# Elliptic Function Solutions in Jackiw-Teitelboim Dilaton Gravity

Authors: Jennie D’Ambroise; Floyd L. Williams
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

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.


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## Body

## 1. Introduction

It is well known that the Einstein gravitational field equations for a vacuum (with a zero matter tensor) are automatically solved by any metricg on a two-dimensional space-time M. A proof of this fact is given in Section 2 of [1], for example. A nontrivial theory of gravity for such an M was worked out in 1984 by Jackiw and 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(1)Rg=2l2,∇i∇jΦ=gijΦl2derived from the action integral(2)Sg,Φ=constant·∫Md2xdet⁡gΦRg-2l2,where R(g) is the constant Ricci scalar curvature of g and the (negative) cosmological constant is Λ=-1/l2. In local coordinates (x1,x2) on M, the Hessian in (1) is given by(3)∇i∇jΦ=∂2Φ∂xi∂xj-∑k=12Γijk∂Φ∂xk,1≤i,j≤2,where Γijk are the Christoffel symbols (of the second kind) of g [1]. The J-T theory has, for example, the (Lorentzian) black hole solution(4)g:ds2=-m2r2-MdT2+dr2m2r2-M,with coordinates (x1,x2)=(T,r), where(5)Λ=-m2,Rg=2m2,ΦT,r=def.mr,with M being 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 numbersa,b≠0 and for a soliton velocity parameter v, we consider the following metric in the variables (x1,x2)=(τ,ρ):(6)ds2=def.a2b2dn2ρ,κa2κ4sn2ρ,κcn2ρ,κdn2ρ,κ-v24dτ2-κ4sn2ρ,κcn2ρ,κdn2ρ,κa2κ4sn2ρ,κcn2ρ,κdn2ρ,κ-v24-1dρ2,where sn(x,κ), cn(x,κ), and dn(x,κ) are the standard Jacobi elliptic functions with modulus κ; 0≤κ≤1 [4]. We will generally assume that(7)v2aκ2>1,κ≠0.As will be seen later, this metric is the diagonalization of a metric constructed from solutions r(x,t),s(x,t) of thereaction diffusion system(8)rt-rxx+2b2r2s=0,st+sxx-2b2rs2=0.We will explicate the solutions r(x,t),s(x,t) in terms of the elliptic function dn(x,κ). Remarkably, the metric in (6) has constant scalar curvature R(g)=8/b2 so that the first equation in (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), which for g in (6) are(9)∇i∇jΦ=Rg2gijΦ=4b2gijΦ,1≤i,j≤2.Here the cosmological constant is Λ=-4/b2.Given the complicated nature of ourg, system (9) is necessarily quite difficult to solve directly. Our method is to construct a series of transformations of variables so that g in (6) is transformed to g in (4). Then we can use the simple solution Φ(T,r)=mr in (5) and other known solutions to work backwards through these transformations of variables to construct Φ(τ,ρ) that satisfies (9). The various details involved, with further remarks that lead to (6), will be the business of Sections 2, 3, and 4.In the end, we obtain the following main result: the metric in (6) solves the first J-T equation of motion (1). Namely, R(g)=8/b2, as we have remarked. Also three linearly independent solutions of the field equations in (1), namely, of the system of equations (9), are given by(10)Φ1τ,ρ=2a2dn2ρ,κ+v24-a22-κ2,Φ2τ,ρ=dnρ,κsinh⁡Aτv24-a2κ4sn2ρ,κcn2ρ,κdn2ρ,κ,Φ3τ,ρ=dnρ,κcosh⁡Aτv24-a2κ4sn2ρ,κcn2ρ,κdn2ρ,κfor(11)A=def.v416-v2a222-κ2+a4κ4,which we assume is nonzero. Given (7), we shall see in Section 4 that A=0 only for a=±(1-1-κ2)v/2κ2 and moreover that the second expression under the radical (i.e., v2/4-⋯) in (10) is positive. For κ=1, A=(v2/4-a2)2>0, but we can have A<0 for some κ<1. Also for κ=1 the solutions in (10) reduce to those given in (60), with (6) given by (61).


