Advances in Mathematical Physics

Advances in Mathematical Physics / 2019 / Article

Research Article | Open Access

Volume 2019 |Article ID 4810904 |

Floyd L. Williams, "Exploring a Cold Plasma-2d Black Hole Connection", Advances in Mathematical Physics, vol. 2019, Article ID 4810904, 11 pages, 2019.

Exploring a Cold Plasma-2d Black Hole Connection

Academic Editor: Stephen C. Anco
Received17 Jan 2019
Accepted24 Jul 2019
Published19 Aug 2019


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.

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] for > 1, which describes the uniaxial propagation of long magnetoacoustic waves in a cold plasma of density with velocity across a magnetic field. Moreover, we present a concrete description of this connection.

We start with a resonance nonlinear Schrödinger (RNLS) equation [35]

Here is a de Broglie quantum potential, and are real numbers with > 1. We will take this to be the same as that in system (1); .

Solutions of the form for real-valued functions are considered. Since are real, it follows directly that (2) is equivalent to the system of equations (= Madelung fluid equations)In Section 2, we review (or slightly generalize, for the sake of completeness) a result in [4, 5] that such solutions (or therefore in (2)) are in correspondence with solutions > 0, of system (1), in case . An approach to the latter system by way of a shallow wave approximation also appears in [4, 5]. In those references , and the parameter there corresponds to our . From (3) it follows that forthe reaction diffusion system (RDS)is solved for the value . here correspond to respectively in the preceding references where again 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 metricof constant Ricci scalar curvature ; see [3, 7, 8]. Constant curvature is a required ingredient for the J-T (Jackiw-Teitelboim) theory of 2d gravity [912]. Since are defined in terms of the solutions , which are in correspondence with solutions of (1) (as we have noted) also will correspond to solutions of the MAS (1). Thus we will also denote by . Details of solutions (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 nondiagonal: in (6). In Section 3 we establish integrability conditions involving and parameters defined in the solutions 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 , which therefore explicates the proposed cold plasma-black hole connection. Using this same transformation, we construct an explicit dilaton such that the pair solves the J-T gravitational field equations, equations that involve a cosmological constant , 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 of the plasma system (1).

2. Formulas for the Cold Plasma Metric

As mentioned in Section 1, 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 . 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 solutions of (3), define for Then one can check that the equations in (1) follow. Conversely, suppose are solutions of system (1). One should clearly define so that as in (7). For the next step, first choose any such that Then the first equation in (1) leads to the first equation in (3) for the pair , and the second equation in (1) implies that the partial derivative of with respect to vanishes. Thus this quantity is a function of only. Choose such that and define Then the pair in (8), (10) satisfies both equations in (3), and the proposed correspondence is established for

Given the definition of in (4), and the prescription for in (6), we arrive at the following formulas, where we set The second expression for here follows from the second equation in (3). By way of the correspondence just established, we can also write (for , which we now assume throughout)

is given uniquely in terms of by definition (8). However, there could be many other choices for in (10) for which the pair solves system (3) and for which (as in (7)). We assert that the metric is not affected by another such choice , in place of . Indeed, so the assertion follows by (11). Note also that for some function of integration . (since already , and . But by the second equation in (3) for , and then also for for some constant “c”. Thus in fact we see also that , and .

The goal now is to express the plasma metric more concretely in terms of concrete solutions of the MAS (1). From [2], traveling wave solutions are given by choosing , and setting

We could also replace here by . denotes a standard Jacobi elliptic function with elliptic modulus [13], and by (11). For our purpose it is convenient to set and therefore write It is also important to note that we can write Namely, and (again) implies the right hand side in (19) is

Some basic facts regarding the standard Jacobi elliptic functions (for any elliptic modulus ) are summed up as follows [13]: Applying the formulas in (12) to given in (16), (18) ( obviously being the main thing to compute) one arrives at the following concrete formulas for the plasma metric: where we use (22) to get that , orand where (by (17)), and is given by the formula (19). From Section 1 we know that this metric has constant Ricci scalar curvature

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

For plasma physics a relevant choice for is the value , so that the plasma density achieves the convenient value as by (16) implies by (18), (22). Similarly, the elliptic functions in the formulas (23) simplify as hyperbolic functions for For , for example, by (19).

3. Diagonalization of the Plasma Metric

In this section we focus on the existence of a change of variables that diagonalizes in (23). 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 (52) below (that simply requires 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.

