Spherically symmetric analytic solutions and naked singularities in Einstein–Aether theory

We analyze all the possible spherically symmetric exterior vacuum solutions allowed by the Einstein–Aether theory with static aether. We show that there are three classes of solutions corresponding to different values of a combination of the free parameters, c14=c1+c4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{14}=c_1+c_4$$\end{document}, which are: 0<c14<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 0< c_{14}<2$$\end{document}, c14<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{14} < 0$$\end{document}, and c14=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{14}=0$$\end{document}. We present explicit analytical solutions for c14=3/2,16/9,48/25,-16\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{14}=3/2, 16/9, 48/25, -16$$\end{document} and 0. The first case has some pathological behavior, while the rest have all singularities at r=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=0$$\end{document} and are asymptotically flat spacetimes. For the solutions c14=16/9,48/25and-16\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{14}=16/9, 48/25\, \mathrm {\, and \,}\, -16$$\end{document} we show that there exist no horizons, neither Killing horizon nor universal horizon, thus we have naked singularities. This characteristic is completely different from general relativity. We briefly discuss the thermodynamics for the case c14=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{14}=0$$\end{document} where the horizon exists.


Introduction
The Lorentz invariance is an exact symmetry of special relativity, quantum field theories and the standard model of particle physics, and also a local symmetry in freely falling inertial frames in general relativity [1]. The Lorentz violation in matter interactions is highly constrained by several precision experiments, see [2] for the latest example, while similar studies in gravity are not as well explored. With this motivation, Jacobson and collaborators introduced and analyzed a general class of vector-tensor theories called the Einstein-Aether (EA) theory [3][4][5][6][7].
The first paper to investigate spherical static vacuum solutions in the EA theory was presented by Eling and Jacobson in 2006 [8]. In that paper, the authors found a family of anaa e-mail: chan@on.br (corresponding author) b e-mail: mfasnic@gmail.com c e-mail: vhsatheeshkumar@gmail.com lytical solutions for the metric functions, up to the inversion of a transcendental function, assuming an aether vector proportional to the timelike Killing field. These solutions do not depend on the parameters c 2 and c 3 of the EA theory but only depend on the combination c 1 + c 4 , which for simplicity we denote as c 14 . such that c 1 + c 4 < 2. For c 1 + c 4 > 2 the coupling constant G becomes negative, implying that the gravity is repulsive, and for c 1 + c 4 = 0 this one is exactly the Newtonian gravitational constant. The authors have shown when c 1 + c 4 = 0 the Schwarzschild solution can be obtained and also explored some solutions for different choices of 0 ≤ c 1 + c 4 < 2. In another paper, Eling, Jacobson and Miller [9] studied a perfect fluid, in order to model a neutron star, and consider the vacuum solution given in the previous paper [8] for any c 1 + c 4 < 2. Almost the complete literature on black holes in EA theory can be found in the papers .
Since this article deals with naked singularities, some introductory remarks are in order. In 1965, Roger Penrose proved the first modern singularity theorem [31] for which he is recognized with a Nobel Prize in physics last year [32]. In this paper, Penrose introduced the notion of closed trapped surface for the first time to prove the gravitational collapse in GR inevitably leads to singularities. The most important physical consequence of the singularity theorem is to know whether these singularities can be observed. Typically the singularities that arise in the solutions of Einstein field equations are hidden within event horizons. The singularities that are not hidden inside the event horizon are called naked singularities. The Cosmic Censorship Conjecture (CCC), introduced by Penrose in 1969 [33], does not permit singularities void of an event horizon as the end state of gravitational collapse for generic regular initial data and for a suitable matter system. Although there is no satisfactory mathematical formulation or the proof of CCC, there are many examples of dynamical collapse models which lead to a black hole or a naked singularity as the collapse end state, depending on the nature of the initial data (see e.g. [34][35][36][37][38][39][40][41] and references therein). Two major themes in this context are either mathematically prove or disprove the CCC or provide counter examples. Naked singularities are of current interest because they have observational properties quite different from a black hole. Besides, theoretically these regions of extreme gravity might have some hints of quantum gravity. If observed, they will provide exciting laboratory for the discovery of new physics. Most up-to-date account on naked singularities can be found in [42]. Some of the observational consequences of naked singularities explored recently can be found in the references [43][44][45][46][47][48].
The paper is organized as follows. The Sect. 2 briefly outlines the EA theory, whose field equations are solved for a general spherically symmetric static metric in Sect. 3. In Sects. 4 and 5 we present the explicit analytical solutions. We briefly discuss black hole thermodynamics in Sect. 6 for the case c 14 = 0 where the horizon exists. We summarize our results in Sect. 7.

