Some 3-dimensional maximally symmetric solutions of Einstein-aether theory

Working in isotropic coordinates, we get some maximally symmetric nonrotating solutions of the Einstein-aether theory in 2+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2+1$$\end{document} dimensions, all in analytical forms. Curvature singularities are not found in the Ricci and Kretschmann scalars, while conical singularities are avoidable by fixing some integration constants. In Schwarzschild type coordinates, most line elements become the metric of a black hole or a massive particle in the 2+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2+1$$\end{document}-dimensional de Sitter or anti-de Sitter spacetime, but in some solutions the aether field cannot be written in closed forms.


Introduction
The Einstein-aether theory is an intriguing vector-tensor theory invented by Jacobson and Mattingly two decades ago [1]. It breaks Lorentz invariance by coupling a unit timelike vector field to the metric. The vector field defines a congruence of timelike curves filling the full spacetime like an omnipresent fluid [2]. For this reason, the field is called the aether field, and the theory is dubbed the Einstein-aether theory. The Einsteinaether theory has been investigated from various aspects in the past two decades, and even more intensively in the past decade. For very recent studies on this theory, see [3][4][5][6][7][8][9][10][11][12][13] as a partial list and references therein.
Ever since its invention, various solutions to the Einsteinaether theory have been worked out in the literature. In 3 + 1 dimensions, there are a wormholelike static aether solution [14,15], some numerical black hole solutions [16,17], as well as analytical solutions of spherical neutral black holes [18,19], spherical charged black holes [20] and slowly rotating black holes [21]. In 2 + 1 dimensions, a family of rotating black hole solutions were discovered in Ref. [22], while their charged counterparts were reported in Ref. [23]. In 1 + 1 a e-mail: twang@phy.ecnu.edu.cn (corresponding author) dimensions, all solutions to the Einstein-aether theory have been presented in Ref. [24].
When deriving the above solutions, it is customary to choose the Schwarzschild type coordinates [14,23], the Eddington-Finkelstein type coordinates [16,[18][19][20], or their rotating analogs [21,22]. In all of these choices, the metric has an angular component of the same form, that is g 22 = ρ 2 . In other words, these references have chosen the same radial coordinate ρ, often named as the areal radius [15]. One exception is Ref. [15], in which the isotropic coordinates are utilized to study spherically symmetric solutions of the Einstein-aether theory in 3 + 1 dimensions. Interestingly, for the wormholelike static aether solution, all components of the metric are explicit functions of the isotropic radial coordinate r [15]. In contrast, for the same solution [25], the radial and temporal components of the metric are implicit functions of the areal radius [14]. It indicates that the isotropic coordinates have an advantage over the Schwarzschild coordinates in the Einstein-aetehr theory. This impression motivates us to extend the work in isotropic coordinates [15] to lower dimensions.
Another motivation of this paper is similar to Ref. [24]. Both classical and quantum gravity become more tractable in lower dimensions [26], so the Einstein-aether theory in lower dimensions might provide a manageable setting for studying the possible role of a preferred frame in quantum gravity. We have to work in 2 + 1 dimensions, because the 1 + 1-dimensional solutions have been derived exhaustively in Ref. [24], and the isotropic coordinates are meaningless in 1 + 1 dimensions.
In 2 + 1 dimensions, a circularly symmetric nonrotating solution can be written in isotropic coordinates as ds 2 = −e 2μ(t,r ) dt 2 + e 2ν(t,r ) dr 2 + r 2 dθ 2 (1) with a unit timelike aether field u a ∂ a = e −μ(t,r ) cosh α(r, t)∂ t + e −ν(t,r ) sinh α(r, t)∂ r . (2) It is easy to check abc u a ∇ b u c = 0, that is to say, the aether field is hypersurface orthogonal [16,22,27]. It is challenging to find all solutions of this form, which will be left for an ongoing exhaustive study. In this paper, as a pathfinding work, we will focus on solutions with α(r, t) = 0 and satisfying a necessary condition of locally maximally symmetric spacetime [22], where a, b = 0, 1, 2 are spacetime indices. The remainder of this paper is organized as follows. Starting with the 2 + 1-dimensional action of Einstein-aether theory and then focusing on the line element (1) and the aether field (2) with α(r, t) = 0, we will write down the Einstein equations, the aether equations and the maximal symmetry condition (3) in Sect. 2, all followed by some necessary comments. General solutions to these equations will be derived and classified in Sect. 3, where Eqs. (21), (22) will be ingored deliberately. Taking the two equations into consideration, we will put the solutions into their final forms in Sect. 4. As a result, there will be five types or fourteen subtypes of solutions, presented orderly in Sects. 4 All of them are maximally symmetric as will be demonstrated in Sect. 5.1, where the Ricci scalar, the Kretschmann scalar and the aether stress-energy tensor will be calculated. In Sect. 5.2, restricted to constant time slices, solutions of the same type yield the same form of spatial line element, which will be employed for illustrations in Fig. 1 and for analyzing conical singularities. After interpreting the physical significance of all solutions in Sect. 6, we will summarize our main results in Sect. 7.
Throughout this paper, we will denote the derivatives with respect to t and r by dots and primes respectively. As we have done in Eqs. (2), (3), the lowercase Latin letters a, b, . . . run over spacetime indices 0, 1, 2 or t, r, θ. In the Einsteinaether theory, there are four coupling constants, notated by c 1 , c 2 , c 3 , c 4 conventionally. When integrating differential equations, we will introduce two types of integration constants. Some are independent of both t and r , and we will denote them as b i (i = 1, 2, 3, . . .). Others are dependent on t but independent of r , so they will be represented by a i (t) (i = 1, 2, 3, . . .), where the argument t shows the timedependence. However, in the case that a i (t) is set to a timeindependent constant, we will suppress its argument t. From now on, the arguments of μ(t, r ), ν(t, r ), α(t, r ), β(t, r ) will be omitted for succinctness.

