Lower order ODEs to determine new twisting type N Einstein spaces via CR geometry

Abstract

In the search for vacuum solutions, with or without a cosmological constant, Λ, of the Einstein field equations of Petrov type N with twisting principal null directions, the CR structures to describe the parameter space for a congruence of such null vectors provide a very useful tool. Work of Hill, Lewandowski and Nurowski has given a good foundation for this, reducing the field equations to a set of differential equations for two functions, one real and one complex, of three variables. Under the assumption of the existence of one Killing vector, the (infinite-dimensional) classical symmetries of those equations are determined and group-invariant solutions are considered. This results in a single ODE of the third order which may easily be reduced to one of the second order. A one-parameter class of power series solutions, g(w), of this second-order equation is realized, holomorphic in a neighborhood of the origin and behaving asymptotically as a simple quadratic function plus lower order terms for large values of w, which constitutes new solutions of the twisting type N problem. The solution found by Leroy, and also by Nurowski, is shown to be a special case in this class. Cartan’s method for determining equivalence of CR manifolds is used to show that this class is indeed much more general. In addition, for a special choice of a parameter, this ODE may be integrated once to provide a first-order Abel equation. It can also determine new solutions to the field equations, although no general solution has yet been found for it.

pp-waves in modified gravity

“…It is shown by Bičák and Podolský [214,215] that the family of Petrov type N gravitational waves with a nonvanishing expansion with a cosmological constant belongs to Kundt family of metrics whereas the expanding solutions belong to the Robinson-Trautman family of metrics. It also possible to construct twisting type N solutions [216][217][218][219]. However, the twisting type N solutions lead to a field equations that are quite involved compared the nontwisting solutions and are not discussed above.…”

CR structures and twisting vacuum spacetimes with two Killing vectors and cosmological constant: type II and more special

“…as imposed by the commutation relations (4). Hence generally, the system (11)(12)(13)(14) are in fact PDEs for the unknown functions L, n and p of the coordinate variables (ζ,ζ, u). For other possible coordinate choices, the metric (1-20) admits the following coordinate freedom ( [17], see Section 2.6):…”

“…Given the algebraically special twisting metric form (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) formulated according to CR geometry, it is important to know how it is related to other pre-existing formalisms that have been extensively studied in the past. Here we quote from [12] (p. 439-441) a most commonly used one by Kerr, Debney et al [18,19,20].…”

“…Now we go back to the metric (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20). Following the same idea as [14] for solving the field equations, we assume that the unknowns p, c and n have no u-dependence, i.e., ∂ 0 p = ∂ 0 c = ∂ 0 n = 0. This assumption avoids the involvement of the function L inside the operator ∂ since now we have, e.g., ∂p = ∂ ζ p. Therefore the system (11)(12)(13)(14) becomes effectively PDEs for the unknowns c, p and n instead of L, p and n. Once c = c(ζ,ζ) is solved, one may further determine a function L without u-dependence from (20) (cf.…”

“…Hence it is worthwhile to consider whether our equations (43-46) can capture some of these physically important solutions. But first we should comment that the CR formalism (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) used here is constructed on just one single shearfree null congruence aligned with a multiple principal null direction , which is unique in type II, III and N spacetimes. However, a type D spacetime possesses two such congruences, each along one of the two doubly degenerate principal null directions, and consequently one cannot treat them at the same time in the CR formalism.…”