Black Holes and Neutron Stars in Vector Galileons
BBlack Holes and Neutron Stars in Vector Galileons
Javier Chagoya a , Gustavo Niz b , and Gianmassimo Tasinato aa Department of Physics, Swansea University, Swansea, SA2 8PP, U.K. b Departamento de F´ısica, Universidad de Guanajuato - DCI, C.P. 37150, Le´on, Guanajuato, M´exico.
Abstract
The direct detection of gravitational waves opens new perspectives for measuring properties of grav-itationally bound compact objects. It is then important to investigate black holes and neutron starsin alternative theories of gravity, since they can have features that make them observationally distin-guishable from their General Relativity (GR) counterparts. In this work, we examine a special caseof vector Galileons, a vector-tensor theory of gravity with interesting cosmological properties, whichconsists of a one parameter modification of the Einstein-Maxwell action. Within this theory, we studyconfigurations describing asymptotically flat, spherically symmetric black holes and neutron stars. Theset of black hole solutions in this theory is surprisingly rich, generalising results found in GR or inrelated scalar-tensor theories. We investigate the properties and conserved charges of black holes, usingboth analytical and numerical techniques, highlighting configurations that are more compact than inGR. We then study properties of neutron stars, showing how the vector profile can influence the starinternal structure. Depending on properties of matter and fields inside the star, neutron stars can bemore massive than in GR, and they can be even more compact than Schwarzschild black holes, makingthese objects observationally interesting. We also comment on possible extensions of our configurationsto magnetically charged or rotating configurations.
The first direct detection of gravity waves opens a new window for astronomy [1], promising to measure withgreat accuracy properties of gravitationally bound compact objects. In this context, it is important to investigategeneral properties of black holes and neutron stars in alternative theories to General Relativity (GR). The aim isto understand whether the physics of compact objects allows one to distinguish among different theories of gravity,which may share similar predictions in the weak gravity regime but present a different behaviour when stronggravity is involved. The simplest and most popular modifications of GR are scalar-tensor theories, which have beenextensively studied in stationary and cosmological situations (see [2] for a comprehensive review). Asymptoticallyflat black holes in scalar-tensor theories are usually similar to those in GR, thanks to powerful no-hair theoremsthat forbid the existence of scalar hair in black holes (see e.g. [3]). Instead, the physics of compact objects such asneutron stars can reveal interesting new effects, allowing one to distinguish them from their GR counterparts (seee.g. [4]).In contrast, vector-tensor theories are relatively unexplored, although they can have important applications forcosmology. The standard Maxwell electromagnetism, vector field models of inflation [5], vectors mediating darkforces in millicharged dark matter scenarios [6], or vector dark energy set-ups [2] are examples of this fact. Recently,models of vector Galileons coupled with gravity have been introduced, where the vector Abelian symmetry is brokenby derivative vector self-couplings, and its longitudinal polarization acquires Galilean interactions in the appropriatedecoupling limit [7]. These set-ups are free of Ostrogradsky instabilities, and have been explored in a variety ofcontexts – see e.g. [8]. In this work, we focus on the physics of spherically symmetric black holes and neutron starsin one of the simplest vector-Galileon scenarios, described by the vector-tensor non-minimal coupling β √− g G µν A µ A ν , (1)in addition to the standard Einstein-Maxwell Lagrangian. Here, G µν is the Einstein tensor, A µ is the vector field,and β is a constant parameter. This term is a special case of these vector Galileons, and the Abelian symmetryis only broken by the non-minimal coupling to gravity. As a consequence, the theory propagates five degrees of a r X i v : . [ g r- q c ] M a r reedom: two tensors (gravity), two vectors (the vector transverse modes), and one scalar (the longitudinal vectorpolarization). The latter turns dynamical by the symmetry breaking term (1). We study asymptotically flat blackholes and neutron stars, in the absence of a cosmological constant, pointing out the differences between thesesystems and their counterparts in scalar-tensor theories, where normally no-hair theorems prevent scalar hair to bedetected asymptotically.The set of black hole solutions in this theory is surprisingly rich, generalising not only the standard solutionsof Einstein-Maxwell theory, but also the black hole configurations of related scalar-tensor theories. When studyingneutron stars, we find that the vector plays an important role for determining the interior configuration of the star,and allows us to find examples of stable neutron stars that can be more massive, and more compact, than theirGR counterparts. Whether the vector field is in the dark sector and plays a role in governing the dark energy ordark matter components, or if it corresponds to a modification of Maxwell’s electromagnetism in the strong gravityregime, our findings may have relevant astrophysical applications.The roadmap for this work is the following: • Section 2 presents our system, some motivations, and the equations of motion to be examined. • Section 3 studies spherically symmetric, asymptotically flat black hole configurations, going well beyond theresults we first presented in [9]. When the parameter β is turned on, we find two distinctive branches ofasymptotically flat black hole solutions. The first branch is characterised by non-trivial profiles for the vectorlongitudinal and transverse polarizations, and the geometry is well described by a Schwarzschild configuration,plus subleading corrections which decay at large distances. Black holes are characterized by Komar chargescorresponding to a mass and a vector charge, similarly to the Reissner-Nordstr¨om configurations in theEinstein-Maxwell theory. An additional integration constant, mainly controlling the profile of the longitudinalvector polarization, is not associated with an asymptotic conserved charge, hence it does not correspondto a black hole scalar hair. In the second branch of solutions, the configuration is more sensitive to theintegration constants associated with vector and scalar profiles; they can modify the leading contributions tothe geometrical black hole features, rendering the geometry very different from the Schwarzschild solution.We also study analytically how black hole configurations behave in the limit of β going to zero, which requiressome care due subtle strong coupling effects, which are nevertheless manageable in our context. • Analytic solutions describing black holes are possible only for special choices of parameters, and more generally,a numerical analysis is needed. Therefore, in subsections 3.3 and 3.4 we numerically study some black holesand their properties, in cases where analytic solutions are absent. We analyse what conditions should besatisfied to obtain physically acceptable configurations. We find regions of the parameter space where regularblack hole configurations exist, and more interestingly, where the compactness of such solutions can be muchlarger than in GR, making this vector Galileon model observationally distinguishable from GR in the blackhole sector. • Section 4 studies neutron stars, with particular attention to the role of the vector profile to specify the starconfiguration. The non-minimal coupling of equation (1) can influence the star’s internal structure, since forcertain values of the parameter β , the vector contributes to the energy momentum tensor and modifies thegeometry. Moreover, neutron stars in this vector Galileon model can be larger and more massive than inGR, and, for certain parameter ranges, more compact. Actually, they could be even more compact than theSchwarzschild black hole for some cases, making these objects observationally interesting. We also commenton possible generalizations to magnetically charged or rotating configurations, which should be of interest inthe case where the vector-tensor coupling (1) is considered a modification to Maxwell’s electromagnetism,describing exotic objects such as magnetars. • Section 5 is devoted to conclusions, with a summary of our main results, and a brief discussion of ways forwardfor testing vector-tensor theories in strong gravity regimes. Moreover, we include appendixes that containtechnical details, or extensions to the material presented in the main text.2
Set-up
We consider a specific case of the theory of vector Galileons [7], which is characterized by the breaking of an Abeliansymmetry due to a ghost-free non-minimal coupling of the vector field to gravity. The action is S = (cid:90) d x √− g (cid:20) κ R − F µν F µν + βG µν A µ A ν + L matter (cid:21) . (2) κ and β are the gravitational and vector Galileon coupling constants, F µν = ∂ µ A ν − ∂ ν A µ , and G µν is the Einsteintensor. The dimensions of the quantities involved are as follows: R , G µν have units of length − , while β and g µν aredimensionless. Using standard QFT conventions, κ has units of length squared, and c = 1. This system is invariantunder inversion A µ → − A µ ; however, the Abelian symmetry, A µ → A µ + ∂ µ ξ (with ξ an arbitrary function) isbroken by the non-minimal coupling between gravity and the vector, weighted by the parameter β .Vector fields can play an important role for cosmology, even hypothetical ones that do not correspond to thecarriers of Maxwell electromagnetic force. The vector-tensor set-up we consider in eq (2) belongs to a class oftheories that have been studied in various works as a theory of dark energy. Scenarios involving dark vector fieldsassociated with millicharged dark matter and dark photons are also sometimes analysed in the context of darkmatter. Here we keep agnostic on the nature of the vector field, and we focus on how it backreacts to the gravitysector, determining the gravitational properties of compact objects. Switching on the non minimal coupling β inthe action (2) leads to a far richer variety of asymptotically flat black holes and neutron stars than configurationsassociated with the Einstein-Maxwell action.The covariant field equations of motion (EOMs) corresponding to (2) are0 = 12 κ G µν − (cid:20) F µρ F νρ − g µν F (cid:21) + β (cid:20) g µν ( D α A α ) − A ( µ D ν ) D α A α + g µν A α D α D β A β + 12 g µν D α A β D β A α + D α (cid:0) A ( ν D µ ) A α + A ( µ D α A ν ) − A α D ( µ A ν ) (cid:1) − D α A ( µ D ν ) A α − (cid:0) A G µν + A µ A ν R − D µ D ν A + g µν (cid:3) A (cid:1)(cid:21) − T µν , (3a)0 = D µ F µν + 2 βG µν A µ , (3b)where T µν = − √− g δ ( √− g L matter ) δg µν . (4)Notice that eqs (3a)-(3b) do not reduce to Maxwell equations when selecting the Minkowski metric and T ( m ) µν = 0.Dynamical gravity sets constraints that the vector field has to satisfy, even around Minkowski’s spacetime .In what follows, we are interested in studying configurations that are static, spherically symmetric, and asymp-totically flat . Asymptotic flatness is imposed in order to analyse in a more transparent way large distance properties(such as charges) of the solutions, and to appreciate the differences with scalar-tensor theories. In the first part ofthe paper, we focus our study to configurations in the absence of a matter energy-momentum tensor (that is, weset L matter = 0 in eq (2)), while in the second part, we study the physics of compact objects corresponding toneutron stars (and we include an L matter describing the stellar internal content).Under our assumptions, we select the following general field Ansatz for determining spherically symmetricconfigurations ds = − (cid:18) − n ( r ) r (cid:19) dt + (cid:18) − m ( r ) r (cid:19) − dr + r dθ + r sin θdϕ , (5a) A = ( A ( r ) , π ( r ) , , . (5b)Notice that we allow for a non trivial profile π ( r ) for the longitudinal polarization of the vector, which is not a puregauge mode in this context, due to the broken Abelian symmetry. One may search for more general static configurations by choosing the possibility of time dependence in the vector field, whilekeeping the vector’s effective energy-momentum tensor and the metric time-independent. However, one of the metric equations (3a)forces the longitudinal component π to be a function of only the radial coordinate, while one of the vector’s equations (3b) implies that A cannot be a function of t and r simultaneously. If time-dependence in A is chosen then a further constrain shows that it can onlybe a constant. β , the symmetry breaking parameter,goes to zero. As in other theories where there is no Birkhoff theorem, the black hole geometries found depend onvarious integration constants, and it is our aim to examine their physical interpretation.The charged configurations that we will discuss may also have applications in astrophysics. In nature, astrophys-ical black holes and neutron stars are believed to be uncharged under Maxwell’s electric field, because electricallycharged black holes would absorb oppositely charged particles from their surrounding, loosing their own charge. Incontrast, if these objects are charged under a dark vector field that is different from electromagnetism – motivatedby dark sector theories as discussed above – then the standard model of particles does not feel its associated longrange force, preventing a discharge of these configurations. In this section, we determine and analyze solutions corresponding to charged black holes with a regular horizon. Tomake context with previous work on vector fields in gravity, the non-minimal vector-tensor coupling β A µ A ν G µν breaks the Abelian symmetry and acts as an effective mass for the vector on a given background. However, it differsfrom the usual mass term m A , thus avoiding Bekenstein’s no-go theorems [11] against the existence of black holesolutions in Proca theories. See the discussion in [9] for further details. In our set-up, regular solutions do existfor general values of the coupling constant β and, both the the vector profile A (associated with an electric field F i ) and the scalar profile π (the longitudinal vector polarization), play an important role in characterising thesesolutions.When the parameter β is set to zero, we recover the Einstein-Maxwell action, where there is a unique static,spherically symmetric, and asymptotically flat solution i.e. the Reissner-Nordstr¨om (RN) black hole configuration ds = − (cid:0) − M/r + κQ /r (cid:1) dt + d r (1 − M/r + κQ /r ) + r dθ + r sin θdϕ (6) A = Qr , (7)which corresponds to a charged black hole. In our case, when the single parameter β is turned on, a much richervariety of black hole configurations exist with integration constants controlling vector and scalar configurations.We started the exploration of this topic in the work [9] by focusing on the specific case β = 1 /
4. In this section weextend the analysis and show the existence and behaviour of solutions for general values of β (see also [12, 13]).There are two branches of solutions for our system. Substituting our Ansatz (5) in the equations of motion(3), we find, among other equations, the following key condition (obtained from the off-diagonal components of theEinstein equations): 2 β π ( r ) { n ( r ) − m ( r ) − [ r − m ( r )] n (cid:48) ( r ) } = 0 . (8)When β (cid:54) = 0, this equation shows the existence of two distinct branches of solutions: one with a non trivial profilefor the vector longitudinal polarization (controlled by the scalar π ( r )); the other with π ( r ) = 0. A similar situation– two different branches of spherically symmetric solutions – occur in many examples of modified gravity scenarios,from Gauss-Bonnet gravity [14] to massive gravity (see for example [10, 15–17]). An interesting feature of our caseis that both branches admit asymptotically flat configurations. This implies that there is no uniqueness theoremfor spherically symmetric, asymptotically flat solutions in this theory. Each branch has distinctive features that westudy separately . The existence of two branches of solutions has been also pointed out in the recent paper [13], which we received while our workwas finalised. (See also [12].) Our results are complementary to those presented in [13], where in addition, we numerically describeproperties of black holes in the branch π (cid:54) = 0, analyse properties of vacuum configurations in the branch π = 0, and study neutron starconfigurations. π (cid:54) = 0, we devoted the rest of the mainsections to the study of this branch, and leave the discussion of the second branch, where π = 0, to the appendix A.In Appendix C we instead discuss differences and similarities among vector-galileon and scalar-galileon black holes. In the case π (cid:54) = 0, the constraint (8) imposes the following relation between the metric components m = n − rn (cid:48) − n (cid:48) . (9)A consequence of this algebraic contraint is that this branch is disconnected from the RN configuration (6), whichdoes not satisfy (9). (RN configurations belong to the second branch of solutions, as we shall see.) Moreover, thepolynomial curvature invariants (e.g. the Ricci and Kretschmann scalars) vanish at large r for arbitrary power-lawasymptotic profiles of the metric components when the relation (9) holds. This hints towards the existence ofasymptotically flat solutions.In addition, the radial component of Einstein equations is algebraic in π , with solution π = r β ( r − n ) (cid:20) A (1 − β ) (cid:18) nA r − n + ( rA ) (cid:48) (cid:19) − rA [1 + 8 β ( n (cid:48) − r − n − ( rA ) (cid:48) (cid:21) . (10)The requirement of having a positive right hand side in equation (10) will impose constraints on our configurations.Together with the Bianchi identity, these conditions reduce (3a)-(3b) to two independent equations for A and n ,which in the absence of matter reduce to ξ (1) vac ≡ β − r (cid:26) r ddr (cid:20) A ( A r ) (cid:48) r (2 n (cid:48) − (cid:21) + 2 A (2 n (cid:48) −
1) + A [3 rA − r ( rA ) (cid:48) ] n (cid:48)(cid:48) (cid:27) − n (cid:48) − n (cid:48)(cid:48) κ − rA ) (cid:48) ] n (cid:48)(cid:48) + 2(2 n (cid:48) − rA ) (cid:48) ( rA ) (cid:48)(cid:48) = 0 , (11a) ξ (2) vac ≡ A (4 β − n (cid:48)(cid:48) + ( rA ) (cid:48) n (cid:48)(cid:48) + (1 − n (cid:48) )( rA ) (cid:48)(cid:48) = 0 . (11b)These second order coupled equations admit asymptotically flat black hole solutions, characterized by three indepen-dent integration constants: a mass, a vector ‘electric’ charge that controls the profile of A ( r ), and an independentscalar parameter which controls the profile of π ( r ).Exact solutions to eqs. (11a)-(11b) describing these black holes are not straightforward to obtain in general.Nevertheless, one can make progress analytically for some values of β , or in appropriate perturbative regimes. Thereare two special cases where one may find complete analytic solutions in the branch π (cid:54) = 0: one for a specific choiceof the coupling constant β and the other for a particular value of the vector charge. Charged black hole solutions for β = 1 / β = 1 / n ( r ) = m ( r ) = M, (12a) A ( r ) = Qr + P, (12b) π ( r ) = (cid:112) Q + 2 P Q r + 2
M P rr − M . (12c)The integration constants M and Q represent the mass and electric charge of the configuration. Moreover, P isa new integration constant, absent in the standard Reissner-No¨ordstrom configuration. It controls the asymptoticprofile of π , which decays as r − / if P (cid:54) = 0, or as r − if P = 0. The dimensions of these constants are: [ M ] = length,[ Q ] = length / [ √ κ ] and [ P ] = 1 / √ κ . With electric charge we mean the charge associated with the F i components of the vector’s field strength; we do not necessarilyidentify it with the charge of Maxwell’s electric field. Schwarzschild metric , but with both profiles for A and π turned on.This is an example of a stealth solution, where the effective energy momentum tensor associated to the vectorfield vanishes and the metric reduces to the GR case. For further details on this solution we refer to [9]. ( Seealso [12, 13].)It is interesting to study how the BH solution departs from the configuration (12a)-(12c), when β is slightlydifferent from β = 1 /
4. To understand this regime, we build a series expansion in the parameter ( β − / β − /
4) is well defined below some critical radialvalue r crit . As long as this critical radius is inside the horizon r h , then we can trust the value of the position ofhorizons calculated perturbatively. The analysis simplifies in regimes where P = 0 and M/Q (cid:28)
1, where to obtain r crit < r h one needs 3 | β − / | A ( r h ) κ (cid:28) . (13)In this case, the corresponding position of the single black hole horizon is given by r h ≈ M + ( β − / κQ M + O [( β − / ] . (14)Therefore, as long as our choice of charges and parameters ensures that inequality (13) is satisfied at r = r h , theposition of the horizon gets indeed shifted when β (cid:54) = 1 /
4, by an amount depending on the vector charge. Althoughthere is no analytical guarantee that higher order terms in the series do not spoil this result, we performed numericalchecks confirming that these findings are correct.As a consequence, we notice that vector Galileons can admit black hole solutions which are more compact thanSchwarzschild, since they have a smaller horizon radius for the same black hole mass (more on this later).
