − = 0. These latter two equations are simplified noting − tr ) h 2 ν ) = 2 F r E − p tt + 2 2 (1 ( ( r µν A E r dr ( 2 µ g f βa µ r − f 2 1 + ∇ 2 ) = 2 + r ] µ 0 ( ( 2 RA ) := βκ h ) + " 1 2 βκ 2 dt ν ν ra ) J r A + [( + k ( µ = 0 implies either that 2 h 2 r A µν − r 2 ( − 2 ] 2 R µ J / 2 = 1. As we will also study asymptotically adS and Lifshitz spacetimes βκ 3 = 36 2 1 A h k 2 κ 1 1 2 − r + 2 C ds 2 = r 2 ) = µν r µ corresponds to the curvature of the base manifold ( Z 2 κ 1 8 χ + The equation = 0 and the metric equation t yielding at the end, At the end we are leftJ with the other nonthe trivial component substitution of [ the Maxwell-Proca equation We choose the latter option.the system The is mathematically consistent. The of a massive vector field. constraint, where corresponds to we allow here also forimportant a to hyperbolic notice and the planar presence base of manifold, Take static and spherically symmetric spacetime while the modified Proca equation reads, where JHEP05(2017)114 (2.10) ]. If we 41 ) while taking ]. then the metric 2.9 41 r ), give a full solution r . An interesting twist ( , the metric constraint β behaves asymptotically k 2 r h eff 3 Λ ) and = Λ = 0 yielding the stealth = 0. Since, as we will argue, r ( µ 1 6= 0) found in [ ∼ − a ]. After finding the asymptotically C h , and in this case, we will be able 41 β to have a standard asymptotic (A)dS and = 1 and we will discuss this in the next = 0. Unlike what has been reported in . f 2 2 β β r µ ]. µ 2 4 allows to have a non trivial electric field. eff 3 / − 37 Λ . Two cases have to be distinguished, whether 6= 0, assuming that – 5 – = β = 1 µ ) forces ∼ − eff → ∞ β ) solving the algebraic equation ( Λ is asymptotically some power of f 2.5 r r ( ) for the symmetry at hand. This is the task we will h k ) drops out, allows us to have a non-trivial electric field 4 and the same spacetime (Schwarzschild) metric. Here, ). This is achieved for 2.3 / = 0 by Chagoya et al. [ 2.8 2.8 = 1 µ β = 0, if ) and ( µ 2.1 , the constraint ( = Λ = 0) that we mentioned above, the authors argue that it is the . We will also see that Lifshitz black hole solutions can be obtained in ]. Again the value vanishes or not. For r β , this latter class of solutions will be complimentary to the former class. ] where one has µ 43 µ which is the interesting asymptotically flat solution found in [ . This opens the possibility of solutions that are asymptotically (A)dS or β 36 2 r ∼ Q/r must go to a constant at infinity. These behaviors are in agreement with the 6= 0 we can get get counterpart solutions to the self tuning de Sitter black holes ) in the asymptotic region f + µ f . All scalar-tensor solutions are however not admitted since we still have to satisfy 2.5 q q ] found in [ In the next two sections, we will report on two generic classes of solutions. The first Before finding explicit solutions to the above system, we will analyze the metric con- 36 6= 0 for 1 = 4 ) = ) = r r 1 ( ( 3 The general solutionThis for case has 1 been = studied 4 for flat solution (for the latter section weC will derive solutionsThe for class the will case bethe general literature for other certain asymptoticallycoupling cases flat constant like solutions exist inthis this class. class for other values of the asymptotically flat solutions or conical geometries (for Λ one is obtained forto a derive fixed the coupling generalThese constant solutions static can 1 solution be = with either black 4 a holes maximal or homogeneous solitons and 2 both dimensional are space. asymptotically AdS. On On the other hand,function for the Proca mass as some power of behavior, even Lifshitz. In thewill (A)dS imply case, that the for effective cosmological constant is fixed in term of the Proca mass as In fact the system issection. completely Finally integrable when for the 4 will Proca be an and analogy Maxwell to field the are system taken studied as instraint two ( [ separate fields there Schwarzschild solution with a constantof electric this field solution happens and for arbitrary because the right hand sidea of ( allow for of [ to the field equationsundertake ( in the rest oftensor the system paper. [ Alreadya in this form wethe see modified the relation Proca to equation the ( scalar- These two master equations, when solved with respect to JHEP05(2017)114 = 0 (3.5) (3.3) (3.4) (3.1) (3.2) q function , . We will # κ k . Taking a 4 1. κ 2 − Λ , 2 2 = 4 − Q the Proca charge 2 = 0 2 2 µ dr . 