## 2. Reaction Diffusion Systems and Derivation of the Metric in (6)

Since metric (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(12)rt-rxx+Br2s=0,st+sxx-Brs2=0in the variables (x,t). This system is a special case of the more general reaction diffusion system (RDS),(13)rt=drrxx+Fr,s,st=dssxx+Gr,s,which occurs in chemistry, physics, or biology, for example, where dr and ds are diffusion constants and F and G are growth and interaction functions. The key point for us is that from solutions r(x,t) and s(x,t) of (12) one can construct a metric g of constant Ricci scalar curvature R(g)=4B by the following prescription [5–7]:(14)ds2=def.g11dt2+2g12dtdx+g22dx2,g11=def.-rxsx,g12=def.12srx-rsx,g22=def.rs.One could also simply start with the definitions in (14), 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/b2, where, for real a, b, and v, with a,b≠0 as in Section 1, r(x,t),s(x,t) given by(15)rx,t=def.abdnax-vt,κexp⁡v24+a22-κ2t-vx2,sx,t=def.-abdnax-vt,κexp⁡-v24+a22-κ2t+vx2are solutions of system (12), which also could be checked directly by Maple. For B=2/b2, (12) is system (8) with solutions (15) promised in Section 1, and g in (14) has the scalar curvature 4B=8/b2 discussed in Section 1. From [4], various formulas like(16)sn2x,κ+cn2x,κ=1,dn2x,κ+κ2sn2x,κ=1,ddxsnx,κ=cnx,κdnx,κ,ddxcnx,κ=-snx,κdnx,κ,ddxdnx,κ=-κ2snx,κcnx,κare available. Using prescription (14), one computes that(17)g11=a2b2a2κ4sn2ax-vt,κcn2ax-vt,κ-v24dn2ax-vt,κ,g12=a2b2v2dn2ax-vt,κ,g22=-a2b2dn2ax-vt,κ.For ρ=def.a(x-vt), so that dρ=a(dx-vdt), g can be expressed more conveniently as(18)ds2=a2b2dn2ρ,κa2κ4sn2ρ,κcn2ρ,κdn2ρ,κ-v24dt2-vadtdρ-dρ2a2.The goal now is to set up a change of variables(t,ρ)→(τ,ρ) so that g in (18) is transformed to (6), where the cross term dτdρ doesnot appear, in comparison with the term dtdρ appearing in (18). For this purpose note first, in general, that for(19)h=Aρdt2+C1ρdρdt+C2ρdρ2the change of variables τ=t+ϕ(ρ) gives dt=dτ-ϕ′(ρ)dρ and dt2=dτ2-2ϕ′(ρ)dτdρ+ϕ′(ρ)2dρ2 