Field equations in the EA theory
The general action of the EA theory is given by where, the first term defined by R is the usual Ricci scalar, and G the EA coupling constant. The second term, the aether Lagrangian is given by where the tensor K ab mn is defined as being the c i dimensionless coupling constants, and λ a Lagrange multiplier enforcing the unit timelike constraint on the aether, and δ a m δ b n = g aα g αm g bβ g βn .
Finally, the last term, L matter is the matter Lagrangian, which depends on the metric tensor and the matter field. In the weak-field, slow-motion limit EA theory reduces to Newtonian gravity with a value of Newton's constant G N related to the EA coupling constant G in the action (1) by [11], Here, the constant c 14 is defined as Note that if c 14 = 0 the EA coupling constant G becomes the Newtonian coupling constant G N , without necessarily imposing c 1 = c 4 = 0. For c 14 > 2 the coupling constant G becomes negative, implying that the gravity is repulsive. The coupling constant vanishes when c 14 = 2, this in turn renders the action undefined. Thus, physically interesting region is c 14 < 2 where the Newtonian limit can be recovered. The field equations are obtained by extremizing the action with respect to independent variables of the system. The variation with respect to the Lagrange multiplier λ imposes the condition that u a is a unit timelike vector, thus while the variation of the action with respect u a , leads to [11] where, and The variation of the action with respect to the metric g mn gives the dynamical equations, where G Einstein Later, when we solve the field equations (11), we do take into consideration the equations (7)-(10) in the process of simplification. Thus in the paper (as in the equations (20)-(23) below) we seem to solve only the dynamical equations, but in fact we are also solving the equations arising from the variations of the action with respect λ and u a .
In a more general situation, the Lagrangian of GR theory is recovered, if and only if, the coupling constants are identically null, e.g., c 1 = c 2 = c 3 = c 4 = 0, considering the Eqs. (3) and (7).

Spherical solutions of EA field equations
We start with the most general spherically symmetric static metric ds 2 = −e 2 A(r ) dt 2 +e 2B(r ) dr 2 +r 2 dθ 2 +r 2 sin 2 θ dφ 2 . (13) In accordance with Eq. (7), the aether field is assumed to be unitary and timelike, chosen as This choice is not the most general and is restricted to the scenario where aether is static. The aether must tip in a black hole solution as it cannot be timelike be aligned with the null Killing vector on the horizon. As that is not the case with our choice, our solutions are valid only outside the Killing horizon. This is good enough for solar system tests and even for astrophysical solutions to describe the exterior spacetime to a source. The timelike Killing vector of the metric (13) is giving by The Killing and the universal horizon [49,50] are obtained finding the largest root of and respectively, where χ a is the timelike Killing vector. In our case, For the general spherically symmetric metric (13), we compute the different terms in the field equations (12) below.
where G aether In order to identify eventual singularities in the solutions, it is useful to calculate the Kretschmann scalar invariant K. For the metric (13), it is given by Before we proceed, we simplify the field equations. Substituting the field Eq. (21) into (20) we can eliminate the term e 2B and find We have two field Eqs. (22) and (25), to solve. Using these two, we can eliminate A only when c 14 = 0 and obtain the following, We can obtain another equation when c 14 = 0, From the Eq. (26) we can note that there are two possibilities: c 14 = 2 and c 14 = 2, of which c 14 = 2 is not permitted. As we mentioned previously, c 14 > 2 implies that the gravity is repulsive and c 14 < 0 means gravity is stronger. Thus, both mathematical and physical justifications lead us to three possible cases of interest: (i) 0 < c 14 < 2 (ii) c 14 < 0 (iii) c 14 = 0. We now analyze these three possible cases in detail.
In order to find explicit solutions for the metric functions we used the Maple 16 algebraic software following the algorithm: we have made a survey of all possible explicit solutions imposing c 14 = m/n, where m and n are integer constants in the interval 0 < n < 100 and 0 < m < 100 varying in step of one and imposing that 0 < c 14 < 2 or c 14 < 0. This survey has taken several hours of computational time in an old CPU generation computer (Intel Duo Core).

