# Exploring a Cold Plasma-2d Black Hole Connection

Authors: Floyd L. Williams
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

Using a resonance nonlinear Schrödinger equation as a bridge, we explore a direct connection of cold plasma physics to two-dimensional black holes. Namely, we compute and diagonalize a metric attached to the propagation of magnetoacoustic waves in a cold plasma subject to a transverse magnetic field, and we construct an explicit change of variables by which this metric is transformed exactly to a Jackiw-Teitelboim black hole metric.


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## Body

## 1. Introduction

In the closing remarks in [1] we indicated briefly a connection of black holes in the Jackiw-Teitelboim model of two-dimensional dilaton gravity to the dynamics of two-component cold collisionless plasma in the presence of an external transverse magnetic field. The purpose of the present paper is to greatly expand those brief remarks in various directions. For example, we explore this connection for plasma metrics derived more generally from Gurevich-Krylov solutions of the associated magnetoacoustic system (MAS) [2] (1)ρ1t+ρ1u1x=0u1t+u1u1x+ρ1x+δ-1ρ1xxρ1-12ρ1xρ12x=0,for δ > 1, which describes the uniaxial propagation of long magnetoacoustic waves in a cold plasma of density ρ1x,t>0 with velocity u1(x,t) across a magnetic field. Moreover, we present a concrete description of this connection.We start with aresonance nonlinear Schrödinger (RNLS) equation [3–5] (2)iΨt+Ψxx+γΨ2Ψ=δΨxxΨΨ.Here|Ψ|xx/|Ψ| is a de Broglie quantum potential, and γ,δ are real numbers with δ > 1. We will take this δ to be the same as that in system (1); i2=-1.Solutions of the formΨ=eR-iS for real-valued functions R(x,t),S(x,t) are considered. Since R,S are real, it follows directly that (2) is equivalent to the system of equations (= Madelung fluid equations)(3)Rt-Sxx-2RxSx=0St+1-δRxx+1-δRx2-Sx2+γe2R=0.In Section 2, we review (or slightly generalize, for the sake of completeness) a result in [4, 5] that such solutions R,S (or therefore Ψ in (2)) are in correspondence with solutions ρ1 > 0, u1 of system (1), in case γ<0. An approach to the latter system by way of a shallow wave approximation also appears in [4, 5]. In those references γ=-1/2, and the parameter B there corresponds to our δ-1. From (3) it follows that for(4)rx,t=defexpRx,tδ-1+Sx,t/δ-1δ-1sx,t=def-expRx,tδ-1-Sx,t/δ-1δ-1the reaction diffusion system (RDS)(5)rt-rxx+Br2s=0st+sxx-Brs2=0is solved for the value B=-γ/(δ-1). r,s here correspond to e(+),e(-) respectively in the preceding references where again γ=-1/2 there. Also see [6]. For δ > 1, the RNLS equation is not reducible to a nonlinear Schrödinger equation but instead to a RDS.Now given the RDS in (5), the point for us is that one can construct from its solutions r,s a pseudo-Riemannian metric(6)g:ds2=g11dt2+2g12dtdx+g22dx2,g11=def-rxsx,g12=def12srx-rsx,g22=defrsofconstant Ricci scalar curvature R(g)=4B(=-4γ/δ-1inourcase); see [3, 7, 8]. Constant curvature is a required ingredient for the J-T (Jackiw-Teitelboim) theory of 2d gravity [9–12]. Since r,s are defined in terms of the solutions R,S, which are in correspondence with solutions ρ1,u1 of (1) (as we have noted) g also will correspond to solutions ρ1,u1 of the MAS (1). Thus we will also denote g by gplasma. Details of solutions ρ1,u1 (as traveling waves expressed in terms of the Jacobi elliptic function dn(x,κ)) and a computation of the metric g=gplasma in terms of them are provided in Section 2. In general of course g is nondiagonal: g12(=g21)≠0 in (6). In Section 3 we establish integrability conditions involving δ and parameters defined in the solutions ρ1,u1 that suffice for the existence of a change of variables by which g assumes a simpler diagonal form.The main results are presented in Section4. There we provide another explicit change of variables that transforms gplasma precisely into a (Lorentzian) J-T black hole metric gbh, which therefore explicates the proposed cold plasma-black hole connection. Using this same transformation, we construct an explicit dilaton Φplasma such that the pair (gplasma,Φplasma) solves the J-T gravitational field equations, equations that involve a cosmological constant Λ, which is shown to have the value 2γ/(δ-1) for the γ and δ in (2), this δ being the same δ in (1). Another plasma-black hole connection revealed in Section 4 is the observation that the Hawking black hole temperature and entropy can be expressed in terms of parameters involved in the description of solutions ρ1,u1 of the plasma system (1).