2 Q κ 2 Q i ) µ dr . κ and + 4 . κ 2 2 ) 4 − i µ r ) can also have asymptotically 2 2 2 − κ Q 4 2 √ 2 2 κ β coupling where we can find the µ Q − ]. However, as we will see in the r + 2 κ + ( β 2 2 , 2 2 41 2 + r + Q r 2 ) 2 2 2 κ 2 ) µ r r r arctan( + ( µ 2 ( ) is given by + 4 2 2 ) with Λ = 2 µ k r r r 2 µ ( 2 2 Λ ) − ( r 4 a 2 Q k 2 6= 0 and we see that − µ ) gives, µ − ), one obtains, 4 2 1 (2Λ – 6 – ] are important as (in the absence of the higher µ C h 2.9 − 3.2 2Λ ( = 41 2) ) and 2 Z a ] r − 0 ] but these solutions are pathological, as our analysis ( (2Λ r Z ) 2 2 2 h a r 43 ) Q 2 1 1 2 1 r a ) into ( 1 ( )[ C C [( 1 C Q r is undetermined and we are left with a degenerate system ( 3.1 2 8 (8 64 4 k k √ − + ] where now the curvature term is replaced by − ) = r Q " r 54 r M ( 3 2 k µ 2 − 1 ) = ]. This is something we will have to keep in mind. The Q r ) we find that, = 0 then C ( 2 1 a 53 ) = 2.7 C r + where we can have such asymptotics [ ( h r β ) will give some Q r ) gives ( a ) = 2.8 r ( a Substituting the solution ( Therefore from now on we stick to 3.1 The spherically symmetricFor adS spherical black symmetry, the holes electric and potential solitons Both of these integrals can be easily found depending on the value of Similarly, using ( which determines the spacetimesolutions as solution classified is in [ nownow of look identical at this form class with of the solutions in static detail for different parameters. particular shows, for the system is degenerate and undetermined formodifies this the case. effective horizonglobal curvature monopole and [ may give a solid deficit angle just like for a Note that when of one equation with two variables Indeed, ( and substituting the electric potential next section, a careful analysisflat shows solutions. that Our generic results casesorder and of term) those of a [ Procasection vector we generically concentrate spoils ongeneral usual, this solution flat, particular and dS value will or of have adS thus the a asymptotics. complete In picture of this the static solutions in this case. only value of JHEP05(2017)114 1 C , is = 0, (3.8) (3.9) (3.6) (3.7) M (3.10) (3.11) M dependence a . The metric 2 Q , 0. Finally, we note (0) = 1 for ) f > 2 rµ . 2 µ 2 . The latter charge √   M ) 4 as part of the 2 and it is equivalent to fixing / 4 µ . This key term will contribute 2 2 − r a 1 µ 2 2 r − M and 2 = 1 arctan ( 2 Q 1 , 2 2(Λ 2 β + and . ) − 2 2 ) contributes similarly at asymptotic 2 ,C 2 Λ) + ) r 2 h charge on the other hand is related to µ 2 2 2 2 2 3.6 2 = 1 − 2 µ . r µ 1 µ for large Q 2 2 2 2 2 1 Q − C Q, Q µ   µ h − C 8 − 2 − 2 2 2 4 (Λ 4 alike Coulomb charge and 2 − µ Λ r − µ 2 2 3 2 1 – 7 – 2 2 and = Q ) µ µ 2 − µ Λ)(Λ C µ 2 − 2 2 2 1 f 4 µ eff Λ term in both − 1 √ − C Q = Λ 2 ( 2 4 r C Λ µ (0) = 2 2 eff + 6 h 2(4 Λ ) 2 3 − r r = 0. Therefore these solutions are always locally adS and ( 2 h µ r 2 2 eff = 0, Q Λ is not physical, as it corresponds to the reparametrisation of time, M (2 ) = 2 − 1 r ( C f ) = = 0 and as a result will influence the regularity of the solution. Indeed r is the Coulomb charge which is a stealth parameter for the spacetime r ( h Q takes the following form, h ) drops off from the field equations. The In order to have the standard form of asymptotically adS solutions we have to fix 2.9 This ensures an identicalthe behavior gauge. for Inparameter since, this case the effective cosmological constant is given by the Proca mass making curvature regular at there is no solid deficit angle. so that, which is always positive or zero. Last but not least we have, and we can see that the effective curvature is always equal to unity, a finite number at we can easily see that if solution. In EM theoryThis this stealth feature electric is charge a would particularin feature give associated ( rise to to thethe RN breaking black of hole solution. gaugethe symmetry effective cosmological due constant to is the fixedthe and presence Proca always of mass negative the for term. latter arctan over Secondly, we remark that solution depends on fourpart integration of constants the overall mass,infinity. since The the constant arctan termi.e. in the ( gauge choice. Laterof we adS will fix metric. it such that at infinity we recover the standard form where we set, for the effective cosmological constant and for the effective horizon curvature term. The function and we have two integration constants, JHEP05(2017)114 . 2 r 2 Q Q = 0, r (3.13) (3.15) (3.12) (3.14) = 0 and , r ) rµ 2 √ r smooths the effects Therefore there are ν 5 A arctan ( µ 6= 2. The solution has no 2 , for which the solution is ], A 2 2 2 43 Λ) µν ) Q Q = 0 we have a regular soliton 2 G − µ 2 2 M µ . − 2 . = 0, the mass of the soliton is not r M − (Λ 2 = 0 and = 0 spacetime is actually singular. It 2 µ = 0, since the solution is singular at M µ 2 − Λ = 0 4, when the Proca mass is corrected by 2 2 M M Λ = 0 √ / M is regular there). Here, we have the nice Q − ( − + 1 falloff. Again, we emphasize that the usual 2 h = 0 becomes a soliton. 