5.3 Analytic approaches5 Aspects of current research5.1 Phase diagrams

5.2 The renormalisation group as a time evolution

  It has been pointed out by Argyres [3], Koike, Hara and Adachi [91] and others that the time evolution near the critical solution can be considered as a renormalisation group flow on the space of initial data. The calculation of the critical exponent in Section  3.3 is in fact mathematically identical to that of the critical exponent governing the correlation length near the critical point in statistical mechanics [123], if one identifies the time evolution in the time coordinate tex2html_wrap_inline2337 and spatial coordinate x with the renormalisation group flow. But those coordinates were defined only on self-similar spacetimes plus linear perturbations. In order to obtain a full renormalisation group, one has to generalize them to arbitrary spacetimes, or, in other words, find a general prescription for the lapse and shift as functions of arbitrary Cauchy data.

For simple parabolic or hyperbolic differential equations, a discrete renormalisation (semi)group acting on their solutions has been defined in the following way [67, 24, 32, 33]: Evolve initial data over a certain finite time interval, then rescale the final data in a certain way. Solutions which are fixed points under this transformation are scale-invariant, and may be attractors. One nice distinctive feature of GR as opposed to these simple models is that one can use a shift freedom in GR (one that points inward towards an accumulation point) to incorporate the rescaling into the time evolution, and the lapse freedom to make each rescaling by a constant factor an evolution through a constant time (tex2html_wrap_inline2337, in our notation) interval.

The crucial distinctive feature of general relativity, however, is that a solution does not correspond to a unique trajectory in the space of initial data. This is because a spacetime can be sliced in different ways, and on each slice one can have different coordinate systems. Infinitesimally, this slicing and coordinate freedom is parameterized by the lapse and shift. In a relaxed notation, one can write the ADM equations as tex2html_wrap_inline3419 functional tex2html_wrap_inline3421, where g is the 3-metric, K the extrinsic curvature, tex2html_wrap_inline2323 the lapse and tex2html_wrap_inline3429 the shift. The lapse and shift can be set freely, independently of the initial data. Of course they influence only the coordinates on the spacetime, not the spacetime itself, but the ADM equations are not yet a dynamical system. If we specify a prescription tex2html_wrap_inline3431 functional (g, K), then substituting it into the ADM equations, we obtain tex2html_wrap_inline3419 functional (g, K), which is an (infinite-dimensional) dynamical system. We are then faced with the general question: Given initial data in general relativity, is there a prescription for the lapse and shift, such that, if these are in fact data for a self-similar solution, the resulting time evolution actively drives the metric to the special form (28Popup Equation) that explicitly displays the self-similarity?

An algebraic prescription for the lapse suggested by Garfinkle [59] did not work, but maximal slicing with zero shift does work [64] if combined with a manual rescaling of space. Garfinkle and Gundlach [62] have suggested several combinations of lapse and shift conditions that not only leave CSS spacetimes invariant, but also turn the Choptuik DSS spacetime into a limit cycle. The combination of maximal slicing with minimal strain shift has the nice property that it also turns static spacetimes into fixed points (and probably periodic spacetimes into limit cycles). Maximal slicing requires the first slice to be maximal (tex2html_wrap_inline3439), but other prescriptions 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 only for correct boundary conditions. Roughly speaking, these boundary conditions require a guess of how far the slice is from the accumulation point tex2html_wrap_inline2639, and answers to this problem only exist in spherical symmetry.

5.3 Analytic approaches5 Aspects of current research5.1 Phase diagrams

image Critical Phenomena in Gravitational Collapse
Carsten Gundlach
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