Eur. Phys. J. C (2020) 80:617 https://doi.org/10.1140/epjc/s10052-020-8178-1

Regular Article - Theoretical Physics

Black hole and naked singularity geometries supported by three-form fields

Bruno J. Barros1,a, Bogdan Dˇanilˇa2,3,b, Tiberiu Harko2,3,4,c, Francisco S. N. Lobo1,d 1 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Edifício C8, Campo Grande, 1749-016 Lisbon, Portugal 2 Astronomical Observatory, 19 Ciresilor Street, 400487 Cluj-Napoca, Romania 3 Department of Physics, Babes-Bolyai University, Kogalniceanu Street, 400084 Cluj-Napoca, Romania 4 School of Physics, Sun Yat-Sen University, Xingang Road, Guangzhou 510275, People’s Republic of China

Received: 20 May 2020 / Accepted: 22 June 2020 / Published online: 11 July 2020 © The Author(s) 2020

Abstract We investigate static and spherically symmet- decade [1]. These form fields naturally emerge in funda- ric solutions in a gravity theory that extends the standard mental theories, such as string theory [2–4] and, thus it is Hilbert–Einstein action with a Lagrangian constructed from only reasonable to explore their existence under effective a three-form field Aαβγ , which is related to the field strength formulations of gravity. Its application in cosmology has and a potential term. The field equations are obtained explic- already proven to be fruitful in explaining the early- and itly for a static and spherically symmetric geometry in vac- late-time acceleration periods of the cosmic history [5–8], uum. For a vanishing three-form field potential the gravi- reheating [9], screening solutions [10], generation of cos- tational field equations can be solved exactly. For arbitrary mological magnetic fields [11], among others. Primordial potentials numerical approaches are adopted in studying the inflation driven by multiple three-form fields was studied in behavior of the metric functions and of the three-form field. [12], considering several potential functions. One appealing To this effect, the field equations are reformulated in a dimen- consequence is that these models present distinct signatures sionless form and are solved numerically by introducing a when compared to the standard inflationary setting with a suitable independent radial coordinate. We detect the forma- scalar degree of freedom, compatible with recent cosmolog- tion of a black hole from the presence of a Killing horizon for ical observations. Inflationary models in extra dimensional the timelike Killing vector in the metric tensor components. braneworld scenarios inhabited by a single three-form have Several models, corresponding to different functional forms also been explored in [13], through the use of dynamical sys- of the three-field potential, namely, the Higgs and exponential tems analysis, and tested against the Planck data. The compu- type, are considered. In particular, naked singularity solutions tation of non-Gaussianities produced by several three-form are also obtained for the exponential potential case. Finally, fields inflation has been examined in [14] through the analy- the thermodynamic properties of these black hole solutions, sis of curvature perturbations employing the δN formalism. such as the horizon temperature, specific heat, entropy and It is well known that in four spacetime dimensions a three- evaporation time due to the Hawking luminosity, are studied form field admits a dual scalar field representation [15,16]. in detail. For example, assuming nonquadratic three-form potentials leads to an equivalent scalar representation exhibiting a non- canonical kinetic term. However, this mapping is nontriv- 1 Introduction ial, and thus for several self-interaction choices, or for any nonminimal coupling, this dual representation breaks down The use of differential three-form fields in the realm of cos- [5]. Nonetheless, even in the cases where the dual scalar mology has been gaining ever more attention over the last description exists, it is often quite complex to deal with and it becomes much more practical and intuitive to work in the form-representative framework. An interesting feature a e-mail: [email protected] of three-forms, is that the standard Maxwell term, F = dA, b e-mail: [email protected] constructed from a massless three-form A, naturally induces c e-mail: [email protected] (corresponding author) a cosmological constant term, and therefore has been used to d e-mail: [email protected] 123 617 Page 2 of 20 Eur. Phys. J. C (2020) 80 :617 address the cosmological puzzle regarding the tiny value of Similarly to these scalar field models, the three-form field  [17]. Furthermore, nonminimal interactions between dark theory we have considered is also formulated in the Einstein energy, driven by a three-form field, and cold dark matter, frame. However, there are fundamental differences between were explored in [18,19] along with the corresponding lin- the three-form field theory and scalar field models in the ear cosmological perturbations. It was shown that even small Einstein or Jordan frames. An interesting result in the Brans– values for the coupling lead to substantial variations on the Dicke-type scalar–tensor theories is that the solutions with growth of matter fluctuations. zero scalar field potential have in general no horizons [36]. Screening solutions with three-form fields conformally Our analytic and numerical investigations show that this is coupled to matter were also studied in [10]. In [20]the not the case in the three-form field theory. Another interest- authors investigate the existence of future abrupt cosmolog- ing result in scalar–tensor theories is that globally regular, ical events, particularly the little sibling of the Big Rip, and asymptotically flat solutions are possible. These solutions the possibility of avoiding these by considering interactions correspond to at least partly negative potentials V (φ), and between the three-form field, portraying dark energy, and they are solitons without horizons and with a regular center dark matter. With the aid of dynamical systems techniques, [37]. However, these results specific to scalar–tensor theories it was found that this can be achieved only by considering cannot be recovered in the three-form fields gravitational the- interactions not directly involving the dark matter species. ory. On the other hand our analytical and numerical investi- The authors also shed some light on how to distinguish the gations do not indicate the possible existence of any globally quadratic and linear dark energy interactions, through the regular solutions. statefinder hierarchy diagnosis and computing the growth This work is outlined in the following manner: in Sect. of matter perturbations, compatible with the observational 2, we present the general formalism of the three-form field SDSS III data. An alternate procedure to avoid these future extension of general relativity. In Sect. 3, considering a static cosmological abrupt events induced by the three-form is and spherically symmetric background, we deduce the grav- the quantization of the aforementioned system [21]. The itational field equations. In Sect. 4, exact vacuum solutions Lagrangian formalism for a single three-form fluid with a with three-form fields are presented, for the specific cases noncanonical Maxwell term was also explored, in compari- of a zero potential and for a constant potential. In Sect. 5, son with k-essence cosmology, in [22]. numerical solutions of the field equations are explored, for More recently, theories embracing three-forms have been the Higgs potential and the exponential potential. The ther- extended to spherically symmetric and static spacetimes, modynamic properties of the black holes solutions obtained in particular, wormhole geometries [23]. More specifically, are studied in Sect. 6. Finally, in Sect. 7, we discuss and solutions were found for the modified field equations, assum- summarize our results. ing a single static and radial-dependent three-form field, where the standard matter fields are allowed one to dwell within the entire wormhole domain without violating the null 2 Einstein gravity with a three-form field: general and weak energy conditions. formalism Indeed, the present work also deals with static three-forms supporting spherically symmetric spacetimes, however, in We start by considering the following action for Einstein the context of black holes and naked singularities. In fact, gravity with a standard three-form field Aαβγ :    black hole solutions are well known in many gravitational √ 4 1 field models and, in particular, in standard scalar–tensor S = d x −g R + LA , (1) 2κ2 extensions of general relativity. For instance, black hole solu- tions were recently found in the scalar–tensor representa- with R being the Ricci scalar, g = det gμν the determi- 2 tion of the hybrid metric-Palatini gravitational theory [24], nant of the metric tensor, κ = 8πG and LA stands for the which is a combination of the metric and Palatini f (R) for- Lagrangian density for our three-form, which reads [1,5] malisms unifying local constraints at the Solar System level 1 2 2 LA =− F − V (A ), (2) and the late-time cosmic acceleration [25–28]. Furthermore, 48 many other exact analytical black hole solutions have been where we have used the notation obtained and studied extensively for nonminimally coupled 2 αβγ δ 2 αβγ scalar fields [29–34] (for a review of the nonsingular general F = Fαβγ δ F and A = Aαβγ A . (3) relativistic solutions with minimally coupled scalar fields see Here V (A2) is the three-form potential and F = dA is the [35]). field strength tensor [15,20], a four-form, whose components Generally the latter solutions have been derived in the can be written as Einstein frame, without the assumption of the existence of any coupling between the scalar field and the Ricci scalar. Fαβγ δ =∇α Aβγδ −∇δ Aαβγ +∇γ Aδαβ −∇β Aγδα, (4) 123 Eur. Phys. J. C (2020) 80 :617 Page 3 of 20 617 with ∇μ being the covariant derivative. The equations of On this background geometry, the invariant A2 is given motion for our three-form can be found by varying the action by αβγ Eq. (1) with respect to A . They read [6,23] β( ) A2 =−6 e r ζ(r)2, (13) μ ∂V ∇μ F αβγ = 12 Aαβγ . (5) ∂(A2) and the term expressing the kinetic energy of the three-form Varying now Eq. (1) with respect to the metric gμν one is provided by     finds the following field equations: 2 2   4  2 F =−6 ζ α + β + + 2ζ , (14) Gμν = κ Tμν, (6) r where Gμν is the Einstein tensor and Tμν is the energy where a prime denotes the derivative with respect to the radial momentum tensor of our form field source: component. δLA The equations of motion (5) can now be written in terms Tμν =−2 + gμνL δgμν A of ζ , using the metric Eq. (12), as ∂ 1 V     = (F ◦ F)μν + 6 (A ◦ A)μν + LA gμν, (7) ∂( 2)    4    4 6 A 2ζ + α + β + ζ + α + β − ζ + 2V,ζ = 0, r r 2 with a circle denoting contraction of all indices but the first, αβγ (15) i.e., (F ◦ F)μν = Fμαβγ Fν . We will construct our three-form with the aid of its dual V,ζ = ∂V/∂ζ vector (one-form) [23], via the Hodge star operator where . Through Eq. (7), using Eq. (12), the components of the μ δ 1 1 μαβγ B = ( A) = √ Aαβγ . (8) energy momentum tensor are then found to be 3! −g 2 t =−ρ = F − + ζ , Inverting the last identity, we express the three-form compo- T t V V,ζ (16) nents in terms of its dual vector 48 F2 √ δ r = − . T = pr = − V, (17) Aαβγ g αβγ δ B (9) r 48 θ = φ = = t , We now build the three-form components by parameterizing T θ T φ p T t (18) Bμ in terms of a radial scalar function ζ = ζ(r), i.e., 2 ρ δ where the F term is given by Eq. (14) and we identify , B = (0,ζ(r), 0, 0)T . (10) pr and p with the energy density, the radial pressure, and Due to the antisymmetric nature of differential forms, once the tangential pressure, respectively, of the three-form. The we attain a solution for ζ(r), all the three-form components components of the Einstein tensor can be written as are automatically determined through Eq. (9). 1 d e−β Through Eqs. (4) and (9), we may rewrite the kinetic term Gt = r e−β − = − rβ − eβ , t 2 1 2 1 (19) in the action Eq. (1)as[7] r dr r e−β Gr = + rα − eβ , 1 2 1 0123 1 μ 2 r 2 1 (20) − F =− F0123 F = (∇μ B ) . (11) r     48 2 2 −β  θ φ e  1 α   G θ = G = α + + α − β . (21) We now consider applications of the general formalism φ 2 r 2 obtained here to the specific case of static and spherically symmetric spacetimes. At this point, we have four variables, namely, α, β, ζ and V , and four independent equations, namely, the equation of motion for ζ , i.e., Eq. (15) and the three field equations 3 Spherically symmetric and static background (setting κ = 1 for simplicity): −β 2 e  β F 3.1 Metric and field equations 1 − rβ − e = − V + ζ V,ζ , (22) r 2 48 e−β F2 Consider a static and spherically symmetric spacetime, given + rα − eβ = − V, 2 1 (23) by the following line element: r    48    −β  2 e  1 α   F 2 =− α(r) 2 + β(r) 2 + 2 θ 2 + 2 θ φ2 . α + + α − β = − V + ζ V,ζ . ds e dt e dr r d sin d 2 r 2 48 (12) (24) 123 617 Page 4 of 20 Eur. Phys. J. C (2020) 80 :617

