Avenues for analytic exploration in axisymmetric spacetimes: Foundations and the triad formalism

Axially symmetric spacetimes are the only vacuum models for isolated systems with continuous symmetries that also include dynamics. For such systems, we review the reduction of the vacuum Einstein field equations to their most concise form by dimensionally reducing to the three-dimensional space of orbits of the Killing vector, followed by a conformal rescaling. The resulting field equations can be written as a problem in three-dimensional gravity with a complex scalar field as source. This scalar field, the Ernst potential, is constructed from the norm and twist of the spacelike Killing field. In the case where the axial Killing vector is twist-free, we discuss the properties of the axis and simplify the field equations using a triad formalism. We study two physically motivated triad choices that further reduce the complexity of the equations and exhibit their hierarchical structure. The first choice is adapted to a harmonic coordinate that asymptotes to a cylindrical radius and leads to a simplification of the three-dimensional Ricci tensor and the boundary conditions on the axis. We illustrate its properties by explicitly solving the field equations in the case of static axisymmetric spacetimes. The other choice of triad is based on geodesic null coordinates adapted to null infinity as in the Bondi formalism. We then explore the solution space of the twist-free axisymmetric vacuum field equations, identifying the known (unphysical) solutions together with the assumptions made in each case. This singles out the necessary conditions for obtaining physical solutions to the equations.

Figure 1

Schematic illustration of the decomposition of a twist-free axisymmetric spacetime with closed orbits. Since ${\omega }^{\mu }=0$, $\overline{\mathcal{S}}$, the quotient space of $\mathcal{M}$ that contains all orbits of ${\xi }^{\mu }$, is also a subspace of $\mathcal{M}$. The fact that the orbits are closed implies that a set of fixed points, namely the axis, must exist if the spacetime is asymptotically flat.

Figure 2

Classification of spacetimes possessing a twist-free, axial Killing vector. The abbreviations used in this figure can be interpreted as follows: Killing vector (KV), Principal null direction (PND), Robinson-Trautman (RT), and Newman-Tamburino (NT).

Figure 3

Orthogonal coordinates $\psi$ (solid lines) and $s$ (dashed lines) plotted for a single black hole of mass $M=1$ in canonical Weyl coordinates $(\rho ,z)$.