Introduction

Partial differential equations such as Einstein’s equations can have solutions with features that repeat themselves on different scales. We call this a self-similar solution. This phenomenon is well know from looking at a snowflake in the microscope where its structure looks similar at different magnifications.

Critical collapse in spherical symmetry

A relatively simple, non-linear dynamical system with these properties are Einstein’s equations in spherical symmetry (the spatial dimensions are symmetric shells and can be characterized just but a radius) coupled to a massless scalar field Choptuik, PRL 70, 9, (1993). This system has two end states: if a wave is strong (or dense) enough it can collapse and form a Schwarzschild black hole, or if it is weaker, it can disperse to flat space time.

In my PhD thesis and a technical paper PRD 71, 104005, (2005) I studied this system in a special coordinate system that allows to evolve the waves until infinity (I used compactified Bondi coordinates). The figure at the top of this page shows a Gaussian wave pulse evolving in logarithmic time $\tau$ (you can clearly see it at $\tau = 0$) as it starts to approach the origin ($r=0$ which corresponds to $x = - \infty$). As it does so it comes close to a self-similar attractor which makes it loose its starting shape and turn into a series of discretely self-similar pulses that come ever closer to the origin as it evolves in time.

The self-similarity can only be seen if the initial Gaussian pulse is chosen very carefully so that its eventual endstate is either a very light black hole or slightly below that – it requires fine-tuning. One can think of the attractor as a limit cycle with one unstable direction perpendicular to the cycle which drives away the solution to collapse or dispersion. In the picture below we need to fine tune an initial data parameter $p$ so that it almost lies in the plane where $p=p^*$ and it can get close to the critical solution before being driven away.

Power law tails

A related effect which occurs in the solutions of nonlinear wave equations are radiation tails. They are related to the fall-off properties of the field at late times ($\propto t^{p}$ with $0 < p \in \mathbb{N}$) and they emerge from primary outgoing radiation that is backscattered. This is a far-field effect which can either be due to a background or from an effective potential. I studied tails for the coupled Einstein - Yang Mills system with a special numerical code. It casts the hyperbolic evolution system as a characteristic initial value problem which is then solved with the method of lines with 6th order discretization in space and 4th order Runge Kutta in time. We find $p = −2$ at future null infinity and $p = −4$ at spatial infinity.