Solutions describing black holes with no vector charge (but with a scalar profile)
A second analytic solution in this branch can be found when the integration constant Q associated with the radialdependence of the vector time-like component vanishes: Q = 0. The solution is given by n ( r ) = m ( r ) = M, (15a) A ( r ) = P, (15b) π ( r ) = 12 (cid:115) P ( r − M ) + 8 M P r r ( r − M ) . (15c)This configuration is geometrically described by a Schwarzschild metric, and it is valid for any β . The solutiondepends on two quantities: the black hole mass, M , and the scalar parameter P . In this uncharged case, this isthe same solution one finds for a scalar-tensor theory with non-minimal scalar-tensor coupling ∂ µ φ ∂ ν φ G µν . Wediscuss the relation with scalar-tensor theories in full length in Appendix C. Besides the two analytic solutions we have described, we can obtain analytic results by considering perturbativeexpansions in appropriate parameters. We consider two examples with interesting physical consequences.
Asymptotic expansion of solutions for large radial distances, and asymptotic charges
We analyse the system of equations at large radial distances. In such regime, to determine the solutions for arbitrary β , we expand our fields as 1 − n ( r ) r = 1 + (cid:15) n ( r ) + (cid:15) n ( r ) + O ( (cid:15) ) , (16a) A ( r ) = P + (cid:15) a ( r ) + (cid:15) a ( r ) + O ( (cid:15) ) , (16b)6here (cid:15) is a dimensionless positive small parameter, controlling our asymptotic expansion in powers of 1 /r . Oneshould stress that the powers can be fractional. The small value of (cid:15) reflects the fact that we are far away from thesource. We find the solutions 1 − n ( r ) r = 1 − (cid:15) Mr + (cid:15) Q κδ r + O ( (cid:15) ) , (17a)1 − m ( r ) r = 1 − (cid:15) Mr + (cid:15) Q κδr + O ( (cid:15) ) , (17b) A ( r ) = P + (cid:15) Qr + (cid:15) P Q κδβr + O ( (cid:15) ) , (17c)where the constant parameter δ is defined by δ = (1 − β ) (cid:0) κβ P − κβP (cid:1) − . (18)Moreover, for the particular case of P = 0, the scalar mode has the following form π ( r ) = (cid:15) Q (cid:112) − /β r + (cid:15) M Q (12 β − r (cid:112) β (8 β −
1) + O ( (cid:15) ) + . . . , (19)while for P (cid:54) = 0, it reads π ( r ) = √ (cid:15) (cid:20) P ( M P + Q ) r (cid:21) / + (cid:15) / (cid:2) M P ( M P + Q ) + Q (cid:0) − β − − P (1 − β ) κδ (cid:1)(cid:3) (cid:112) r P ( M P + Q ) + O ( (cid:15) ) . (20a)(Recall the dimensions of the quantities: [ M ] = [ √ κQ ] = [ r ]). As in the exact solution for β = 1 / M and Q are integration constants representing the mass and vector charge respectively, whereas theparameter P controls the asymptotic profile of the longitudinal vector mode π .This series expansion shows that there are two cases for which a Schwarzschild metric is exactly recovered in theweak-field limit: one is for β = 1 / Q = 0, both discussed in Section 3.1 above. In all other cases,there are corrections which start at second in an expansion in inverse powers of the radius. This implies that thesecorrections are suppressed at large distances with small deviations from GR predictions. As a consequence, thecomputation of Komar charges related with the geometry and the conserved vector current J µ = ∇ ν F µν give thesame results as for the standard Reissner-Nordstr¨om configuration (see e.g. [18]): the black holes are charaterisedby the black hole mass M and the vector ‘electric’ charge Q (with the meaning explained in footnote 3) . Thescalar parameter P , on the other hand, is not related with a scalar charge associated with any Gauss law that isvalid asymptotically. Nevertheless, this parameter is important for determining properties of the solution, as theexistence and position of horizons. Small β expansion In theories where breaking a symmetry introduces additional degrees of freedom, it is interesting to ask whetherthere is a well-defined limit where the symmetry is recovered. The vDVZ discontinuity of the Fierz-Pauli massivegravity [19], or the strong coupling problem in Horava gravity [20] are good examples of how delicate such limitcan be, since the theories become strongly coupled. In our scenario, the equations are sufficiently manageablethat a full analytic study can be carried on. A na¨ıve β → π (cid:54) = 0 is notwell-defined, because the scalar longitudinal component π ( r ) of the vector field becomes imaginary. However, aconsistent β → can be defined, if the vector charge Q simultaneously goes to zero in an appropriate way.One should distinguish two cases: β < β >
0. In the case of β <
0, the fields A and π admit the followingsolution for small β A = P + (cid:112) | β | Qr + O ( | β | / ) , (21a)7 = 12 (cid:115) Q ( r − M ) + 8 M P r r ( r − M ) + O ( | β | / ) . (21b)This configuration is valid for any Q and P as long as r > M . On the other hand, if β > A = P + (cid:112) β Qr + O ( | β | / ) , (22a) π = 12 (cid:115) − Q ( r − M ) + 8 M P r r ( r − M ) + O ( | β | / ) , (22b)which requires P (cid:54) = 0 in order to keep π real for any r > M . For small β , the metric reduces to a Schwarzschildconfiguration, plus small corrections1 − n ( r ) r = 1 − Mr + βQ κ r + O ( | β | / ) , (23a)1 − m ( r ) r = 1 − Mr + βQ κ r − β M Q κr + O ( | β | / ) , (23b)Hence, we learn that a vanishing β limit is connected with a Schwarzschild configuration (and not with a RNsolution), accompanied with a non-trivial profile for π ( r ), that in any case is non-physical, being a gauge mode insuch limit. In Appendix A we compare this small β case with the same limit for the other branch of solutions with π = 0. The analytical considerations we made in the previous subsections hint towards the existence of a variety of asymp-totically flat charged black holes, whose properties depend on β , and on the integration constants involved. In thissection, we numerically explore features of black hole solutions in cases where analytical solutions are not avail-able. In Appendix B we analyze the asymptotic behaviour of solutions for large r , and even though the analysisis designed for the neutron star configurations that we will discuss in Section 4, the findings remain essentiallyvalid also for the black holes discussed here. One of the main results of Appendix B is that asymptotically regularconfigurations exist only with a relatively small absolute value | β | . For excessively large values of this parameter,the right-hand-side of eq (10) becomes negative at large values of the radial coordinate; hence the scalar π turnscomplex and the solutions become unphysical. For this reason, in this subsection we choose the representative value β = − / β = − / β = 1 / β = − /
4. We first impose initial conditions on the profile of g tt , A and their derivative at some small value of the radius r = r i . We choose this radius at scale r i ≡ Km,and integrate the equations outwards and inwards. We do not need to integrate g rr , since it is determined by thealgebraic constraint (9). The initial conditions for g tt and A can be expressed in terms of three constants, M , P and Q , in the following way g tt ( r i ) = 1 − M r i (24) A ( r i ) = P + Q r i , (25) A (cid:48) ( r i ) = − Q r i . (26)Notice that the values of these integration constants do not necessarily coincide with the black hole Komar chargesat large radial distances that were discussed in Section 3.2. Indeed, we have seen that the metric components andthe time-like vector profile A receive subleading corrections which scale as higher inverse powers of the radius(at most as 1 /r ). They are sufficient to modify the radial values of the charges. Nevertheless, we numericallychecked that the mass M of the configuration is barely affected by the radial evolution, thus we fix its initial8alue to M = 4 . M (cid:12) ), that approximativelycorresponds to the asymptotic value of this quantity. To analyze the black hole properties, we vary the quantities Q , P , and explore under which conditions we obtain configurations corresponding to regular black holes. We lookfor geometries that are regular everywhere, besides from an essential singularity at the origin, which is covered byan event horizon located at the radial coordinate r h where g rr ( r h ) = 0 . (27)Moreover, regular configurations require that the scalar field is well defined outside the horizon, without turningcomplex (as discussed above). Our results are summarised in Figure 1, where we identify four qualitatively distinctregions of parameters. - - - [ - /[ ]] l og [ /[ ]] - - - - - - - -
50 log [ - /[ ]] - l og [ - /[ ]] Figure 1: This figure represents the black hole behaviour in terms of the initial conditions P and Q imposed at r i =10 Km. In the horizontal and vertical axis we use logarithmic scales: the normalisation factor 10 − is introducedfor convenience, and the [ P ] and [ Q ] factors are used to make the argument of the logarithm dimensionless. Eachpoint in the diagram represents one solution: by varying P and Q , we identify four qualitatively different examplesof configurations. The color notation indicates how the solution behaves: orange (R1) → no event horizon, red(R2) → singularity at large r , grey (R3) → complex π , and blue (R4) → regular solutions. All these solutions have M = 4 . M (cid:12) . In the left panel we show the case when P and Q have the same sign, and in the right panel the casewhen they have opposite signs. The brown line in the right panel is the analytic limit M P + Q = 0 obtained fromeq (20a) for solutions with real π . See the main text for more details. R1 (orange): Asymptotically flat metrics with a naked singularity at the origin:
The shaded orange re-gion corresponds to solutions that do not have an event horizon. Their inwards evolution shows that g rr doesnot vanish at any r . Instead the configuration diverges as r →
0, where the Ricci scalar also diverges,indicating that it is an essential singularity. For large r , the solutions are asymptotically flat. R2 (red): Naked singularities at the origin and at a finite radius:
In the red shaded region we indicatesolutions that do not have an event horizon, and are not well behaved asymptotically, since the Ricci scalaralso diverges at finite large values of r . R3 (grey): Black hole solutions with complex π : The region indicated in grey includes solutions where π becomes complex. We determine them by sampling the value of π at different distances from the source, forall the solutions with an event horizon. R4 (blue): Black hole metrics with real π everywhere: This region is represented with blue points. It in-cludes all the regular BH solutions with an event horizon, and with an everywhere well defined scalar π .9his numerical analysis shows that the condition of having a regular configuration – equipped with a horizonand an everywhere well defined value of the scalar field – singles out a well defined region in the space of parameters P , Q . The boundaries among these regions in parameter space might contain interesting physics, as we will see inthe next subsection.While in this subsection we studied the dependence of the black hole properties from the integration constants P , Q defined at small radial distances, we also repeated the analysis leading to Figure 1 considering boundaryconditions evaluated far from the origin, where the corresponding integration constants are more directly relatedwith black hole charges. We present the details of this discussion in Appendix E. The black hole compactness, that we refer to by the letter C , is an important property to characterise black holes.It is defined in terms of the ratio of the black hole mass versus the position of the horizon (in Planck units). Forthe most famous (and relevant for our discussion) black hole solutions its corresponding values are C = 12 Schwarzschild BH , (28)12 ≤ C ≤ . (29)Given that detection of gravitational waves from black hole mergers can allow one to reconstruct geometricalproperties of black holes as their size and mass, it is important to examine whether our set-up leads to values of C that can be different from the above ones.The answer is affirmative, and we examine two aspects of the question. First, we focus on the β = − / r h that are small, suggesting a potentially large C . This is confirmed in Figure 2 where it is shown that by choosingappropriately the values of the available integration constants, we can get configurations with large compactness C – up to twice the value of RN black holes – in the proximity of the boundaries between regions. Mr h log(10 − Q )log P Mr h Figure 2: Vector Galileon black holes with high compactness. For all the solutions in this plot the black hole massis M ≈ . M (cid:12) . The colour map at the bottom is intended to help to visualise the borders of this region and thezones of high compactness, the empty part corresponds to R1 in the notation of Section 3.3, where the solutionsdo not have an event horizon.A second possibility for studying black hole compactness consists in fixing the asymptotic integration constantsto representative values and vary the coupling constant β , while numerically searching for the values of maximumcompactness. We present the results of this search in Figure 3, which shows that maximal compactness depends ina non-trivial way on the value of β . For a choice log(10 − Q/ [ Q ]) = 10 .
