Advances in Mathematical Physics
Volume 2017 (2017), Article ID 2154784, 10 pages
https://doi.org/10.1155/2017/2154784

Research Article

Elliptic Function Solutions in Jackiw-Teitelboim Dilaton Gravity

Jennie D’Ambroise¹ and Floyd L. Williams²

¹Department of Mathematics/CIS, SUNY Old Westbury, P.O. Box 210, Old Westbury, NY 11568, USA
²Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA

Correspondence should be addressed to Jennie D’Ambroise; jdambroise@gmail.com

Received 24 April 2017; Accepted 14 May 2017; Published 31 July 2017

Academic Editor: Andrei D. Mironov

Copyright © 2017 Jennie D’Ambroise and Floyd L. Williams. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 metric on a two-dimensional space-time . A proof of this fact is given in Section of [1], for example. A nontrivial theory of gravity for such an was worked out in 1984 by Jackiw and Teitelboim (J-T). This involves in addition to a scalar field on called a dilaton field; see [2, 3]. The pair is subject to the equations of motion

derived from the action integral

where is the constant Ricci scalar curvature of and the (negative) cosmological constant is . In local coordinates on , the Hessian in (1) is given by

where are the Christoffel symbols (of the second kind) of [1]. The J-T theory has, for example, the (Lorentzian) black hole solution

with coordinates , where

with 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 numbers and for a soliton velocity parameter , we consider the following metric in the variables :

where , , and are the standard Jacobi elliptic functions with modulus ; [4]. We will generally assume that

As will be seen later, this metric is the diagonalization of a metric constructed from solutions of the reaction diffusion system

We will explicate the solutions in terms of the elliptic function . Remarkably, the metric in (6) has constant scalar curvature 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 in (6) are

Here the cosmological constant is .

Given the complicated nature of our , system (9) is necessarily quite difficult to solve directly. Our method is to construct a series of transformations of variables so that in (6) is transformed to in (4). Then we can use the simple solution in (5) and other known solutions to work backwards through these transformations of variables to construct 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, , 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

for

which we assume is nonzero. Given (7), we shall see in Section 4 that only for and moreover that the second expression under the radical (i.e., ) in (10) is positive. For , , but we can have for some . Also for 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 , consider the system of partial differential equations

in the variables . This system is a special case of the more general reaction diffusion system (RDS),

which occurs in chemistry, physics, or biology, for example, where and are diffusion constants and and are growth and interaction functions. The key point for us is that from solutions and of (12) one can construct a metric of constant Ricci scalar curvature by the following prescription [5–7]:

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 . Our interest is in the choice , where, for real , , and , with as in Section 1, given by

are solutions of system (12), which also could be checked directly by Maple. For , (12) is system (8) with solutions (15) promised in Section 1, and in (14) has the scalar curvature discussed in Section 1. From [4], various formulas like

are available. Using prescription (14), one computes that

For , so that , can be expressed more conveniently as

The goal now is to set up a change of variables so that in (18) is transformed to (6), where the cross term does not appear, in comparison with the term appearing in (18). For this purpose note first, in general, that for

the change of variables gives and and

The condition that the cross term does not appear is therefore that satisfies

Apply this to (18):

Now, by (16), and . If the term in parenthesis in (22) was zero, this would therefore force the inequality . That is, if , which is the assumption in (7), then and therefore the denominator term in parenthesis in (22) is nonzero, which means that is a continuous function and (22) therefore indeed has a solution , with assumption (7) imposed. Also, the coefficient of in (20) is

where for convenience we write sn, cn, and dn for and for

Then

which is the coefficient of in (20) by (23). Then, by (21), (20) reads

which is (6). That is, we have verified that the change of variables with subject to condition (22) (which in fact makes 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 modulus ,

and (18) and (6) simplify:

which are the line elements and , respectively, in [6]; here corresponds to the notation there. Also the cosmological constant in [6] corresponds to our . Similarly, and in (15) reduce to the dissipative soliton solutions and , respectively, in (2.32) of [6], apart from the factor . One can also explicitly determine 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 variables so that by (16)

Also by (16),

for

Again by (7), indeed . By (29) and (30), we see that we can write (6) as

Next let , as in (3.18) of [6], but where our in (31) generalizes their , and for convenience let

in (32). Then in (32) assumes the form

which generalizes the Schwarzschild form (3.19) of [6], since for we have that in (33).

For the change of variables and   with , the Schwarzschild in (34) goes to

which in turn goes to

by way of the change of variables and  . We need one final observation: in general a metric of the form

say , can be transformed to the J-T form (4); namely,

by way of the change of variables . Apply this to (36) with playing the role of there:

for

Using definition (33) for and , which is definition (31), one computes that

in (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 Section 3 (according to remarks in Introduction) lead to the linearly independent solutions , , in (10) of the dilaton field equations in (9). There the metric elements are given by (6). For in (24),

and are given by (3) for . The Christoffel symbols 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 solution in (5) for metric (4) in the variables , there are solutions

We work backwards the changes of variables in Section 3 for and , for example, to see how one arrives at the first two solutions and in (10) in the variables .