## 2. Formulas for the Cold Plasma Metric

As mentioned in Section1, a correspondence between solutions of system (1) and system (3) (or, equivalently, of the RNLS equation (2)) will be reviewed in this section, under the assumption that γ<0. We also find an initial, general formula for the plasma metric; see (12). A concrete formula then follows as concrete solutions of system (1) are considered. The end result is given in (23).For the correspondence, one direction is quite straightforward: Given solutionsR,S of (3), define (7)ρ1=-2γe2R>0,u1=-2Sxfor γ<0. Then one can check that the equations in (1) follow. Conversely, suppose ρ1>0,u1 are solutions of system (1). One should clearly define (8)R=12log⁡ρ1-2γso that ρ1=-2γe2R>0 as in (7). For the next step, first choose any S0 such that -2S0x=u1. Then the first equation in (1) leads to the first equation in (3) for the pair R,S0, and the second equation in (1) implies that the partial derivative of (9)-S0t+S0x2-γe2R+δ-1Rxx+Rx2with respect to x vanishes. Thus this quantity is a function ϕ(t) of t only. Choose h(t) such that h′(t)=ϕ(t) and define (10)Sx,t=S0x,t+ht.Then the pair R,S in (8), (10) satisfies both equations in (3), and the proposed correspondence (ρ1>0,u1)↔(R,S) is established for γ<0.Given the definition ofr,s in (4), and the prescription for g in (6), we arrive at the following formulas, where we set (11)β=def+δ-1:g22x,t=-e2Rx,t/β,g12x,t=-1βe2Rx,t/βSxx,tβg11x,t=e2Rx,t/βRx2-1β2Sx2x,tβ=e2Rx,t/βSt-β2+Rxx+2Rx2+γ-β2e2Rx,tβ.The second expression for g11(x,t) here follows from the second equation in (3). By way of the correspondence (ρ1>0,u1)↔(R,S) just established, we can also write (for γ<0, which we now assume throughout) (12)g22x,t=ρ1x,t/β2γg12x,t=ρ1x,t/βu1x,t/β-4γβg11x,t=ρ1x,t/β-2γ14ρ1,xρ12x,tβ-14β2u12x,tβ.R is given uniquely in terms of ρ1 by definition (8). However, there could be many other choices for S in (10) for which the pair (R,S) solves system (3) and for which u1=-2Sx (as in (7)). We assert that the metric g isnot affected by another such choice S′, in place of S. Indeed, (13)-2S′x=u1=-2Sx⇒S′x=Sx,so the assertion follows by (11). Note also that (14)S′-Sx=0⇒S′x,t=Sx,t+ctfor some function of integration c(t). S′xx=Sxx (since already S′x=Sx), and S′t=St+c′(t). But by the second equation in (3) for (R,S′), and then also for (R,S)(15)0=St+c′t+1-δRxx+Rx2-Sx2+γe2R=c′t⇒ct=c,for