2 ) is everywhere bounded, finite at 2 = 1 µ + r µ 6 ) drops out, if /r β 2 2 – 8 – 3.6 M 3 µ − r M − 2 2 2 3.12 2 µ µ − 2 2 2 2 ), reads, which has asymptotic adS geometry. We also see here ) = Q Λ r Q 2 . µ ( term in ( = 0 (even if − 2 h 3.10 2 2 (0) = 1. A solid angle deficit (or excess) would have meant r   µ µ f ) r 6 2 2 4 µ 2 arctan 2 − − − 1 as in ( µ 2 2 − 2 2 Λ 1 Q µ is associated to the breaking of U(1) gauge invariance and it can 2 2 2(Λ C + 2 2 Q + 2 must be discarded in the case r 2 does not influence the spacetime metric. Q 2 r + Q = 2   r 2 ) 2 Q r 3 µ ( 2 h ). 3.8 ) = ) = r r ( ( Note that although in the case of the soliton Let us look into more detail the solution for Note that included and fixed h f 5 three distinct sub-classes of solutions with adS asymptotic withinzero. this class. Indeed, one can deduce the solitonsee mass eq. ( from the asymptotic behavior at large On the other hand for the effective curvature is zerowe although we have have a a non spherical stealth horizon. solution For other which values for of With this choice we get a stealth Schwarzschild-adS solution [ The Proca charge take arbitrary values. There arespecial. however In particular particular values the of last term in ( on the mass we havean a electrically black charged hole RN (withwould black adS seem hole asymptotics). where that This the in iscurvature Proca interaction radically theory the different and from situation for M is regularized. The full spacetime solution, with mass solid angle deficit as wethat have spacetime is singularresult for that the metric issolution completely for regular arbitrary and Proca hence mass that for the addition ofof the the curvature-vector Proca interaction mass term, term giving a regular solution with adS asymptotics. When we switch properties to the sphericalThis or is planar because adS static thedecays black latter at holes infinity depending as onCoulomb the a charge value mass of term with a 1 The resulting solution is always an asymptotically adS black hole. It has very similar JHEP05(2017)114 2 µ (3.16) (3.17) as 2 = 0 and this Q µ . Although such ] has only three 2 41 , m ) − r . ( ) 1, the arctangent term h = 2 2 = 0, − µ 2 ) 2 2 µ κ 2 ]. Asymptotically however 1 = , Q C p ) 41 r 2 κ r . − . 32 2 2 dy (4 is fixed in terms of Ω 2 Λ) , therefore one can formally define ) + d 1 2 2 2 − r /r C µ 2 ) = arctan ( ) 1 2 2 r dx 2 2 + ( µ ( 2 (0) = 1 and therefore the solution has − ∼ 2 2 ) 2 Q 4) r f µ = 0. We get, dr − 1 − − (Λ + 2 C 2 2 µ eff (4 ) µ 2 2 + 5 , f Q (8 r 2 2 ( 1 2 – 9 – ( 2 √ 32 ) dr Q 2 f 2 ] 2 ( 2 dt 1 µ = 0 = 4 π 2 Q + C 55 r 2 1 κ − 2 . Another particular limit is to take p − 32 C dt = ∞ + ) (4 ∼ − r + 2 ( eff r r M M h ds 2 2 M → − − r − = ]. We note that in presence of a Λ-term the solution behaves 2 for ) = 41 = 0. This agrees with the result of [ r ds 4 the last term has the form ( ) = 1 = 0 however we see that r r r h ( ∼ h M f → ∞ r ]. . For 1 one could consider an imaginary Proca mass term, r 53 0 while − ). For . These solutions can be trivially obtained but will have singular asymptotics, and ) will be replaced by a hyperbolic arctangent which will explode exponentially at r = → It is easy to see that choosing hyperbolic geometries, If we additionally set Λ = 0, the solution becomes, There are a number of special values for the coupling constants. Choosing Λ = 2 3.6 3.12 κ h 3.2 Planar horizonLet black us holes suppose now that the horizon’s geometry is locally flat, for a term would behigher discarded order due terms to does not instabilityshows guarantee in this that usual intuition. the Proca However, solution a theory,singular quick has here, for analysis always the de in negative presence this Sitter case effective of asymptotics. curvature and as a result is always and this explainsintegration why constants. the asymptotically flat solution reportedin in ( [ finite hence we do not discuss them further. To get an arctangent term and de Sitter asymptotics In order to make it asymptotically flat, the constant regular curvature at space will have a solid deficitsolution angle [ removed from the sphere similar to the global monopole solution was found inasymptotically as [ a conical geometry [ for large The solution has genericallyas an event horizon. However it also has unusual asymptotics the effective mass of the soliton as will kill the effective cosmological constant Λ in ( JHEP05(2017)114 ) and (3.21) (3.19) (3.20) (3.22) (3.23) (3.18) 2.9 arbitrary. one would β β from ( 2 k r 2 ) 4 2 µ 1 2 Q ) C 2 . µ 2 (8 i ) 2 2 2 µ − − 2 2 Q Q − r M ( . (Λ 2 2 2 r . Therefore it is instructive and r r 4 1 2 − 2 2 = 2 2 − µ 1 µ 4 Q β 2 2 ) ) . 