Mimetic gravity in $(2+1)$-dimensions

Mimetic gravity in $(2+1)$-dimensions is discussed. In particular, some physical properties of stationary black hole solutions of this theory in the presence of charge or angular momentum are investigated.

In this paper, we consider three dimensional black holes in the context of mimetic gravity. The theory of mimetic gravity was proposed a few years ago, as an alternative description for the dark matter puzzle [21]. This theory has arisen a lot of attractions in the past few years both from the cosmological viewpoint [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38] as well as black holes physics [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. However, till now black hole solutions of mimetic gravity in (2 + 1)-dimensions have not been constructed. Our purpose here is to explore static and stationary analytical black hole solutions of mimetic gravity in three dimensional spacetime and investigate their properties. We shall consider several cases including whether or not there is a coupling to the Maxwell field or whether or not there is an angular momentum associated with the spacetime. We study the effects of the mimetic field on the casual structure and physical properties of the solutions and disclose that, in contrast to the three dimensional solution of general relativity, in mimetic gravity a curvature singularity emerges at r = 0 even in the absence of Maxwell field. This essential singularity might be due to the extra longitudinal degree of freedom of the gravitational field encoded by the mimetic field. Surprisingly, the curvature singularity disappears by adding an angular momentum to the spacetime. This paper is organized as follows. Section 2 is devoted to introducing the basic field equations of mimetic gravity in (2+1)-dimensions. For simplicity we first ignore the coupling to the Maxwell field in this section and construct three dimensional black hole solutions. In Sec. 3, we take into account the Maxwell field and explore charged mimetic black holes in three dimensions. In Sec. 4, we add an angular momentum to the back hole and investigate rotating (2 + 1)-dimensional solution of mimetic gravity. We summarize our results in Sec. 5.

Field equations and solutions
We start with the following action where R is the Ricci scalar, λ is the Lagrange multiplier, and l is related to the cosmological constant by −Λ = l −2 . F µν = ∂ µ A ν − ∂ ν A µ is the electromagnetic field tensor and A µ is the gauge potential. In the above action, ǫ = ±1, in which the positive and negative sign refer to, respectively, spacelike and timelike nature of vector ∂ µ φ. We adopt (−, +, +) as our signature and set 8πG N = 1 throughout this work. The equations of motion can be derived from varying action (2.1), yielding being the Maxwell energy-momentum tensor. Equation (2.5) restricts the evolution of the mimetic field φ and indicates that the scalar field is not dynamical by itself, nevertheless it makes the longitudinal degree of freedom of the gravitational field dynamical [21]. It was argued [21] that if one assume g µν = g µν (φ,g µν ), in such a way that g µν = ǫ(g αβ ∂ α φ∂ β φ)g µν , then one recovers (2.5) immediately (see also [36,39] Our aim here is to derive static (2 + 1)-dimensional black hole solutions of the above field equations. We assume the metric of spacetime as where an additional degree of freedom is incorporated in the line element through the metric function g(r) which is expected to reflect an extra degree of freedom of gravitation encoded by the mimetic field φ. With metric (2.9), the constraint Eq. (2.5) transforms to f (r)φ ′2 = ǫ, which has solution of the form where we have chosen the positive sign and set the integration constant equal to zero, without loss of generality. Equation (2.10) shows the explicit dependence of the mimetic field on the metric function and reveals that it is not an independent dynamical variable. Let us first consider the uncharged solution. In this case the field equations (2.7) and (2.8), have the following solutions where b is the constant of integration which incorporates the impact of the mimetic field into the solutions. In general b could be either positive or negative. For b = 0, our solutions reduce to the (2 + 1)-dimensional BTZ black holes of Einstein gravity [7]. The horizon is located at r + = l √ M where f (r + ) = 0, while g(r + ) diverges. However, r = r + is a coordinate singularity and both Kretschmann and Ricci scalars have finite values at r + . The sign of |r 2 − M l 2 | depends on whether one considers interior solution (r < r + ) or exterior solution (r > r + ). Expanding g(r) for large r leads to Therefore as r → ∞, we have g(r) ≈ 1 + b, which implies that the remnant of the mimetic field φ contributes to the metric function g(r) through constant b. Nevertheless the asymptotic behavior of the solutions is still AdS since the constant 1 + b can be absorbed by redefinition of the time at the asymptotic region. The (tt) component of the metric is given by where its expansion for large r is given by This confirms that in large r limit, we have −g tt = g rr . On the other hand when r → 0, we have g(r) = 1 and −g tt = g rr = f (r). The infinite redshift surface can be obtained by setting g tt (r) = 0, which yields where i = 1, 2 and − for r s 1 > r + and + for r s 2 < r + . This means that we have 2 infinite redshift surfaces and the black hole horizon is located between them, r s 2 < r + < r s 1 . Besides, r s 1 exists provided b 2 < 1.
We now calculate the scalar curvatures of the spacetime. It is a matter of calculations to show that the Ricci scalar and the Kretschmann invariant are given by Thus, as r → 0, both Ricci and Kretschmann invariants diverge, they are finite at r = 0 and go to R = −6/l 2 and R µνρσ R µνρσ = 12/l 4 as r → ∞. Therefore, the spacetime has an essential singularity at r = 0. This is in contrast to the (2 + 1)-dimensional black holes of Einstein gravity where r = 0 is not a curvature singularity but, rather, a singularity in the causal structure [8]. Indeed, for Einstein gravity where b = 0, the curvature invariants have constant values anywhere, namely R = −6/l 2 and R µνρσ R µνρσ = 12/l 4 , and the curvature singularity disappears.
In the next section, we shall consider the (2 + 1)-dimensional charged black hole of mimetic gravity.