some constant “c”. Thus in fact we see also that S′t=St, and S′=S+c.The goal now is to express the plasma metricg more concretely in terms of concrete solutions ρ1>0,u1 of the MAS (1). From [2], traveling wave solutions are given by choosing α3>α2≥α1≥0,u0>0, and setting(16)ρ1x,t=defα1+α3-α1dn2α3-α11/22x-u0tβ,κu1x,t=defu0+Cρ1x,t;C=+α1α2α31/2,κ=α3-α2α3-α11/2.We could also replaceu0+C/ρ1 here by u0-C/ρ1. dn(x,κ) denotes a standard Jacobi elliptic function with elliptic modulus κ [13], and β=def+δ-1>0 by (11). For our purpose it is convenient to set (17)a0=def+α3-α11/22β>0,v=defu0β>0and therefore write (18)ρ1x,t=α1+4a02β2dn2a0x-βvt,κ.It is also important to note that we can write (19)C=α11/24a02β21-κ2+α11/24a02β2+α11/2.Namely, (20)1-κ2=def1-α3-α2α3-α1=α2-α1α3-α1and (again) (21)4a02β2=defα3-α14a02β21-κ2+α1=α2,4a02β2+α1=α3implies the right hand side in (19) is α11/2α21/2α31/2=defC.Some basic facts regarding the standard Jacobi elliptic functionssn(x,κ),cn(x,κ),dn(x,κ) (for any elliptic modulus κ) are summed up as follows [13]: (22)sn2x,κ+cn2x,κ=1,dn2x,κ+κ2sn2x,κ=1,ddxsnx,κ=cnx,κdnx,κ,ddxcnx,κ=-snx,κdnx,κ,ddxdnx,κ=-κ2snx,κcnx,κ,snx,1=tanhx,cnx,1=dnx,1=sechx.Applying the formulas in (12) to u1,ρ1 given in (16), (18) (g11 obviously being the main thing to compute) one arrives at the following concrete formulas for the plasma metric: (23)g22x,t=α1+4a02β2dn2a0x-vt,κ2γg12x,t=-vγa02β2dn2a0x-vt,κ-vα14γ-C4γβg11x,t=4a02β2-a02κ42γsn2cn2a0x-vt,κ+v28γdn2a0x-vt,κ+16α1κ4a04β2sn2cn2a0x-vt,κ+C2/β28γα1+4a02β2dn2a0x-vt,κ+v2α18γ+vC4γβ,where we use (22) to get that ρ1x2(x,t)=64κ4a06β4(sncndn)2(a0(x-βvt),κ), or(24)ρ1x2x,tβ=64κ4a06β4sncndn2a0x-vt,κ,and where v=defu0/β (by (17)), and C is given by the formula (19). From Section 1 we know that this metric has constant Ricci scalar curvature (25)Rg=-4γβ2=def-4γδ-1.Our convention for scalar curvature is spelled out on page 182 of [12], for example. For other authors, see [9]; for example, there is asign difference: Our R(g) would correspond to their -R(g).For plasma physics a relevant choice forα1,α2 is the value 1, so that the plasma density ρ1 achieves the convenient value 1 as x→∞:κ=1 by (16) implies (26)ρ1x,t=1+4a02β2sech2a0x-βvtby (18), (22). Similarly, the elliptic functions sn,cn in the formulas (23) simplify as hyperbolic functions for κ=1. For α1=κ=1,C=(4a02β2+1)1/2, for example, by (19).