2 r M − µ ) 2 = µ β 1 r 2 ( 2 = 0 while keeping the coupling C 2Λ − h . − 1 2 2 (4 2 h C µ 3 µ µ 2 2 2 2 – 10 – (Λ 2 2 r 1 r ) Q 2 2 2 C 2 − + r µ µ 2 2 4 4 − 4 ). We will not undertake this task here. We saw also 1 ) Q 2 r Q Λ + 3 r − 2 2 r 1 C ( r 2 8 2.8 2 C h 2 (2Λ µ 4 µ 2Λ 2 − µ 2 2 the central singularity will be dressed by an event horizon Λ , we are able to obtain the general spherical, hyperbolic or = 0 we have a black hole horizon dressing the singularity − − r Q µ Q 1 4 1 2 − 2 128 M 1 + Λ using ( r 2 C M = 2 a 4 µ ) = µ 2 β = 0 and arbitrary r 4 ) = ) = ( 1 r r a could not be set to zero for ( ( C h f and we get the solution, 1 1 ) = ) = C ) the metric functions take the form, r r C ( ( but it does not change the properties of the solution. This is because it is ) it is straightforward to obtain the electric potential, h f 3.4 2 µ 3.3 2 − is positive. This is contrary to the usual RN solution which is singular for small 0 h = 0. This is due to the Proca charge which now is of the form of an imaginary RN In order to have adS asymptotics as before we must impose, r have to resort to somethen numerical numerically integration. solve for Indeed onethat would the solve constant for easier to study nowWe the will particular see the case asymptotic of nature of the solutions does not generically change and we again always the imaginary charge that is dominant at smaller 4 Solutions for For the special coupling planar, static solution. In order to obtain the general solution for arbitrary Again we see thatat even if charge. We cansign have of an Λ effective positive or negative curvature term depending on the black holes. The asymptotics are locally adS. effectively fixing similar to that of aneffective adS curvature RN term geometry, however, is withwhile of imaginary the an charge. effective cosmological undetermined Additionally, is the signeven always at negative. fixed the by The absence imaginary the of chargesince Lagrangian term mass parameters, means that Although we have a planar geometry for the horizon surface, the black hole potential is whereas from ( Integrating ( JHEP05(2017)114 ]. a X 56 , (4.6) (4.7) (4.1) (4.2) (4.3) (4.4) (4.5) y , ) 0 = Λ = 0, βy ry = 0 gives an . µ = 0. Indeed, − 1 y 1 C = 0 (1+4 C ) by introducing 1+ . 2 ry 2.8 βy ydy = = 0 ) by noting, 0 β cannot yield a solution = 0 with + 4 2 X 4 f X 2 y β βy − βy ) to get, dr . (1 +4 . ) β 4.4 4 1 + 8 βy βκ Λ) β − β 2 1 + 2 1+8 4 , ) and the full solution is known for ]. ) 2 2 − ) r + 2 36 2 2 4.6 2 βy y y µ a ) µ is just a constant given by the power in 8 βκ ( r . 2 )( ( 0 y + βX r = 0 (see the Lifshitz section). Finally, if + 2 r k h 2 a + 2 6= 0, numerical solutions have been reported in [ ( 2 ra κ (1 + 4 + 1 2 y X r C Λ = 2 then Λ) ) = – 11 – βκ β Z µ and replace it in ( r β y 4 r ( ( − 2 r 1 2 Λ) f 2 there are three generic cases given by 2 X 2 = β − µ y + 6 βa 2 2 X µ − 2 r ( M r 2 2 2 ) = case, r µ r + − ( ( β ) is now given by, 2 + ) we find βκ r X ), it is easy to see that pure Proca theory 2 2.7 ) = βκ ) form a closed system of equations on three functions: 4.5 r 4.5 4 ( is any power of )( = 0 for which the above equation is separable. In these cases, for X h 4.5 r as follows, )–( ( a κ = 0. We can already anticipate the form of − is exactly zero, and the resulting solution is nothing but the stealth ) + h 2 y 4.4 ), reduce to, / κ β 3 4 /y 2.9 X and − ) and ( (1 X βκ 4.4 . In what follows we will discuss each of the cases in detail, first the former two ) and ( r +2 β dr ), ( 2 Λ and finally r 2.8 = 0, it is convenient to bring down the order of the equation ( 2 β 4.3 . Notice that if µ 1 a For the moment we have not achieved much from the change of variables, as the field C Indeed, from eqs. ( = 0, the full system reduces to a single ODE ( = 2 1 6 does not completely cancel out in the above system of equations; unless 1 2 C while, unphysical metric. Nevertheless, in the case where C arbitrary and then the latter In addition to theµ vanishing is constant then 1 configuration on the Schwarzschild AdS spacetime [ a in this latter case, using ( Eqs. ( and question. With this observation oneto can the verify system that except any in constant particular cases like Indeed, in this case,solve, ( after straightforward calculations, the master equations we have to On the other hand we set for the variables The metric function obtain asymptotically flat of adS solutions. In order to achieve the solution of the system JHEP05(2017)114 . ) y y can 4.7 or a (4.9) (4.8) r 4, we (4.13) (4.10) (4.11) (4.12) r / 1 ). ), since we coordinates. β > 4.8 4.8 r = 0 from ( is a constant and X = 0. Namely for as a function of y 2 ) and ( a ) we find, βy 4.9 4.9 coordinates where 4 there are no real roots. , + 4 y ) / 1 βκ = 0 does not give physically βy from ( y β + 2 2 < β < . , r , 2 ydy . 2 Λ γ 1 γ γ 2 γ γ µ β − ), yielding, 2 2 ( = βy 4 is not covered by eq. ( 1+ . 