Charged mimetic black holes in 3D
In the presence of coupling to the Maxwell field, the gauge potential and the non-vanishing component of the electromagnetic tensor become where prime indicates derivative with respect to r. Then it is easy to show that the Maxwell equation (2.4) transforms to which has the following solution where q is an integration constant which is indeed the electric charge of the black hole. Substituting metric (2.9), condition (2.10), and the electric field (3.3) into Eq. (2.7), we find These equations have the following solutions where b 1 is again an integration constant which reflects the imprint of the mimetic field on the spacetime. Clearly for b 1 = 0, one recovers the (2 + 1)-dimensional charged black holes of general relativity [9]. One can easily check that these solutions also satisfy Eq. (2.8) for the mimetic scalar field. The horizons are given by the roots of f (r h ) = 0. Depending on the parameters this equation may have at most two real roots corresponding to inner and outer horizon of the black hole. Of course, one may choose the parameters such that the solutions also describe extremal black hole with one horizon, or a naked singularity (see Fig. 1). The integrand in expression (3.8) also diverges at r h indicating that we encounter singularity. However, r = r h is only a coordinate singularity and both Kretschmann and Ricci scalars have finite values on the horizon. The integral in g(r) function cannot be analytically performed for arbitrary values of r, however, it is instructive to study the large r limit of g(r). In this case, one can write Therefore, as r → ∞ we have −g tt = g rr , and the remnant of the mimetic field contributes to the metric function g tt through constant b 1 . It is also interesting to take a close look on the roots of g tt = 0 which define the infinite redshift surfaces. Since there is no any physical reason to avoid negative b 1 , thus one can choose either b 1 > 0 or b 1 < 0. For b 1 ≤ 0, the infinite redshift surfaces r s i coincide with the horizons r h = r ± , namely the roots of f (r h ) = 0. However, for b 1 > 0, the function g tt admits an additional root r s 3 = b 1 /2 (see Fig. 2). Note that r s 3 can be either larger, equal or smaller than r − depending on the values of b 1 . It is easy to show that, for all values of b 1 , both Ricci and Kretschmann scalars diverge at r = 0, they are finite at r = 0 and as r → ∞ they tend to constant values, R = −6/l 2 and R µνρσ R µνρσ = 12/l 4 , similar to 3D solutions of Einstein gravity. This analysis confirms that there is a curvature singularity at r = 0, regardless of the value of b 1 . The behaviour of the electric field for the large r is also given by We have also plotted the behavior of the electric field F tr in Fig. 3. This figure shows that the electric field diverges for small r and goes to zero in large r limit. For b 1 > 0, the electric field has a maximum at finite r which can be easily seen from expression (3.11). Expression (3.11) also shows that, compared to the case of Einstein gravity, the electric field of three dimensional charged mimetic black hole get modified due to the presence of mimetic field.

