Elliptic function solutions in Jackiw-Teitelboim dilaton gravity

We present a new family of solutions for the Jackiw-Teitelboim model of two-dimensional gravity with a negative cosmological constant. Here, a metric of constant Ricci scalar curvature is constructed, and explicit linearly independent solutions of the corresponding dilaton field equations are determined. The metric is transformed to a black hole metric, and the dilaton solutions are expressed in terms of Jacobi elliptic functions. Using these solutions we compute, for example, Killing vectors for the metric.


I. INTRODUCTION
It is well-known that the Einstein gravitational field equations for a vacuum (with a zero matter tensor) are automatically solved by any metric g on a two-dimensional space-time M . A proof of this fact is given in section 2 of [1], for example. A non-trivial theory of gravity for such an M was worked out in 1984 by R. Jackiw and C. Teitelboim (J-T). This involves in addition to g a scalar field Φ on M called a dilaton field; see [2], [3]. The pair (g, Φ) is subject to the equations of motion derived from the action integral where R(g) is the constant Ricci scalar curvature of g, and where the (negative) cosmological constant is Λ = −1/l 2 . In local coordinates (x 1 , x 2 ) on M , the Hessian in (1.1) is given by where Γ k ij are the Christoffel symbols (of the second kind) of g [1]. The J-T theory has, for example, the (Lorentzian) black hole solution = mr, (1.5) with M a black hole mass parameter. Here and throughout we note that our sign convention for scalar curvature is the negative of that used in [2], [3], and by others in the literature. The purpose of this paper is the following. For real numbers a, b = 0 and for a soliton velocity parameter v we consider the following metric in the variables (x 1 , x 2 ) = (τ, ρ): = a 2 b 2 dn 2 (ρ, κ) a 2 κ 4 sn 2 (ρ, κ)cn 2 (ρ, κ) dn 2 (ρ, κ) − v 2 4 dτ 2 − κ 4 sn 2 (ρ, κ)cn 2 (ρ, κ) dn 2 (ρ, κ) a 2 κ 4 sn 2 (ρ, κ)cn 2 (ρ, κ) where sn(x, κ), cn(x, κ), dn(x, κ) are the standard Jacobi elliptic functions with modulus κ; 0 ≤ κ ≤ 1 [4]. We will generally assume that κ = 0, v 2aκ 2 > 1. (1.7) As will be seen later, this metric is the diagonalization of a metric constructed from solutions r(x, t), s(x, t) of the reaction diffusion system r t − r xx + 2 b 2 r 2 s = 0 (1.8) We will explicate the solutions r(x, t), s(x, t) in terms of the elliptic function dn(x, κ). Remarkably, the metric in (1.6) has constant scalar curvature R(g) = 8/b 2 so that the first equation in (1.1) holds. The main work of the paper then is to solve the corresponding system of partial differential equations (the dilaton field equations) in (1.1), which for g in (1.6) are Here the cosmological constant is Λ = −4/b 2 . Given the complicated nature of our g, the system (1.9) is necessarily quite difficult to solve directly. Our method is to construct a series of transformations of variables so that g in (1.6) is transformed to g in (1.4). Then we can use the simple solution Φ(T, r) = mr in (1.5), and other known solutions, to work backwards through these transformations of variables to construct Φ(τ, ρ) that satisfies (1.9). The various details involved, with further remarks that lead to (1.6), will be the business of sections 2, 3, and 4.
In the end, we obtain the following main result: The metric in (1.6) solves the first J-T equation of motion (1.1). Namely R(g) = 8/b 2 , as we have remarked. Also three linearly independent solutions of the field equations in (1.1), namely of the system of equations (1.9), are given by which we assume is non-zero. Given (1.7) we shall see in section 4 that A = 0 only for a = ±(1 − √ 1 − κ 2 )v/2κ 2 and moreover that the second expression under the radical (i.e. v 2 /4 − · · · ) in (1.10) is positive. For κ = 1, A = (v 2 /4 − a 2 ) 2 > 0, but we can have A < 0 for some κ < 1. Also for κ = 1 the solutions in (1.10) reduce to those given in (4.19), with (1.6) given by (4.20).