Combining the first two field equations, Eqs. (22) and (23), 4 Exact vacuum solutions with three-form fields one finds   β In the present section we will consider some solutions of α + β =−re ζ V,ζ . (25) the system of the vacuum field equations (22)–(24), respec- Using Eqs. (22) and (24) yields tively, which must be solved together with Eq. (15), once the    functional expression of the three-form potential V is fixed. 2  β  1 α   1 − rβ − e = α + + α − β . (26) r 2 r 2 4.1 First case: V = 0 Moreover, from Eq. (22) we obtain     2 If the three-form field potential V identically vanishes, V ≡  β F 2 rβ = 1 − e 1 + − V + ζ Vζ r . (27) 48 0, then Eq. (25) can be immediately integrated, giving These expressions will be useful below. α =−β, (36)

3.2 Dynamical system formulation where we have set, without loss in generality, the arbitrary integration constant equal to zero. Then Eq. (26) becomes

The field equation (22) can be rewritten as   2 −α   α + α 2 = 1 − e . (37) 2 r 2 d −β F 2 re = 1 − V − ζ V,ζ − r , (28) dr 48 By introducing a new variable α = ln f ,Eq.(37) becomes and can immediately be integrated to give 2 f 2 f  − + = , 2 2 0 (38) −β 2M (r) r r e = 1 − eff , (29) r with the general solution given by ( ) where the effective mass Meff r is defined as c    ( ) = + 1 + 2, r 2 f r 1 c2r (39) 1 F 2 r Meff (r) = V − ζ V,ζ − r dr, (30) 2 0 48 where c1 and c2 are arbitrary constants of integration. Since in and satisfies the mass continuity-type equation the limit F2 → 0 the Schwarzschild solution of standard gen-   eral relativity must be recovered, it follows that c =−2M, ( ) 2 1 dMeff r 1 F 2 =− = V − ζ V,ζ − r . (31) where M is the mass of the gravitating body, while c2 dr 2 48 can be interpreted as the cosmological constant. Hence, we From Eq. (23) we obtain the expression of α: have recovered the Schwarzschild–de Sitter solution, with

3 2 the cosmological constant naturally included,  r F /48 − V + 2Meff (r) α = . (32) 2 ( − ( )/ ) α −β 2M r 1 2Meff r r e = e = 1 − − r 2. (40) r With the use of Eq. (25), Eq. (15) can be reformulated as ζ  Equation (15), giving the evolution of the function ,  2 rζ V,ζ + ζ V,ζ ζ /2  becomes ζ + − ζ − G (r,ζ) ζ + V,ζ = , − ( )/ 0 (33) r 1 2Meff r r  2  2 ζ + ζ − ζ = 0, (41) r r 2 where we have denoted 2 ζ V,ζ and it has the general solution G (r,ζ) = + 2 − ( )/ r 2 [1 2Meff r r] C2   ζ(r) = C1r + , (42) 1 + F2/48 − V + ζ V,ζ r 2 2 × − . r 2 − ( )/ (34) 1 2Meff r r where C1 and C2 are arbitrary constants of integration. With 2 Finally, for the function F2 we obtain the use of Eq. (42), we obtain finally for F the expression    2 2 =− 2 = , 4 rζ V,ζ F 216C1 constant (43) F2 =− − ζ + ζ  . 6 − ( )/ 2 (35) r 1 2Meff r r as expected. On the other hand with the use of the field equa- 2 Here we have presented the relevant equations, which set tion (23) we obtain F = 144c1 =−288M. By comparing 2 the stage for exploring solutions, both exactly and numeri- the two expressions√ for F we obtain for C1 the representa- cally. tion C1 = 4M/3. 123 Eur. Phys. J. C (2020) 80 :617 Page 5 of 20 617