and(20)h=Aρdτ2+-2ϕ′ρAρ+C1ρdτdρ+Aρϕ′ρ2-C1ρϕ′ρ+C2ρdρ2.The condition that the cross term dτdρ does not appear is therefore that ϕ(ρ) satisfies(21)ϕ′ρ=C1ρ2Aρ.Apply this to (18):(22)ϕ′ρ=-v2a·a2κ4sn2ρ,κcn2ρ,κdn2ρ,κ-v24-1.Now, by (16), dn2(x,κ)=1-κ2sn2(x,κ)=sn2(x,κ)+cn2(x,κ)-κ2sn2(x,κ)=cn2(x,κ)+(1-κ2)sn2(x,κ)≥cn2(x,κ) and sn2(x,κ)≤sn2(x,κ)+cn2(x,κ)=1⇒sn2(x,κ)cn2(x,κ)/dn2(x,κ)≤1. If the term in parenthesis in (22) was zero, this would therefore force the inequality v2/4a2κ4≤1. That is, if v/2aκ2>1, which is the assumption in (7), then v2/4a2κ4>1 and therefore the denominator term in parenthesis in (22) is nonzero, which means that ϕ′(ρ) is a continuous function and (22) therefore indeed has a solution ϕ(ρ), with assumption (7) imposed. Also, the coefficient of dρ2 in (20) is(23)a2b2dn2Qv24a2Q2-ab2vdn2v2aQ-b2dn2=-b2v2dn24Q-b2dn2,where for convenience we write sn, cn, and dn for sn(ρ,κ),cn(ρ,κ),dn(ρ,κ) and Q for(24)Qρ,κ=def.a2κ4sn2ρ,κcn2ρ,κdn2ρ,κ-v24.Then(25)Q+v24=a2κ4sn2cn2dn2⟹1+v24Q=a2κ4sn2cn2dn2Q⟹-b2dn2-b2v2dn24Q=-a2b2κ4sn2cn2Q,which is the coefficient of dρ2 in (20) by (23). Then, by (21), (20) reads(26)h=a2b2dn2Qdτ2-a2b2κ4sn2cn2Qdρ2,which is (6). That is, we have verified that the change of variables τ=t+ϕ(ρ) with ϕ(ρ) subject to condition (22) (which in fact makes ϕ′(ρ) a continuous function, again assuming (7)) transforms the reaction diffusion metric in (18) to the diagonal metric in (6).In the special case when the elliptic modulusκ=1,(27)snx,1=tanh⁡x,cnx,1=dnx,1=sech⁡xand (18) and (6) simplify:(28)ds2=a2b2sech2⁡ρa2tanh2⁡ρ-v24dt2-vadtdρ-dρ2a2,ds2=a2b2sech2⁡ρa2tanh2⁡ρ-v24dt2-tanh2⁡ρa2tanh2⁡ρ-v24-1dρ2,which are the line elements (3.12) and (3.14), respectively, in [6]; a here corresponds to the notation k there. Also the cosmological constant Λ0 in [6] corresponds to our 2Λ=-8/b2:b2=8/(-Λ0). Similarly, r and s in (15) reduce to the dissipative soliton solutions q+ and q-, respectively, in (2.32) of [6], apart from the factor b. One can also explicitly determine ϕ(ρ) in (22).