1 / ) ) 1 y γ γ 4.6 r dy γ γ dr = 2 + 4 β − 2 − κ = 1 − 4 2 ). Note that for our choice of . While integrating ( β dy β 4 ra 2 βy γ µ 4 4 β y ∞ + 1 + 1 + + 1 + + 1 − Z ), and the case s + – 12 – + y y will map to a different solution in y y = 0. However then we see that r 1 coordinates. The effective expression in terms of ( ( y 1 + 8 = 4.8 2 y → y + 1 + y γ = = = y + 2 y + to 0 0 r r M )( a a r 1 and set , β y a 1 da ” solutions are excluded. Indeed then 2 r γ 4 r then − = Λ = 0 or + 2 β > ) to get ( µ ∞ β β 1 4 ) = 4 + 1 + + ) we obtain, r ( y − Λ greatly simplifies eq. ( ) we can also find a differential equation on h → 2.7 β r 4.3 = ( 2 = 2 1 and the roots are both negative. Then a straightforward integration = 1, for y 2 κ µ + are now understood as explicit functions of 0 we will have two real roots while for 0 ) and ( y a < γ < 4.8 + 2 β < β 1 and 4 r ) gives, 1 or Let us assume now that 4 Using ( 4.8 depends on the solutions of the quadratic equation 1 + 8 β > and we see that when so that have that 0 of ( where We can express the physicalIn metric particular, and from the eq. vector ( field in terms of the new coordinate consistent solutions, as we mentionedthe before. region Let we us want now to solve study. step by step andvariable, determine Each root corresponds tosingularity. an Therefore, each endpoint interval of in Since spacetime, we are either seeking asymptotic blackasymptote infinity hole infinity. solutions in we Also will notedivided seek that regions by the in the case factor (1 is fixed by the requirement which is not allowed.by Otherwise coordinate we transforming have a from y separable ODE which can4 be explicitly solved The assumption First of all we see that “power of 4.1 The case JHEP05(2017)114 . ). 0 r will 3 = 1 2 √ n 4.13 (4.17) (4.18) (4.19) (4.14) (4.16) (4.15) 2 0 3 0. The to be a a 2 0 1. ≡ , n is related h βr y > 0 . a min r γ +1 2 γ < γ < 3 ≥ ) 2 r k = 0 while they are γ y and take ) ) 2 2 y , µ n 3 n ( µ dy h k ) = 2 0 , but + 2 big enough the solution a y r + 1 + y r ( k γ 2 0 y γ 2 -coordinates. Substituting y y y M 2 ln I ( 1) y 1+ 2 1) βa 1 2 − 9 γ − µ − ( 64 − 2 γ 2 0 κ 3 = 2 n , as it can be seen from ( 4 a ) + 0 3 0 = 0). Indeed fixing 2 ) γ r a 2 2 + 4 y y ((2 corresponds to a singularity hidden y 16 ( − y 2 dr M f 3 → − ) + + n =0 min y a ln but the solution is always asymptotically k X + 1 6 y = ( r , covers not all k 1 y h y y I 1 ( 3 , 2 0 + 1 + γ γ a y dy γ γ n − 0 2 1+ 1 ), we obtain for the metric function 1 + ) ) 2 βr – 13 – Z γ γ + 2 y takes the generic form, 2 y 2 = + 3 4 y = 0 for example gives us, 4.10 − 3 1 y 2 y I 1) 2 µ y we find that 3 4 M 3 2 1 + − + 1 + 1 + y 1 + − y y n simply rescales the radial coordinate, while 1 + ( ( 1 + ) into ( 6= 0 we get instead adS asymptotics in the same fashion as 0 ,I (2 2 ) r 2 0 µ γ γ y ) + y γ γ ) 1 ln − = 2 and 2 2 y ( 4.13 1+ 1 y βa ( 0 r ( ) ) ) = n 0 2 a h y γ γ h ( + 0 − ) = . For example h y y n ) = ) = ) and ( ( y y = h ( + 1 + 1 + ( 1 µ f y y I are constants of integration. It is now straightforward to see that 4.12 h ( ( 0 y a dy . ) we also notice that the coordinate Z 6= 0. In order to get an explicit solution we can set h are some numerical coefficients. Note then that for and = i µ = 0 is a singularity for the integrals the interval we want to focus on is 0 = 4.12 a r 1 y f I Switching on Proca mass The integration constant and Proca mass plays the roleyield of higher negative an powers effective in cosmological the constant. expression for Higher powers of giving an asymptotically flatgives black hole (even with in the previous section. Indeed we have asymptotes a Schwarzschild solutionNewtonian and falloff. is therefore Taking asymptotically flat with the correct finite power series of where to gauge choice.From Indeed ( for large As we will see below,by this an is event horizon. not a TheadS problem, solutions for since are always asymptoticallypositive flat integer for greater than 1. Simple integration then shows that the above integrals are Since remaining singularities of the integrals are excluded from this region for 0 where we introduced the notation, which shows that the metricthe expressions acquires ( a homogeneous form in where JHEP05(2017)114 . , ) 6 a 1 h f A 4.6 . It , such (4.20) (4.21) 1 M r = 0 the µ tends to 1 5 n ) picks up an 1.10 1.08 4 4.18 = 0 corresponding 1.06 ) = 1, we find the κ 1.04 ∞ ( is the AdS asymptotic 1.02 h = 0, the equation ( 3 , there is a black hole 4 while as µ = 0 1 / 1.00 M 1 C 1.0 0.8 0.6 0.4 0.2 0.0 . . As in the case C non integer, for example, for = 4 etc. . . It is interesting to 2 2 ! , n n 3) 0, see the right panel of figure √ 3 < 1 → ∞ − √ eff 2 M term for 1.2 1.0 0.8 0.6 0.4 0.2 = 0 (the left panel); b) for negative 3 log(2 , r /y M − = 0 case with eff = 1 case. Here, for r 1 κ M β is shown in figure