= = du V,ζ + ζ V,ζ ζ /2 1 V,ζ 4.2 The constant potential V V0 constant =− u + G (η, ζ) ζ − , 3 4 4 (51) dη η (1 − 2ηMeff ) η η = = In the case of the constant potential V V0 constant, where Eq. (33) take the same form as in the case of the vanishing ζ V,ζ scalar potential of the three-form field, Eq. (41), and its solu- G (η, ζ) = 2η2 + 2 2 (1 − 2ηM ) tion is given again by Eq. (42). As for F , given by Eq. (35), eff  2 =− 2 2 2 we obtain again F 216C1 . By integrating the mass η + F /48 − V + ζ V,ζ × 2 − (52) continuity equation, Eq. (31), we obtain η2 (1 − 2ηM )   eff ( ) = 1 + 9 2 3 + , Meff r V0 C1 r c1 (44) and 6 2    2 ζ V,ζ where c1 is an arbitrary constant of integration. Hence for F2 =−6 4η − ζ − 2η2u , (53) −β η ( − η ) e we immediately obtain 1 2 Meff   −β c1 1 9 respectively. In order to obtain the above equations we have e = 1 − − V + C2 r 2. (45) ζ/ =−η2 ζ/ η r 6 0 2 1 used the mathematical relations d dr d d , and Since also in the constant potential case the general rela- d2ζ d2ζ dζ = η4 + 2η3 , (54) tion α + β = 0 holds, we obtain the metric tensor coefficient dr 2 dη2 dη eα as   respectively. The system of Eqs. (48), (49), (50) and (51) α c 1 9 e = 1 − 1 − V + C2 r 2. (46) must be integrated with the initial conditions at infinity, given r 6 0 2 1 ( ) = (0) α( ) = ζ( ) = ζ ( ) = by Meff 0 Meff , 0 0, 0 0, and u 0 u0, The solution is again of the Schwarzschild–de Sitter type, respectively. 2 with the potential V0 generating, together with F , an effec- tive cosmological constant. On the other hand, the arbitrary 5.1 The Higgs potential: V (ζ ) = μ2ζ 2 + νζ4 integration constant c1 is undetermined by the field equa- tions, and must be chosen from physical considerations. If 5.1.1 General considerations the integration constant C1 and the constant potential V0 van- ish, the metric reduces to the standard Schwarzschild form. The Higgs-type potential Moreover, there are no restrictions on the integration constant V (ζ ) = μ2ζ 2 + νζ4 (55) c1, whose sign and physical interpretation remains arbitrary. plays a fundamental role in elementary particle physics. From a physical point of view we may assume that −μ2 represents 5 Numerical solutions of the field equations the mass of the three-form field associated to the gravitational interaction. For the strong interaction case the Higgs self- The system of three Eqs. (31), (32) and (33) for the three coupling constant ν takes the value ν ≈ 1/8[38], a value α ζ unknown functions Meff , and , representing a strongly which follows from the analysis of accelerator experiments. nonlinear system of differential equations, determines the But of course in the case of the gravitational models in the vacuum solutions of the three-form field model gravity. In presence of a three-form field the values of both μ2 and ξ order to integrate the equations we introduce a new indepen- may be very different from those suggested by elementary η dent variable , defined as particle physics. 1 In the case of the Higgs-type potential of the three-form η = . (47) r field the vacuum gravitational field equations take the form Then we can reformulate the gravitational field equations     dζ dM 1 F2 1 as the following first order dynamical system: = u, eff = + μ2 + 3νζ2 ζ 2 , (56) dη dη 2 48 η4 ζ d dα F2/48 − μ2 + νζ2 ζ 2 + 2η3 M = u, (48) =− eff , (57) dη η η3 ( − ηM )   d 1 2 eff 2 2 2 2 2 dMeff 1 F 1 du 3μ + 10νζ ζ 1 2 μ + 2νζ = + ζ V,ζ − V , (49) =− u + G (η, ζ) ζ − , 4 3 4 4 dη 2 48 η dη η (1 − 2ηMeff ) η η α 2/ − + η3 (58) d =−F 48 V 2 Meff , 3 (50) dη η (1 − 2ηMeff ) 123 617 Page 6 of 20 Eur. Phys. J. C (2020) 80 :617

Fig. 2 Specific case of the Higgs potential: Variation of the metric Fig. 1 Specific case of the Higgs potential: Variation of the metric −β η ν = . tensor coefficient eα as a function of the coordinate η,forν = 0.01 and tensor coefficient e as a function of the coordinate ,for 0 01, and μ2 μ2 = . μ2 = . for different values of μ2: μ2 = 0.0001 (solid curve), μ2 = 0.00012 for different values of : 0 0001 (solid curve), 0 00012 μ2 = . μ2 = . (dotted curve), μ2 = 0.00013 (short dashed curve), μ2 = 0.000138 (dotted curve), 0 00013 (short dashed curve), 0 000138 μ2 = . (dashed curve), and μ2 = 0.000141 (long dashed curve), respectively (dashed curve), and 0 000141 (long dashed curve), respectively where G(η, ζ ) and F2 are given by

2 μ2 + 2νζ2 ζ 2 G (η, ζ) = 2η2 + 2 (1 − 2ηM )  eff  η2 + F2/48 + μ2 + 3νζ2 ζ 2 × − 2 2 η (1 − 2ηMeff ) (59) and    2 2 μ2 + 2νζ2 ζ 2 F2 =−6 4η − ζ − 2η2u , (60) η (1 − 2ηMeff ) respectively. Equations (56)–(58) must be considered with Fig. 3 Specific case of the Higgs potential: Variation of the effective −5 the initial conditions at infinity Meff = 1, ζ(0) = 10 , mass Meff as a function of the coordinate η,forν = 0.01, and for u(0) = 40, and α(0) = 0, respectively. In Figs. 1, 2, 3 and different values of μ2: μ2 = 0.0001 (solid curve), μ2 = 0.00012 η (dotted curve), μ2 = 0.00013 (short dashed curve), μ2 = 0.000138 4 we present the variations with respect to of the metric 2 α −β (dashed curve), and μ = 0.000141 (long dashed curve), respectively tensor coefficients e , e , of the effective mass Meff and of the radial scalar function ζ . The variation of the metric tensor coefficient eα is repre- sented in Fig. 1. The metric function monotonically decreases from its constant, Minkowskian value at infinity, to zero, a value reached for finite values of η, and which defines the singular surface of the black hole, or its event horizon. The position of the event horizon is strongly dependent on the numerical values of μ2. A similar behavior characterizes the metric tensor component e−β , whose variation with respect to η is represented in Fig. 2. The inverse of the metric ten- sor component decreases linearly from infinity to the event horizon of the black hole, where the metric tensor becomes singular. Similarly to eα, the variation of e−β is significantly influenced by the numerical values of μ2. The changes in the Fig. 4 Specific case of the Higgs potential: Variationof the radial scalar ζ η ν = . effective mass M are plotted in Fig. 3. The mass increases function as a function of the coordinate ,for 0 01, and for eff different values of μ2: μ2 = 0.0001 (solid curve), μ2 = 0.00012 rapidly from its initial value at infinity to a maximum value, (dotted curve), μ2 = 0.00013 (short dashed curve), μ2 = 0.000138 reached before the event horizon, an effect due to the pres- (dashed curve), and μ2 = 0.000141 (long dashed curve), respectively ence of the three-form field, and its mass–energy contribution 123 Eur. Phys. J. C (2020) 80 :617 Page 7 of 20 617

−β Fig. 5 Variation of the metric tensor coefficient eα, for the Higgs Fig. 6 Variation of the metric tensor coefficient e , for the Higgs 2 potential, as a function of the coordinate η,forμ2 = 0.000095, and potential, as a function of the coordinate η,forμ = 0.000095, and for different values of ν: ν = 0.04 (solid curve), ν = 0.08 (dotted for different values of ν: ν = 0.04 (solid curve), ν = 0.08 (dotted curve), ν = 0.12 (short dashed curve), ν = 0.16 (dashed curve), and curve), ν = 0.12 (short dashed curve), ν = 0.16 (dashed curve), and ν(0) = 0.20 (long dashed curve), respectively ν(0) = 0.20 (long dashed curve), respectively to the mass of the central object. After reaching its max- imum value the effective mass decreases before reaching the event horizon. However, for some particular values of μ2, the mass becomes approximately constant beginning for some finite value of η. This indicates that the model enters very quickly in an approximate Schwarzschild regime, with −β e ≈ 1−2Meff η, with Meff a function of the parameters of the Higgs-type potential, and of the initial conditions for ζ . This dependence on the initial conditions and on the param- eters of the potential also determines the modifications of the position of the event horizon of the black hole, with respect to its standard general relativistic value. The dependence of the Fig. 7 Variation of the effective mass Meff , for the Higgs potential, as scalar radial function ζ is depicted in Fig. 4. The behavior of a function of the coordinate η,forμ2 = 0.000095, and for different ζ indicates a complex dynamics, with ζ increasing initially, values of ν: ν = 0.04 (solid curve), ν = 0.08 (dotted curve), ν = 0.12 (short dashed curve), ν = 0.16 (dashed curve), and ν(0) = 0.20 (long reaching a maximum value, and then becoming again zero dashed curve), respectively at the event horizon. For some values of the Higgs potential the variation has a quasi-oscillatory behavior, characterized by an alternation of local maxima and minima. The behavior of the geometric and physical quantities in vacuum for the three-form supported black holes also depend sensitively on the numerical values of the self-coupling con- stant ν of the Higgs potential. In Figs. 5, 6, 7 and 8 we present the behavior of the metric tensor components, of the effec- tive mass and of the scalar function ζ for different values of ν. To integrate the gravitational field equations we have fixed the values of μ2 = 0.000095, ζ(0) = 0, u(0) = 40, Meff (0) = 1, and α(0) = 0, respectively, and we have varied the values of ν. The metric tensor components eα and e−β , represented in Fig. 8 Variation of the radial scalar function ζ , for the Higgs potential, Figs. 5 and 6, decrease from their Minkowski values at infin- 2 η as a function of the coordinate η,forμ = 0.000095, and for different ity to zero, corresponding to a finite value of , indicating the values of ν: ν = 0.04 (solid curve), ν = 0.08 (dotted curve), ν = 0.12 presence of a singularity corresponding to the formation of (short dashed curve), ν = 0.16 (dashed curve), and ν(0) = 0.20 (long a black hole. Their numerical values depend effectively on dashed curve), respectively the numerical values of ν, which also strongly influence the 123 617 Page 8 of 20 Eur. Phys. J. C (2020) 80 :617