Solutions for 0 < c 1< 2 and c 1< 0
The non-trivial solutions in EA theory exist for 0 < c 14 < 2 which help us understand the physical effects of aether. As we mentioned earlier, we have two field Eqs. (22) and (25), to solve. We first use (25) to express B as From this equation we can notice that if 2 + c 14 r A = 0, we would get A = − 2 c 14 r which would not lead to any solution. Therefore, we proceed by assuming 2+c 14 From the Eqs. (26) and (29), we have and Substituting B and A from these equations into the field Eqs. (20)-(21) we get only an unique equation Solving Eq. (32) for B we have Now, differentiating (33) and substituting into (30) we get (31) which implies consistency.

Solution for c 14 = 3/2
Solving the Eq. (31) we can obtain an analytical solution for A for a particular case c 14 = 3/2. Making a transformation u(r ) = A we can reduce the second order differential equation in A into a first order differential equation in u, thus This equation is easily integrated giving three solutions which one is real and two are imaginaries. The real one is given by where β is an arbitrary integration constant. Since r 2 < 27β 2 , we notice that this solution is not real in all spacetime but only in the region near to the center, meaning that this solution can not represent the exterior of a source. Thus, it will not be analyzed in more detail.

Solution for c 14 = 16/9
Solving the Eq. (31) we can obtain an analytical solution for A for another particular case c 14 = 16/9. Making a transformation u(r ) = A we can reduce the second order differential equation in A into a first order differential equation in u, thus This equation is easily integrated to get two solutions given by where α is an arbitrary integration constant and δ = ±1.
Integrating once this equation and assuming the new arbitrary constant of integration to be zero, without any loss of generality, we obtain When we calculate the limit r → +∞ we get e 2 A → 0 when δ = +1 and e 2 A → 2 √ 2 when δ = −1. Thus, since we must have a flat spacetime at this limit, hereinafter, we assume δ = −1. Besides, we can also notice that for α > 0 the metric function (38) becomes imaginary for r < 8α. Thus, we assert that this solution could be used as the exterior vacuum solution of a static system, where r > r = 8α and r would be the radius of the static system. In this way we can eliminate the pathological imaginary part of the interior spacetime that has no physical meaning. The case α = 0 gives us a flat spacetime since A = 3 4 ln(2) and B = 0. Thus, we will assume, hereinafter, α ≤ 0.
We can now obtain A just differentiating equation (37). Thus, With the analytical solution A we can get B from Eq. (33) giving We can also obtain A and B analytically. When we calculate the limit r → +∞ we get e 2B → 1. We obtain B just differentiating equation (40), that is, In order to check if the analytical equations for A , B, A and B , we substitute them into the field Eqs. (20)- (22) and we show that they are identically satisfied. Thus, we can get analytically the metric components g rr = e 2B , g tt = e 2 A and the Kretschmann scalar using Eq. (24). The Kretschmann scalar obtained is From the Kretschmann scalar, we can get the singularities which are at Since the radial coordinate is always positive, the second singularity does not exist, since α ≤ 0. So, the singularity at r sing1 is physical and which is independent of α. Substituting this Equation into (18) and (19) we get We can see easily again these equations do not have any root with α < 0, hence, there exists no horizon, neither Killing nor universal horizon. In order to compare this result with the Schwarzschild spacetime, we will assume here, for the sake of simplicity, α = −1. From the Schwarzschild we have where M is the mass of the particle. Substituting these metric functions into the Eq. (24) we have the well known Kretschmann scalar for the Schwarzschild metric From Eqs. (38) and (40) we get A and B for α = −1. From this equation we can get g rr = e 2B and g tt = e 2B analytically.
The Kretschmann scalar is obtained from Eq. (42) K for α = −1, thus In the Figs. 1, 2 and 3 we show the comparison of the present work quantities g rr , g tt and K with the Schwarzschild metric ones.