– 14 – 0 − µ r attains the special value 1 6 1 a h f + . Fixing the gauge such that β ' 1 = 0 and a) M f µ r = ' 5 h eff M 1.06 = 0, the event horizon exists, see the left panel of figure 4 1.04 M . 1.02 3. The details of the integral calculation are presented in appendix / 3 1.00 = 3 and then an additional 1 = 0 and non zero respectively. For example, expression ( goes to infinity then → ∞ n = 1 µ goes to infinity. 0.0 0.5 0.4 0.3 0.2 0.1 -0.1 r n β β 2 for 2 . Black hole solutions for /y 1 = 0 is always hidden behind the event horizon. A black hole solution for non-zero It is possible to work out other explicit solutions for y behaviour at 4.2 Lifshitz black holes:We have already topological found analytical solutions withto a asymptotically planar base AdS manifold black holes in the 4 The horizon does existgravitational force. in We this also would case,at like but to stress the that far the curvature away singularity observer taking would place measuresingularity is repulsive covered with the event horizon. The effect of nonzero Note that even for is interesting thatsolution choosing with negative negative value asymptotic effective for mass, the bare mass asymptotic behaviour as follows, where note that as we get that the value of and the result is depicted in figure Figure 1 that the effective mass at infinity is negative (the right panel). flat or adS for extra 1 0.2 0.6 0.4 1.0 0.8 JHEP05(2017)114 (4.23) (4.22) = 1 is adS z , permits the ν solutions. They A µ y A µν G ) = 1 in order to ensure 6 r a ( , h f . F i 2 2 2 y dx r →∞ + r ydy ]. In the present case, we will see + ) y 3.0 2 1 6.0 β 59 4 , 2.9 dx = 0 yields constant + 2 − , leading to the AdS asymptotic behaviour. 58 β 5.5 2 1 2.8 4 µ r βy (1 h + 2.7 2 5.0 + 2 2 F = β 2.6 2 2 dr r is such that lim – 15 – r 4.5 dr 1.0 0.8 0.6 0.4 0.2 0.0 − + ] F 2 βy 4.0 F dt + 2 z = 0 and non-zero ]. On the other hand, the massive character of the Proca 2 3.5 β r 57 2 M − − 3.0 = [1 2 ds 8 6 4 2 10 . Black hole solution for denotes the dynamical exponent that is responsible for the anisotropy ( z Lifshitz spacetimes take the form, In the last decade, there has been some interest in extending the ideas underlying where as a special case). The metric function obstruction to generate blacksolutions holes. found In in fact, the toan literature extra our for source knowledge, the materialized all Einstein-Proca bythat the a model the Lifshitz U(1) require nonminimal gauge black coupling the field hole which introductionemergence [ plays of of the Lifshitz role black of holes a without mass the term, need of additional fields. constant, and instead requireFrom the the introduction advent of of Lifshitz sourceEinstein spacetimes, or gravity it with higher-order was a gravity clear negativeto theories. that cosmological engineer a Lifshitz constant massive spacetime constitutes Proca [ anfield field in excellent coupled spite toy to of model being completely compatible with the Lifshitz asymptotics may be a strong the AdS/CFT correspondence toanalogy field with theories the AdS/CFT with correspondence,Lifshitz an the spacetime, anisotropic gravity enjoys scaling dual a symmetry. metric,nates scaling commonly scale In symmetry with known different for as weight. whichwith Because the of AdS, spatial this anisotropy, can and Lifshitz not spacetimes temporal in be coordi- contrast sustained by pure Einstein gravity with eventually a cosmological Unlike in the previous subsectionin here, fact 1 correspond to Lifshitz spacetimes which we will turn to now. Figure 2 becomes separable yielding JHEP05(2017)114 4 as / µ ) one , = 1 (4.29) (4.31) (4.24) (4.25) (4.26) (4.28) (4.33) (4.30) (4.32) (4.27) µ β 4.4 is fixed in 0 in ( X 1 , β − ) 2 β 2 2 2 . ) while coordinate , dx ,, = 0 coordinates, 1 +1) + − y γ 4.22 +1) γ +1) 1 . +1 2 1 γ γ γ 2(2 − b > 1) ) 2(2 dx 2(2 ): b − ) ( ) , provided that we fix 2 γ b 2 b b γ r 3 I 4.25 2(2 + 1) [ excluding the option , + ) + 1) b + 1) γ 1) ∞ ( 1) , so that 1 γ , r γ (2 b 1 1 +1 1 2 − − γ (2 M + 1 − − − z Λ) ] (2 = 0 imposes a constraint on 1) Λ) 2 2 , β β γ β β β r ∪ γ . γ γ 2 6 2 β [ κ 2 2 2 − ) for large (4 2 1 6 +1 . − 2 dr given by ( 2 z M = (1 + , in the following way, − 2 − 1 µ 2 γ 1 1) (3 (1 + + 1) 0 z 1) 2 (1 + 0 r 2 0 2 4.27 1) − a z µ β a − 1) X − 2 +1 µ γ 1) 0 1 2 − − ∞ γ ) with − r β (2 − 2( − r − β ] γ Λ) 1 4( 2(2 γ 2 γ β so that it has the same asymptotic behavior as 2. We start by resolving ( ∈ 1 + (2 β β ) + and (4 2.5 2(2 – 16 – / b 2(2 2 b + ) with 2 ) = β 1 0 ) b ) ) = 1 s = 2 1) r b ) correspondingly. Note that the mass term in the − z dt b r z > 1) (3 M 2 ( 1) z 2 ( 1) r − 4.23 2 r µ 2 µ F − − γ is an integration constant. The constant γ 4.26 − − 2( +1 β γ 0 (2 z M γ is not constant we have to proceed as we did in the last 2 2 γ thus obtaining in turn, (4 X (2 r ) = (2 y ) = γ 2 r γ b s ( γ ( − a ) and ( 1 z h (1 + 1 = db z 4.25 1 + 2 , where 0 z 0 2 = (1 + = (1 + 1 