Table 1 The position of the event horizon ηS for selected values of the initial conditions and Higgs model parameters

2 ζ0 u0 μ νηs

10−5 40 0.0001 0.01 0.88 10−5 40 0.00012 0.01 0.46 10−5 40 0.00013 0.01 0.49 10−5 40 0.000138 0.01 0.61 10−5 40 0.000141 0.01 0.76 10−5 40 0.0001 0.015 0.59 10−5 40 0.0001 0.02 0.51 10−5 40 0.0001 0.015 0.48 10−5 40 0.0001 0.015 0.46 10−5 45 0.0001 0.01 0.75 10−5 50 0.0001 0.01 0.66 Fig. 9 Distribution of the event horizons of the three-form field sup- μ2 ν 10−5 55 0.0001 0.01 0.58 ported black holes as a function of the parameters and of the ζ ={, . , . , . , . }× −4 − Higgs potential for 0 1 1 105 1 21 1 315 1 42 10 ,and 10 5 55 0.0001 0.01 0.54 −2 u0 ={1, 2, 3, 4, 5}×10 10−4 40 0.0001 0.01 0.78 5 × 10−4 40 0.0001 0.01 0.56 10−3 40 0.0001 0.01 0.48 5.1.2 Interpolating functions 5 × 10−5 40 0.0001 0.01 0.44 In order to facilitate the further investigations of the prop- erties of the three-form field supported black holes in the following we will present some explicit analytic expressions position of the event horizon. The effective mass, shown in for the basic physical and geometrical quantities, obtained Fig. 7, increases from its value at infinity towards a maximum from the interpolation of the numerical results. For the effec- (η) value reached far away from the event horizon. The value of tive mass function Meff we assume a general expression the effective mass also strongly depends on the self-coupling of the form constant ν. For some numerical values of ν the mass becomes M (η) = A + B η + C η2, (61) roughly a constant beginning from a finite η, and the geom- eff MH MH MH etry near the compact object becomes quasi-Schwarzschild, where the coefficients A , B , and C are functions of with the effective mass of the black hole strongly dependent MH MH MH (μ2,ν,ζ , u ). In the following analysis we will concentrate on the values of ν. The radial scalar function ζ , represented 0 0 mostly on the dependence on the initial conditions, and hence in Fig. 8, also shows an effective dependence on ν, reach- we will fix the parameters of the Higgs potential as μ2 = ing, similarly to the effective mass, a finite value at the event − − 10 4, and ν = 10 2, respectively. Then we obtain horizon of the black hole. √ The numerical values of the event horizon ηS are pre- ζ v =− . − . ζ − . 0 sented, for a selected value of the initial conditions and of A 1 915 3031 75 0 509 327 u0 the model parameters, in Table 1. In the adopted system of ζ + 0 + . − . 2, units the position of the Schwarzschild singularity corre- 142182 0 1444u0 0 0016u0 (62) u0 sponds to ηS = 1/2. The position of the event horizon rs = /η of the black hole is found to be rs rg S, where rg is the with the correlation coefficient R2 = 0.999, Schwarzschild gravitational radius of the object, defined as r = GM/c2 M = . + . ζ − . × 7ζ 2 g , where is the total mass of the object. For BMH 70 627 81785 6 0 5 312 10 0 example, the physical position of the event horizon of the ζ − . × 6 0 − . + . 2, black hole supported by a three-form field with Higgs poten- 2 98 10 3 27u0 0 036u0 (63) u0 tial having ηS = 0.88 is located at rs = 1.13rg, indicating a black hole more extended that its Schwarzschild counterpart. with R2 = 0.989, and For a three-form field black hole with η = 0.44, the event S ζ horizon is located at r = 2.27r . The three-dimensional dis- 7 0 s g CMH =−294.508 + 916417ζ0 − 1.4889 × 10 tribution of the event horizons of the black holes supported u0 + . − . ζ − . 2, by three-form fields is represented in Fig. 9. 12 86u0 12456 8 0u0 0 129447u0 (64) 123 Eur. Phys. J. C (2020) 80 :617 Page 9 of 20 617 with R2 = 0.99. The general expression of the metric tensor potential is obtained in string-type theories and in four- component eα is dimensional effective Kaluza–Klein theories from the com- α(η) 2 pactification of the higher dimensions. Moduli fields and non- e = Aα + Bα η + Cα η , (65) H H H perturbative effects in quantum field theory such as gaugino 2 with the coefficients AαH , BαH , CαH given, for fixed μ and condensation can also generate exponential-type potentials ν, as functions of (ζ0, u0) by for scalar fields [39]. The role of the exponential potential √ has been intensively investigated especially in the framework ζ0 ζ0 Aα = 0.847 − 125.916ζ − 36.981 + 9656.83 of scalar field cosmological and gravitational models and for H 0 u u 0 0 many field configurations, including the inhomogeneous and + . − . 2, 0 0069u0 0 000059u0 (66) homogeneous scalar fields [40–48]. In the presence of an with R2 = 0.999, exponential potential the gravitational field equations (48)– (51) take the form = . + . ζ − . × 6ζ 2 BαH 0 62 2036 19 0 4 162 10 0   2 ζ dζ dM 1 F λζ 1 − × 6 0 − . + . 2, = u, eff = + V (λζ − 1) e , (73) 102912 10 0 0942u0 0 000678u0 η η 0 η4 u0 d d 2 48 (67) dα F2/48 − V eλζ + 2η3 M =− 0 eff , (74) η η3 ( − η ) with R2 = 0.999, and d 1 2 Meff λζ λζ ζ du V0λ (1 + λζ/2) e 1 λV0e =− . + . ζ − . × 6 0 =− u + G (η, ζ) ζ − , CαH 2 754 90221 2 0 1 96 10 dη η3 (1 − 2ηM ) η4 η4 u0 eff + . − . ζ − . 2, (75) 0 00058u0 882 929 0u0 0 00193u0 (68) with R2 = 0.998. where For the metric tensor component eβ the general interpo- λV ζ eλζ lating function can be taken as G (η, ζ) = 2η2 + 0 2 (1 − 2ηMeff ) β(η) 2   e = Aβ + Bβ η + Cβ η , (69) H H H η2 + F2/48 + V (λζ − 1) eλζ × − 0 2 2 with the coefficients Aβ H , Bβ H , and Cβ H given as functions η (1 − 2ηMeff ) of (ζ0, u0) by √ (76) ζ ζ = . + . ζ + . 0 − 0 Aβ H 1 653 730 135 0 67 886 35382 and u0 u0 + . − . 2,    0 0324u0 0 000353u0 (70) λ ζ λζ 2 2 =− η − V0 e ζ − η2 , 2 F 6 4 2 u (77) with R = 0.999, η [1 − 2ηMeff ] ζ = . − . ζ + . × 6ζ 2 + 0 Bβ H 12 688 16943 1 0 9 355 10 0 737250 respectively. u0 2 3 −1.12u0 + 0.0296u − 0.000243u , (71) 0 0 5.2.1 Naked singularity solutions with R2 = 0.976, and ζ The description of the state and structure of ordinary mate- = . + ζ + . × 6 0 − . Cβ H 59 249 212369 0 3 0549 10 2 506u0 rial systems, forming an initial regular distribution, after the u0 gravitational collapse, is one of the most important theoret- +2762.06ζ u + 0.0206u2, (72) 0 0 0 ical and observational problems in general relativity. There with R2 = 0.999. are two questions one should consider when investigating the gravitational collapse. The first question is to find the ini- λζ 5.2 The exponential potential: V (ζ ) = V0e tial conditions of the gravitational collapse that lead to the formation of a black hole. On the other hand a careful inves- As a second example of black hole solutions supported by tigation of the gravitational collapse shows that it does not a three-form field with non-zero potential, we consider the end always with the creation of a black hole. Depending on λζ case of the exponential-type potential, V (ζ ) = V0e , where the initial conditions, another type of object, called a naked V0 and λ are constants. There are many physical processes singularity, can also be born as the final state of the collapse in string theory and elementary particle physics described [49–54]. For reviews of the naked singularity problem see by this type of potential. For example, an exponential-type [55,56], respectively. 123 617 Page 10 of 20 Eur. Phys. J. C (2020) 80 :617