II. REACTION DIFFUSION SYSTEMS AND DERIVATION OF THE METRIC IN (1.6)
Since the metric (1.6) is one of the main objects of interest we indicate in this section its derivation. For a constant B consider the system of partial differential equations r t − r xx + Br 2 s = 0 (2.1) s t + s xx − Brs 2 = 0 in the variables (x, t). This system is a special case of the more general reaction diffusion system (RDS) that occurs in chemistry, physics, or biology, for example, where d r , d s are diffusion constants, and F, G are growth and interaction functions. The key point for us is that from solutions r(x, t), s(x, t) of (2.1) one can construct a metric g of constant Ricci scalar curvature R(g) = 4B by the following prescription [5], [6], [7]: = rs One could also simply start with the definitions in (2.3), apart from the preceding references that employ Cartan's zweibein formalism [8], and use a Maple program (tensor), for example, to check directly that indeed R(g) = 4B. Our interest is in the choice B = 2/b 2 , where for real a, b, v, with a, b = 0 as in section 1, r(x, t), s(x, t) given by are available. Using the prescription (2.3) one computes that = a(x − vt), so that dρ = a(dx − vdt), g can be expressed more conveniently as The goal now is to set up a change of variables (t, ρ) −→ (τ, ρ) so that g in (2.7) is transformed to (1.6) -where the cross term dτ dρ does not appear, in comparison with the term dtdρ appearing in (2.7). For this purpose note first, in general, that for The condition that the cross term dτ dρ does not appear is therefore that φ(ρ) satisifies (2.10) Apply this to (2.7): (2.11) Now by (2.5), dn 2 (x, κ) = 1−κ 2 sn 2 (x, κ) = sn 2 (x, κ)+cn 2 (x, κ)−κ 2 sn 2 (x, κ) = cn 2 (x, κ)+(1−κ 2 )sn 2 (x, κ) ≥ cn 2 (x, κ) and sn 2 (x, κ) ≤ sn 2 (x, κ) + cn 2 (x, κ) = 1 ⇒ sn 2 (x, κ)cn 2 (x, κ)/dn 2 (x, κ) ≤ 1. If the term in parenthesis in (2.11) were zero, this would therefore force the inequality v 2 /4a 2 κ 4 ≤ 1. That is, if v/2aκ 2 > 1, which is the assumption in (1.7), then v 2 /4a 2 κ 4 > 1 and therefore the denominator term in parenthesis in (2.11) is non-zero, which means that φ ′ (ρ) is a continuous function and (2.11) therefore indeed has a solution φ(ρ), with the assumption (1.7) imposed. Also, the coefficient of dρ 2 in (2.9) is where for convenience we write sn,cn,dn for sn(ρ, κ), cn(ρ, κ), dn(ρ, κ) and Q for which is the coefficient of dρ 2 in (2.9) by (2.12). Then by (2.10),(2.9) reads which is (1.6). That is, we have verified that the change of variables τ = t + φ(ρ) with φ(ρ) subject to condition (2.11) (which in fact renders φ ′ (ρ) a continuous function, again assuming (1.7)) transforms the reaction diffusion metric in (2.7) to the diagonal metric in (1.6).

III. TRANSFORMATION OF THE METRIC IN (1.6) TO A J-T BLACK HOLE METRIC
Now that the existence of the metric in (1.6) has been described in the context of a reaction diffusion system (namely (1.8)), the strategy of this section is to set up a series of changes of variables, as indicated in the introduction, that transforms it to the simpler J-T form (1.4). Other applications, of independent interest, can flow from this -apart from our main focus to solve the system (1.9). A general method to go from (1.6) to (1.4) has been developed by the first named author. Alternatively, one can generalize part of the argument in [6] that leads at least to a Schwarzschild form, as we do here, and then argue a bit more to obtain the J-T form -the final result being expressed by equations (3.11)-(3.13) below.
= |b|r 2 0 /2, the Schwarzschild g in (3.6) goes to which in turn goes to by way of the change of variables t = bT , r − = r 1 /b. We need one final observation: In general a metric of the form say A 1 = 0 can be transformed to the J-T form (1.4), namely by way of the change of variables r = x + B1 2A1 . Apply this to (3.8) with x playing the role of r 1 there: for Using definition (3.5) for α, β and r 2 0 = v 2 4 − a 2 κ 4 , which is definition (3.3), one computes that in (3.11).