6, log( P/ [ P ]) = 27 .
5, and M = 4 . M (cid:12) , thecompactness is of order C (cid:39) β = − / - β M / r h - - β r h ( K m ) Figure 3: Compactness for different β ’s at fixed P = 3 . × cm / gr / / s , Q = 4 . × cm / gr / / s and M = 4 . M (cid:12) . The points at r h = 0 do not correspond to black holes but to naked singularities at the origin. Neutron stars (NS) represent a promising arena for studying gravity in a strong field regime, and for revealingdepartures from GR in modified gravity scenarios (see e.g. [4, 21] for reviews). For example, in the context ofscalar-tensor Brans-Dicke theories, Damour and Esposito-Farese have pointed out the phenomenon of spontaneousscalarization of NS [22]. This effect can lead to large deviations from GR predictions, even in theories that satisfyPPN constraints in a weak field regime. The physics of neutron stars have also been studied within the Horndenskiscalar-tensor framework. In particular, for a non-minimal derivative coupling of the scalar field to the Einsteintensor, by tuning the scalar field coupling constants and charge it has been numerically shown that it is possibleto obtain NS’s with masses up to approximately 2.5 M (cid:12) [23, 24]. In this case, the fields outside the NS’s aredescribed by the stealth Schwarzschild solution. NS’s are also known in Einstein-Dilaton-Gauss-Bonnet gravity(see [25] and references therein), which is another sub-sector of Horndeski where the scalar field couples linearly tothe Gauss-Bonnet term. See [23, 26] for reviews.In this section we study NS solutions in our vector Galileon scenario, with the main aim to find NS systems withdistinctive properties that make them distinguishable from GR. We show that the additional vector field profileassociated with A can lead to consistent NS configurations that are more massive and more compact than theirGR counterparts. We focus on static, electrically charged configurations, and towards the end of this section wealso comment on the possibility to include rotation and magnetic fields. This is interesting in the case where thenon-minimal vector-tensor coupling is considered as modification of Maxwell electromagnetism in strong gravityregime.In order to analyse NS configurations, we need to first determine solutions for the field profiles in the exterior ofthe star. These configurations are then smoothly connected with interior solutions, which depend on the equation ofstate of the internal matter. Interior solutions are determined by hydrostatic equilibrium configurations controlledby the matter which forms the star, and by the theory of gravity under consideration – in our case a vector Galileonmodel. For definiteness, we focus on the π (cid:54) = 0 branch of solutions of the constraint condition (8). In this branch,the geometrical exterior solutions are well described by the Schwarzschild geometry, plus corrections that rapidlydecay at large distances, even in presence of non-trivial vector and scalar profiles, as we have discussed in theprevious sections. Moreover, the analysis of Appendix B shows that, for this branch of solutions, the value of β cannot be too large in absolute value, otherwise the scalar field profile π becomes complex in the exterior of the star.To avoid this, we make the same choice β = − / β we find physically interesting NS solutions with sizeable differences from theirGR counterparts. We checked that if β is taken positive one does not find configurations that significantly differentfrom GR.Before proceeding, it is worth to remark that in the main text we do not consider direct couplings between thevector Galileon field and regular matter. In other words, we assume that the only interaction between the star’s Notice that the analysis of Appendix B uses values of neutron star properties – radius, mass – corresponding to neutron star objectssimilar to GR. As we will learn in this Section, for objects with more exotic mass or radius one can find unphysical cases where thescalar turns complex even for β = − / χ ( r ) as ρ ( r ) = ρ (cid:18) χ + K Γ − χ Γ (cid:19) , (30a) p ( r ) = Kρ χ Γ , (30b)with ρ = c n m b = c . × gr / cm . n is the baryon number density and m b the baryon mass. Thepolytropic constant K and exponent Γ take values appropriate to adjust the observed masses of NS. In GR, thevalues Γ = 2 .
34 and K = 0 . M = 1 . ± .
04 Solarmasses [28] and PSR J0348+0432 with M = 2 . ± .
04 Solar masses [29]). We have verified that the qualitativefeatures of our configurations do no change if a more accurate equation of state is considered.The covariant equations of motion (3) in the case of the neutron star configurations we are considering simplifyin the following way. In the interior of the star, there are two algebraic conditions which determine the profiles forthe metric component m ( r ) and the scalar field π ( r ), given by m ( r ) = n − rn (cid:48) − n (cid:48) , (31a) π ( r ) = r β ( r − n ) (cid:20) r (1 − n (cid:48) ) p − A (4 β − (cid:18) nA r − n + ( rA ) (cid:48) (cid:19) − rA [1 + 8 β ( n (cid:48) − r − n − ( rA ) (cid:48) (cid:21) . (31b)Moreover, we have dynamical equations for the remaining quantities, which have the same structure as in theexterior of the object, but with contributions associated with the star energy momentum tensor, namely0 = ξ (1) vac − r ρ (2 n (cid:48) − r − n + 2 pr (2 n (cid:48) − (cid:20) (2 n (cid:48) −
1) (4 n − r + 2 rn (cid:48) )2 n − r − rn (cid:48)(cid:48) (cid:21) , (32a)0 = ξ (2) vac . (32b)where ξ (1) vac and ξ (2) vac are defined in eq (11a) and (11b). Together with the equation of state (30a-30b), and theenergy-momentum tensor conservation, ∇ µ T µν = 0, these equations further determine the time-like component A of the vector, the 00-component of the metric (the function n ( r )), and the density and pressure of matter.We numerically solve these equations imposing appropriate boundary conditions, and choosing the value β = − / n ( r = 0) = 0, requires A (cid:48) (0) = 0. We are left with only one free parameter to fully determine the initialcondition at the origin: A ( r = 0) ≡ A c . To work with dimensionless quantities, it is convenient to parametrisethe initial conditions for A c as a c = A c / (10 cm / gr / / s) , so that a c is dimensionless and its logarithm of order 1. Our aim is to determine how NS configurations dependon a c , and how the results differ from standard GR configurations. We match the interior solution at a starradius R (cid:63) with a regular, asymptotically flat exterior solution for the field equations. The radius R (cid:63) correspondsto the value of the radial coordinate where the internal pressure vanishes. Figure 4 shows our results in terms ofmass-radius (M-R) curves for different initial conditions a c . Vector Galileons allow neutron star masses and radiilarger than in GR, whose permitted configurations are represented with a black dashed line in Figure 4. The fieldprofiles are well defined for any a c ≥
5, and our configurations are stable according to the static stability criterion dM/dρ c > a (cid:38)
10 we get masses larger than the upper bounds theoretically estimated in GR for anyEoS [31]. Given the additional degree of freedom, it is expected that modified gravity models may help to relax theGR mass bounds [32]. We re-adjust the value K = 0 . (cid:72) Km (cid:76) MM (cid:159) n m b n m b K (cid:61) (cid:71)(cid:61) Β(cid:61)(cid:45) a (cid:32) c Ρ c Figure 4: Mass-radius curves for neutron stars in vector Galileons for β = − /
4. We use a polytropic EoS with K = 0 .
014 and Γ = 2 .