α Fig. 10 Variation of the metric tensor coefficient e as a function of Fig. 11 Variation of the metric tensor coefficient e−β as a function of the coordinate η for the case of the exponential three-form potential the coordinate η for the case of the exponential three-form potential = λζ λ =− −3 = . λζ −3 V V0e ,for 10 , and for different values of V0: V0 0 001 V = V0e ,forλ =−10 , and for different values of V0: V0 = 0.001 = . = . (solid curve), V0 0 0012 (dotted curve), V0 0 0013 (short dashed (solid curve), V0 = 0.0012 (dotted curve), V0 = 0.0013 (short dashed = . = . curve), V0 0 0014 (dashed curve), and V0 0 0015 (long dashed curve), V0 = 0.0014 (dashed curve), and V0 = 0.0015 (long dashed curve), respectively curve), respectively

The second question one must also necessarily consider is the question if the physically realistic collapse solutions of the Einstein gravitational field equations that indicate the for- mation of naked singularities do really correspond to existing natural objects, which can be observed by astrophysical or astronomical methods. If detected observationally, the exis- tence of the naked singularities would be counterexamples of the Cosmic Censorship Hypothesis, proposed by Roger Pen- rose [57]. The Cosmic Censorship Hypothesis conjectures that curvature singularities are always covered in asymptoti- cally flat spacetimes by event horizons. In fact, one can for- mulate the Cosmic Censorship Hypothesis in a strong sense Fig. 12 Variation of the effective mass Meff as a function of the coordi- η = λζ (in a geometry that is physically appropriate naked singular- nate for the case of the exponential three-form potential V V0e , λ =− −3 = . ities cannot form), and in a weak sense (if naked singular- for 10 , and for different values of V0: V0 0 001 (solid curve), V0 = 0.0012 (dotted curve), V0 = 0.0013 (short dashed curve), ities do really exist, they are securely covered by an event V0 = 0.0014 (dashed curve), and V0 = 0.0015 (long dashed curve), horizon, and therefore they cannot be detected by far-away respectively observers). There have been many attempts to prove the Cos- mic Censorship Hypothesis (see [58] for a review of the early investigations and results in this field). For the possibilities of observationally identifying naked singularities see [59], and the references therein. For a certain range of parameters and of initial conditions, naked singularity solutions of the gravitational field equa- tions in the presence of a three-form field with exponential potential can also be obtained. In Figs. 10, 11, 12 and 13 we present the behavior of the metric tensor coefficients eα, e−β , of the effective mass Meff , and of the radial scalar function ζ for the initial conditions M(0) = 1, α(0) = 0, ζ(0) = 10−5, and u(0) = 10−4, respectively. For λ we have adopted the ζ value λ =−10−3, and we have slightly varied the numerical Fig. 13 Variation of the radial scalar function as a function of the coordinate η for the case of the exponential three-form potential V = values of V . λζ −3 0 V0e ,forλ =−10 , and for different values of V0: V0 = 0.001 As one can see from Figs. 10 and 11, the metric tensor (solid curve), V0 = 0.0012 (dotted curve), V0 = 0.0013 (short dashed = . = . coefficients are monotonically increasing functions of η, and curve), V0 0 0014 (dashed curve), and V0 0 0015 (long dashed they are singular only at the origin η →∞, or, equivalently, curve), respectively 123 Eur. Phys. J. C (2020) 80 :617 Page 11 of 20 617

Fig. 14 Variation of the metric tensor coefficient eα as a function of Fig. 15 Variation of the metric tensor coefficient e−β as a function of the coordinate η for the case of the exponential three-form potential the coordinate η for the case of the exponential three-form potential λζ −10 λζ −10 V = V0e ,forV0 = 9.9 × 10 , and for different values of λ: V = V0e ,forV0 = 9.9 × 10 , and for different values of λ: λ =−40 (solid curve), λ =−120 (dotted curve), λ =−200 (short λ =−40 (solid curve), λ =−120 (dotted curve), λ =−200 (short dashed curve), λ =−280 (dashed curve), and λ =−360 (long dashed dashed curve), λ =−280 (dashed curve), and λ =−360 (long dashed curve), respectively curve), respectively r → 0. The correspondent massive object does not have an event horizon, and therefore it corresponds to a naked singu- larity, with the only singular point located at the center. The effective mass of the naked singularity, presented in Fig. 12, becomes negative at infinity, and takes a constant, negative value up to the singular center of the naked singularity. Hence the metric of this exotic object can be represented as

 α( ) −β( ) 2Meff V0,λ,ζ0,ζ (0) e r = e r = 1 + . (78) r The numerical values of the (negative) effective mass are determined by the initial conditions of the gravitational field equations at infinity, as well as by the parameters of the Fig. 16 Variation of the effective mass Meff as a function of the coor- λζ exponential potential. The solutions of the gravitational field dinate η for the case of the exponential three-form potential V = V0e , −10 equations depend sensitively on these parameters. The radial for V0 = 9.9 × 10 , and for different values of λ: λ =−40 (solid λ =− λ =− scalar function ζ , shown in Fig. 13, diverges at the center of curve), 120 (dotted curve), 200 (short dashed curve), λ =−280 (dashed curve), and λ =−360 (long dashed curve), respec- the naked singularity. Its behavior is also dependent on the tively initial conditions used to solve the gravitational field equa- tions, and on the parameters of the exponential potential. initial value at infinity, and becomes a constant, having the 5.2.2 Black hole solutions same numerical value from infinity to the event horizon of the black hole. Hence the metric is of the Schwarzschild type, The static spherically symmetric vacuum gravitational field with equations in the presence of a three-form field also admit  black hole-type solutions. In Figs. 14, 15, 16 and 17 we α( ) −β( ) 2Meff V0,λ,ζ(0), ζ (0) e r = e r = 1 − , (79) present the results of the numerical integration of the gravita- r tional field equations for M(0) = 1, α(0) = 0, ζ(0) = 10−2,  −1 −10 ζ (0) = 10 , V0 = 9.9 × 10 , and different values of λ. with the effective mass, and the position of the event hori- As one can see from Figs. 14 and 15, the metric tensor zon depending on the initial conditions at infinity, and on coefficients eα and e−β decrease monotonically from their the parameters of the exponential potential. The radial scalar Minkowskian values at infinity to zero, a value reached for a function increases rapidly from infinity when approaching finite value of η = ηS. Hence the compact object possesses the event horizon, and takes a finite value for η = ηS. an event horizon, and is thus a black hole. The effective mass We can obtain an interpolating expression for the effective Meff , depicted in Fig. 16, decreases very quickly from its mass function Meff in the case of the exponential potential, 123 617 Page 12 of 20 Eur. Phys. J. C (2020) 80 :617

6 Thermodynamic properties of black holes

In the present section we consider the thermodynamic prop- erties of the black hole solutions supported by a three-form field. In particular we will concentrate on the surface grav- ity of the black holes, their Hawking temperature, their spe- cific heat, their entropy and their Hawking luminosity. In our investigation of the vacuum field equations in the three-form fields model we have adopted the simplifying assumption that the effective mass function and the lapse function eα are functions of the radial coordinate r only. Hence the space- time is static and a timelike Killing vector tμ exists [60,61]. In the present section, for the sake of clarity, we will restore Fig. 17 Variation of the radial scalar function ζ as a function of the coordinate η for the case of the exponential three-form potential V = the physical units in all mathematical expressions. λζ −10 V0e ,forV0 = 9.9 × 10 , and for different values of λ: λ =−40 (solid curve), λ =−120 (dotted curve), λ =−200 (short dashed 6.1 Brief summary of black hole thermodynamics curve), λ =−280 (dashed curve), and λ =−360 (long dashed curve), respectively For a static black hole that possesses a Killing horizon the definition of the surface gravity κ˜ is given by [60,61] μ ν ν t ∇μt = t κ,˜ (82) μ where t is a Killing vector, and ∇μ denotes the covariant derivative with respect to the metric. For a static, spherically symmetric geometry, with the line element given by