## 3. Diagonalization of the Plasma Metric

In this section we focus on the existence of a change of variables that diagonalizesg=gplasma in (23). Such a simplified version of g would be of quite an advantage as a goal is to eventually map g to a black hole metric. It will be shown that the two conditions spelled out in (52) below (that simply requires v2 to be sufficiently large) suffice to insure such a diagonalization. These conditions are prototypical in the sense that similar ones will be set up in Section 4 to insure that g indeed is mapped to a black hole metric.Consider the initial change of variables(x,t)→(ρ,t) for ρ=defa0(x-vt).(27)x=ρa0+vt⇒dx=dρa0+vdt,dx2=dρ2a02+2va0dρdt+v2dt2,dxdt=dρdta0+vdt2.Then by (6), (28)g=g11+2vg12+v2g22dt2+2a0g12+vg22dρdt+g22a02dρ2,the point being that (29)gijx,t=gijρa0+vt,there depends only on ρ (not on t) by the formulas in (23): We can therefore write (30)g=Aρdt2+C1ρdρdt+C2ρdρ2:Aρ=defg11+2vg12+v2g22ρ,C1ρ=def2a0g12+vg22ρ,C2ρ=defg22ρa02.It follows by Section 2 of [1] (or by a direct, independent argument) thatif there exists a function ϕ(ρ) such that (31)ϕ′ρ=C1ρ2Aρ (an integrability condition, as was referenced in the introduction), then the change of variables τ=t+ϕ(ρ) reduces g to the diagonal form (32)g=Aρdτ2+C2ρ-C12ρ4Aρdρ2.By (30), (33)C12ρ4-AρC2ρ=g122-g11g22ρa02=-detgρa02and since (34)C2-C124A=-C12/4-AC2A, (32) can be written as (35)g=Aρdτ2--detgρ/a02Aρdρ2.Now C1(ρ),A(ρ) in particular are continuous functions (they are actually C∞ functions) so that if A(ρ) is nonvanishing the integrability condition ϕ′(ρ)=C1(ρ)/2A(ρ) can be satisfied. We have assumed that A(ρ)≠0 of course in deriving (35), an assumption that we shall explore presently.From (23), (2vg12+v2g22)(ρ) reduces to the single term -2vC/4γβ. Therefore by (30) and (23) again(36)Aρ=g11ρ-2vC4γβ=4a02β2-a02κ42γsn2ρ,κcn2ρ,κ+v28γdn2ρ,κ+16α1β2κ4a04sn2ρ,κcn2ρ,κ+C2/β28γα1+4a02β2dn2ρ,κ+v2α18γ-vC4γβ.One could use (23) a third time, or more simply use (12) to compute (37)g11g22-g122ρ=-ρ1x2x,t/β16γ2and hence conclude by (24) that (38)detgρ=-4κ4a06β4γ2sn2cn2dn2ρ,κ.Equation (35) then assumes the concrete form (39)g=Aρdτ2-1Aρ4κ4a04β4γ2sn2cn2dn2ρ,κdρ2for A(ρ) given by (36). In the special case when α1 is chosen to be 0, we get C=0(by definition (16)) so that A(ρ) simplifies greatly to (40)Aρ=4a02β2dn2ρ,κa02κ4sn2cn2dn2ρ,κ-v24,say for γ=-1/2, again as in [4, 5]. Also by (39) (41)g=Aρdτ2-4a02β2κ4sn2cn2ρ,κa02κ4sn2cn2dn2ρ,κ-v24-1dρ2=4a02β2dn2ρ,κa02κ4sn2cn2dn2ρ,κ-v24dτ2-κ4sn2cn2dn2ρ,κa02κ4sn2cn2dn2ρ,κ-v24-1dρ2,which is exactly the diagonal metric that we focused on in the paper [1]. See definition (6) there, where the notation a,b corresponds to a0,2β here, respectively, with v there the same as v here, a soliton velocity parameter. We see, as indicated in the introduction, that indeed the consideration of the plasma metric here with α1 in (16) allowed to benonzero can lead to a vast generalization of some of the work in [1].We turn now to the lingering question of conditions that will imply thatA(ρ)≠0, and thus validate formula (39). In the special case just considered, for A(ρ) in (41), the single condition (42)v24a02κ4>1suffices, as shown in [1], an argument