34 . Each line corresponds to solutions with the same central value a c indicated by thecolour bar on the top left corner of the plot, and each point along these lines corresponds to a solution with centraldensity ρ c indicated by the sidebar. The dark dashed line is the M - R curve adjusted in GR to fit the maximumobserved mass of a NS, it is computed using a polytropic EoS with K = 0 . .
34. This plot showsthat for β = − / m ( R ∗ ) larger than in GR. Circled pointscorrespond to unphysical configurations where the scalar field turns complex inside the star radius. a c The initial condition corresponding to the value of the vector profile at the origin, a c , represents the key parameterfor controlling the NS solutions. As we decrease a c , maintaining the same values for the EoS parameters, we findthat the scalar π turns complex, hence the configuration has to be discarded.At first sight, Fig. 4 would seem to suggest that NS solutions exist for arbitrarily large values of a c . However,this is not always the case. In Fig. 5 we explore a larger range of values of a c for fixed central NS densitiesapproximately up to the critical density associated with maximal mass in the M - R curves. We find that theproperties of solutions with high a c depend on ρ c , the value of the density at the center of the star. For lowdensities, there are equilibrium configurations in the complete range of a c that we explored. Instead, for relativelyhigh densities we find an upper bound on a c beyond which the radius of the star cannot be defined in the standardway, since we do not find any radial position R (cid:63) where the pressure of matter vanishes, hence the star has no naturalboundary.There exist additional constraints on the allowed range for a c , which arise when studying in detail the externalfield configurations. When a low central density ρ c is chosen all the solutions can be matched to healthy exteriorsolutions. But when ρ c is increased, this is not the case. The black points in the left panel of Fig. 5 correspond tointerior solutions that can not be matched to an asymptotically flat exterior spacetime, the black lines in the rightpanel show that the scalar field profile π diverges for these solutions.Interestingly, by calculating the NS mass using asymptotic properties of the exterior solutions, we always find aNS mass larger than in GR. We exemplify this fact in Fig. 6. As well as in the black hole case, the compactness of neutron stars is an important property that can be usedfor observationally characterizing these objects. In this subsection we show that, despite the constraints one hasto satisfy, the exterior solutions of NS lie in a region of the parameter space where the NS compactness can beincreased with respect to GR. 13 (cid:61) (cid:71)(cid:61)
Β(cid:61)(cid:45) Ρ c (cid:61) n m b0 Ρ c (cid:61) n m b0 Ρ c (cid:61) n m b0 Ρ c (cid:61) n m b0 Ρ c (cid:61) n m b0 (cid:64) a c (cid:68) m (cid:72) R (cid:42) (cid:76)(cid:144) R (cid:42) Ρ c (cid:61) n m b0 Π (cid:144) Π N Ρ c (cid:61) n m b0 (cid:144) R (cid:42) Π (cid:144) Π N Figure 5: Range of interest for a c . Each point in the left panel corresponds to a static configuration computed withthe central density shown in the plot. At high densities there is a maximum a beyond which star configurationsdo not exist. Red points represent solutions for which π is complex, blue points regular solutions, and black pointssolutions that cannot be matched to a regular exterior solution. The right panel shows the interior and exteriorscalar field profiles for configurations with the same central density, each profile is associated to a point in the leftpanel. For a c (cid:46) m ( R ∗ ) and R ∗ remain constant. For a c (cid:38) and low densities, m ∗ and R ∗ increase with a c ,but the ratio m ∗ /R ∗ remains constant.In GR the exterior solution of a NS is exactly Schwarzshild. The compactness is then determined by dividingthe Schwarzschild mass (in units of distance) over the radius of the star, and since R is always larger than theSchwarzschild radius the compactness is always less than that of a Schwarzschild black hole, i.e., 1 / M/r s >M/R .In vector Galileons, as shown in the previous sections, the asymptotic mass gets contributions from the vectorfield outside the star which can assume a non-trivial profile. Therefore, in order to obtain the compactness of a NSwe first extract M from the exterior solution evaluated at a large r – where the asymptotic Schwarzschild solutionholds – and then divide it by the radius of the star. The result is always smaller than the compactness of a vectorGalileon black hole. This is expected, but interestingly, the compactness of such black holes can be larger than thatof a Schwarzschild BH, opening the possibility for NS in vector Galileon theories that are more compact than GRblack holes.In Fig. 7 we represent the results of a numerical analysis aimed to exemplify the discussion above. Each point inthe plot corresponds to a global solution (i.e., a NS plus the exterior solution) with initial conditions indicated by r (cid:72) Km (cid:76) G c (cid:45) m (cid:72) r (cid:76)(cid:144) M (cid:159) r (cid:72) Km (cid:76) Π Π N Figure 6: Interior (black line) and exterior solutions (grey line) for β = − /
4. Solid lines are for a (0) = 6 . a (0) = 12 .
0. The central density is ρ c = 5 . n m b . The left panel shows m ( r ) in units of solarmasses, both lines reach asymptotic constant values. The dotted line is the GR solution.14 .5 (cid:180) (cid:180) (cid:180) (cid:72) Km (cid:76) MR n m b n m b K (cid:61) (cid:71)(cid:61) a c Ρ c Figure 7: Compactness of NS. This quantity is computed with the Komar mass of the exterior solutions thatmatch each NS configuration with the central value of ρ and A indicated by the color bars in the plot. Onlystable configurations are shown. The black solid line shows the compactness of the same configurations in GR forpolytropic stars modelled so that the M-R curve gives a maximum mass around 2 M (cid:12) . For the vector Galileonswe chose K = 0 .
014 and Γ = 2 .
34. The dashed horizontal lines show GR limits for the compactness of sphericallysymmetric and static solutions:
M/R = 0 . M/R = 4 / ρ c is given by the point colour and the bar to the right of the plot,while A c is given by the dashed line on which the point resides and the colour bar inside the plot. The remaininginitial conditions are g tt (0) = 1, g (cid:48) tt (0) = 0, and A (cid:48) (0) = 0. The interior of the NS is modelled using a polytropicEoS with K = 0 .
014 and Γ = 2 .
34. This choice leads for M - R curves with M max ≥ M (cid:12) and real π . For thesolutions circled in red, π becomes complex inside the star. From each interior solution we obtain the radius R ofthe respective star, which is given in the x -axis of the plot, and a set of initial conditions for the exterior solution.From the exterior solution we determine M . Knowing R and M we calculate the compactness of each star, and theresults are shown in the y -axis of the plot.Besides the compactness of NS in vector Galileons, Fig. (7) also displays the following reference lines: the dashedhorizontal lines show GR limits for the compactness of spherically symmetric and static solutions: M/R = 0 . M/R = 4 / K = 0 . .
