2.4 Type I

In type I critical phenomena, the same phase space picture as in Section 2.1 applies, but the critical solution is now stationary or time-periodic instead of self-similar or scale-periodic. It also has a finite mass and can be thought of as a metastable star. (Type I and II were so named after first and second order phase transitions in statistical mechanics, in which the order parameter is discontinuous and continuous, respectively.) Universality in this context implies that the black hole mass near the threshold is independent of the initial data, namely a certain fraction of the mass of the stationary critical solution. The dimensionful quantity that scales is not the black hole mass, but the lifetime tp of the intermediate state where the solution is approximated by the critical solution. This is clearly
-1- tp = − λ ln|p − p∗| + constant. (13 ) 0

Type I critical phenomena occur when a mass scale in the field equations becomes dynamically relevant. (This scale does not necessarily set the mass of the critical solution absolutely: There could be a family of critical solutions selected by the initial conditions.) Conversely, as the type II power law is scale-invariant, type II phenomena occur in situations where either the field equations do not contain a scale, or this scale is dynamically irrelevant. Many systems, such as the massive scalar field, show both type I and type II critical phenomena, in different regions of the space of initial data [34Jump To The Next Citation Point].

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