Starting with the (39) version of (4), we have by (40), with given by (41). Here (for )

By the final change of variables in Section 3, we see that

The change of variables and   preceded the change , so that

since . We had and for , which gives

for the Schwarzschild version of our metric in (34). Next let to get

where we have disregarded the multiple in (47) and have used . Finally, the first change of variables in Section 3 gives

by definition (40). If we disregard the multiple in (49) and use by definitions (31) and (33), we obtain from (49) the first solution

in (10). More work is required of course to obtain the second solution there.

First, we note that, by (40) and (41),

which is in (10). Also, for , , the quantity under the other radical in (48) is

where, by definition (40),

again by definitions (31) and (33). That is, since by definition (33), the quantity in (52) (which is under the radical in (48) for ) is

We let denote the latter bracket here. By (48), (51), and (54), we see that (for now) , if we disregard the multiple .

We find an alternate expression for , which is simpler and shows that , given (7). Again we write sn, cn, and dn for , and we make use of (16).

where we noted in Section 2 that . Hence

by (7), again as in (31), and we see that , since for being a real number. Moreover, we have established the desired expression for in (10). Clearly one can replace the hyperbolic sine in the preceding discussion by the hyperbolic cosine in (43) to obtain the third solution in (10). To finish other claims made in Section 1, we check that in (11) only for . We continue to assume (7) of course.

The quartic equation has roots and with and , respectively. (7) requires that , which forces the inequalities

of which the first one reads , with the left-hand side here being , a contradiction. That is, we cannot have which means that . Also we check that the solutions are linearly independent: assume for constants that

Differentiate this equation with respect to and evaluate the result at :

since and . The choice then gives , since , and differentiation of the equation and at gives . Using again we see by (10) that and hence also .

Note that if and , for example, then even though (so that (7) is satisfied), we have that .

Again in the special case when the elliptic modulus , we have in (11) that and (by (7) or (31)), and ; by (27), (55) . Here (directly) , again as . Thus, by (10) and (28),

(where we have disregarded the multiple in ) are dilaton field solutions for the metric

The solutions in (60) are also new.

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

Recall that a smooth vector field on an -dimensional Riemannian manifold is called a Killing vector field (or an infinitesimal motion of ) if, for arbitrary smooth vector fields on ,

If is an expression of in terms of local coordinates on , then (62) is equivalent to the system of equations

for [8, 9]. In the special (diagonal) case with for and with , the Killing equations (63) simplify to the following three equations:

As was have shown in [10], every solution of the field equations in (1) gives rise to a corresponding Killing vector field by way of the local prescription

with being a permutation symbol. preserves both and . For in (4) and for the fields 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 in (10) with given by (6). By (42), . Since could be replaced by a scalar multiple of itself (e.g., ), we shall disregard the absolute value of here, and given (9), we shall take (instead of ). For and , (65) then assumes the generic form

where we take in (64).

For the first solution

in (10), the computation of the corresponding Killing vector field is trivial: by (16) and (66), , and of course . Since by (42), the Killing equations in (64) are satisfied and we see that

for . Computations for the other two solutions and in (10) are more involved. The result is the following, where again

in definition (11).

For ,

For one has quite similar formulas for and except that (as expected) the roles of the hyperbolic sine and hyperbolic cosine in (70) are interchanged: the factor for in (70) is replaced by , and, similarly, for in (70) is replaced by .

One can also find the following alternative expressions for the Killing vector field components for :

for in (11). Corresponding alternative expressions for 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

for each of as we indicated in the sentence following (65) about preserving .

6. Some Closing Remarks

For the metric in (6), whose derivation was discussed in Section 2, we have obtained as a main result explicit linearly independent solutions , , in (10) of the corresponding system of dilaton field equations in (9). We have also computed the associated Killing vector fields that leave both and 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 is 1, and simplifies to the expression given in (61).

For defined in (24), it was shown in the short argument following (22) that if for some , then necessarily :

in contrast to the standing assumption (7). To better understand the meaning of this inequality note first by (42) that so exhibits a horizon singularity at

again by (24). Keep in mind that is a velocity parameter of a dissipative soliton (also called a dissipaton) as in (15), for example, especially for , as we have remarked at the end of Section 2. Inequality (73) is the statement therefore that for an arbitrary elliptic modulus , with , the velocity of a black hole dissipaton cannot exceed the limiting value . This statement was deduced in [6, 7], for example, in the special (but important) case of .

In Section 3, by a series of explicit transformations of variables, moreover was transformed to a Jackiw-Teitelboim black hole metric of the simple form (4), namely, to 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 and black hole entropy .

We point out, for the record, that the general solutions of all 2D dilaton gravity models are known. For example, see Section 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 a resonant 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 field B. 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

is achieved. Here and are rescaled space and time variables, and B has an expression in terms of a suitable power series expansion in the parameter . has the form , where is a velocity potential and is the mass density of the plasma. Also note the remarks in [15]. A key point of interest for us is that, for

in the variables (), the reaction diffusion (RD) system (12) is satisfied; and are denoted by and in [14]. On page 186 of [1], it is shown that, conversely, given solutions and   of the RD system (12), one can naturally construct a RNLS solution. By (15), with by (76), we can take

All of this means that we can apply prescription (14) to construct a metric of constant Ricci scalar curvature , as we did in (18), where the notation there is now taken to mean . Moreover, our results show that can be transformed to a - 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.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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