Research Article | Open Access
Jennie D’Ambroise, Floyd L. Williams, "Elliptic Function Solutions in Jackiw-Teitelboim Dilaton Gravity", Advances in Mathematical Physics, vol. 2017, Article ID 2154784, 10 pages, 2017. https://doi.org/10.1155/2017/2154784
Elliptic Function Solutions in Jackiw-Teitelboim Dilaton Gravity
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.
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 , 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 motionderived from the action integralwhere is the constant Ricci scalar curvature of and the (negative) cosmological constant is . In local coordinates on , the Hessian in (1) is given bywhere are the Christoffel symbols (of the second kind) of . The J-T theory has, for example, the (Lorentzian) black hole solutionwith coordinates , wherewith 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 ; . We will generally assume thatAs will be seen later, this metric is the diagonalization of a metric constructed from solutions of the reaction diffusion systemWe 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) areHere 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 byforwhich 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 equationsin 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 , 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 byare 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 , various formulas likeare available. Using prescription (14), one computes thatFor , 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 forthe change of variables gives and andThe 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) iswhere for convenience we write sn, cn, and dn for and forThenwhich is the coefficient of in (20) by (23). Then, by (21), (20) readswhich 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 ; here corresponds to the notation there. Also the cosmological constant in  corresponds to our . Similarly, and in (15) reduce to the dissipative soliton solutions and , respectively, in (2.32) of , 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  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.
Next let , as in (3.18) of , but where our in (31) generalizes their , and for convenience letin (32). Then in (32) assumes the formwhich generalizes the Schwarzschild form (3.19) of , since for we have that in (33).
For the change of variables and with , the Schwarzschild in (34) goes towhich in turn goes toby way of the change of variables and . We need one final observation: in general a metric of the formsay , 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: forUsing definition (33) for and , which is definition (31), one computes thatin (39).
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 solutionsWe 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 thatsince . We had and for , which givesfor the Schwarzschild version of our metric in (34). Next let to getwhere we have disregarded the multiple in (47) and have used . Finally, the first change of variables in Section 3 givesby definition (40). If we disregard the multiple in (49) and use by definitions (31) and (33), we obtain from (49) the first solutionin (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) iswhere, 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 ) isWe 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 . Henceby (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 inequalitiesof 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 thatDifferentiate 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 metricThe solutions in (60) are also new.
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 equationsfor [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 , every solution of the field equations in (1) gives rise to a corresponding Killing vector field by way of the local prescriptionwith 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 , 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