## 3. Transformation of the Metric in (6) to a J-T Black Hole Metric

Now that the existence of the metric in (6) has been described in the context of a reaction diffusion system (namely, (8)), the strategy of this section is to set up a series of changes of variables, as indicated in the introduction, which transforms it to the simpler J-T form (4). Other applications, of independent interest, can flow from this, apart from our main focus to solve system (9). A general method to go from (6) to (4) has been developed by the first named author. Alternatively, one can generalize part of the argument in [6] which leads at least to a Schwarzschild form, as we do here, and then argue a bit more to obtain the J-T form, with the final result being expressed by (39)–(41) below.Start with the change of variablesr=|a|dn(ρ,κ) so that dr=-κ2|a|sn(ρ,κ)·cn(ρ,κ)dρ by (16) ⇒(29)dr2r2=κ4sn2ρ,κcn2ρ,κdρ2dn2ρ,κ.Also by (16),(30)a2κ4sn2ρ,κcn2ρ,κdn2ρ,κ=a2κ21-dn2ρ,κ1-sn2ρ,κdn2ρ,κ=a2-a2dn2ρ,κκ2-κ2sn2ρ,κdn2ρ,κ=a2-r2κ2+dn2ρ,κ-1r2/a2=a2-r2κ2-1+r2/a2r2/a2=a2-r2r2a2κ2-1+r2⟹a2κ4sn2ρ,κcn2ρ,κdn2ρ,κ-v24=a2-r2r2a2κ2-1+r2-v24=a4κ2-1r2+2a2-a2κ2-r2-v24=a4κ2-1r2+2a2-a2κ2-r2-r02-a2κ4for(31)r02=def.v24-a2κ4>0.Again by (7), v2/4a2κ4>1⇒v2/4>a2κ4⇒ indeed r02>0. By (29) and (30), we see that we can write (6) as(32)g=b2-r2r2+r02+a2κ4+κ2-2+a41-κ2r2dτ2+r2+r02+a2κ4+κ2-2+a41-κ2r2-1dr2.Next letx=def.(2r2+r02)/r04, as in (3.18) of [6], but where our r02 in (31) generalizes their r02, and for convenience let(33)α=def.a2κ4+κ2-2,β=def.a41-κ2in (32). Then g in (32) assumes the form(34)g=b2-r044r04x2-1-αr022r02x-1-βdτ2+116r08r044r04x2-1+α2r02r02x-1+β-1dx2=b2r044-r04x2-1+2αr02r02x-1+4βr04dτ2+r04x2-1+2αr02r02x-1+4βr04-1dx2which generalizes the Schwarzschild form (3.19) of [6], since for κ=1 we have that α=β=0 in (33).For the change of variablest=A0τ and  r-=A0x with A0=def.|b|r02/2, the Schwarzschild g in (34) goes to(35)g=-4b2r-2-1+2αr022br--1+4βr04dt2+4b2r-2-1+2αr022br--1+4βr04-1dr-2,which in turn goes to(36)g=-4b2r12-b2+2αr022bbr1-b2+4βb2r04dT2+4r12b2-b2+2αr022bbr1-b2+4βb2r04-1dr12by way of the change of variables t=bT and  r-=r1/b. We need one final observation: in general a metric of the form(37)g1=-A1x2+B1x+C1dT2+A1x2+B1x+C1-1dx2,say A1≠0, can be transformed to the J-T form (4); namely,(38)g1=-A1r2+C1-B124A1dT2+A1r2+C1-B124A1-1dr2,by way of the change of variables r=x+B1/2A1. Apply this to (36) with x playing the role of r1 there: (39)g=-A1r2+C1-B124A1dT2+A1r2+C1-B124A1-1dr2for(40)A1=def.4b2,B1=def.4αbr02b,C1=def.-b2-2αb2r02+4βb2r04.Using definition (33) for α,β and r02=v2/4-a2κ4, which is definition (31), one computes that(41)C1-B124A1=-b2r04v416-a2v222-κ2+a4κ4in (39).


