Exploring a cold plasma-2d black hole connection

Using a resonance nonlinear Schrodinger 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 magneto-acoustic 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-Teitelbiom black hole metric.


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  + ( 1  1 )  = 0  1  +  1  1  +  ) 2 ]  = 0, for  > 1, which describes the uni-axial propagation of long magnetoacoustic waves in a cold plasma of density  1 (, ) > 0 with velocity  1 (, ) across a magnetic field.Moreover, we present a concrete description of this connection.We start with a resonance nonlinear Schrödinger (RNLS) equation [3], [4], [5] Ψ  + Ψ  + |Ψ| 2 Here |Ψ| is a de Broglie quantum potential, and ,  are real numbers with  > 1.This  we will take to be the same as that in the system (1) ;  2 = −1.
In Section 2, we review (or slightly generalize, for the sake of completeness) a result in [4], [5] that such solutions ,  (or therefore Ψ in (2)) are in correspondence with solutions  1 > 0,  1 of the system (1), in case  < 0. An approach to the latter system by way of a shallow wave approximation also appears in [4], [5].In those references  = − .,  here correspond to  (+) ,  (−) respectively in the preceding references where again  = − 1 2 there.Also see [6].For  > 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 ,  a pseudo Riemannian metric of constant Ricci scalar curvature () = 4 (= −4 (−1) in our case); see [3], [7], [8].Constant curvature is a required ingredient for the J-T (Jackiw-Teitelboim) theory of 2d gravity [9], [10], [11], [12].Since ,  are defined in terms of the solutions ,  , which are in correspondence with solutions  1 ,  1 of (1) (as we have noted)  also will correspond to solutions  1 ,  1 of the MAS (1).Thus we will also denote  by   .Details of solutions  1 ,  1 (as traveling waves expressed in terms of the Jacobi elliptic function (, )) and a computation of the metric  =   in terms of them are provided in Section 2. In general of course  is non-diagonal:  12 (=  21 ) ≠ 0 in (6).In Section 3 we establish integrability conditions involving  and parameters defined in the solutions  1 ,  1 that suffice for the existence of a change of variables by which  assumes a simpler diagonal form.
The main results are presented in Section 4.There we provide another explicit change of variables that transforms   precisely into a (Lorentzian) J-T black hole metric  ℎwhich therefore explicates the proposed cold plasma -black hole connection.Using this same transformation, we construct an explicit dilaton Φ  such that the pair (  , Φ  ) solves the J-T gravitational field equations -equations that involve a cosmological constant Λ, which is shown to have the value for the  and  in (2), this  being the same  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  1 ,  1 of the plasma system (1).

Formulas for the cold plasma metric
As mentioned in Section 1, a correspondence between solutions of the system (1) and the system (3) (or, equivalently, of the RNLS equation (2)) will be reviewed in this section, under the assumption that  < 0. We also find an initial, general formula for the plasma metric; see (11).A concrete formula then follows as concrete solutions of the system (1) are considered.The end result is given in (17).
Our convention for scalar curvature is spelled out on page 182 of [12], for example.For other authors, see [9] for example, there is a sign difference: Our () would correspond to their −().

Diagonalization of the plasma metric
In this section we focus on the existence of a change of variables that diagonalizes  =   in (17).Such a simplified version of  would be of quite an advantage as a goal is to eventually map  to a black hole metric.It will be shown that the two conditions spelled out in (33) below (that simply require  2 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  indeed is mapped to a black hole metric.
Then by (6) for () given by (23).In the special case when  1 is chosen to be 0, we get  = 0 (by definition ( 12)) so that () simplifies greatly to ], say for  = −1/2, again as in [4], [5].Also by ( 25) which is exactly the diagonal metric that we focused on in the paper [1].See definition (6) there, where the notation ,  corresponds to  0 , 2 here, respectively, with  there the same as  here -a soliton velocity parameter.We see, as indicated in the introduction, that indeed the consideration of the plasma metric here with  1 in ( 12) allowed to be non-zero 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 that () ≠ 0, and thus validate formula (25).In the special case just considered, for () in ( 26), the single condition  2 4 0 2  4 > 1 suffices, as shown in [1] -an argument there being based on the inequality , which means that we can write (28) as In particular if () = 0 for some , then hold (the first one being the single condition prior to (27) for the special case  1 = 0), then () cannot vanish at any point .Of course if  1 = 0 then (again)  = 0 by definition and the second condition in (32) is the triviality 0 ≥ 0. If  1 > 0, then since  3 >  2 ≥  1 ,  > 0 by (12) and That is, conditions for the non-vanishing of () are (for with the single condition  2 > 4 0 2  4 if  1 = 0.

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 (, ) → (, ) of the plasma metric   in (25) with coordinates (, ) to a J-T black hole metric  ℎ , given in (46) below, with coordinates (, ), is presented in equation (48).For a suitable dilaton field Φ, the pair ( ℎ , Φ) solves J-T field equations that involve a negative cosmological constant Λ that we express (interestingly enough) in terms of  and  in the RNLS equation ( 2),  there (as pointed out in the introduction) being the same  in the cold plasma system (1).The Hawking black hole temperature and entropy are also expresses in terms of  and  , and in terms of the parameters  1 ,  0 , ,  that describe the plasma density  1 (, ) in ( 14), and hence also the velocity  1 (, ) in (12).The mapping is used, moreover, to construct an explicit dilaton field Φ  , and we show that the pair (  , Φ  ) 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 page 4 there, in the second equation for  2 in (28) the  2 should read  2 .
2. On page 10 the factor  / for  in (76) should read  −/ .With (25) now established on solid ground, via the assumptions (33), we first set up a critical change of variables by which   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 which might be seen as limited progress, but the point is the latter quotient term, which we denote by (), very fortunately by long division has a zero remainder.To see this, note that , where,  3 −  2 =  2 ( 3 −  1 )(by equation ( 12)) = 4 2  0 2  2 (by( 13)) 8 2 , which is precisely the numerator () of () since  2 =   1  2  3 by (12).However (again by (13))  3 =  1 + 4 0 2  2 and therefore we see also that ) () where () is the denominator of () ⇒ () = ( (by (15)) leads to This expression substituted in (37) gives the useful result that () is simply a quadratic polynomial in  ∶ with  given by (15), as usual.
Equations (36), (39) show that   has the form Therefore since  1 ≠ 0, it follows directly that the change of variables transforms  to the J-T Lorentzian form By (40), one can calculate that Before we declare  ℎ to be an authentic J-T black hole metric, we would first like to have that < 0 since, for example, we would look for an event horizon by setting , where already  1 > 0 by definition (40), as  < 0. We saw that () was non-vanishing provided that  2 was sufficiently large; see the conditions in (33).
Similarly we check that also  1 −   ] > 0, M being a mass parameter given by (42).