Solution for c 14 = 48/25
Solving the Eq. (31) we can obtain an analytical solution for A for another particular case c 14 = 48/25. Making a transformation u(r ) = A we can reduce the second order differential equation in A into a first order differential equation in u, thus 2ru + 4ru 2 + 4u + 48 25 12r (−2r + 27γ ) where γ is an arbitrary integration constant. Substituting Eq. (52) into (33) we can calculate the limit r → +∞ we get e 2B → 1 only when γ < 0. Thus, since we must have a flat spacetime at this limit, hereinafter, we assume γ = −|γ |. Thus, Substituting Eq. (53) into (33) we get Integrating the Eq. (53) using Mathematica for |γ | = 1 , we obtain A which is given by 1 (a, b; c; x) is a hypergeometric function. Substituting this Eq. into (18) and (19) we found that these equations do not have any root, hence, there exists no horizon, neither Killing nor universal. Thus, we have naked singularities. We obtain B just differentiating equation (40). In order to check the consistency of A , B, A and B , we substitute them into the field Eqs. (20)- (22) and show that they are identically satisfied. Now, we can get analytically the metric components g rr = e 2B , g tt = e 2 A and the Kretschmann Calculating the limit r → +∞, we obtain g tt = e 2 A → 1, as we can see in this figure. Calculating the limit from left r → r sing1 = 0 we obtain g tt = e 2 A → +∞, as we can see in this figure scalar using Eq. (24). The Kretschmann scalar is calculated but not shown because it is too long. From the Kretschmann scalar we can get the singularity which are given by In order to compare this result with the Schwarzschild spacetime, we use the Eqs. (47) and (48). For the sake of simplicity we assume also |γ | = 1. The Kretschmann scalar is obtained from Eq. (24) analytically. In the Figs. 4, 5, and 6 we show the comparison of the present work quantities g rr , g tt and K with the Schwarzschild metric ones. Taking c 14 < 0 implies that the coupling constant of EA theory is stronger than the Newtonian gravitational constant [see Eq. (5)]. Although this possibility is ruled out by experimental data [51], we include this case just for the sake of completeness. Solving the Eq. (31) we can obtain an analytical solution for A for another particular case c 14 = −16. Making a transformation u(r ) = A we can reduce the second order differential equation in A into a first order differential equation in u, thus 2ru + 4ru 2 + 4u − 16r 2 u 3 = 0. (57)

This equation is easily integrated giving
where κ is an arbitrary integration constant. The solution with κ > 0 give us an imaginary solution, thus it will not be considered here. The solution with κ < 0 is gives us real solution. Thus in the following equation we assume that κ = −|κ|.
Substituting Eq. (59) into (33) we can calculate the limit r → +∞ we get e 2B → 1 only when κ < 0. Thus, we have a flat spacetime at this limit. Thus, is a hypergeometric function. Substituting this equation into (18) and (19) we found that these equations do not have any root, hence, there exists no horizon, neither Killing nor universal. Thus, we have naked singularities. In the Figs. 7, 8, and 9 we show the comparison of the present work quantities g rr , g tt and K with the Schwarzschild metric ones.

Solutions for c 14 = 0
Taking c 14 = 0 implies that the coupling constant of EA theory is the same as the Newtonian gravitational constant From the above, we have the solution where μ and ν are some constants of integration which can be chosen appropriately. With this solution for A and B, we can cast the metric (13) in the Schwarzschild form, that is, which yields the standard Schwarzschild metric of GR when μ = 1/2 and ν = M. Substituting A and B equations into (18) and (19) we get which means the Killing horizon is at r K H = 2M, and implies the universal horizon is at r U H = 2M. We can see again that the Killing and the universal horizons coincide just as in GR.

A note on black hole thermodynamics
It is clear from the equations above that they are exactly the same as in GR. However, for a non-static aether, one may have to redefine surface gravity because of the contribution of aether vector field [54].

Conclusions
In the present work, we have analyzed all the possible exterior vacuum solutions with spherical symmetry allowed by the EA theory with static aether. We show that there are three classes of explicit analytic solutions corresponding to different values for c 14  also picks Newtonian gravitational constant as coupling.
As expected, the Killing horizon coincides with the universal horizon and is the same as the event horizon of the Schwarzschild spacetime in GR.
In the 2006 paper, Eling and Jacobson [8] analyzed the behavior of the static aether solutions for 0 ≤ c 14 < 2. They asserted that regular black holes cannot have static aether fields since the Killing vector is null, not timelike on the horizon as always is the case with static aether. As far as we know, this the only work in EA theory that mentions the existence of naked singularity. What we show here is the corollary, meaning the static aether leads to absence of any kind of horizon, suggesting a possible formation of naked singularities, which could circumvent the need for switching from timelike vector to null vector at the horizon by totally avoiding the existence of the horizon itself. Of course, to rigorously prove that the singularity is naked, it would be still necessary to prove that future directed null geodesics can emanate from the singularity.
Finally, we note that the unique case where we have black hole is for c 14 = 0 which furnishes the GR limit. For this the gravitational mass is the same as the Schwarzschild mass [see Eq. (65)]. We have calculated the Killing and the universal horizons and studied their thermodynamic properties. Besides, for the other solutions (c 14 = 16/9, 48/25 and c 14 < 0) we found that there are no horizons, neither Killing nor universal. This suggests that we can have naked singularities.

Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: This paper is a pure theoretical one, thus we do not need nor supplementary observational neither laboratorial data.] Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Funded by SCOAP 3 .