r a r − Z − ) 2 ) β b − 0 ) will decay only for b 0 2 ( r = ( /b has the Lifshitz spacetime, a r r = a 3 0 h I 2 4.23 X = 1 ds given by ( y = ). . Similarily, µ r X 3 ) and the integration constants and z In the general case where 4.26 for large The solution asymptotes thein Lifshitz ( solution ( In general the integration can be performed numerically. where Note that we have chosenr the coordinate In order to obtain the metric we need to coordinate transform to section. Here we take fortransforming simplicity with Lifshitz metric ( (see figure The full metric then takes the form ( The Proca field reads then, On the other hand the metric constraint ( finds that such a way that where the Lifshitz exponent is given by the correct Lifshitz asymptotic. For our case here, setting constant JHEP05(2017)114 r µ 4.0 6= 0. 1 C ) and for 3.5 ] where it ) explicitly 4.27 41 1.1 4 and / 3.0 6= 1 β 2.5 = 1. On the left panel the M 2.0 a, y=const a, y≠const 4, have a singularity which is / 1.5 ). We found static black holes but = 1 β = 1, Λ = 2, 1.1 8 6 4 2 . On the left panel the radial component of β 10 12 r . The massive vector nature of the solution 3 r – 17 – = 0 and 1 ) for large 4 we found regular soliton solutions by putting the for the same solutions. / . This is in contrast to the claim of [ r ,C 4 4.27 2.0 while they are asymptotically flat in the absence of = 1 = 0 = 0, aside from β κ µ M 1.8 4 could sustain asymptotically flat black holes. It is important / to zero, see section , see section = 1 β 1.6 M β ). The solutions we found are generically of adS asymptotics in the is depicted as a function of the radial coordinate for the solution ( 2.9 ), which asymptotes ( 1.4 h 4.31 h, y=const h, y≠const . Black hole solutions for ) and ( 1.2 2.8 Furthermore, for the case of 5 however always hidden behind an event horizon.solutions Therefore are in more these regular extended Proca thanwith theories in electric field standard we Einstein have a Maxwellto RN (EM) electric solution theory. charge. which is Indeed In however in singular otherhelps GR for words, in small within mass regularizing this compared spacetime modified EM solutions theory giving the in vectorial certain mass cases term gravitational particle like An imaginary Proca mass term would permit de Sitterintegration asymptotics. constant regularises the spacetime metricmay and need the to solutions use are numericsOther particle to solutions like see if lumps we this of found property matter. for can One be extended for (and Λ) but, generic was found that only to note that itasymptotics is exist, due to in the contrasttheory, higher the to order Proca the vector mass case galileon term term of plays that the pure solutions role Proca with of theory. regular an effective In negative the cosmological constant. extended Proca solutions of extended Procaalso vector for the tensor first theory timereducing ( soliton the solutions. system We studied toeqs. the two general ( problem master for equations, ( presence one of algebraic a and Proca a mass non term linear ODE, see the vector field is shown as a function of 5 Conclusions In this paper we have undertaken a complete analysis of spherically symmetric and planar Figure 3 metric function the solution ( 10 20 15 JHEP05(2017)114 ν A µ A , since ]. The µν 4 58 G 4, the three . / a contribution β . On the other h r = 1 M ]. These are some β 14 – 11 – 18 – and the location of the horizon 2 Q should not be independent. In particular, the stealth M and ]. The Lifshitz coefficient is set by the coupling 2 56 , Q, Q 55 , because of the presence of the arctan-term in the metric functions. ) is independent of the unique integration constant, this latter may 2 Q 4.28 ] have a vanishing mass, and this is not in contradiction with the first law is expressed in terms of 58 Q In EM theory electric and magnetic solutions are identical due to the electromagnetic The physical interpretation of the integration constants present for the adS as well as Caprini and Christos Tsagas for interestingLavinia discussions Heisenberg on for cosmological making magnetic fields some and critical remarks in the introduction. the research program, Programme nationalFrance de and cosmologie from et the Galaxies projectFoundation of for the DEFI Basic CNRS/INSU, InFIniTI Research 2017. Grantby EB No. grant was RFBR 1130423 supported 15-02-05038. from infunded MH FONDECYT by part is and Proyectos by CONICYT- partially from Russian Research supported CONICYT. Council U.K. This DPI20140053. project CC is thanks also Chiara partially Acknowledgments EB and CC are gratefulsupport to during the the IMAFI course institute ofnancial in this support Talca-Chile during work. for the hospitality MH initial and stages thanks of financial the this work. LabEx P2IO EB and of CC Paris