Consider the initial change of variables for Then by (6), the point being that here depends only on (not on ) by the formulas in (23): We can therefore write

It follows by Section 2 of [1] (or by a direct, independent argument) that if there exists a function such that (an integrability condition, as was referenced in the introduction), then the change of variables reduces to the diagonal form By (30), and since (32) can be written as Now in particular are continuous functions (they are actually functions) so that if is nonvanishing the integrability condition can be satisfied. We have assumed that of course in deriving (35), an assumption that we shall explore presently.

From (23), reduces to the single term Therefore by (30) and (23) again

One could use (23) a third time, or more simply use (12) to compute and hence conclude by (24) that Equation (35) then assumes the concrete form for given by (36). In the special case when is chosen to be 0, we get

(by definition (16)) so that simplifies greatly to say for , again as in [4, 5]. Also by (39) which is exactly the diagonal metric that we focused on in the paper [1]. See definition there, where the notation corresponds to 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 in (16) allowed to be nonzero 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 , and thus validate formula (39). In the special case just considered, for in (41), the single condition suffices, as shown in [1], an argument there being based on the inequality which we shall use again here. For simplicity of notation we shall write , suppressing the variables

By (36), Multiplication by and division by lead to terms here. The first and fifth term (in order) combined simplify to which means that we can write (44) as In particular if for some , then by (43) and the fact that That is, (47) says that if vanishes at some point Now assume that Then by (48) In other words, if both conditions hold (the first one being the single condition prior to (43) for the special case , then cannot vanish at any point Of course if then (again) by definition and the second condition in (50) is the triviality If , then since by (16) and That is, conditions for the nonvanishing of are (for with the single condition if

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 of the plasma metric in (39) with coordinates to a J-T black hole metric , given in (74) below, with coordinates , is presented in (76). For a suitable dilaton field , the pair solves J-T field equations that involve a negative cosmological constant 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 expressed in terms of and , and in terms of the parameters that describe the plasma density in (18), and hence also the velocity in (16). The mapping is used, moreover, to construct an explicit dilaton field , and we show that the pair 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 there, in the second equation for in the should read .(2)On page the factor for in should read .

With (39) now established on solid ground, via the assumptions (52), 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 From (22), (53) which by (39) gives for with related by (53) Plug the last three elliptic functions expressions in (54) into (36). A modest amount of work renders the result 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 which is precisely the numerator since by (16). However (again by (17)) and therefore we see also that where is the denominator of , where (by (19)) leads to This expression substituted in (56) gives the useful result that is simply a quadratic polynomial in with given by (19), as usual.

Equations (55), (62) show that has the form Therefore since , it follows directly that the change of variables

transforms to the J-T Lorentzian form By (63), one can calculate that Before we declare to be an authentic J-T black hole metric, we would first like to have that since, for example, we would look for an event horizon by setting , where already by definition (63), as We saw that was nonvanishing provided that was sufficiently large; see the conditions in (52). Similarly we check that also provided is sufficiently large; see the conditions in (67), (73) below.

Note that , in particular. That is Thus if , then , which is the first condition in (52). Suppose in fact that Then since , that isTherefore to insure that in (65), it is enough to require the condition on the remaining terms there, as one continues to keep (19) in mind: happens in particular ifwhich is automatic if With conditions (67), (73) in place to guarantee that , we express in (64) as a J-T black hole metric in the form for , M being a mass parameter given by (65).

We have noted that (67) implies, in particular, the first condition in (52).

The composite change of variables therefore transforms the cold plasma metric in (39) to the black hole metric in (74). This composition is immediately computed: where (by (53), (63), (22)) That is,

Going back to (39) again, which we write as we note that and in (76) are in fact related: a fact that will be useful later.

solves the J-T field equations where is a negative cosmological constant, is a dilaton field, and the Hessian is given by for the (second) Christoffel symbols of In the present case and The equations of motion (79) are derived from the J-T action integral for a two-dimensional spacetime and a gravitational coupling constant

For the normalized Planck constant, one has the general formulas for the black hole Hawking temperature and Bekenstein-Hawking entropy [14]: In particular, and are also expressed in terms of solution data for the plasma system (1).

Given in (76), (81), define by The Christoffel symbols for and (in the form expressed in (77) and (74)) can be computed using Maple, for example. The nonvanishing ones are given, respectively, by

By our notation/definition, , which permits one to relate the Christoffel symbols of and in (85): the left hand side of these equations being the Christoffel symbols of Using the equations in (86), we can, in the end, relate the Hessian acting on and in (84), (81); see (80):