Action and equations
Incorporating a cosmological constant , the action for Einstein-aether theory in 2 + 1 dimensions can be written as [23] in which ∇ denotes the covariant derivative, indices a, b, c, d = 0, 1, 2, and the tensor Here c 1 , c 2 , c 3 , c 4 are dimensionless coupling constants. 1 In Eq. (4), λ is the Lagrangian multiplier to ensure that the aether field u a is a unit timelike vector. Note that 1/(16π G) is an overall factor in this action, thus the Einstein equations are Rg ab + g ab = T ae ab (6) in our conventions of notation [23], where T ae ab is the stressenergy tensor of the aether field.

Five classes of general solutions
Equations (7), (16) are clumsier than the other equations.
In this subsection, we will attack them with Eqs. (17), (20), (21), (22) and (25). As an outcome, we will write down the general solutions of μ and ν. Equation (25) enables us to briefly rewrite Eq. (7) as Combining it with Eqs. (16), (20), we can obtain ν 2 −μν +ν re 2ν = e 2μ re 2ν + r μ ν + μ (30) in which both μ and ν have been carefully cancelled. Substituting Eqs. (21), (22) into (30) and then multiplying it with e −μ−2ν /r , we arrive at with b 1 being a time-independent positive constant. Substituting (22) into Eq. (29) and then multiplying it with re −2μ , we find Because each step of our derivation here is reversible, one can reverse it to reproduce (7), (16) from Eqs. (31), (32). From now on, we will pay attention to Eqs. (31), (32) which are similar to each other and more tractable. In fact, they can be integrated in closed forms to give the general solutions of μ and ν. We will present their solutions exhaustively in the rest of this subsection.
In the simplest case, we consider Inserting it into Eq. (31), we get a constraint which can be integrated as with σ being ±1 or 0, while a 5 (t) is an undetermined function of t. Keep in mind that b 1 is a positive constant. The analytical solution of this equation takes different forms depending on the signs of a 2 (t) 2 − and σ a 5 (t) 2 . There are four cases to be considered. Let us present them one by one. 2 should be positive on the right hand side of Eq. (36), and the solution of Eq.
Substituting this solution into Eq. (31), it is easy to obtain the expression and that of Eq. (31), (40) Inserting it into Eq. (31), we get the expression In this case, the general solution of Eq. (31) is

