Coordinate choices for the dynamical systems picture

2.5 Coordinate choices for the dynamical systems picture

The time evolution of Cauchy data in GR can only be considered as a dynamical system if the ADM evolution equations are complemented by a prescription for the lapse and shift. To realise the phase space picture of Section 2.1, the critical solution must be a fixed point or limit cycle. We have seen how coordinates adapted to the self-similarity can be constructed, but is there a prescription of the lapse and shift for arbitrary initial data, such that, given initial data for the critical solution, the resulting time evolution actively drives the metric to a form (5

) that explicitly displays the self-similarity?

Garfinkle and Gundlach [85 ] have suggested several combinations of lapse and shift conditions that leave CSS spacetimes invariant and turn the Choptuik DSS spacetime into a limit cycle (see [91 , 81 ] for partial successes). Among these, the combination of maximal slicing with minimal strain shift has been suggested in a different context but for related reasons [188 ]. Maximal slicing requires the initial data slice to be maximal ( $a Ka = 0$ ), but other prescriptions, such as freezing the trace of $K$ together with minimal distortion, allow for an arbitrary initial slice with arbitrary spatial coordinates.

All these coordinate conditions are elliptic equations that require boundary conditions, and will turn CSS spacetimes into fixed points (or DSS into limit cycles) only given correct boundary conditions. Roughly speaking, these boundary conditions require a guess of how far the slice is from the accumulation point $t = t∗$ , and answers to this problem only exist in spherical symmetry. Appropriate boundary conditions are also needed if the dynamical system is extended to include the lapse and shift as evolved variables, turning the elliptic equations for the lapse and shift into hyperbolic or parabolic equations.

Turning a CSS or stationary spacetime into a fixed point of the dynamical system also requires an appropriate choice of the phase space variables $Z(xi)$ . To capture CSS (or DSS) solutions, one needs scale-invariant variables. Essentially, these can be constructed by dimensional analysis. The coordinates $xi$ and $τ$ are dimensionless, $le−τ$ has dimension length, and $g μν$ has dimension $l2$ . The scaling for the ADM and any matter variables follows.

Even with a prescription for the lapse and shift in place, a given spacetime does not correspond to a unique trajectory in phase space. Rather, for each initial slice through the same spacetime one obtains a different slicing of the entire spacetime. A possibility for avoiding this ambiguity would be to restrict the phase space further, for example by restricting possible data sets to maximal or constant extrinsic curvature slices.

Another open problem is that in order to talk about attractors and repellers on the phase space we need to define a norm on a suitable function space which includes both asymptotically flat data and data for the exact critical solution. The norm itself must favour the central region and ignore what is further out and asymptotically flat if all black holes of the same mass are to be considered as the same end state.