there being based on the inequality (43)sn2x,κcn2x,κdn2x,κ≤1,which we shall use again here. For simplicity of notation we shall write sn,cn,dn, suppressing the variables ρ,κ.By (36), (44)-2γAρdn2=4a02β2a02κ4sncndn2-v24-v2α14dn2+vC2βdn2-16α1κ4a04β2sncn/dn2+C2/β2dn24α1+4a02β2dn2.Multiplication by 4a02β2 and division by D=def4[α1+4a02β2dn2]>0 lead to 6 terms here. The first and fifth term (in order) combined simplify to (45)4a04β2κ4sn2cn2dn24a02β2dn2D/4,which means that we can write (44) as (46)-2γAρdn2=4a02β2κ4sn2cn2dn2·4a04β2dn2D/4-a02β2v2-v2α14dn2+vC2βdn2-C2Dβ2dn2.In particular if A(ρ)=0 for some ρ, then (47)v2a02β2+α14dn2-vC2βdn2+C2Dβ2dn2=4a04β2κ4sn2cn2dn2·4a02β2dn2D/4≤4a04β2κ4·16a04β2dn2D≤4a04β2κ4by (43) and the fact that α1≥0⇒D≥16a02β2dn2. That is, (47) says that (48)v2a02β2≤-v2α14-vC2β1dn2+4a04β2κ4-C2Dβ2dn2≤-v2α14-vC2β1dn2+4a04β2κ4if A(ρ) vanishes at some point ρ. Now assume that (49)v2α14-vC2β≥0.Then v2a02β2≤4a04β2κ4 by (48):v2≤4a02κ4. In other words, if both conditions (50)v2>4a02κ4,α1v≥2Cβhold (the first one being the single condition prior to (43) for the special case α1=0), then A(ρ)cannot vanish at any point ρ. Of course if α1=0 then (again) C=0 by definition and the second condition in (50) is the triviality 0≥0. If α1>0, then since α3>α2≥α1,C>0 by (16) and (51)v≥2Cα1β⇒v2≥4C2α12β2>0.That is, conditions for the nonvanishing of A(ρ) are (for α1≠0)(52)v2>4a02κ4,v2≥4C2α12β2,with the single condition v2>4a02κ4 if α1=0.


## 4. Mapping the Cold Plasma Metric to a Black Hole Metric

We are in a position now to proceed towards a formulation of the main results. A precise mapping(τ,ρ)→(τ,r) of the plasma metric gplasma in (39) with coordinates (τ,ρ) to a J-T black hole metric gbh, given in (74) below, with coordinates (τ,r), is presented in (76). For a suitable dilaton field Φ, the pair (gbh,Φ) solves J-T field equations that involve a negative cosmological constant Λ that we express (interestingly enough) in terms of γ and δ in the RNLS equation (2), δ there (as pointed out in the introduction) being the same δ in the cold plasma system (1). The Hawking black hole temperature and entropy are also expressed in terms of γ and δ, and in terms of the parameters α1,a0,v,κ that describe the plasma density ρ1(x,t) in (18), and hence also the velocity u1(x,t) in (16). The mapping is used, moreover, to construct an explicit dilaton field Φplasma, and we show that the pair (gplasma,Φplasma) is also a solution of the J-T field equations, a solution in terms of Jacobi elliptic functions, which thus is another extension of results in [1].Two trivial corrections of errors in [1] are in order:(1)
On page4 there, in the second equation for ds2 in (28) the dt2 should read dτ2.(2)
On page10 the factor eS/β for s in (76) should read e-S/β.With (39) now established on solid ground, via the assumptions (52), we first set up a critical change of variables by which gplasma assumes a considerably more manageable form, a form which is a step away from a J-T black hole metric. This key change of variables is given by (53)x=defa02β2κ2γsn2ρ,κ.From (22), (53) (54)dx=2a02β2κ2γsncndnρ,κdρ,dx2=4a04β4κ4γ2sn2cn2dn2ρ,κdρ2,sn2ρ,κ=γxa02β2κ2,cn2ρ,κ=1-γxa02β2κ2,dn2ρ,κ=1-γxa02β2,which