dr2 ds2 =−˜σ 2(r) f (r)c2dt2 + + r 2d2, (83) f (r) we can adopt a suitable normalized Killing vector defined as μ t = (1/σ˜∞, 0, 0, 0). Then the surface gravity of the black hole is found to be [61]     σ˜ c4 2GM(r)  κ˜ = hor 1 −  . (84) σ˜ 2  ∞ 4GMhor c hor Fig. 18 Variation of the position of the event horizons of the The subscript hor requires that all physical quantities must be three-form field supported black holes as a function of the evaluated on the outer apparent horizon. If the function σ˜ ≡ parameters V and λ of the exponential potential for ζ = 0 0 1, and M = constant, from the above definition we reobtain {0.0004, 0.0064, 0.0124, 0.0184, 0.0244} and u0 ={4, 8, 16, 32, 64} the standard result of the surface gravity of a Schwarzschild black hole, which is given by [60] (exp) (η) ≈ , 4 Meff AMH (80) c κ˜ = . (85) 4GMhor (exp) λ with the coefficient AMH given, for fixed V0 and ,by The temperature TBH of the black hole is found to be √ h¯ ζ ζ T = κ,˜ (86) (exp) ≈ . + . ζ + . 0 − . 0 BH 2πck AMH 0 74 9 78 0 15 812 146 854 B u0 u0 + . − . 2, where kB is Boltzmann’s constant. Equivalently, in the vari- 0 00134u0 0 0000528u0 (81) able r = rg/η we obtain for the Hawking temperature of the black hole the expression with R2 = 0.997.   (η)  The distribution of the position of the black holes event TH 2 dMeff  TBH = 1 + η  M (η ) dη horizon supported by a three-form field with exponential eff S η=ηS λ = θ(η)| , potential are represented, for fixed V0 and ,inFig.18. TH η=ηS (87) 123 Eur. Phys. J. C (2020) 80 :617 Page 13 of 20 617 where

hc¯ 3 TH = , (88) 8πGkB M0

M0 is the standard general relativistic mass of the black hole, and we have denoted   1 dM (η) θ(η) = 1 + η2 eff . (89) Meff (η) dη

The specific heat CBH of the black hole is defined as   dM dM dr  CBH = =  , (90) Fig. 19 Variation of the dimensionless Hawking temperature of the dT dr dT BH BH r=rhor three-form field supported black holes as a function of the event horizon radius ηS for the case of the Higgs-type three-form field potential, for μ2 = −4 ν = −2 ∈[ , ] ζ and it takes the dimensionless form 10 , 10 , u0 40 60 , and for different values of 0: −5 −5 −5 ζ0 = 10 (solid curve), ζ0 = 2 × 10 (dotted curve), ζ0 = 4 × 10  −5 −5 η η  (short dashed curve), ζ0 = 8×10 (dashed curve), and ζ0 = 16×10 M0 dMeff ( ) d  CBH =  . (91) (long dashed curve), respectively T dη dθ H η=ηS

The Hawking entropy SBH of the black hole is found to or, in an equivalent dimensionless form, be  η S (η)   1 dMeff τ (ηS) =−τH dη, (98) rhor dM rhor 1 dM η2θ 4 (η) dη S = = dr, (92) 0 BH T T dr rin BH rin BH where we have denoted or, in a dimensionless form, c4 τ = . (99)  H π 2σ 4 η 8 G M0TBH S 1 dMeff (η) S (ηS) = CH dη. (93) BH θ (η) dη 0 6.2 Thermodynamics of the Higgs-type black holes The black hole luminosity due to Hawking evaporation can be found to be With the help of the interpolating function for the effective mass the Hawking temperature of a black hole supported by dM a three-form field is obtained explicitly: L =− =−σ A T 4 , (94) BH dt BH BH  + η2 + η  1 (BMH 2CMH )  TBH (ηS) = TH  . (100) where σ is a parameter depending on the adopted physical A + B η + C η2 MH MH MH η=ηS model, while The variations of the Hawking temperature of the three- = π 2 form field black holes are represented in Fig. 19. ABH 4 rhor (95) The Hawking temperature depends on the position of the is the surface area of the event horizon. Then for the black event horizon of the three-form black hole, and it monotoni- η η = . ≈ . hole evaporation time τ we find cally increases in time with S.For S 0 80, TBH 1 6TH , while for ηS = 0.60, TBH ≈ 1.2TH . The standard Hawk-   −8 t fin 1 t fin dM ing temperature is of the order of TH = 6.169 × 10 × τ = dt =− . (96) 2 4 (M /M0), and for astrophysical-type three-form field black t 4πσ t r T in in hor BH holes, having large masses, the shifts in the position of the Hence for the black hole evaporation time τ we obtain the event horizon produce negligible effects. expression The specific heat of the Higgs-type three-form field black hole can be found to be    t fin 1 t fin dM τ = dt =− , (97) C (η ) = C (B + 2C η) × πσ 2 4 BH S H MH MH tin 4 tin rhorTBH  2 [AMH + η (BMH + CMHη)] 123 617 Page 14 of 20 Eur. Phys. J. C (2020) 80 :617

Fig. 20 Variation of the dimensionless specific heat of the three-form Fig. 21 Variation of the dimensionless Hawking entropy of the three- field supported black holes as a function of the event horizon radius ηS form field supported black holes as a function of the event horizon μ2 = −4 for the case of the Higgs-type three-form field potential, for 10 , radius ηS for the case of the Higgs-type three-form field potential, for ν = −2 ∈[ , ] ζ ζ = −5 2 −4 −2 10 , u0 40 60 , and for different values of 0: 0 10 (solid μ = 10 , ν = 10 , u0 ∈[40, 60], and for different values of ζ0: ζ = × −5 ζ = × −5 −5 −5 −5 curve), 0 2 10 (dotted curve), 0 4 10 (short dashed ζ0 = 10 (solid curve), ζ0 = 2 × 10 (dotted curve), ζ0 = 4 × 10 ζ = × −5 ζ = × −5 −5 −5 curve), 0 8 10 (dashed curve), and 0 16 10 (long dashed (short dashed curve), ζ0 = 8×10 (dashed curve), and ζ0 = 16×10 curve), respectively (long dashed curve), respectively

   3 BMH 2AMHη + 4CMHη − 1   +η ( η − ) + 2 η3 CMH 6AMH 2 2CMH  + 2 η2 , BMH (101) and its variation with respect to the event horizon is repre- sented in Fig. 20. The specific heat of the three-form black holes shows a complicated behavior. After initially increasing as a function of ηS, CBH reaches a maximum, and then it monotonically decreases towards a minimum value reached at η)S ≈ 0.90. η ∈ ( . , . ) In the range S 0 50 0 65 there is an increase in the Fig. 22 Variation of the dimensionless evaporation time of the three- numerical values of CBH as compared to the standard general form field supported black holes as a function of the event horizon η relativistic case, so that, for ηS = 0.65, C ≈ 1.45CH . radius S for the case of the Higgs-type three-form field potential, for BH 2 −4 −2 μ = 10 , ν = 10 , u0∃[40, 60], and for different values of ζ0: The variation of the black hole entropy as a function of −5 −5 −5 ζ0 = 10 (solid curve), ζ0 = 2 × 10 (dotted curve), ζ0 = 4 × 10 the event horizon, as given by Eq. (93), is represented in −5 −5 (short dashed curve), ζ0 = 8×10 (dashed curve), and ζ0 = 16×10 Fig. 21. The behavior of the Hawking entropy of the three- (long dashed curve), respectively form field black holes has a similar behavior like their specific heat. The entropies are monotonically increasing functions objects remains very high, with a three-form field black hole for small ηS, they reach a maximum, and they decrease for having one solar mass completely evaporating via Hawking larger values of ηS. The maximum values of the entropy are radiation in around 1062 − 1063 years. of the order SBH ≈ 1.5SH . With the use of the general Eq. (98), the ratio of the black hole evaporation time and of the Hawking evaporation time 6.3 Exponential potential-type black holes is represented in Fig. 22. As a function of ηS the evaporation time monotonically decreases from a maximum value τBH ≈ In the presence of an exponential potential of the three-form 120τH , reached for ηS ≈ 0.55, to a value of τBH ≈ 20τH for field, the metric of the black holes are quasi-Schwarzschild, ηS ≈ 0.85. Since the standard Hawking evaporation time of with the effective mass a constant for most of the range of the −27 3 a black hole is of the order τH ≈ 4.8 × 10 × (M0/g) , variation of η. Hence the thermodynamical properties of the it turns out that the evaporation time for three-form field black holes can be obtained by the thermodynamic properties supported black holes can be one or two orders of magnitude of the Schwarzschild black holes, with the mass substituted higher. Even so, the evaporation time for astrophysical size by the effective mass Meff , as given by Eq. (80). For the 123 Eur. Phys. J. C (2020) 80 :617 Page 15 of 20 617