IV. DERIVATION OF THE SOLUTIONS (1.10) OF THE FIELD EQUATIONS (1.9)
The main result is derived in this section. Namely, we indicate how the series of changes of variables in section 3 (according to remarks in the introduction) lead to the linearly independent solutions Φ (j) (τ, ρ), j = 1, 2, 3 in (1.10) of the dilaton field equations in (1.9). There the metric elements g ij are given by (1.6): For Q(ρ, κ) in (2.13) = a 2 b 2 dn 2 (ρ, κ)Q(ρ, κ), g 12 = g 21 = 0, and the ∇ i ∇ j Φ are given by (1.3) for (x 1 , x 2 ) = (τ, ρ). The Christoffel symbols Γ k ij in (1.3) (which could be computed by Maple, for example) will not be needed for the derivation of (1.10), although they could be used to verify these solutions. Obviously any dilaton solution could be replaced by any non-zero multiple of itself. In the following then, we can disregard such multiples if we wish to.
By the final change of variables r = r 1 + B1 2A1 in section 3, we see that The change of variables t = bT , for the Schwarzchild version of our metric in (3.6). Next let x = (2r 2 + r 2 0 )/r 4 0 to get where we have disregarded the multiple |b|/b = ±1 in (4.6) and have used that sinh(|b|x/b) = (|b|/b) sinh(x). Finally, the first change of variables r = |a|dn(ρ, κ) in section 3 gives by definition (3.12). If we disregard the multiple |b|/r 2 0 in (4.8) and use that r 2 0 + α def.
is an expression of Y in terms of local coordinates (x 1 , . . . , x n ) on M , then (5.1) is equivalent to the system of equations for 1 ≤ j, k ≤ n [8], [9]. In the special (diagonal) case with g ij = 0 for i = j, and with n = 2, the Killing equations (5.2) simplify to the following three equations: As was shown in [10], every solution (g, Φ) of the field equations in (1.1) gives rise to a corresponding Killing vector field Y = Y (g, Φ) by way of the local prescription with ǫ ij = a permutation symbol. Y preserves both g and Φ. For g in (1.4) and for the fields Φ in (1.5) and (4.2), the corresponding Killing vector fields are given in equations (16), (17), (18) of [11], for example. Our interest of course is in the case of the three solutions (g, Φ (j) ) in (1.10) with g given by (1.6). By (4.1), |detg| = a 2 b 2 κ 2 |sncn|dn. Since Y i could be replaced by a scalar multiple of itself (for example, −Y i ), we shall disregard the absolute value of sncn here, and given (1.9) we shall take l = b 2 (instead of |b| 2 ). For ǫ 11 = ǫ 22 = 0, ǫ 12 = −1 = −ǫ 21 (5.4) then assumes the generic form where we take (x 1 , x 2 ) = (τ, ρ) in (5.3).

VI. SOME CLOSING REMARKS
For the metric g in (1.6), whose derivation was discussed in section 2, we have obtained as a main result explicit linearly independent solutions Φ (j) , j = 1, 2, 3, in (1.10) of the corresponding system of dilaton field equations in (1.9). We have also computed the associated Killing vector fields Y (g, Φ (j) ) that leave both g and the Φ (j) invariant; see (5.7), (5.9), (5.10) and the remarks that follow (5.9) and (5.10). The dilaton fields simplify to the expressions given in (4.19) in the special case when the elliptic modulus κ is 1,and g simplifies to the expression given in (4.20).
In section 3, by a series of explicit transformations of variables, g moreover was transformed to a Jackiw-Teitelboim black hole metric g J−T of the simple form (1.4) -namely to g J−T given by (3.11), with accompanying data given by (3.12), (3.13). Here again assumption (1.7) was imposed. An advantage of the parameterization (3.11) is that, for example, simple formulas exist [10], [12] for thermodynamic quantities such as the Hawking temperature T H and black hole entropy S.
We point out, for the record, that the general solutions of all 2d dilaton gravity models are known. For example, see section 3 of the paper [13] of T.Klosch and T.Stroble. However (again) we have constructed very explicit elliptic solutions for the specific model of interest -solutions that we feel do not follow directly at all as a corollary of the results of [13].
In view of an interesting suggestion of the referee, whom we thank (and also for the references [14], [15]), we add some final remarks that provide a brief review of a connection of the J-T model to cold plasma physics. This connection is facilitated by way of a resonant nonlinear Schrödinger (RNLS) equation.
The authors in [14] consider a system of nonlinear equations that describe the dynamics of two-component cold collision-less plasma in the presence of an external magnetic field B. For uni-axial plasma propagation, this system is reduced to a system that describes the propagation of nonlinear magneto-acoustic waves in a cold plasma with a transverse magnetic field. By way of a shallow water approximation of the latter system, a reduction of it to a RNLS equation of the form i ∂ψ ∂t ′ + ∂ 2 ψ ∂x ′ 2 − 1 2 |ψ| 2 ψ = (1 + β 2 ) 1 |ψ| is achieved. Here x ′ = βx, t ′ = βt are rescaled space and time variables, and B has an expression in terms of a suitable power series expansion in the parameter β 2 . ψ has the form ψ = √ ρ e −iS where S(x ′ , t ′ ) is a velocity potential and ρ is the mass density of the plasma. Also note the remarks in [15]. A key point of interest for us is that for r def.
= − √ ρ e S/β < 0, = 2β, (6.4) in the variables (x ′ , τ def. = βt ′ ), the reaction diffusion (RD) system (2.1) is satisfied; r, s are denoted by e (+) , e (−) in [14]. On page 186 of [1] it is shown that, conversely, given solutions r > 0, s < 0 of the RD system (2.1), one can naturally construct a RNLS solution. By (2.4), with b = 2β by (6.4), we can take All of this means that we can apply the prescription (2.3) to construct a metric g plasma of constant Ricci scalar curvature R = 4B def. = 2 β 2 , as we did in (2.7) where the notation t, ρ there is now taken to mean τ, a(x ′ − vτ ). Moreover our results show that g plasma can be transformed to a J-T black hole metric of the form (1.4). Thus we can account for a J-T black hole connection in cold plasma physics. Our results also provide for elliptic solutions of the corresponding dilaton field equations.
The authors declare that there is no conflict of interest regarding the publication of this paper.