Equation (26) is not independent
In passing we mention that Eq. (26) is not an independent equation, because it can be derived from Eqs. (16), (17), (29) as follows. After multiplying Eq. (29) with e −2μ−2ν /r , we differentiate it with respect to r . Eliminatingν andν in the resulted equation with Eq. (17), we find eitherν = 0 or In both situations, one can directly verify Eq. (26) with the help of Eq. (16).
In case A, according to the constraint (34), a 2 (t) should be a constant if a 1 (t) = 0, so we can safely exclude the subcase A1. In cases B, C, D and E, a 1 (t) andȧ 2 (t) cannot vanish simultaneously, otherwise the expressions of e μ will reduce to zero and the metric (1) will fail to describe a 2 + 1dimensional spacetime. Therefore, we need not consider subcases B4, C4, D4 and E4.
For convenience, we combine Eqs. (21), (22) into the forṁ Inserting Eq. (33) into it, we geṫ In the present subcase, this equality dictates a 3 (t) = −1 and a 2 (t) = 0, which further implies = 0. Consequently, inserting Eq. (33) into Eqs. (21), (22), we obtainȧ Up to a transformation of time coordinate and a redefinition of b 1 , the metric and the aether field can be written as There is no suitable value for a 3 (t) to meet Eqs. (34) and (47) in this subcase.
In this subcase, Eq. (34) is well satisfied. Inserting Eq. (33) into Eqs. (21), (22), we obtain in which a 2 (t) = ± √ / . Up to a coordinate transformation, the metric and the aether field can be written as Then from a 2 (t) 2 − < 0 we can infer that > 0. Up to a coordinate transformation and a redefinition of b 1 , the metric and the aether field can be written as Up to a transformation of time coordinate, the metric and the aether field are of the form In particular, when a 2 (t) = 0, the metric and the aether field are with > 0.

Subtype
After transforming the time coordinate, we can write the metric and the aether field as Inserting Eqs. (39), (40) into Eq. (22), we find in this subcase Up to a coordinate transformation and a redefinition of b 1 , the metric and the aether field can be written in the form for ≥ 0, and a similar form with sinh 2 √ t replaced by − cos 2 √ − t for < 0.

Subtype
Up to a transformation of time coordinate, the metric and the aether field are of the form Especially, if a 2 (t) = 0, the metric and the aether field can be written as with < 0.

Subtype
Up to a transformation of time coordinate, the metric and the aether field are of the form r 2 sinh 2 √ t sinh −2 a 5 ln r b 1 + a 6 (t) dr 2 + r 2 dθ 2 , for ≥ 0, and a similar form with − tanh √ t / √ , Inserting Eqs. (41), (42) into Eq. (22), we find in this subcase Up to a coordinate transformation and a redefinition of b 1 , the metric and the aether field are of the form for ≥ 0, and a similar form with sinh 2 √ t replaced by − cos 2 √ − t for < 0.

Subtype
The metric and the aether field are of the form up to a transformation of time coordinate.
In the special limit a 2 (t) = 0, the metric and the aether field can be written as with < 0.
Up to a coordinate transformation and a redefinition of b 1 , the metric and the aether field can be written as for ≥ 0, and a similar form with sinh 2 √ t replaced by − cos 2 √ − t for < 0.

Subtype
In an appropriate time coordinate, the metric and the aether field can be written as In the limit a 2 (t) = 0, the metric and the aether field are Up to a transformation of time coordinate, the metric and the aether field are of the form for ≥ 0, and a similar form with − tanh √ t / √ , − t respectively for < 0.

Maximal symmetry and curvature singularities
Maximally symmetric solutions, such as the Friedmann-Lemaître-Robertson-Walker (FLRW) metric and the Bañados-Teitelboim-Zanelli (BTZ) black hole [29], are of great interest in physics. For the BTZ black hole, the spacetime is maximally symmetric. In the FLRW metric, only the space is maximally symmetric. As we have mentioned, Eq. (3) is a necessary condition for maximal symmetry, thus it is still uncertain whether the obtained solutions are maximally symmetric at last. In this subsection, we will confirm that all of the solutions presented in Sect. 4 are maximally symmetric. By definition, maximally symmetric spacetimes are those with the maximal number of independent Killing vectors. In 2 + 1 dimensions, this number is three. However, given the metric of a spacetime, it is often difficult to find all of the Killing vectors. Here we turn to a convenient criterion in [30]. As argued in [30], if the Riemann tensor satisfies R abcd = 1 6 R(g ac g bd − g ad g bc ) (80) with the Ricci scalar R a constant, the metric will be maximally symmetric. In Eq. (80) we have fixed the dimension of spacetime to 2 + 1. After some tedious but straightforward computations, one can check Eq. (80) and Second, in the special case T ae ab = 0, Eqs. (3) and (80) are trivial results of the Einstein equations. Generally, they put a constraint on the stress-energy tensor, Inserting Eq. (81), we find which can be checked with solutions in Sect. 4 straightforwardly. From Eq. (80), one can directly prove a relation between the Ricci and Kretschmann scalars Together with Eq. (81), it suggests that the solutions we have obtained are free of curvature singularities. As we will illustrate in the coming subsection, they are not free of conical singularities unless appropriate values are set to parameters a 3 and a 5 .