## 4. Derivation of Solutions (10) of the Field Equations (9)

The main result is derived in this section. Namely, we indicate how the series of changes of variables in Section3 (according to remarks in Introduction) lead to the linearly independent solutions Φ(j)(τ,ρ), j=1,2,3, in (10) of the dilaton field equations in (9). There the metric elements gij are given by (6). For Q(ρ,κ) in (24),(42)g11=def.a2b2dn2ρ,κQρ,κ,g12=g21=0,g22=a2b2dn2ρ,κ-κ4sn2ρ,κcn2ρ,κdn2ρ,κQρ,κ-1,and ∇i∇jΦ are given by (3) for (x1,x2)=(τ,ρ). The Christoffel symbols Γijk in (3) (which could be computed, e.g., by Maple) will not be needed for the derivation of (10), although they could be used to verify these solutions. Obviously any dilaton solution could be replaced by any nonzero multiple of itself. In the following then, we can disregard such multiples if we wish to.In addition to the dilaton solutionΦ(1)(T,r)=def.mr in (5) for metric (4) in the variables (T,r), there are solutions(43)Φ2T,r=def.m2r2-Msinh⁡mMT,Φ3T,r=def.m2r2-Mcosh⁡mMT.We work backwards the changes of variables in Section 3 for Φ(1)(T,r) and Φ(2)(T,r), for example, to see how one arrives at the first two solutions Φ(1)(τ,ρ) and Φ(2)(τ,ρ) in (10) in the variables (τ,ρ).Starting with the (39) version of (4), we have m2=A1=4/b2 by (40), with M=-(C1-B12/4A1) given by (41). Here mM=B12-4A1C1/2 (for m=2/|b|) ⇒(44)Φ1T,r=2br,Φ2T,r=A1r2+C1-B124A1sinh⁡B12-4A1C12T.By the final change of variables r=r1+B1/2A1 in Section 3, we see that A1r2=A1r12+B1r1+B12/4A1⇒(45)Φ1T,r1=2br1+B12A1,Φ2T,r1=A1r12+B1r1+C1sinh⁡B12-4A1C12T.The change of variables t=bT and  r-=r1/b preceded the change r=r1+B1/2A1, so that(46)Φ1t,r-=2bbr-+B1bA1,Φ2t,r-=4r-2+B1br-+C1sinh⁡B12-4A1C12bt,since A1b2=def.4. We had t=A0τ and r-=A0x for A0=def.|b|r02/2, which gives(47)Φ1τ,x=br02x+B1bA1,bbΦ1τ,x=br02x+B1bA1,Φ2τ,x=b2r04x2+B1bb2r02x+C1sinh⁡B12-4A1C14·br02bτ,for the Schwarzschild version of our metric in (34). Next let x=(2r2+r02)/r04 to get(48)Φ1τ,r=b2r2+r02r02+B1bA1,Φ2τ,r=b22r2+r022r04+B1bb22r2+r02r02+C1sinh⁡B12-4A1C14r02τ,where we have disregarded the multiple |b|/b=±1 in (47) and have used sinh(|b|x/b)=(|b|/b)sinh(x). Finally, the first change of variables r=|a|dn(ρ,κ) in Section 3 gives(49)Φ1τ,ρ=br022a2dn2ρ,κ+r02+αbr02,by definition (40). If we disregard the multiple |b|/r02 in (49) and use r02+α=def.v2/4+a2(κ2-2) by definitions (31) and (33), we obtain from (49) the first solution(50)Φ1τ,ρ=2a2dn2ρ,κ+v24+a2κ2-2in (10). More work is required of course to obtain the second solution there.First, we note that, by (40) and (41),(51)B12-4A1C1=-4A1C1-B124A1=16r04v416-a2v222-κ2+a4κ4⟹B12-4A1C14r02τ=v416-a2v222-κ2+a4κ4·τ,which is Aτ in (10). Also, for r=|a|dn, dn=dn(ρ,κ), the quantity under the other radical in (48) is(52)b2r042a2dn2+r022+B1bb2r022a2dn2+r02+C1=4a4b2r04dn4+4a2b2r02+B1bba2r02dn2+b2+B1bb2+C1,where, by definition (40),(53)B1bba2r02=4a2b2αr04,b2+B1bb2+C1=b2+2b2αr02-b2-2αb2r02+4βb2r04=4βb2r04⟹4a2b2r02+B1bba2r02=4r02a2b2+4a2b2αr04=4a2b2r04r02+α=4a2b2r04v24+a2κ2-2again