by (39) gives for A(x)=A(ρ) with x,ρ related by (53) (55)g=Axdτ2-dx2Ax.Plug the last three elliptic functions expressions in (54) into (36). A modest amount of work renders the result (56)Ax=2γx2β2-2a02κ2+v22x+a02β2v22γ+v2α18γ-vC4γβ+-2α1γx2β2+2α1κ2a02x+C28γβ21-4γx+α1+4a02β2,which might be seen as limited progress, but the point is the latter quotient term, which we denote by Q(x), very fortunately by long division has a zero remainder. To see this, note that (57)-4γx+α3x+α24γ=-4γx2+α3-α2x+α2α34γ,where (58)α3-α2=κ2α3-α1byequation16=4κ2a02β2by17⇒α12β2-4γx+α3x+α24γ=-2α1γx2β2+2α1κ2a02x+C28γβ2,which is precisely the numerator nxofQ(x) since C2=defα1α2α3 by (16). However (again by (17)) α3=α1+4a02β2 and therefore we see also that (59)nx=α12β2x+α24γdxwhere d(x) is the denominator of Qx⇒Q(x)=(α1/2β2)(x+α2/4γ)=α1x/2β2+α1α2/8γβ2, where (60)α1α2=C2α3=C2α1+4a02β2=α14a02β21-κ2+α1 (by (19)) leads to (61)Qx=α1x2β2+α18γβ24a02β21-κ2+α1.This expression substituted in (56) gives the useful result that A(x) is simply a quadratic polynomial in x:(62)Ax=2γx2β2-2a02κ2+v22-α12β2x+α18γβ24a02β21-κ2+α1+a02β2v22γ+v2α18γ-vC4γβwith C given by (19), as usual.Equations (55), (62) show that gplasma has the form (63)g=-A1x2+B1x+C1dτ2+A1x2+B1x+C1-1dx2forA1=def-2γβ2>0,B1=def2a02κ2+v22-α12β2,C1=def-α18γβ24a02β21-κ2+α1-a02β2v22γ-v2α18γ+v4γβα14a02β21-κ2+α14a02β2+α11/2.Therefore since A1≠0, it follows directly that the change of variablesr = d e f x + B 1 / 2 A 1 transforms g to the J-T Lorentzian form (64)gbh=-A1r2+C1-B124A1dτ2+dr2A1r2+C1-B12/4A1.By (63), one can calculate that (65)C1-B124A1=v4γβα14a02β21-κ2+α14a02β2+α11/2-a02β2v22γ+a04κ4β22γ+a02κ2v2β24γ+v4β232γ-3v2α116γ-3α1232γβ2+α1a02κ24γ-α1a022γ.Before we declare gbh to be an authentic J-T black hole metric, we would first like to have that C1-B12/4A1<0 since, for example, we would look for an event horizon by setting A1r2+C1-B12/4A1=0:r=-(C1-B12/4A1)/A1, where already A1>0 by definition (63), as γ<0. We saw that A(ρ) was nonvanishing provided that v2 was sufficiently large; see the conditions in (52). Similarly we check that also C1-B12/4A1<0 provided v2 is sufficiently large; see the conditions in (67), (73) below.Note thatk≤1⇒2κ2+κ4≤2+1<4, in particular. That is (66)22-κ2>κ4⇒8a022-κ2>4a02κ4.Thus if v2>8a02(2-κ2), then v2>4a02κ4, which is the first condition in (52). Suppose in fact that (67)v2>8a022-κ2+6α1β2.Then since γ<0, (68)v2β232γv2<v2β232γ8a022-κ2+v2β232γ6α1β2=v2a02β22γ-v2a02κ2β24γ+3v2α116γ;that is(69)v4β232γ-v2a02β22γ+v2a02κ2β24γ-3v2α116γ<0.Therefore to insure that (C1-B12)/4A1<0 in (65), it is enough to require the condition (70)vC4γβ+a04κ4β22γ-3α1232γβ2+α1a02κ24γ-α1a022γ<0on the remaining terms there, as one continues to keep (19) in mind: (71)vC4β+a04κ4β22-3α1232β2+α1a02κ24-α1a022>0happens in particular if(72)vC4β-3α1232β2-α1a022≥0,which is automatic if α1=0:(73)v≥α1C3α18β+2a02β,α1≠0. With conditions (67), (73) in place to guarantee that C1-B12/4A1<0, we express gbh in (64) as a J-T black hole metric in the form (74)gbh=-m2r2-Mdτ2+dr2m2r2-Mfor m2=A1=-2γ/β2>0,M=-[C1-B12/4A1]>0, M being a mass parameter given by (65).We have noted that (67) implies, in particular, the first condition v2>4a02κ4 in (52).The