Hawking temperature of the exponential-type black hole we three-form field. In order to investigate the gravitational prop- obtain erties of the model we have considered the simplest case, corresponding to a vacuum static and spherically symmet- hc¯ 3 hc¯ 3 ric geometry. In this case the system of gravitational field T = ≈ . (102) BH π (η ) (exp) ζ 8 GkB Meff S 8πGkB A (V0,λ,ζ0, u0) equations depend on the scalar radial function , the radial MH δ component of the dual vector B of Aαβγ , and on the arbitrary Hence the Hawking temperature of the black hole is depen- potential V (ζ ) of the three-form field. dent on the parameters of the potential, and of the initial Even within the simple vacuum spherically symmet- conditions at infinity of the field ζ . For the specific heat of ric static model the field equations of the theory become the black hole we obtain extremely complicated. However, the exact solution of the field equations can be obtained in the case of a constant poten- 8πk G 8πk G ( ) C =− B M ≈− B A exp (V ,λ,ζ , u ) . (103) tial. In this case the metric functions can be obtained in a form BH ¯ eff ¯ MH 0 0 0 hG hG similar to the Schwarzschild–de Sitter geometry, but with There is a dependence of the specific heat on the potential the solution containing two arbitrary integration constants parameters, and on the initial values for ζ . The negative sign c1 and c2. Similarly to the metric, the radial scalar function indicates that, as a black hole loses mass, and hence energy, also depends on two arbitrary constants. Depending on the its temperature increases. For the entropy of the black hole choice of these constants, several types of compact geome- we can write down the standard expression, tries can be obtained. We can first reproduce the standard Schwarzschild–de Sitter geometry, in which the three-form 3 π 3 = kBc = kBc 2 field generates a cosmological constant. However, different SBH ABH rhor 4hG¯ hG¯ choices of the constants are also possible, with the solutions 2 πkBGM 1 corresponding to the Schwarzschild–anti de Sitter geometry, = 0 . (104) ¯ η2 ( ,λ,ζ , ) or, more interestingly, to naked singularities having no event hc S V0 0 u0 horizon, and with the singularity located at the center r = 0 The numerical value of the black hole entropy is deter- of the massive object. mined by the mass of the central compact object, as well as For arbitrary non-constant potentials V (ζ ) in order to of the initial condition at infinity of the radial scalar function obtain solutions of the vacuum field equations one must use ζ . Finally, for the rate of the mass loss we have the relation numerical methods. In order to investigate numerically the dM hc¯ 4 1 static spherically symmetric Einstein field equations in the = , (105) dt 15360πG2 M2 presence of a three-form field we have reformulated them in a dimensionless form, and, moreover, we have introduced as which yields for the lifetime of the black hole the expression the independent variable η the inverse of the radial coordinate 5120πG2 M3 5120πG2 M3 η = rg/r. In this representation the numerical integration t = 0 M3 = 0 M3 hc¯ 4 eff hc¯ 4 eff procedure of the field equations is significantly simplified. 2 3   On the other hand to proceed with the numerical integration 5120πG M (exp) 3 = 0 A (V ,λ,ζ , u ) . (106) we need to fix the numerical values of the effective mass, hc¯ 4 MH 0 0 0 of the radial scalar field ζ , and of its derivative ζ  at infin- The corrections to the black hole lifetime are given by the ity. In our present numerical approach we have assumed that (exp) ( ,λ,ζ , ) third power of the function AMH V0 0 u0 , determined at infinity the geometry is asymptotically flat, and therefore by the parameters of the potential and the initial conditions at the metric tensor components take their Minkowskian values infinity. However, the evaporation time of the black holes is when r →∞. But at infinity the values of ζ , and of ζ  may not significantly influenced by the potential parameters, and be arbitrary (but small), and this choice is consistent with the initial conditions. the interpretation of the radial scalar field as a cosmological constant. When integrating numerically the gravitational field equa- 7 Discussion and final remarks tions two possible behaviors are detected. The presence of a singular behavior at finite η = ηS in the field equations, or, In the present paper we have investigated the possible exis- more precisely, in the variation with the distance of the met- tence of massive compact astrophysical objects, described ric tensor coefficients, is explained as indicating the existence by black hole and naked singularity-type geometries, in the of an event horizon. We detect the presence of the horizon α(η ) −β(η ) framework of the three-form field gravitational theory, in from the conditions e S = 0 and e S = 0, respec- which the standard Hilbert–Einstein action of general rel- tively. Consequently, a singularity at finite η corresponds to ativity is extended by the addition of the Lagrangian of a a black hole-type massive astrophysical object. The physical 123 617 Page 16 of 20 Eur. Phys. J. C (2020) 80 :617 mass of the black hole corresponds to the effective mass of One important question in the physics of the naked singu- the three-form field model, which is found to be the sum of larities is if the energy conditions are satisfied for this type the standard mass of the black hole plus the energy contribu- of objects. For example, in [53] it was shown that in the case tion from the scalar component of the three-form field. The of a mixture of a null charged strange quark fluid and radia- second situation is related to the location of the singularity tion collapsing in a Vaidya spacetime a naked singularity can at the center of the massive object r = 0 only, with the cor- be formed, with the matter satisfying the weak, strong and responding object representing a naked singularity, an object dominant energy conditions. The formation of a naked sin- not covered by an event horizon. gularity or of a black hole depends on the initial distribution We have considered two classes of numerical solutions of of the density and velocity, and on the constitutive nature of the gravitational field equations in the three-form field theory, the collapsing matter. In the case of the naked singularities corresponding to two choices of the three-form field potential generated by the three form fields one could also formulate V (ζ ). The potentials we have chosen for our investigations a number of energy conditions, similar to the case of ordi- are the Higgs-type potential, and the exponential-type poten- nary matter, as ρ>0, pr > 0, p > 0, ρ>pr > 0, and tial, respectively. In the case of the Higgs potential we have ρ>p > 0. However, from a physical point of view, these restricted our investigations to initial conditions that lead to conditions refer to some effective quantities constructed from the formation of an event horizon, and consequently of black the three form field, and generally they are not satisfied for holes. The locations of the event horizons are strongly depen- arbitrary three form field configurations. dent on the values at infinity of the radial scalar field and of its The problem of the existence in nature of the naked sin- derivative (the initial conditions), indicating the existence of gularities is closely related to their stability properties. The an important and subtle relation between the initial values of Reissner–Nordström naked singularity with |Q| > M is the three-form field at infinity, and the black hole properties. unstable under small linear perturbations, and a similar insta- The location of the event horizon is also strongly dependent bility occurs for the rotating (Kerr) solution with angular on the parameters μ2 and ν of the potential Higgs-type poten- momentum a > M [62]. These results suggest that these tial, indicating a complex multi-parametric dependence of spacetimes cannot be the endpoint of physical gravitational the position of the event horizon, and of the black hole prop- collapse. In [63] it was shown that as a result of the gravita- erties. The event horizon for three-form field models with tional collapse of a spherically symmetric scalar field the Higgs potential can be located at distances of the order of formation of the naked singularities occurs, but this phe- 2rg from the massive object center, at distance much higher nomenon is unstable. However, as compared to the above- than those of the standard Schwarzschild black holes, indicat- mentioned types of naked singularities, those formed in the ing the formation of more massive black holes than predicted presence of a three form field are of a modified Schwarzschild by standard general relativity. In the case of the Higgs poten- type, with the sign of the term M/r changed from negative tial the numerical results can be fitted well by some simple to positive. The problem of their stability/instability is a very analytical that depend on the initial conditions at infinity. interesting one, due to the different nature of the metric. It In the case of the exponential potential we have identi- is well known that the Schwarzschild black hole is linearly fied numerically two distinct classes of solutions, depend- stable under gravitational and electromagnetic perturbations ing on the initial values of ζ and ζ  at infinity, and on the [64], so that one may conjecture that a similar property holds potential parameters. These two classes correspond to naked for a three form field supported by Schwarzschild-type naked singularity-type solutions, with the singularity located at the singularities. Hence the investigation of the stability prop- center, and no event horizon present, and to standard black erties of the three form field naked singularities may lead holes, characterized by the presence of an event horizon. In to different results as compared to the Reissner–Nordström, both cases the numerical solutions for the metric functions Kerr or scalar field collapse cases. can be approximated well by a Schwarzschild-type form with We have also investigated in detail the thermodynamic α −β e = e = 1 ± 2Meff /r, where Meff depends on the initial properties of the three-form field black hole solutions conditions at infinity, and on the parameters of the expo- obtained by numerical methods. The Hawking temperature is nential potential. The positive sign indicates the presence of an interesting and essential physical property of black holes. a naked singularity, while the negative sign corresponds to The horizon temperature of the three-form black holes indi- an object with an event horizon. The analytical fittings of cates a strong dependence on the initial conditions at infinity the numerical results are extremely useful in the study of of the radial scalar field, and of the properties of the radial the thermodynamic properties of the three-form field black scalar field potential. This is very different from the proper- holes. They also greatly simplify the study of the dynamics ties of the standard general relativistic Hawking temperature, and motion of matter particles around black holes and naked which depends only on the mass of the black hole, and is singularities. independent of the asymptotic conditions at infinity. Simi- lar properties characterize the behavior of the specific heat, 123 Eur. Phys. J. C (2020) 80 :617 Page 17 of 20 617 entropy and evaporation time of the three-form field black scalar field (decomposed as two real scalar fields), the exis- holes. The numerical results show that these quantities are tence of the dual description from a single three-form field also strongly dependent on the initial conditions of the radial could be arguable. However, it is important to note that this scalar field ζ at infinity. dual nature, three-form ↔ scalar field, breaks down in some Moreover, in the three-form field gravitational theory the cases, even for fairly simple self-interactions [5], depending black hole evaporation times may be very different as com- on the choice for the potential V (A2). This fact, however, is pared to the similar results in standard general relativity. But not problematic, and, on the contrary, simply suggests that we should note that our results on the thermodynamics of three-forms can provide us with new physics upon richer black holes, obtained for only two radial scalar field poten- cosmological and astrophysical settings, undoubtedly worth tials and for a limited range of initial conditions at infinity exploring. may be considered of qualitative nature only. But even at this In particular, one may mention Bose–Einstein conden- qualitative level they show the complexity of the compact sates consisting of ultralight bosons, and which can form objects supported by the three-form fields, and of the inter- localized and coherently oscillating stellar-type configura- esting physical and astrophysical processes related to them. tions [73]. For bosons having a higher mass, the bounded con- In particular, the analytic representations of the numerical figurations may have typical masses and sizes of the order of results may be applied for the study of the electromagnetic magnitude similar to those of the neutron stars. Such objects properties of the thin accretion disks existing around black formed from a primordial scalar field are known as boson holes. These properties may help in discriminating the three- stars (for reviews on the structure and properties of boson form field black holes from other similar theoretical objects stars see [71] and [72]). Usually boson stars are constructed as well as from their general relativistic counterparts, and from a complex scalar field coupled to gravity, with the scalar for allowing to obtain observational constraints on the model field φ (t, r) decomposed into two real scalar fields φR (t, r) parameters. and φI (t, r) so that φ (t, r) = φR (t, r)+iφI (t, r) [72]. The The no-hair theorem [65–68] is an important result in evolution and properties of a boson star can be obtained from black hole physics. It states that asymptotically flat black the action [72] holes do not allow for the presence of external nontrivial    √ 1 scalar fields, with non-negative field potential V (φ).The S = d4x −g R + L , (107) BS κ2 SF results obtained in our present investigations indicate that 2 in its standard formulation the no-hair theorem cannot be where extended to the static spherically symmetric solutions of the    1 μν ¯ 2 three-form field gravitational theory. All the numerical black L =− g ∇μφ ∇νφ + V |φ| , (108) SF 2 hole solutions we have obtained are asymptotically flat, and three-form fields with positive radial scalar field potentials where φ¯ is the complex conjugate of the field, while V (|φ|2) exist around them. Similar results have been obtained for is the scalar field potential depending only on the magnitude the black hole solutions in the hybrid metric-Palatini gravity of the scalar field. There is a formal analogy between scalar theory [24], suggesting that the no-hair theorems may not be field models, and the three form field model investigated in valid in some modified gravity theories. But the answer to the present paper. In both cases the action is constructed from the question if such properties are a result of the particular the standard Hilbert–Einstein term plus the field Lagrangian, choice of the three-form field radial scalar potentials, of the which in our approach is given by Eq. (2). Hence if one initial conditions and of the scalar potential parameters, or could map in the complex plane the three-form Lagrangian that they are some generic properties of the theory, requires LA → LSF, then the present three form field extension of further and detailed investigations, at both theoretical and Einstein gravity would become equivalent with a scalar field computational levels. model, and the corresponding solutions, assuming that the Different types of physical fields may play an important mapping three forms field → scalar field does exist, would role in astrophysics and cosmology, especially as potential describe boson stars of different types. constituents of the dark energy and dark matter components In our analysis we have considered as a particular case of the Universe. In particular the role of the scalar fields of the three-form field potential the Higgs-type potential, has been intensively investigated. From a qualitative point given by Eq. (55). In the case of boson stars the gravitation- of view, since there is a dual representation of the three- ally bounded configuration with the quartic self-interaction |φ|2 = 2 |φ|2 + λ |φ|4 form theory to a scalar field [15,16] one may explore partic- potential V m 2 was investigated ular astrophysical settings describing objects such as oscil- in [74]. The physical properties of such a potential can  = λ 2 / π 2 latons [69,70] or maybe, their complex relatives, boson stars be parameterized by the quantity MPlanck 4 Gm , [71,72], in the scalar representative frame. In the latter case, where MPlanck is the Planck mass. The maximum mass of a since boson stars are usually constructed from a complex boson star with quartic self-interaction potential is given by 123 617 Page 18 of 20 Eur. Phys. J. C (2020) 80 :617