by definitions (31) and (33). That is, since β=a4(1-κ2) by definition (33), the quantity in (52) (which is under the radical in (48) for r=|a|dn) is(54)4a4b2r04dn4+4a2b2r04v24+a2κ2-2dn2+4a4b21-κ2r04=4a2b2r04a2dn4+v24+a2κ2-2dn2+a21-κ2.We let B(ρ) denote the latter bracket here. By (48), (51), and (54), we see that (for now) Φ(2)(τ,ρ)=B(ρ)sinhAτ, if we disregard the multiple 4a2b2/r04=2ab/r02.We find an alternate expression forB(ρ), which is simpler and shows that B(ρ)>0, given (7). Again we write sn, cn, and dn for sn(ρ,κ),cn(ρ,κ),dn(ρ,κ), and we make use of (16). (55)Bρ=def.dn2a2dn2+v24-2a2+a2κ2+a21-κ2dn2=dn2a21-κ2sn2+v24-2a2+a2κ2+a21-κ2dn2=dn2-a2κ2sn2+v24-a21-κ2+a21-κ2dn2=dn2v24-a2κ2sn2+a21-κ2dn21-dn2=dn2v24-a2κ2sn2+a21-κ2dn2κ2sn2=dn2v24-a2κ2sn2dn2-1+κ2dn2=dn2v24-a2κ2sn2dn2-κ2sn2+κ2=dn2v24-a2κ4sn2dn21-sn2=dn2v24-a2κ4sn2cn2dn2,where we noted in Section 2 that sn2cn2/dn2≤1. Hence(56)v24-a2κ4sn2cn2dn2≥v24-a2κ4>0by (7), again as in (31), and we see that B(ρ)>0, since dn(ρ,κ)≠0 for ρ being a real number. Moreover, we have established the desired expression for Φ(2)(τ,ρ) in (10). Clearly one can replace the hyperbolic sine in the preceding discussion by the hyperbolic cosine in (43) to obtain the third solution Φ(3)(τ,ρ) in (10). To finish other claims made in Section 1, we check that in (11) A=0 only for a=±(1-1-κ2)v/2κ2. We continue to assume (7) of course.The quartic equationA=0 has roots a=±(1+1-κ2)v/2κ2 and ±(1-1-κ2)v/2κ2 with a2=(2-κ2+21-κ2)v2/4κ4 and (2-κ2-21-κ2)v2/4κ4, respectively. (7) requires that a2<v2/4κ4, which forces the inequalities(57)2-κ2+21-κ2<1,2-κ2-21-κ2<1,of which the first one reads 1-κ2+21-κ2<0, with the left-hand side here being ≥0, a contradiction. That is, we cannot have a=±(1+1-κ2)v/2κ2 which means that a=±(1-1-κ2)v/2κ2. Also we check that the solutions are linearly independent: assume for constants c1,c2,c3 that(58)c1Φ1τ,ρ+c2Φ2τ,ρ+c3Φ3τ,ρ=0.Differentiate this equation with respect to τ and evaluate the result at (τ,0):(59)c2Acosh⁡Aτv2+c3Asinh⁡Aτv2=0,since dn(0,κ)=1 and sn(0,κ)=0. The choice τ=0 then gives c2=0, since A,v≠0, and differentiation of the equation c3Asinh⁡(Aτ)|v|/2=0 and at τ=0 gives c3=0. Using again dn(0,κ)=1 we see by (10) that Φ(1)(τ,0)=def.v2/4+a2κ2>0 and hence also c1=0.Note that ifv=a=2 and κ=1/2, for example, then even though v/2aκ2=2>1 (so that (7) is satisfied), we have that A=-12<0.Again in the special case when the elliptic modulusκ=1, we have in (11) that A=(v2/4-a2)2>0 and A=v2/4-a2>0 (by (7) or (31)), and B(ρ)=sech2ρv2/4-a2tanh2ρ=sech2ρv2-4a2tanh2ρ/4; by (27), (55) ⇒B(ρ)=1/2sechρv2-4a2tanh2ρ. Here (directly) tanh2ρ≤1⇒v2-4a2tanh2ρ>0, again as v2>4a2. Thus, by (10) and (28),(60)Φ1τ,ρ=2a2sech2⁡ρ+v24-a2,Φ2τ,ρ=sinh⁡v24-a2τsech⁡ρv2-4a2tanh2⁡ρ,Φ3τ,ρ=cosh⁡v24-a2τsech⁡ρv2-4a2tanh2⁡ρ(where we have disregarded the multiple 1/2 in B(ρ)) are dilaton field solutions for the metric(61)ds2=a2b2sech2⁡ρa2tanh2⁡ρ-v24dτ2-tanh2⁡ρa2tanh2⁡ρ-v24-1dρ2.The solutions in (60) are also new.