composite change of variables(τ,ρ)→(τ,x)→(τ,r) therefore transforms the cold plasma metric gplasma in (39) to the black hole metric gbh in (74). This composition is immediately computed: (τ,ρ)→(τ,r) where (by (53), (63), (22)) (75)r=defx+B12A1=defa02β2κ2γsn2ρ,κ+2a02κ2+v2/2-α1/2β2-4γ/β2=a02β2γ1-dn2ρ,κ-a02β2κ22γ-v2β28γ+α18γ.That is, (76)r=ψρ=defa02β2dn2ρ,κ-γ+a02β22γ2-κ2+v2β2-α1-8γ.Going back to (39) again, which we write as (77)gplasma=Aρdτ2+BρAρdρ2,Bρ=def-4a04β4κ4γ2sn2cn2dn2ρ,κ,we note that B(ρ) and ψ(ρ) in (76) are in fact related: (78)ψ′ρ2=-Bρ,a fact that will be useful later.g b h solves the J-T field equations (79)Rg+2Λ=0∇i∇jΦ+ΛΦgij=0,where Λ is anegative cosmological constant, Φ is adilaton field, and the Hessian is given by (80)∇i∇j=∂2∂xi∂xj-∑k=12Γijk∂∂xk,for the (second) Christoffel symbols Γijk of g. In the present case (x1,x2)=(τ,r) and (81)Λ=-m2=def2γβ2=def2γδ-1,Φτ,r=defmr.The equations of motion (79) are derived from the J-T action integral (82)IJ-Tg,Φ=12G∫M2d2xdetgΦRg+2Λfor a two-dimensional spacetime M2 and a gravitational coupling constant G.Forℏ=h/2π the normalized Planck constant, one has the general formulas for the black hole Hawking temperature TH and Bekenstein-Hawking entropy SBH [14]: (83)TH=ℏGπM=ℏGπ-C1-B124A1,SBH=2πℏGM=2πℏG-C1-B124A1.In particular, TH and SBH are also expressed in terms of solution data for the plasma system (1).Givenψ,Φ in (76), (81), define Φplasma by (84)Φplasmaτ,ρ=Φτ,ψρ.The Christoffel symbols for gplasma and gbh (in the form expressed in (77) and (74)) can be computed using Maple, for example. The nonvanishing ones are given, respectively, by (85)Γ121=12A′ρAρ,Γ112=-12AρA′ρBρ,Γ222=12B′ρBρ-A′ρAρ,Γ121=-m2r-m2r2+M,Γ112=--m2r2+Mm2r,Γ222=m2r-m2r2+M.By our notation/definition,A(ρ)=-m2ψ(ρ)2+M⇒A′(ρ)=-2m2ψρψ′(ρ), which permits one to relate the Christoffel symbols of gplasma and gbh in (85): (86)Γ121τ,ρ=Γ121τ,ψρψ′ρΓ112τ,ρ=-Γ112τ,ψρψ′ρBρΓ222τ,ρ=12B′ρBρ+Γ222τ,ψρψ′ρ,the left hand side of these equations being the Christoffel symbols of gplasma. Using the equations in (86), we can, in the end, relate the Hessian acting on Φplasma and Φ in (84), (81); see (80): (87)∇1∇1Φplasmaτ,ρ=∇1∇1Φτ,ψρ∇1∇2Φplasmaτ,ρ=∇1∇2Φτ,r=0∇2∇2Φplasmaτ,ρ=-Bρ∇2∇2Φplasmaτ,ψρ,where for the last formula here we use (78) and its implication 2ψ′ρψ′′(ρ)=-B′(ρ).Since the pair(gbh,Φ) solves the J-T field equations (79) (as we have noted, for the cosmological constant Λ=-m2=def2γ/β2), one concludes from (87) that the pair (gplasma,Φplasma) also solves the J-T field equations, where by (76), (84) Φplasma is given explicitly by (88)Φplasmaτ,ρ=ma02β2dn2ρ,κ-γ+a02β22-κ22γ+v2β2-α1-8γ.For convenience, we iterate that the plasma metric for the dilaton field Φplasma in (88) is given by (36), (39):(89)gplasma=Aρdτ2-4κ4a04β4Aργ2sn2cn2dn2ρ,κdρ2for (90)Aρ=4a02β2-a04κ42γsn2cn2ρ,κ+v28γdn2ρ,κ+16α1β2κ4a04sn2cn2ρ,κ+C2/β28γα1+4a02β2dn2ρ,κ+v2α18γ-vC4γβ,C=α11/24a02β21-κ2+α11/24a02β2+α11/2.In the special, but important, case when the elliptic modulus κ is chosen to be 1, Φplasma in (88) reduces to the equation(91)Φplasmaτ,ρ=ma02β2sech2ρ-γ+a02β22γ+v2β2-α1-8γand gplasmain (89) simplifies to(92)gplasma=Aρdτ2-4a04β4Aργ2tanh2sech4ρdρ2for (93)Aρ=4a02β2-a042γtanh2sech2ρ+v28γsech2ρ+16α1β2a04tanh2sech2ρ+C2/β28γα1+4a02β2sech2ρ+v2α18γ-vC4γβ,C=α14a02β2+α11/2,by (22). Lastly as the only restriction on α1 is α1≥0, one can certainly specialize the choice α1=0 in (91), in which case C=0  and Aρ in (92), (93) simplifies to (94)Aρ=4a02β2-a042γtanh2sech2ρ+v28γsech2ρ.Also for the choices α2=α1=1 (relevant for the plasma physics) we have noted in formula (26) the simpler expression for the Gurevich–Krylov traveling wave solution of the plasma density ρ1(x,t) (and hence of its velocity u1(x,t) by (16)) in terms of the hyperbolic secant.


## 5. Closing Remarks

The transformation formula (76) is the key connection between the cold plasma metric gplasma in (39) attached to the solutions (16), (18) of the magnetoacoustic system (1) and the J-T black hole metric gbh in (74), a connection nicely facilitated by the general correspondence set up in Section 2 between solutions of (1) and of the resonance Schrödinger equation (2). The dilaton field Φplasma constructed in (88) provides for a solution (gplasma,Φplasma) of the J-T field equations (79) (in terms of Jacobi elliptic functions), where the cosmological constant Λ is negative and is expressed in (81) in terms of the parameters γ,δ in the Schrödinger equation (2), γ being assumed negative (which was exploited in Section 2 in the aforementioned correspondence of solutions of (1) and (2), and δ>1 being the same δ in (1)).The conditions established in (67), (73) insure positivity of the black hole mass M in (74). In particular, the expressions in (83) for Hawking temperature and entropy are meaningful. However, one could relax these conditions and obtain, more generally, by way of (76) a connection between cold plasma and 2d black holes with a naked singularity (for M<0) and black hole vacua, with M=0. It should be possible, extending methods presented here, to construct additional dilaton fields Φ such that the pairs (gplasma,Φ) also solve the J-T field equations (79), this and other matters the author could consider in future work.The construction of the plasma metric (89), which eventually was transformed to the black hole metric (74), was based on solutions of the RDS (5) generated by solutions of the RNLS equation (2), or equivalently (as shown in Section 2) by solutions of the MAS (1), given the general prescription (6). Apart from the traveling wave, one-soliton solutions of (2), or of (5), that arise here for κ=1, one could consider multisoliton solutions as well. In Section 6 of [3], Lee and Pashaev apply the Hirota bilinear representation to construct, for example, an explicit two-dissipaton solution of the RDS (5). This means that in principle one could construct the corresponding constant scalar curvature metric (again by way of (6)), attempt to diagonalize it, and map the diagonalized form to a black hole metric. Obviously that task would be more difficult than that which was dealt with here, but it would be an extremely interesting, fascinating task to pursue as it would likely provide insight, for example, on the collision of black holes in the plasma, or possibly multi-black hole solutions in J-T gravity.



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