34, so that the M-R curve gives a maximum mass around2 M (cid:12) . The blue shaded region outlines the compactness of the same neutron stars solutions when we ignore thecontributions of A µ outside the star, i.e. we construct the global solution taking the same set of vector GalileonNS configurations corresponding to the points in the plot, but using a Schwarzschild configuration for the exteriormetric. The red shading extends this to solutions with higher values of the central density, these solutions areunstable. As the plot shows, the solutions in the shaded region can never be more compact than a Schwarzschildblack hole; this option is possible only for vector Galileons, thanks to the interactions between gravity and thevector field in the exterior of the star.To conclude our discussion on neutron stars, we comment on possible future developments, in case the non-15inimal vector-tensor couplings we consider is used to parameterize modifications of Maxwell electromagnetism instrong gravity regimes. While in this section we considered a situation in which matter in the NS interior does notdirectly couple to the vector (but only indirectly through gravity) we could directly couple A µ with internal currents.It is known that for standard electromagnetism such couplings can modify the equilibrium configurations [33] sinceelectric currents modify the equations of hydrostatic equilibrium. A complete treatment of this topic for vectorGalileons is under investigation, and we include preliminary results in Appendix G . We find that, in certain cases,the inclusion of an interaction term A µ J µ does not spoil the existence of neutron stars solutions in this model, andcan improve the properties of the scalar field π , in such a way that configurations with complex π (that we had todiscard in our analysis above) can be turned into configurations with real scalar π . Other possible developments leftfor the future include the addition of rotation and magnetic fields (with F ij (cid:54) = 0) to explore possible connections withmagnestars. We make preliminary steps to find configurations with magnetic fields for vector Galileons in AppendixF, while rotating exterior configurations (for small values of the rotation parameter) are discussed in [9, 12]. In thefuture, we plan to study whether I - Love - Q relations [34, 35] – which relate the moment of inertia I , the tidal Lovenumber, and the quadrupole moment Q of the star – get modified in the context of vector Galileons. We studied black hole and neutron star configurations in a vector-tensor theory of gravity, a special case of vectorGalileons, described by a simple, one parameter modification of the Einstein-Maxwell action, corresponding to thefollowing non-minimal vector-tensor coupling β √− g G µν A µ A ν , with G µν the gravitational Einstein tensor and β a coupling constant. Such coupling term can be used in the contextsof vector inflation, or a vector governing dark energy or dark matter. Alternatively, one may think of this vector asa way to parameterise deviations from the standard Maxwell’s electromagnetism in regimes of strong gravity. Thephysics of black holes reveal surprisingly rich properties, that generalize the standard Reissner-Nordstr¨om solutionof Einstein-Maxwell theory of gravity, but also black hole configurations of related scalar-tensor theories. There existtwo disconnected branches of static, spherically symmetric, asymptotically flat solutions, whose features we studiedin detail. Solutions have a mass M and a vector charge Q , and are also characterized by an integration constant P mainly controlling the profile of the longitudinal scalar polarisation of the vector ( P is not however associated to aconserved asymptotic charge). The existence and position of horizons depend in a non-trivial way by the parametersinvolved. In some cases analytic configurations are available, while for most values of the parameters involved weextract the behaviour of solutions through a numerical analysis. We find that in certain regions of parameter spaceblack holes can be more compact than in General Relativity, providing a distinctive feature of black holes in thisvector-tensor set-up. In an Appendix, moreover, we study in detail differences and similarities of our black holeconfigurations with solutions of scalar-tensor theories.With respect to the neutron stars, our study shows that the vector profile plays an important role in determiningthe star configuration, both for controlling its internal hydrostatic equilibrium configuration, and for determiningthe external gravitational solution which is generally not described by a pure Schwarzschild geometry. The prop-erties of neutron stars in this vector-tensor theory are quite rich; they can be larger and more massive than theirGR counterparts and, for certain parameter choices, more compact. In some cases, they might be even morecompact than Schwarzschild black holes, making this objects observationally interesting for the prospect of gravitywave detection. We also comment on possible generalizations to magnetically charged or rotating configurations,presenting in appendixes some preliminary calculations on this respect. This can be of interest especially whenthis vector-tensor non-minimal coupling is used to describe deviations from Maxwell theory, for describing exoticcompact objects as magnetars.Future interesting developments will include a study of rotating exterior solutions for neutron stars and blackholes – with arbitrarily large rotation parameter – and an analysis of possible generalizations of I-Love-Q relationsfor neutron stars in vector Galileon theories. We hope to report soon on these topics. Acknowledgments
JC is supported by STFC and CONACyt grants 263819 and 179208. GN is supported by CONACyT grants:179208, 269652, Fronteras de la Ciencia 281, and by DAIP-Universidad de Guanajuato grant 1,046/2016.16 ppendixesA Black hole configurations for the branch π = 0 In this Appendix we discuss black hole configurations in the second branch of solutions satisfying the constraint(8) with π = 0. This branch contains the Reissner-N¨ordstrom configuration in the limit β = 0. However, fornon-vanishing β , the profile for A changes the geometry more drastically than in the other branch π (cid:54) = 0, leadingto modifications of the GR geometry at large distances.The algebraic constraint (8) vanishes identically for this branch, and the equations of motion (3a)-(3b) give twoindependent differential equations for A and one of the metric functions n and m . It is, however, more convenientto present the system as a set of three differential equations that are first order in derivatives of metric, namely0 = r κ ( r − m ) A (cid:48) − (cid:0) r − rβκA − n (cid:1) m (cid:48) , (33a)0 = A (cid:48) [ r ( n ( − m (cid:48) ) + r (2 − m (cid:48) + n (cid:48) )) + m (8 n − r (3 + 2 n (cid:48) ))]+ ( r − n ) (cid:0) − βA m (cid:48) + r ( r − m ) A (cid:48)(cid:48) (cid:1) , (33b)0 =8 n + 2 rn (cid:0) − βκA + 8 rβκA A (cid:48) + r κA (cid:48) − n (cid:48) (cid:1) − r (cid:0) rβκA A (cid:48) + r κA (cid:48) − n (cid:48) + 4 βκA n (cid:48) (cid:1) + 2 m (cid:2) rβκA ( r − n ) A (cid:48) +( r − n ) (cid:0) r κA (cid:48) − n (cid:48) (cid:1) + 2 βκA ( r − n + 2 rn (cid:48) ) (cid:3) . (33c)The first (third) equation is algebraic in n ( m ), and the result can be plugged back in the remaining two equationsto find the two independent, but not first order in derivatives of the metric functions, equations for m ( n ) and A .We can not determine an analytical solution for this branch. We can proceed with studying solutions usingperturbative expansions, as done for the first branch π (cid:54) = 0 in the main text. Asymptotic expansion for large values of the radial coordinate
For r (cid:29) /r regime for n ( r ), A ( r ), and m ( r ). We find1 − n ( r ) r =1 − (cid:15) M + κβP ( M P + 2 Q )] r (1 − P βκ ) + (cid:15) κ r (1 − P βκ ) (cid:110) Q (1 − β ) − κP β (cid:2) Q + 4 (cid:0) M P + 2 M P Q − Q (cid:1) β (cid:3) − κ P β ( M P + Q )( M P + 2 Q ) (cid:111) + O ( (cid:15) ) , (34a)1 − m ( r ) r =1 − M (cid:15)r + (cid:15) Q κ r (1 − κβP ) + O ( (cid:15) ) , (34b) A ( r ) = P + Q(cid:15)r − P Q (2 M P + Q ) β(cid:15) κ r (1 − κβP ) + O ( (cid:15) ) . (34c)In contrast to the case π (cid:54) = 0, the asymptotic geometry in this branch is more sensitive to the vector charge, whichaffects the geometry contributing already at first order in the expansion parameter (cid:15) . It would be interesting tostudy in detail whether asymptotic scalar charges can characterise black hole configurations in this branch. Noticehowever that we could define a stronger weak-field limit, imposing that the parameter P , which enters at r orderin the expansion for A , is small: P → (cid:15) P . In this case, the corrections to the geometry associated with the vectorcontributions (parameters P and Q ) are pushed to second order in an (cid:15) expansion, and the geometry would becharacterized by a Komar mass controlled by the parameter M . Small β expansion This branch is smoothly connected to the RN solution in the limit β →
0. The simplest way to see this is to considerthe following ansatz: 1 − n ( r ) r = 1 − Mr + βf ( r ) + β f ( r ) + . . . , − m ( r ) r = 1 − Mr + βg ( r ) + β g ( r ) + . . . , ( r ) = (cid:112) βa ( r ) + β / a ( r ) + . . . , (35)(36)and solve for the functions f , f , g , g , a , a , . . . . At leading order in β the RN solution is generated: g = f = Q M p r , (37) a = P − Qr . (38)This leading order solution perturbatively reconstruct the first term in the RN geometry. At next-to-leading orderwe find corrections only to the time component of the metric: g = a = 0 , (39) f = − M P M p r + 2 P QM p r − Q M p r . (40)It is possible to solve to higher orders in β , where corrections to A and g rr are present too, however the analyticexpressions are not particularly interesting. We verified that solving to the next orders in β does not lead to newintegration constants: any integration constants that arise at each higher order, can be reabsorbed in the definitionsof M , P and Q . B Exterior solutions in the branch π (cid:54) = 0 In this appendix we numerically study exterior solutions for neutron star configurations, for different values of theparameter β in the branch π (cid:54) = 0. Although we have in mind neutron stars, the same analysis remains valid outsidethe horizon of black hole configurations. Hence we study numerical solutions to eqs. (11a,11b) for A ( r ) and n ( r )for representative values of β , and use eqs. (9,10) to determine π ( r ) and m ( r ). We use c.g.s. units to ease thecomparison to the literature on NS. To find numerical solutions, we set initial conditions at a typical radius ofneutron stars, R ∗ = 12 Km, and take values for n ( R ∗ ), n (cid:48) ( R ∗ ), A ( R ∗ ) and A (cid:48) ( R ∗ ) within a range motivated bythe interior numerical solutions that we investigate in the main text. From now on, all quantities with a subscript‘ ∗ ’ denote quantities evaluated at R ∗ , for example A ∗ = A ( R ∗ ). In order to work with dimensionless quantities,we introduce a normalisation for A : a = A / (10 cm / gr / / s), with a dimensionless. This choice is made tomatch the order of magnitude of the quantity 1 / √ κ = 6 . × cm / gr / / s which turns out to have the samedimensions. Analogously, we also include a normalization for the scalar, π N = 10 cm / gr / / s, and plot thedimensionless quantity π/π N .We plot our results for a representative choice of parameters in the Figure 8, and comment on their physicalconsequences below. • Fig. 8a represents the metric components g − rr and g tt as function of the radial coordinates, for different choicesof β (the colour code for each line is explained at the bottom right panel of Fig. 8). When β (cid:54) = 1 /
4, thesequantities are not equal. Asympototically, g rr → g tt does not tend to unity atlarge r . Nevertheless, we can always redefine the time coordinate in such a way that g tt = 1 for large r , so tohave an asymptotic configuration corresponding to Minkowski space written in spherical coordinates. • Fig. 8b represents the radial evolution of mass of the object in units of Solar masses, for different choices of β .The asymptotic values of these quantity correspond to the Komar mass. While for β = 1 / m ( r )stops evolving outside the star, for different β ’s the vector field continues to source the Einstein tensor andconsequently the gravitational mass is not fixed at R ∗ . As a result, the mass observed asymptotically in themodels with β < / β have to be discarded since m takes negative asymptotic values. Indeed, we have checked that for large values of β the solutions developessential singularities at large values of r outside the star surface, hence they are physically not interesting. • Fig. 8c represents the radial profile of the time-like component of the vector field, a = A / (10 cm / gr / / s).For any β (cid:54) = 1 / A decays faster than for β = 1 / r ; that is, it decays faster than 1 /r . We do notfind any hint of singular behavior in the numerical solutions for A .18
00 200 300 400 5000.70.80.91.01.11.2 r (cid:72) Km (cid:76) (cid:56) g rr (cid:45) , g tt (cid:60)
20 30 40 50 60 70 80123 r (cid:72) Km (cid:76) G c (cid:45) m (cid:144) M (cid:159) (cid:72) Km (cid:76) a
20 30 40 50 60 70 800246 r (cid:72) Km (cid:76) Π (cid:144) Π N Β (cid:61)
Β (cid:61) Β (cid:61)
Β (cid:61)(cid:45) Β (cid:61)(cid:45) Figure 8: Clockwise from top-left, we represent various quantities as functions of radius r : a) g tt (dashedlines) and g − rr (continuus line) components of the exterior metric; b) m ( r ) measured in solar masses; c) a ( r ) = A ( r ) / (10 cm / gr / / s); d) the dimensionless quantity | π ( r ) /π N | . The colour code in each plot is indi-cated in the bottom right panel. The horizontal line in the top left panel a) is a reference to make it easier to verifyvisually that g rr → n ∗ = 2 . , n (cid:48)∗ = 0 , a ∗ = 4 , a (cid:48) ∗ = − .