## 5. Killing Vector Fields for Solutions (6) and (10)

Recall that a smooth vector fieldY on an n-dimensional Riemannian manifold (M,g) is called aKilling vector field (or an infinitesimal motion of M) if, for arbitrary smooth vector fields X,Z on M,(62)YgX,Z=gY,X,Z+gX,Y,Z=0.If Y=∑i=1nYi∂/∂xi is an expression of Y in terms of local coordinates (x1,…,xn) on M, then (62) is equivalent to the system of equations(63)∑i=1ngki∂Yi∂xj+gji∂Yi∂xk+∂gjk∂xiYi=0for 1≤j,k≤n [8, 9]. In the special (diagonal) case with gij=0 for i≠j and with n=2, the Killing equations (63) simplify to the following three equations:(64)2g11∂Y1∂x1+∂g11∂x1Y1+∂g11∂x2Y2=0,g11∂Y1∂x2+g22∂Y2∂x1=0,∂g22∂x1Y1+2g22∂Y2∂x2+∂g22∂x2Y2=0.As was have shown in [10], every solution (g,Φ) of the field equations in (1) gives rise to a corresponding Killing vector field Y=Y(g,Φ) by way of the local prescription(65)Yi=lϵijdet⁡g∂Φ∂xjwith ϵij being a permutation symbol. Y preserves both g and Φ. For g in (4) and for the fields Φ in (5) and (43), the corresponding Killing vector fields are given in (16), (17), and (18) of [11], for example. Our interest of course is in the case of the three solutions (g,Φ(j)) in (10) with g given by (6). By (42), |det⁡g|=a2b2κ2|sncn|dn. Since Yi could be replaced by a scalar multiple of itself (e.g., -Yi), we shall disregard the absolute value of sncn here, and given (9), we shall take l=b/2 (instead of |b|/2). For ϵ11=ϵ22=0 and ϵ12=-1=-ϵ21, (65) then assumes the generic form (66)Y1=2a2bκ2snρ,κcnρ,κdnρ,κ-1-∂Φ∂ρ,Y2=2a2bκ2snρ,κcnρ,κdnρ,κ-1∂Φ∂τ;Y=Y1∂∂τ+Y2∂∂ρ,where we take (x1,x2)=(τ,ρ) in (64).For the first solution(67)Φ1τ,ρ=2a2dnρ,κ+v44-a22-κ2in (10), the computation of the corresponding Killing vector field Y is trivial: by (16) and (66), Y1=2/b, and of course Y2=0. Since ∂g11/∂τ=∂g22/∂τ=0 by (42), the Killing equations in (64) are satisfied and we see that(68)Y=2b∂∂τfor (g,Φ(1)). Computations for the other two solutions Φ(2) and Φ(3) in (10) are more involved. The result is the following, where again(69)A=116v4-16v2a2+8v2a2κ2+16a4κ4in definition (11).ForΦ(2)(τ,ρ), (70)Y1=1/4sinh⁡Aτv2+-v2κ2-4a2κ4-8a2κ2sn2ρ,κ+8a2κ4sn4ρ,κ+4a2κ2a2bdn2ρ,κv2+-v2κ2-4a2κ4sn2ρ,κ+4a2κ4sn4ρ,κ,Y2=A/4cosh⁡Aτv2+-v2κ2-4a2κ4sn2ρ,κ+4a2κ4sn4ρ,κa2bκ2snρ,κcnρ,κdnρ,κ.For Φ(3)(τ,ρ) one has quite similar formulas for Y1 and Y2 except that (as expected) the roles of the hyperbolic sine and hyperbolic cosine in (70) are interchanged: the factor sinh⁡Aτ for Y1 in (70) is replaced by cosh⁡Aτ, and, similarly, cosh⁡Aτ for Y2 in (70) is replaced by sinh⁡Aτ.One can also find the following alternative expressions for the Killing vector field components forΦ(2)(τ,ρ):(71)Y1=v2+4a2κ2cn2ρ,κ-4a2κ2sn2ρ,κsinh⁡Aτ4a2bdnρ,κv2-4a2κ4cn2ρ,κsn2ρ,κdn-2ρ,κ,Y2=4Acosh⁡Aτv2-4a2κ4cn2ρ,κsn2ρ,κdn-2ρ,κ16a2bκ2cnρ,κsnρ,κfor A in (11). Corresponding alternative expressions for Φ(3)(τ,ρ) are similar to (71) except that the roles of the hyperbolic sine and hyperbolic cosine are interchanged. By a direct check one sees that the dilaton fields computed in (10) are indeed invariant along the corresponding Killing directions. That is, they satisfy(72)∂Φi∂τY1+∂Φi∂ρY2=0for each of i=1,2,3 as we indicated in the sentence following (65) about Y preserving Φ.