acknowledge support Saclay from for fi- duality. Here, the theorysolutions should we be have found studied,the anew. breaks higher Also order electromagnetic it curvature duality term would andof with be the the magnetic interesting Horndeski issues to Maxwell that term study may [ the be combination worth of pursuing in the near future. turns to beblack proportional holes to [ thedue electric to charge. the presence Hence ofthe the the Proca electric field electrically charge. ( Lifshitz Incorrespond the to charged present the case mass described of in the section Lifshitz black holes. from the constant Finally, the Lifshitz casementioned, is the Proca also field interesting isand for fundamental the in the existence order following of toresulting a peculiarity. give Lifshitz the horizon As correct is black Lifshitz ensured hole we asymptotic, by solution already introducing is an described extra by Maxwell a field unique [ integration constant which solutions yields a finite expression inthat spite of may the blow presence up of the ata nonminimal the term well-defined horizon. regularized Preliminary Euclideanintegration calculations action constants indicate for that, the inparameter adS order to solution have with hand, the mass of the adS solutions will receive in addition to the constant only one vector field [ for the Lifshitz black holethis solutions paper. deserve In a fact separate thebe study, complete which discussed thermodynamic is elsewhere. analysis beyond of However, the the we scope solutions would reported of like here to will point out that the Wald entropy of our solitons. We also found Lifshitz spacetime solutions which have the characteristic to require JHEP05(2017)114 ) , y ) (A.6) (A.4) (A.5) (A.1) (A.2) (A.3) y +1) )(3 + 2 x 1 )(3+2 1 . 3 y y . √ γ − + 1 +1 = 0 2 5 2 γ 3 x x 2( y y 1 ) 3 3 2 2 − 3 2 γ C √ √ 3(1+2 3(1 + 2 , 1) + h p p − y log − x − 1 y 3 , + 1 + 3 = 1, ) , such that y 1 2 8 y √ √ y x ( κ ( 9 1 2 3+4 2( 1 2 γ I − + 2 γ + 3 + 4 2 3 + 2 log µ x x ) ) 2 0 y ) can be integrated explicitly. Up γ 3 κ r x log a − 1 ), we find, Substituting the expres- 4 3 0 1 1 √ − 3 1 + 1 r − +1 A.3 2 . y 8 x x √ dy 2 4.10 3 3 2(1+ 9 / )+ + 1 1 ) + √ √ Z y + y y + ! 2 ( ( . ) 2 3 2 1 1 + 2 log 2 log I x y ) = dy 2 y 2 0 ) ) into ( + + 1 y ) 3 − )(3+2 y a ( 2 r y y 1 0 2 Z √ y – 19 – x 2 + 2 2 x 2(1

4.13 βr = 2 y − + = 2 2 + (1+2 )(3 + 2 x x dx 3(1 ,I x y p − 2 1 2 3 r M ) and ( 1 1 2 + Z 2 ), we obtain for the metric function 21 + 10 2 3 2 1 3 ,I − = = = ) γ γ ) 2 4.12 γ γ (1 + 2 + + 4.10 − y 1) 2 2 x 1+ 1 1) y y ) = p ) ) x − − − r γ γ 2 ( 2 + y x h dy x − y 3(1 dx dx )(3 + 2 (3 ) into ( (3 4 3 one can find an explicit expression for the above integrals. Indeed 2 2 y Z 4 / x ) x + ) + 1 + 1 + 2 4 2 2 and we find, 4 ) 2 + 2 y y 4.13 x / x 2 ( ) = ( = 1 x − y − (1 + 2 ( y x β = 1 dy (1 1 − 7 I (1 p γ Z Z ) and ( 1 4(1 12 Z + 2 log = = = ) = 1 4.12 I y ) = ( y 1 ( I 2 I and to the constant of integration, they read, To evaluate the above expressions we introduce a new variable With this change of the variable, the integrals in eq. ( in this case where we introduced the notations, For the value Substituting the expressions ( sions ( A Integration of the metric function for the case JHEP05(2017)114 , ] ] ]. (A.7) 177 Phys. J. , D 7 , , 505 . Lect. Notes , SPIRE 0 , IN ]. [ → → ∞ Class. Quant. , y y arXiv:1404.2955 Phys. Rev. Astrophys. J. [ , SPIRE , arXiv:0801.1547 Phys. Rept. [ IN for , ][ (1979) 1745 (2015) 3 20 ) for y 892 log( PoS(QG-PH)020 3 is, ]. , Modified Gravity and Cosmology 8 3 √ 2 3 9 y I ]. arXiv:1303.5083 ]. [ ∼ ∼ ]. J. Math. Phys. SPIRE ) ) , and IN y y ( ( SPIRE 1 ][ 2 2 Primordial magnetogenesis I Planck 2013 results. XXIII. Isotropy and statistics SPIRE IN Lect. Notes Phys. SPIRE – 20 – IN , ][ Vector-Metric Theory of Gravity IN ]. ]. ][ [ ]. ]. (2014) A23 Energy Momentum Tensor of the Electromagnetic Field Conservation Laws and Preferred Frames in Relativistic Conservation Laws and Preferred Frames in Relativistic ,I 571 SPIRE SPIRE SPIRE ! SPIRE IN IN An introduction to the Vainshtein mechanism ), which permits any use, distribution and reproduction in IN [ IN [ 3) [ ][ (1979) 726 √ 3 arXiv:1304.7240 − 20 [ √ 2 arXiv:1405.1612 ]. arXiv:1106.2476 [ ]. [ (1972) 775 log(2 CC-BY 4.0 (1977) 1691 Conservation of Charge and the Einstein-Maxwell Field Equations Null Electromagnetic Fields In The Generalized Einstein-maxwell Field Characteristic Surfaces And Characteristic Initial Data For The (1976) 1980 From Lovelock to Horndeski’s Generalized Scalar Tensor Theory SPIRE This article is distributed under the terms of the Creative Commons Gravity and Scalar Fields 177 ,I SPIRE 1 + Einstein-æther gravity: A Status report Astron. Astrophys. IN 17 ) IN , [

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