Spatial slices and conical singularities
Restricted to a constant time slice t = t 0 , the line element (1) reduces to which describes a spatial surface as long as e 2ν(t,r ) > 0. To illustrate this surface, we will embed it in a 3-dimensional Euclidean or Minkowski space ds 2 = ±dz 2 + d 2 + 2 dθ 2 , contingent on the value of terms in the square bracket in Eq. (86). Taking e ν(t,r ) > 0 without loss of generality, the parametric equations for the surface are determined by = re ν(t 0 ,r ) , Here the upper and the lower signs are in accordance with Eq.
(86). The signs should be appropriately chosen to guarantee that z 2 ≥ 0. For example, if e 2ν(t 0 ,r ) − re ν(t 0 ,r ) 2 > 0, then one have to choose the plus sign, and ( , θ, z) are exactly cylindrical coordinates in a 3-dimensional Euclidean space.
Once the parametric equations are gained from Eq. (87), we can depict the surface straightforwardly by plotting the parametric equations in 3D spaces. Indeed, we find e 2ν(t,r ) > 0 for all of the solutions in Sect. 4, hence the constant time slices of these solutions are all spatial. In previous sections, we have classified the solutions into five types, tagged as A to E, and we have divided each type into several subtypes. Fortunately, for solutions of the same type but different subtypes, the function e 2ν(t,r ) depends on r in the same form. Therefore, the same type of solutions have the same form of spatial slice. Applying the above procedure to these solutions, we have successfully drawn the 3D pictures of their spatial slices in Fig. 1. In the figure, surfaces embedded in the Euclidean space are drawn in color without meshes. In contrast, meshed surfaces in black and white are embedded in the Minkowski space. Note that even for the same solution, the sign of e 2ν(t 0 ,r ) − re ν(t 0 ,r ) 2 depends not only on the value of a 3 or a 5 , but also on the value of r . Therefore, as clear in the figure, different parts of one surface are probably embedded in different spaces. In Fig. 1, we have not depicted the spatial slice for type A solutions in the cases |1 + a 3 | = 1, which is a 2-dimensional flat plane.
There is an alternative method to draw the figure. For all solutions in Sect. 4, one can rewrite the spatial line elements in terms of as for a 3 = −1; Comparing each line element with Eq. (86), it is not hard to derive dz/d and therefore the surface equation, which can be plotted to produce Fig. 1.
The line elements in Eq. (88) are helpful for explaining many features in Fig. 1. Here we make use of them to analyze conical singularities in this figure. In the figure, it is apparent that there are usually conical singularities at = 0 in solutions of types A, B and C. To better understand such singularities, let us take the limit → 0 of dl 2 A , dl 2 B and dl 2 C . Assuming θ ranges from 0 to 2π , we find in this limit the cone angle is 2π |1 + a 3 | (for type A solutions) or 2π |a 5 | (for solutions of type B or C). This elucidates the absence of conical singularities in subfigures with |1 + a 3 | = 1 or |a 5 | = 1. When |1 + a 3 | < 1 or |a 5 | < 1, the cone angle is less than 2π , so there is a conical singularity at the point = 0, near which the surface is embedded in the Euclidean space. Otherwise, if |1 + a 3 | > 1 or |a 5 | > 1, then the cone angle is greater than 2π , and there is a conical singularity at the point = 0 embedded in the Minkowski space. To get rid of conical singularities in solutions of types A, B and C, we can set a 3 = 0 or −2, a 5 = ±1 or change the range of θ equivalently. From the expression of dl 2 E , it is clear that the point = 0 is not a conical singularity but a spike extending to infinity.

