2.7 The analogy with critical phase transitions

Some basic aspects of critical phenomena in gravitational collapse, such as fine-tuning, universality, scale-invariant physics, and critical exponents for dimensionful quantities, can also be identified in critical phase transitions in statistical mechanics (see [208] for an introductory textbook).

From an abstract point of view, the objective of statistical mechanics is to derive relations between macroscopic observables A of the system and macroscopic external forces f acting on it, by considering ensembles of microscopic states of the system. The expectation values ⟨A ⟩ can be generated as partial derivatives of the partition function

∑ −H (microstate,μ,f) Z (μ, f) = e (14 ) microstates
Here the μ are parameters of the Hamiltonian such as the strength of intermolecular forces, and f are macroscopic quantities which are being controlled, such as the temperature or magnetic field.

Phase transitions in thermodynamics are thresholds in the space of external forces f at which the macroscopic observables A, or one of their derivatives, change discontinuously. We consider two examples: the liquid-gas transition in a fluid, and the ferromagnetic phase transition.

The liquid-gas phase transition in a fluid occurs at the boiling curve p = p (T ) b. In crossing this curve, the fluid density changes discontinuously. However, with increasing temperature, the difference between the liquid and gas density on the boiling curve decreases, and at the critical point (p∗ = pb(T∗),T∗) it vanishes as a non-integer power:

γ ρliquid − ρgas ∼ (T∗ − T ) . (15 )
At the critical point an otherwise clear fluid becomes opaque, due to density fluctuations appearing on all scales up to scales much larger than the underlying atomic scale, and including the wavelength of light. This indicates that the fluid near its critical point is approximately scale-invariant (for some range of scales between the size of molecules and the size of the container).

In a ferromagnetic material at high temperatures, the magnetisation m of the material (alignment of atomic spins) is determined by the external magnetic field B. At low temperatures, the material shows a spontaneous magnetisation even at zero external field. In the absence of an external field this breaks rotational symmetry: The system makes a random choice of direction. With increasing temperature, the spontaneous magnetisation m decreases and vanishes at the Curie temperature T ∗ as

|m | ∼ (T ∗ − T )γ. (16 )
Again, the correlation length, or length scale of a typical fluctuation, diverges at the critical point, indicating scale-invariant physics.

Quantities such as |m | or ρliquid − ρgas are called order parameters. In statistical mechanics, one distinguishes between first-order phase transitions, where the order parameter changes discontinuously, and second-order, or critical, ones, where it goes to zero continuously. One should think of a critical phase transition as the critical point where a line of first-order phase transitions ends as the order parameter vanishes. This is already clear in the fluid example. In the ferromagnet example, at first one seems to have only the one parameter T to adjust. But in the presence of a very weak external field, the spontaneous magnetisation aligns itself with the external field B, while its strength is to leading order independent of B. The function m (B, T) therefore changes discontinuously at B = 0. The line B = 0 for T < T∗ is therefore a line of first order phase transitions between directions (if we consider one spatial dimension only, between m up and m down). This line ends at the critical point (B = 0,T = T∗) where the order parameter |m | vanishes. The critical value B = 0 of B is determined by symmetry; by contrast p∗ depends on microscopic properties of the material.

We have already stated that a critical phase transition involves scale-invariant physics. In particular, the atomic scale, and any dimensionful parameters associated with that scale, must become irrelevant at the critical point. This is taken as the starting point for obtaining properties of the system at the critical point.

One first defines a semi-group acting on micro-states: the renormalisation group. Its action is to group together a small number of adjacent particles as a single particle of a fictitious new system by using some averaging procedure. This can also be done in a more abstract way in Fourier space. One then defines a dual action of the renormalisation group on the space of Hamiltonians by demanding that the partition function is invariant under the renormalisation group action:

∑ ∑ ′ e−H = e−H . (17 ) microstates microstates′
The renormalised Hamiltonian is in general more complicated than the original one, but it can be approximated by the same Hamiltonian with new values of the parameters µ and external forces f. (At this stage it is common to drop the distinction between µ and f, as the new µ’ and f ’ depend on both µ and f.) Fixed points of the renormalisation group correspond to Hamiltonians with the parameters at their critical values. The critical values of many of these parameters will be 0 or ∞, meaning that the dimensionful parameters they were originally associated with are irrelevant. Because a fixed point of the renormalisation group can not have a preferred length scale, the only parameters that can have nontrivial values are dimensionless.

The behaviour of thermodynamical quantities at the critical point is in general not trivial to calculate. But the action of the renormalisation group on length scales is given by its definition. The blowup of the correlation length ξ at the critical point is therefore the easiest critical exponent to calculate. The same is true for the black hole mass, which is just a length scale. We can immediately reinterpret the mathematics of Section 2.3 as a calculation of the critical exponent for ξ, by substituting the correlation length ξ for the black hole mass M, T − T ∗ for p − p ∗, and taking into account that the τ-evolution in critical collapse is towards smaller scales, while the renormalisation group flow goes towards larger scales: ξ therefore diverges at the critical point, while M vanishes.

In type II critical phenomena in gravitational collapse, we should think of the black hole mass as being controlled by the functions P and Q on phase space defined by Equation (27View Equation). Clearly, P is the equivalent of the reduced temperature T − T∗. Gundlach [104Jump To The Next Citation Point] has suggested that the angular momentum of the initial data can play the role of B, and the final black hole angular momentum the role of m. Like the magnetic field, angular momentum is a vector, with a critical value that must be zero because all other values break rotational symmetry.

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