Dimensional reduction of gravity and relation between static states, cosmologies, and waves

Abstract

We introduce generalized dimensional reductions of an integrable (1+1)-dimensional dilaton gravity coupled to matter down to one-dimensional static states (black holes in particular), cosmological models, and waves. An unusual feature of these reductions is that the wave solutions depend on two variables: space and time. They are obtained here both by reducing the moduli space (available because of complete integrability) and by a generalized separation of variables (also applicable to nonintegrable models and to higher-dimensional theories). Among these new wavelike solutions, we find a class of solutions for which the matter fields are finite everywhere in space-time, including infinity. These considerations clearly demonstrate that a deep connection exists between static states, cosmologies, and waves. We argue that it should also exist in realistic higher-dimensional theories. Among other things, we also briefly outline the relations existing between the low-dimensional models that we discuss here and the realistic higher-dimensional ones.

Multiexponential models of (1+1)-dimensional dilaton gravity and Toda-Liouville integrable models

“…In some cases, the potentials in Lagrangian (2) obtained from a higher-dimensional theory are given by the sum of exponentials of linear combinations of the scalar fields and the dilaton field ϕ. 8 In our previous work [23], we studied the constrained Liouville model in which the system of equations of motion (4), (6), and (7) is equivalent to the system of independent Liouville equations for linear combinations of fields q n ≡ F +q (0) n , where F ≡ log |f |. The easily derived solutions of these equations must satisfy constraints (5), which was the most difficult part of the problem.…”

“…Nevertheless, Eqs. (10) become integrable, and constraints (11) can be solved if the N -component vectors v n ≡ (a mn ) are pseudoorthogonal, as proposed in [16]- [18], [20], [23]. We now consider more general nondegenerate matrices a mn and define the new scalar fields x n :…”

“…We represent its general solution in a form that is equivalent to the standard form [36] but is more compact and more suitable for constructing one-dimensional solutions generalizing the solutions studied in [23]. The starting point is the Liouville (A 1 Toda) equation ∆ 2 (X) =ε 2 ≡ ε 1 (see [23], [36], [38], [39]). Calculating the derivatives of ∆ 2 (X) with respect to the variables u and v, we can easily prove Eqs.…”

“…Some of them can be solved using modern mathematical methods of soliton theory (see, e.g., [1], [2], [11], [19]). We note that the theories in dimension 1 + 0 (cosmologies) and 0 + 1 (static states and, in particular, black holes) may be integrable although their (1+1)-dimensional "parent" theory is not integrable without a deformation (see an example in [23]). …”

“…See, e.g.,[8],[11],[12],[17]-[23]; a review and further references can be found in[23],[26],[27] 4. The group of internal symmetries and the conditions for integrability of two-dimensional dynamical systems with exponential right-hand sides were studied in[33]-[35].…”