27 Km − .See the main text for an analysis of the plots. • Fig. 8d represents the radial profile for the absolute value of the scalar field | π | . While for small values of β such radial profile is smooth, it develops singularities when β is sufficiently large (of order 10 in absolute value),since such function becomes negative at finite values of r . This should be connected with the singularities inthe geometry that we find for large values of β .The results obtained indicate that | β | cannot be arbitrarily large, otherwise the scalar field becomes complex. C Difference between vector Galileon and scalar Horndenski theories
In this appendix we discuss how black hole configurations in the vector-tensor (VT) system (2) we are consideringdiffer from scalar-tensor (ST) theories as Horndeski. A comparison between these theories is not straightforwardgiven the distinct nature of the fields involved. For example, in general ∇ µ A ν (cid:54) = ∇ ν A µ unless A µ = ∂ µ φ for somescalar field φ . Furthermore, the number of equations of motion is in general different in the vector and scalar cases.The main message of this appendix will be that, when a comparison is possible, the vector Galileon model allows forphysically interesting generalisations of the black hole solutions found in the Horndenski scalar case, also clarifyingsome of the properties of the latter. First difference: the set of equations
Black hole configurations have been extensively studied in scalar-tensor theories. A particularly interesting subclassof scalar-tensor theories is known as the
Fab Four [36], and has the property that it is the only subset of Horndeski’sgravity where the cosmological constant can be self-tuned, meaning that Minkowski and cosmological solutions are19llowed for any value of the bare cosmological constant. Within the Fab Four, there are minimal subsets that allowfor self-tuning solutions while keeping some simplicity in their analytical treatment. An example is the action S = (cid:90) d x √− g (cid:20) ˜ κ ( R − −
12 (˜ αg µν − ηG µν ) ∇ µ φ ∇ ν φ (cid:21) + S m , (41)where ˜ κ , ˜ α and η are coupling constants and φ a real scalar field. in the language of Fab Four, this corresponds toa combination of the George and
John
Lagrangians that has been studied in cosmological and static contexts (seefor example [23, 37]). When focussing on static spacetimes, the choice ˜ α = 0 and a scalar field profile with lineartime dependence φ ( t, r ) = P t + ψ ( r ) (42)leads to a black hole solution with a Schwarzschild metric.Eq. (42) is our starting point for a concrete comparison to the vector-tensor model: if we write A µ = ∂ µ φ , then ∂ r φ ( t, r ) = ∂ r ψ ( r ) plays the role of π ( r ), and ∂ t φ ( t, r ) = P plays the role of a constant time like component A ofthe vector field.At first sight, one might think that under these identifications the vector system (2) is equivalent to (41) with˜ α = 0 (remember that F µν vanishes for a constant A = P ). However, this is not the case because the vector equationof motion derived from (2) is more restrictive than the scalar equation derived from (41): ∇ µ ( G µν ∂ ν φ ) = 0, wherewe set ˜ α = 0. While in the ST case only the vanishing of the radial component of G µν ∂ ν φ is required (by the ( t, r )component of the metric equations), in the VT case the time component of G µν ∂ ν φ is required to vanish as well, bythe vector equations of motion. Under the spherically symmetric ansatz (5a), and with A = P , this last conditionis P m (cid:48) ( r ) = 0. Assuming P (cid:54) = 0, the only option left is m ( r ) =constant, which leads to the stealthy Schwarzschildsolution when in vacuum (see Section 3.1). Hence in vacuum, and with the simplest, constant profile for the vector A component, ST and VT have the stealth Schwarzschild solution in common. The difference among the twosystems arise in presence of matter: while in the ST case solutions with ansatz (42) can be found, in the VT with A = P (cid:54) = 0 the constraint m (cid:48) = 0 forbids the existence of regular solutions.Therefore, in the specific case A = P the vector model is more restricted than the scalar model. On the otherhand, the advantage of the VT model is that, as shown in the previous sections, we can turn on a non-trivial,radial dependent profile for A and still find physically interesting vacuum solutions, including Schwarzschild, aswell as physical solutions in presence of matter (as discussed in our analysis of neutron star configurations). Thesesolutions do not have an equivalent in the scalar case: a static vector field with non-trivial profiles for the time andradial components is not obtainable in a ST theory (see also [12] for a related discussion). Second difference: black hole hairs
Another physically relevant difference with respect to scalar-tensor theories lies in the “hair” of the spacetimesolutions. In ST, there is a no-hair theorem for the shift-symmetric sub-sector of Horndeski gravity [38], a sub-sector that includes action (41). Under the assumptions that the spacetime is static, spherically symmetric andasymptotically flat, and the scalar field is a function only of r , regular BH solutions do not support non-trivialprofiles of the scalar field (see [3, 23] for a concise review of the theorem and possible ways to circumvent it).Relaxing these assumptions BH’s with scalar hair have been found [39], although this hair is dubbed “secondary”since it is not independent of the other charges (mass and electric charge for static BH’s). Solutions with primaryhair have been found in bi-scalar extensions of Horndeski gravity [40].It is worth to clarify whether these no-hair theorems have any consequence for the vector model studied inthis paper. When A µ = ∂ µ φ the answer is yes, and the only BH that we can obtain is Schwarzschild. This isconsistent with ST theories of the form (41) where, as discussed earlier, the only solution for ˜ α = 0 with a scalarfield linear in t is the Schwarzschild metric. As soon as we allow for a non-trivial profile of A , we break the relation A µ = ∂ µ φ and we are able to find asymptotically flat BH solutions with primary hair, without introducing a timedependence in the vector field. This corresponds to the vector charge, denoted with Q in the main text, analogouslyto Reissner-Nordstr¨om black holes. Recall that on the branch of solutions with a non-trivial profile for a scalar field π (cid:54) = 0, we have an independent integration constant (denoted with P in the main text) which plays an importantrole in controlling the scalar profile. However, this integration constant is not associated with any conserved charge,and does not obey a Gauss law: hence it can not be considered a scalar hair.20 nother class of vector Galileon black hole solutions For the sake of further comparison with the scalar tensor case, we can investigate what happens removing the term F µν F µν in (2). This corresponds to a system that we do not consider in the rest of the paper, but which has thesame Lagrangian structure as (41) with ˜ α = 0. In vacuum, for A = P we have the stealth Schwarzschild solution.A more general solution with non-trivial A exists, given by n ( r ) = m ( r ) = 2 M, (43a) A ( r ) = P + 2 P Qr , (43b) π ( r ) = 2 P ( M P + Q ) r ( r − M ) . (43c)Asymptotically, the vector field profiles are A ( r ) = P + Qr − Q P r + O ( r − ) , (44a) π ( r ) = (cid:112) P ( M P + Q ) (cid:18) r / + 2 Mr / + O ( r − / ) (cid:19) . (44b)The origin of this solution is again in the vector equation of motion, which restricts the geometry to be Schwarzschild;and in the absence of the restriction A µ = ∂ µ φ for some scalar field φ . D Charged black hole solutions for β ≈ / In this Appendix we provide more details on the arguments sketched in Section 3.2. We examine a region of interestin the available parameter space nearby the value β = 1 / π (cid:54) = 0. To search for solutions in thisregion, we express our fields as n ( r ) = n +( β − / n ( r )+ . . . , and similarly for A , m ; then we solve the equations(11a)-(11b) order by order in β − /
4. We find1 − n ( r ) r = 1 − Mr + ( β − /
4) 8 Q κr ( κP −
4) + O [( β − / ] , (45a)1 − m ( r ) r = 1 − Mr + ( β − / (cid:18) − Mr (cid:19) Q κr ( κP −
4) + O [( β − / ] , (45b) A ( r ) = P + Qr + ( β − /
4) 4
P Q κr ( κP −
4) + O [( β − / ] , (45c) π can be determined from (10), resulting in the same expression as (12c), plus O ( β − / r h , located at the points where g rr = 0, we need toinvestigate whether these configurations can be trusted along the entire range of radial direction. To do so, we canestimate the size of higher order corrections O [( β − / n ], n ≥
2, to the previous formulae. The corrections oforder O [( β − / ] are not difficult to calculate, and either by numerical analysis, or analytically but using someapproximations ( P and M small). We find that for some values of the parameters M, P and Q the terms quadraticin ( β − /
4) become larger than terms linear on this quantity: there exists a critical value r c of the radial coordinatebelow which our expansion becomes unreliable. If r h ≤ r c , our estimate of the position of horizons is unreliable.A full expression for r c in (45b) is complicated and not particularly illuminating. On the other hand, we can getintuition of its structure by considering the simplified case P = 0 and M/Q <<
1. Then we find that our seriesexpansion is well defined up to second order in ( β − / | β − / | A ( r ) κ (cid:28) . (46)We then find an event horizon at the position r h ≈ M + ( β − /
4) 4 κQ M − κM P + O [( β − / ] . (47)as long as our choice of charges and parameters involved ensures that inequality (46) is satisfied at r = r h . Expression(47) shows that the position of the horizons get indeed shifted when β (cid:54) = 1 /
4, by an amount depending also on21he charges involved. Although there is no guarantee that the third and higher order terms in the series expansion(45a)-(45c) are also consistent under the conditions discussed above, we numerically checked that these findings arecorrect.