1/2 Mmax ≈ 0.22 MPlanck/m [72,74]. It is also interesting Acknowledgements We would like to thank the anonymous ref- to point out that in the Thomas–Fermi limit the scalar field eree for comments and suggestions that helped us to improve our becomes equivalent with a fluid, which in the low or moderate manuscript. BJB is supported by Fundação para a Ciência e a Tec- nologia (FCT, Portugal) through the grant PD/BD/128018/2016. BD ∝ ρ2 density limit has the equation of state P [74]. is partially supported by a grant from the UEFISCDI PNIII-P1-1.2- By assuming that the three form field-scalar field corre- PCCDI-2017-266 “SAFESPACE” Contract, Nr. 16PCCDI/01.03.2018. spondence is valid, by parameterizing the Higgs-type poten- FSNL acknowledges support from the Fundação para a Ciência e a tial with the help of the parameter  = ν/4πGμ2,the Tecnologia (FCT) Scientific Employment Stimulus contract with ref- erence CEECIND/04057/2017. BJB and FSNL also thank funding maximum√ mass√ of a stable three form field star is given by from the research grants No. UID/FIS/04434/2019 and No. PTDC/FIS- 3 Mmax ∝ ν/ 4πGμ , which could lead to masses of the OUT/29048/2017. same order of magnitude as the Chandrasekhar limit for neu- ≈ Data Availability Statement This manuscript has no associated data tron stars, Mmax 3M . However, the similarity (or equiva- or the data will not be deposited. [Authors’ comment: The present work lence) between three-form field stars and boson stars strongly is theoretical, and no data have been generated.] 2 relies on the fulfilling of the condition pr ∝ ρ , which, with the use of Eqs. (16) and (17) becomes Open Access This article is licensed under a Creative Commons Attri- bution 4.0 International License, which permits use, sharing, adaptation,   distribution and reproduction in any medium or format, as long as you 2 2 2 give appropriate credit to the original author(s) and the source, pro- F − V ∝ F /48 − V + ζ V,ζ . (109) vide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article If this differential equation has a solution for ζ , then the are included in the article’s Creative Commons licence, unless indi- cated otherwise in a credit line to the material. If material is not corresponding three form field stellar model has similar prop- included in the article’s Creative Commons licence and your intended erties to a boson star. On the other hand, if no such solution use is not permitted by statutory regulation or exceeds the permit- for the radial function ζ exists, then in the absence of ordi- ted use, you will need to obtain permission directly from the copy- nary matter the modified Einstein gravity in the presence of right holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. the three form fields admits only black hole-type solutions. Funded by SCOAP3. We would like to point out that the above conclusions about the relations between boson and three form stars are mostly of qualitative nature, and in order to fully clarify these issues a detailed investigation of the properties of the three form References field stars is required. From a technical point of view one can identify the formation of a star-like object from the con- 1. T.S. Koivisto, N.J. Nunes, Three-form cosmology. Phys. Lett. B α(η ) −β(η ) ditions e S = 0 and e S = 0, respectively. 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