## 6. Some Closing Remarks

For the metricg in (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 (10) of the corresponding system of dilaton field equations in (9). We have also computed the associated Killing vector fields Y(g,Φ(j)) that leave both g and Φ(j) invariant; see (68), (70), and (71) and the remarks that follow (70) and (71). The dilaton fields simplify to the expressions given in (60) in the special case when the elliptic modulus κ is 1, and g simplifies to the expression given in (61).ForQ(ρ,κ) defined in (24), it was shown in the short argument following (22) that if Q(ρ,κ)=0 for some ρ, then necessarily v2/4a2κ4≤1:(73)v≤2aκ2in contrast to the standing assumption (7). To better understand the meaning of this inequality note first by (42) that Q(ρ,κ)=0⇒g11=0 so g exhibits a horizon singularity at(74)a2κ4sn2ρ,κcn2ρ,κdn2ρ,κ=v24:snρ,κcnρ,κdnρ,κ=±v2aκ2again by (24). Keep in mind that v is a velocity parameter of a dissipative soliton (also called adissipaton) as in (15), for example, especially for κ=1, as we have remarked at the end of Section 2. Inequality (73) is the statement therefore that for an arbitrary elliptic modulus κ, with 0<κ≤1, the velocity of a black hole dissipaton cannot exceed the limiting value |vmax|=def.2|a|κ2. This statement was deduced in [6, 7], for example, in the special (but important) case of κ=1.In Section3, by a series of explicit transformations of variables, g moreover was transformed to a Jackiw-Teitelboim black hole metric gJ-T of the simple form (4), namely, to gJ-T given by (39), with accompanying data given by (40) and (41). Here again assumption (7) was imposed. An advantage of parameterization (39) is that, for example, simple formulas exist [10, 12] for thermodynamic quantities such as the Hawking temperature TH 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 Section3 of the paper [13] of Klösch and Strobl. However (again), we have constructed very explicit elliptic solutions that do not follow directly from the results of [13].Reviewing [14, 15], we have added 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 aresonant nonlinear Schrödinger (RNLS) equation.The authors in [14] consider a system of nonlinear equations that describe the dynamics of two-component cold collisionless plasma in the presence of an external magnetic fieldB. For uniaxial plasma propagation, this system is reduced to a system that describes the propagation of nonlinear magnetoacoustic waves in 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(75)i∂ψ∂t′+∂2ψ∂x′2-12ψ2ψ=1+β21ψ∂2ψ∂x′2ψis achieved. Here x′=βx and t′=βt are rescaled space and time variables, andB has an expression in terms of a suitable power series expansion in the parameter β2. ψ has the form ψ=ρe-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(76)r=def.ρeS/β>0,s=def.-ρeS/β<0,B=def.12β2=2b2,b=def.2β,in the variables (x′,τ=def.βt′), the reaction diffusion (RD) system (12) is satisfied; r and s are denoted by e(+) and e(-) in [14]. On page 186 of [1], it is shown that, conversely, given solutions r>0 and  s<0 of the RD system (12), one can naturally construct a RNLS solution. By (15), with b=2β by (76), we can take(77)rx′,τ=2αβdnax′-vτ,κexp⁡v24+a22-κ2τ-vx′2,sx′,τ=-2αβdnax′-vτ,κexp⁡-v24+a22-κ2τ+vx′2.All of this means that we can apply prescription (14) to construct a metric gplasma of constant Ricci scalar curvature R=4B=def.8/b2=def.2/β2, as we did in (18), where the notation t,ρ there is now taken to mean τ,a(x′-vτ). Moreover, our results show that gplasma can be transformed to a J-T black hole metric of the form in (4). Thus we can account for a J-T black hole connection in cold plasma physics. Our results also provide elliptic solutions of the corresponding dilaton field equations.



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