Stationary cylindrically symmetric spacetimes with a massless scalar field and a nonpositive cosmological constant

Abstract

The general stationary cylindrically symmetric solution of Einstein-massless scalar field system with a non-positive cosmological constant is presented. It is shown that the general solution is characterized by four integration constants. Two of these essential parameters have a local meaning and characterize the gravitational field strength. The other two have a topological origin, as they define an improper coordinate transformation that provides the stationary solution from the static one. The Petrov scheme is considered to explore the effects of the scalar field on the algebraic classification of the solutions. In general, these spacetimes are of type I. However, the presence of the scalar field allows us to find a non-vacuum type O solution and a wider family of type D spacetimes, in comparison with the vacuum case. The mass and angular momentum of the solution are computed using the Regge-Teitelboim method in the case of a negative cosmological constant. In absence of a cosmological constant, the curvature singularities in the vacuum solutions can be removed by including a phantom scalar field, yielding non-trivial locally homogeneous spacetimes. These spacetimes are of particular interest, as they have all their curvature invariants constant.

Higher-dimensional Buchdahl and Janis–Robinson–Winicour transformations in the Einstein–Maxwell system with a massless scalar field

“…This Levi-Civita solution with a massless scalar hair was obtained in [29]. Then, by the JRW transformation (2.21) and (2.22) from the above solution, we obtain…”

Rotating Cylinders with Anisotropic Fluids in General Relativity

“…In the important case of p = wρ, where w = w 1 = w 2 = w 3 , the general solution available at w 3 = −1 corresponds to a cosmological constant Λ = ρ/κ = −p/κ . We thus arrive at the Lewis wellknown solution, which is traditionally written in other notations [1,2] and is also discussed in [22,23]; note also its generalizations with a massless scalar field [6,8] and scalar fields with exponential potentials [7].…”

“…where κ = 8πG is the gravitational constant, R the Ricci scalar, and T the trace of the SET. In what follows we will mostly use the equations in the form (8), but it is also helpful to join the constraint equation from (7), which is the first integral of the others and contains only first-order derivatives of the metric functions:…”

“…Up to now, we worked with an arbitrary radial coordinate x. To solve the Einstein equations (8), it is helpful to choose the harmonic "gauge"…”

“…Among motivations of such studies one can mention the relative simplicity of the gravitational field equations as compared to more realistic axial symmetry (to say nothing on the general nonsymmetric case) and the possible existence of linearly extended structures like cosmic strings in the Universe. In addition, cylindrical symmetry is the simplest one that admits rotation and is most suitable for studying rotational phenomena in GR, so not too surprising is the wealth of existing stationary (that is, assuming rotation) exact solutions to the Einstein equations with various sources of gravity: the cosmological constant Λ [3][4][5]; scalar fields with or without self-interaction potentials [6,7], including Λ as a constant potential [8]; dust in rigid or differential rotation [9][10][11], electrically charged dust [12], dust with a scalar field [13], perfect fluids with various equations of state, mostly p = wρ, w = const (in usual notations) [14][15][16][17][18]; some examples of anisotropic fluids [19][20][21] etc, see also references therein as well as reviews [22,23].…”