E Black hole numerical solutions: the behavior of the charges
When we discussed the parameter space for black hole solutions we used the parameters P and Q defined fornumerical convenience as P = A ( r i ) + r i A (cid:48) ( r i ) , (48) Q = − A (cid:48) ( r i ) r i . (49)These parameters correspond to the integration constants P and Q only when they are evaluated at r i → ∞ .Numerically, we used r i = 10 Km. Here we show in what region of the parameter space this r i is large enough for P and Q to accurately represent P and Q . To do so in a way that is easy to visualise we proceed as follows:1. For each solution in Fig. 1 (which is reproduced in the left panel of Fig. 9) we solve from r i = 10 Kmoutwards, up to r a = 10 Km, where we can guarantee that the fields have reached their asymptotic values.2. At r = r a we compute P and Q for each solution and we redo Fig. 1 parametrising each solution in terms of P and Q . This is shown in the right panel of Fig. 9, the colouring in this plot has the same meaning as in theleft panel. The solutions in red in the left panel do not have a corresponding point in the right panel sincethey do not have an asymptotic region where P and Q can be defined3. By comparing the left and right panels of Fig. 9 we get a qualitative idea of how different are ( P , Q ) and( P, Q ).For most of the parameter space where regular solutions exist (R4), the “charges” evaluated close to the source area good approximation for the asymptotic values of P and Q . Stronger asymptotic effects could be present in thesub-regions of R4 where the blue points in the right panel of Fig. 9 are shifted with respect to the left panel, a moredetailed numerical and analytical study of the solutions in those regions would be required to explore this. F Solutions with magnetic fields
In this Appendix we present solutions to the system of vector Galileons (2) in presence of a ϕ -component of A µ inthe branches π = 0 and π (cid:54) = 0 (which we interpret as ‘magnetic field’ since it turns on the spatial components F ij ).In both cases, our procedure consists in turning on A ϕ and identifying the metric components that are necessaryto solve the equations of motion at the lowest order in an asymptotic approximation. We show that (2) admits aricher set of solutions when magnetic configurations are investigated. Branch π (cid:54) = 0 In this case the component of the metric that we need to turn on to compensate for A ϕ is g rϕ .1 − n ( r ) r = 1 − (cid:15) Mr , (50a)1 − m ( r ) r = 1 − (cid:15) Mr , (50b) g rϕ ( r, θ ) = (cid:15) / √ B (cid:112) M P + QP β(cid:15) / κ cos θ √ r (1 + P βκ ) , (50c) A ( r ) = P + (cid:15) Qr , (50d) π ( r ) = (cid:15) / √ (cid:112) M P + QP √ r , (50e)22 - - [ - /[ ]] l og [ /[ ]] - - - [ - Q /[ Q ]] l og [ P /[ P ]] Figure 9: Black hole behaviour in terms of the vector field charges. The axes in the left panel are parametrised interms of quantities evaluated close to the source, P and Q , while in the right panel the axes represent the asymptoticquantities Q and P . The blue solid line corresponds to P = Q /r i , and the brown solid line to M = P/Q . Thereasons to show these lines are: for solutions below the blue line we naively expect that Q is large enough to causea difference between P and P , or, in other words, that an asymptotic expansion of the form P + Q/r is not yetvalid at r i . As for the brown line, when P and Q have opposite signs – which is not the case in this plot – solutionsbelow the line | M | = | P | / | Q | are expected to have a complex π . This does not happen for this set of solutions, butinterestingly enough, for large P and Q the same line signals the transition between regular solutions and solutionswithout an event horizon. In these plots we fixed M = 4 . M (cid:12) . The asymptotic mass M is not fixed, when thevector charges are large the difference between M and M is at most 5%, i.e. 1 − M/ (4 . M (cid:12) ) ≈ . A ϕ ( r, θ ) = (cid:15) B cos θ . (50f)The metric is still static, but the cross term G rϕ induces a deformation of the spatial line element that breaksspherical symmetry, as can be confirmed by computing the invariants R µν R µν and R µναβ R µναβ and noticingthat they depend on r and θ . Indeed, the Kretschmann scalar is the same as for the Schwarzschild geometry toleading order in (cid:15) , the angular dependence enters at next-to-leading order and gives a contribution of the form w ( θ ) /r , where w ( θ ) is a function only of θ that diverges near θ = 0 and θ = π , signalling that near the poles ourapproximation is only valid for r → ∞ . Branch π = 0 Here we need to turn on the metric components g tϕ and g r,θ , and the effect of the magnetic field on the other fieldsis only seen at quadratic and higher orders in (cid:15) . For simplicity, we set P = 0, but we checked that an approximatedsolution can be found with small P as well. For large P , the time component of the metric acquires a complicatedangular dependence. 1 − n ( r ) r = 1 − (cid:15) Mr + (cid:15) κ B + Q (1 − β )2 r , (51a)1 − m ( r ) r = 1 − (cid:15) Mr + (cid:15) κ Q + B (1 + 2 β ) − B βh ( θ )2 r , (51b) h ( θ ) = csc θ + cos θ log cot ( θ/
2) (51c) g rθ ( r, θ ) = (cid:15) κ β B (cid:0)
11 cos θ − θ ) + 4 log tan ( θ/
2) sin θ (cid:1) r sin θ , (51d) g tφ ( r, θ ) = (cid:15) κ βB Q cos θr (51e)23 ( r ) = (cid:15) Qr , (51f) A ϕ ( r, θ ) = (cid:15) B cos θ . (51g)Once again, the curvature invariants depend on r and θ and near the poles the approximation is only valid if r → ∞ . This branch is connected to the magnetic RN solution in the limit β → G Currents inside the star: the basic formalism
The system (2) can be generalised in order to explore the consequences of adding a current density term, so thatthe action now reads S = (cid:90) d x √− g (cid:20) κ ( R − − F µν F µν + βG µν A µ A ν + j µ A µ + L matter (cid:21) . (52)For simplicity, the current density is chosen to be proportional to the matter density, j µ = √ κγρ ( r ) U µ , (53)where U µ is the 4-velocity of the perfect fluid described by T µν . In the rest frame of the perfect fluid – the samethat we use to write the components of T µν – the only non-vanishing component of the 4-velocity is U t = (cid:112) g tt ,where g tt comes from (5). κ is introduced for convenience to make the arbitrary constant γ dimensionless: U µ isdimensionless, [ ρ ] = [ c ] · mass / lenght = [ κ − ][ (cid:96) − ], and j µ has to have dimensions of [ A µ ][ (cid:96) − ] = [ κ − / ][ (cid:96) − ],these dimensions are given by √ κρ . The dimensions of j µ can be written as (electric charge) / length , therefore thequantity c j /ρ ( r ) = c √ κγU is a mass-to-charge ratio characteristic of the solutions under consideration. Since U <<
1, the maximum mass-to-charge ratio is attained at the center of the star and given by c √ κγ .Let us derive the equations of motion. The term √− gj µ is constant under variation of the metric. Therefore, j µ A µ only contributes to the vector equation of motion:0 = D µ F µν + 2 βG µν A µ + j ν . (54)Note that, whatever the form of j µ – i.e., forget (53) for a moment, the angular components of this equation demand j θ = j ϕ = 0. In addition, in the branch π (cid:54) = 0 we need to satisfy (9) (the only assumptions behind this equation,are that T µν is a perfect fluid described in its rest frame and that the metric is diagonal). Once (9) is substitutedin (54), the r -component of such equations demands j r = 0. Thus, j is the only new function introduced in (52).Restoring (53), and supplementing our system with the EoS (30b), we have a complete set of equations todetermine A ( r ) , n ( r ) and χ ( r ). The only difference with respect to the system analysed in Sec. 4 is the contributionof j to the vector equation.Although not necessary –since all the information is encoded in the system of equations described above – itis illuminating to combine the Bianchi identity with the vector equation in such a way that the physical differencebetween the systems described by actions (2) and (52) is highlighted:0 = j A (cid:48) − p (cid:48) − n − r n (cid:48) r − r n ( p + ρ ) . (55)Eq. (55) shows that the inclusion of a current density allows the vector field to affect directly the equation thatdetermines the matter equilibrium configurations. This equation is known as the hydroelectrostatic equilibriumequation .Further physical insight can be gained by rewriting the t -component of the vector equation of motion as0 = j ( r ) − r (cid:112) − g tt g rr (cid:2) r √− g tt g rr F tr (cid:3) (cid:48) − β g tt A r [ r (1 − g rr )] (cid:48) , (56)where F tr = − g tt g rr A (cid:48) . 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