Fully pseudospectral solution of the conformally invariant wave equation on a Kerr background

Abstract

We study axisymmetric solution to the conformally invariant wave equation on a Kerr background by means of numerical and analytical methods. Our main focus is on the behaviour of the solutions near spacelike infinity, which is appropriately represented as a cylinder. Earlier studies of the wave equation on a Schwarzschild background have revealed important details about the regularity of the corresponding solutions. It was found that, on the cylinder, the solutions generically develop logarithmic singularities at infinitely many orders. Moreover, these singularities also ‘spread’ to future null infinity. However, by imposing certain regularity conditions on the initial data, the lowest-order singularities can be removed. Here we are interested in a generalisation of these results to a rotating black hole background and study the influence of the rotation rate on the properties of the solutions. To this aim, we first construct a conformal compactification of the Kerr solution which yields a suitable representation of the cylinder at spatial infinity. Besides analytical investigations on the cylinder, we numerically solve the wave equation with a fully pseudospectral method, which allows us to obtain highly accurate numerical solutions. This is crucial for a detailed analysis of the regularity of the solutions. In the Schwarzschild case, the numerical problem could effectively be reduced to solving (1 + 1)-dimensional equations. Here we present a code that can perform the full 2 + 1 evolution as required for axisymmetric waves on a Kerr background.

The characteristic initial value problem for the conformally invariant wave equation on a Schwarzschild background

“…These singularities are expected to be further transported to I + , where they are superimposed to the physical information about outgoing radiation there. This effect is confirmed by the numerical studies in [8,13], and it provides another example of the above-mentioned causality violation at infinity.…”

“…Another possibility is to apply similar coordinate transformations starting from Schwarzschild coordinates. A generalisation of this idea was employed in [13], where a conformal compactification for a part of the Kerr solution was constructed, which was initially expressed in Boyer-Lindquist coordinates. In the nonrotating limit, we immediately obtain suitable coordinates for the Schwarzschild region S + .…”

“…We now come back to the conformally invariant wave equation on different background spacetimes as discussed in [6][7][8]13]. In the Minkowski case, it turned out that solutions are globally regular, provided a single regularity condition is satisfied by the initial data.…”

“…Similarly, the investigations of the Maxwell equations on a Schwarzschild background in [21,22] show that the electromagnetic field has logarithmic singularities, unless the initial data satisfy regularity conditions. Furthermore, in a series of papers [6][7][8]13], the behaviour of solutions to the conformally invariant wave equation (i.e. the zero-rest-mass equation for spin zero) has been studied on Minkowski, Schwarzschild and Kerr backgrounds.…”

Symmetry operators for the conformal wave equation in rotating black hole spacetimes

“…provided it also scales as Φ = Ω w Φ , with the conformal weight w = 1 − D/2 . This wave equation (1) has also found use in a recent study of the asymptotic structure of Kerr spacetime via conformal compactification [12].…”