Physical interpretation
From Eqs. (6) and (84), we observe that all line elements obtained in this paper are solutions of the equations value of e f f was unknown before we obtained the solutions, but rewriting Eq. (6) in such a form helps to understand the physical significance of the solutions. Enlightened by the above observation, we transform the coordinates t, r into τ , ρ, and put solutions of subtypes B2, C2, D2, E2 in a unified form In the parameter region e f f < 0, which is permissible in subcases C2, D2, E2, this line element has been discussed in Ref. [29]. In subcase C2, σ a 2 5 > 0, the line element describes an anti-de Sitter spacetime if a 2 5 = 1, or a similar spacetime with a conical singularity otherwise. In subcase D2, σ a 2 5 < 0, the line element (90) represents a static BTZ black hole of mass a 2 5 /(8G), whose vacuum state [29] is described by the subtype E2 solution (σ a 2 5 = 0). In the parameter region e f f > 0, which is possible in subcases B2, C2, D2, E2, the above line element can be interpreted as the 3-dimensional Schwarzschild-de Sitter solution with a mass 1 − σ a 2 5 /(8G) [31,32], akin to the solutions in flat spacetime [33,34]. We should remind that the expression (91) for u a is well-defined only if σ a 2 5 − − a 2 2 ρ 2 ≥ 0. This inequality can be satisfied by any value of ρ in subcases C2 and E2, but only by ρ 2 ≥ σ a 2 5 / − a 2 2 in subcases B2 and D2. Therefore, in subcases B2 and D2, the solution covers only a patch of the spacetime mentioned here.
Likewise, the subtype A4 solution can be put in the form Clearly it corresponds to a 3-dimensional Schwarzschild-de Sitter spacetime with the mass 1 − (1 + a 3 ) 2 /(8G) and the effective cosmological constant e f f = / > 0.
In the subtype A2 solution, the effective cosmological constant e f f is zero. The physical significance of this solution can be better understood by transforming into In the special case that κ(τ ) is a constant, the solution describes a 2 + 1-dimensional Rindler spacetime. Generally, it describes a spacetime with a uniform surface gravity κ(τ ) [35]. We find = 1 and thus e f f = for all of the remaining solutions, i.e., solutions of subtypes B1, B3, C1, C3, D1, D3, E1, E3. After some tedious algebras, we are able to rewrite the line elements of these solutions in the form whose physical significance is similar to metric (90). However, the aether field cannot be expressed explicitly in the Schwarzschild type coordinates τ and ρ in all solutions. For instance, in subcase B3, the metric is related to Eq. (94) by the coordinate transformation ρ = a 5 √ cosh √ t cosh −1 a 5 ln r b 1 + a 6 (t) , Unless a 6 (t) takes very special forms, the coordinate τ is not an explicit function of t and r , and thus the aether field (58) cannot be expressed in τ and ρ in a closed form. This situation is similar to the wormholelike static aether solution, which can be written in a closed form in the isotropic coordinates but not in the Schwarzschild coordinates.
Although there is a timelike Killing field ∂ τ in most solutions, we emphasize that the aether field u a ∂ a is seldom parallel to the Killing field.
We should warn that the above interpretation of our solutions are not unique, because the interpretation of a gravitational solution is observer-dependent. A well-known example is the de Sitter solution, which can be interpreted either as a static spacetime by a static observer or as an expanding spacetime by an comoving observer. Solutions of subtypes B1, C1, D1, E1 can be rewritten in a form generalizing the 2 + 1-dimensional FLRW metric ds 2 = −dt 2 +ã(t) 2 dr 2 σ a 2 5 + kr 2 +r 2 dθ 2 , where k = sign a 2 (t) 2 − . Here again 1 − σ a 2 5 /(8G) can be interpreted as the mass located at the origin, inducing a conical singularity. In subcase B1, the scale factorã(t) = 1 √ cosh √ t . In subcases C1, D1 and E1, a(t) 2 = 1 sinh 2 √ t for > 0, whileã(t) 2 = − 1 cos 2 √ − t if < 0.

Summary
In this paper, we have found all maximally symmetric solutions of the form (1) for Einstein-aether theory, with α(r, t) = 0 in the aether field (2). The solutions are classified into fourteen subtypes and presented orderly in Sects. 4 There are several issues to investigate in the future. First, one can study the perturbations of these solutions. Although 2+1 Einstein gravity does not have local degrees of freedom, the aether field should introduce extra degrees of freedom. Second, making use of these solutions, it will be interesting to study possible roles of the aether field in quantum gravity. Third, the authors are seeking for more general solutions of the form (1) without the maximal symmetric condition and the ansatz α(r, t) = 0.

Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: This paper is a theoretical work, and all of the data are shown as figures and formulae in this paper.] 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 . SCOAP 3 supports the goals of the International Year of Basic Sciences for Sustainable Development.