Rotating mimetic black holes in 3D
Now we consider spinning solutions of mimetic gravity in (2 + 1)-dimensions. We take the spinning metric as where functions f (r), g(r) and N (r) are determined by solving the field equations. Clearly, in the limiting case where the mimetic gravity reduces to Einstein gravity, one expects to have g(r) = 1 and N (r) = −1/2r 2 [7]. The (tφ) component of the field equations (2.7) yields rg(r)N ′′ (r) + 3g(r)N ′ (r) − rN ′ (r)g ′ (r) = 0, (4.2) which implies g(r) = r 3 N ′ (r). Combining with (tt) component of the field equation we arrive at with the following solution Finally, the (rr) component of the field equations (2.7) can be solved to give One can easily show that these solutions fully satisfy the field equations (2.7) and (2.8). In the limiting case where b 0 = 0, the obtained solutions reduce to the (2 + 1)-dimensional rotating black hole solutions of general relativity [7]. When J = 0, they restore the solutions (2.11) and (2.12) provided we define b = 2b 0 M l 2 . It is a matter of calculations to check that f (r) vanishes for two values of r given by where r + is the black hole horizon. In order for the solution to describe a black hole, one must have In the extremal case where |J| = M l, the two roots coincide and we have r + = r − = l M/2. The surface of infinite redshift can be given by finding the positive real roots of the following equation with following solution . In general the location of r ext s depends on the value of b 0 M l 2 and it is easy to check that it is located out of horizon, namely r ext s > r + where r + = l M/2 is the horizon radius of extremal case.
Next we study the scalar invariants. It is easy to check that as r → 0, the Ricci and Kretschmann invariants behave as     ĄLet us remind that for b 0 = 0 the curvature invariants have finite values at the origin for both static and rotating solutions, namely R = −6/l 2 and R µνρσ R µνρσ = 12/l 4 [8]. In mimetic gravity, however, we observed that while for static solutions they diverge at r = 0, but for rotating solution they have finite values at r = 0. In other words, imprint of mimetic field to the spacetime structure in mimetic theory of gravity makes the spacetime singular at the origin r = 0, while the combination of angular momentum J and mimetic field remove the curvature singularity at the origin.
The behavior of the metric functions f (r), g(r) and N (r) for rotating mimetic black holes in three dimensions are plotted in Figs. 4-8. From Fig. 4, we see that depending on the metric parameters, our solutions can represent black hole with one horizon, two   horizon or naked singularity. Figs. 5 and 6 reveal that the metric function g(r) tends to a constant value far from the black hole. The value of this constant depends on b 0 , M and l but independent of J. The intersection of all curves in Fig. 6 is the point in which g(r) = 1 which occurs at r = J/ √ 2M independent of b 0 . We have also depicted the behavior of N (r) in Figs. 7 and 8 where it can be seen that N (r) → ∞ as r → 0, while for large values of r we have N (r) → 2b 0 .

Summary
To sum up, using a scalar mimetic field for isolation the conformal degree of freedom of the gravitational field in a covariant way, it has been demonstrated that the scalar field can encode an extra dynamical longitudinal degree of freedom to the gravitation field which can play the role of mimetic dark matter even in the absence of particle dark matter [21]. In this paper, we have focused on mimetic gravity in (2 + 1)-dimensional spacetime and constructed various static and spinning black hole solutions of this theory. In contrast to the three dimensional solutions of Einstein gravity which has only causal singularity and scalar invariants are constant everywhere, in mimetic gravity the spacetime admits a curvature singularity. We confirmed that both Ricci and Kretschmann curvatures of three dimensional mimetic black holes diverge at r = 0 even in the absence of Maxwell field. Interestingly, when the angular momentum is added to the spacetime, the singularity at r = 0 disappears and the scalar invariants become constant.