6.3. Gravitational turbulence

The general solution to the vacuum Einstein equations will describe what we would intuitively view as gravitational turbulence'. This would, presumably, manifest some of the chaotic features displayed by the homogeneous Mixmaster Universe. In this section we shall draw some parallels between ideas thai have been pursued to obtain a ir athematical description of hydrodynamic turbulence and the problem of describing 'gravitational turbulence'.

First, it is instructive to recall the three famous criteria laid down by Hadamard [86] in 1923 to test whether solutions of differential equations are physically realistic. He stipj»ates that such mathematical solutions should (i) exist, (ii) be unique and, (iii) be stable. All three criteria are involved in descriptions of 1 ydrodynamic and gravitational turbulence. Leray [87] has developed a theory of hydrodynamic turbulence which associates this phenomenon with a break-down of the Navier-Stokes equations"!"; that is, turbulence signals that solutions cease to exist for some t > 0 after regular Cauchy data was set at l = 0. In general relativity, under reasonable physical conditions, all geodesies are either (and maybe both) future or past incomplete (3|. However, the interpretation of this incompleteness is not gravitational turbulence but a space-time singularity. Interestingly, general relativity does not appear to meet the first Hadamard criterion.

it is not obvious that twe second and third Hadamard criteria are i desirable feature of a mathematical theory of turbulence that should describe the intermittent and chaotic behaviour observed in real systems. Could the Einstein equations exhibit non-uniqueness like the equation ! In one and two dimensions it is known th.«t solutions exist for all r>0. but the existence problem remains unsolved in three dimensions. Wirmus rcsul's on local existence are known, see ref, [88| for details.

which possesses two distinct solutions with boundary condition x(0) = 0, that is,

Ordinary differential equations will avoid this non-uniqueness problem if they obey a local Lifshitz condition [891. Collins and Stewart [90] have shown that the homogeneous cosmological models are described by a system of ordinary differential equations obeying this condition. The system (6.28) does not obey the Lifshitz condition at the origin. (For the general case where the field equations are partial derivatives see ref. [34].) The third Hadamard criterion is not satisfied by any system with a non-zero dynamical entropy, in particular by the Mixmaster Universe.

Another famous theory of turbulence which has practical implications (unlike Leray's), is that due originally to Landau [91] and extended by the ideas of Hopf [92]. Here fluid turbulence is envisaged as a manifestation of the generic behaviour of the Navier-Stokes equation or some other underlying equation. The flow is controlled by some parameter, /a, which may be the Reynold's number. The flow could begin with a stationary velocity field but as ju. increases it will become quasi-periodic, gradually picking up further frequency components, vh as the velocity field evolves v(t)-> f(v{t)-> f(vrt, i>2t)-> ■ ■ f(vi*, ... vit,...) and in general approaches

v(x, t)- X flm(jr)exp(im(rm + a)}        (6.29)

where all the v{ are irrationally related. The velocity field (6.29) is so complicated as to be essentially random. What this is saying, in the language of differential equations, i; that when only ore frequenc\ component (^i) appears the phase space trajectory is spiralling into an ordinary one-dimensional limit cycle bu* when the second component (v2) appears the attractor becomes more complicated - a two-dimensional torus, r2 - thereafter when r irrationally related frequency components appear the attractor is rr, a product of r independent periodic cycles. The transition between different basins of attraction is described by Hopf's bifurcation theory. However, the predictions of this theory - never- ending quasi-periodic behaviour - appear to be at variance with observation [93]; something more complicated is happening. Ruelle and Takens [1] showed that when the attracior reaches r4 the behaviour becomes far more complex, the attractor becomes 'strange' and has the non-integral dimension of a Cantor set. The motion is highly chaotic and exhibits sensitive dependence on initial conditions. Quasi-periodic motions are unstable and strange attractors are a generic result of their instability

Is it possible that similar ideas could be developed for a description of g ivitational turbulence b\ applying certain types of stability analysis to the Einstein equations? If one represents the Einstein equations in autonomous form as

where fi is some suitably chosen parameter or set of parameters controlling the behaviour, then a degeneration to increasingly complicated and chaotic behaviour would occur as (i changed. An appropriate candidate for /x would be some dimensionless measure of the three-curvature anisotropy or, for the homogeneous case, some parameter which varied along the Bianchi sequence of cos­mological models. At first the behaviour would resemble the Kasner («-statioraty) solution but as the curvature parameter grew there would eventually be a bifurcation to a Mixmaster chaotic regime. Since the Mixmaster model does not appear to contain a strange attractor in the technical (1| sense it is possible that perturbations of it, or more general space-times, will. An interesting bifurcation problem m.ght be the following: The Einstein equations for the diagonal Bianchi VIII and IX models are given by equations of the form (4.4-6),



where {A. n, v} are structure constants (A = v = n - 1 in type IX but p = -1 and A = v ~ 1 for type VIII). Now consider the (6.31)-(6.33) reduced to the form (6.30) and examine its behaviour as the parameters {A, fi,»/} vary in the range (-», if one wishes to have a physical interpretation of such a procedure then one might imagine the cases {A, p, v} s* {1,1, ±1} to represent a spatial variation in »he 'best-fit' homogeneous model as one moves about the Universe. The result of a source of white-noise on the system, or more specifically, on the Mixmaster return map (4.20), might also be interesting.

This type of picture has some points in common with that of Belinskii et al. [31] wherein the general solution is imagined to possess no interesting spatial degrees of freedom. In essence this envisions that a time-slice through the general cosmological solution to the Einstein equations will not exhibit chaotic behaviour as one moves from one point of the slice to its neighbour.

One of the principal motivations to search for chaotic behaviour in the Einstein equations is the discovery that p wide class of chaotic systems exhibit universal behaviour. This discovery has provided the theory of t urbulence with a firm mathematical basis which does not depend on the exact mathematical model being employed. If the mappings generated by t!.< Einstein equations could be shown to possess universal properties important deductions about "gravitational turbulence' or the general solution to the field equations would be possible.