The subtraction in the denominator for yields the sensitivity to initial conditions associated with Mixmaster dynamics (MD). Misner  described the same behavior in terms of the model's volume and anisotropic shears. A multiple of the scalar curvature acts as an outward moving potential in the anisotropy plane. Kasner epochs become straight line trajectories moving outward along a potential corner while bouncing from one side to the other. A change of corner ends a BKL era when . Numerical evolution of Einstein's equations was used to explore the accuracy of the BKL map as a descriptor of the dynamics as well as the implications of the map [145, 164, 166, 20]. Rendall has studied analytically the validity of the BKL map as an approximation to the true trajectories .
Later, the BKL sensitivity to initial conditions was discussed in the language of chaos [6, 125]. An extended application of Bernoulli shifts and Farey trees was given by Rugh  and repeated by Cornish and Levin . However, the chaotic nature of Mixmaster dynamics was questioned when numerical evolution of the Mixmaster equations yielded zero Lyapunov exponents (LE's) [76, 45, 111]. (The LE measures the divergence of initially nearby trajectories. Only an exponential divergence, characteristic of a chaotic system, will yield positive exponent.) Other numerical studies yielded positive LE . This issue was resolved when the LE was shown numerically and analytically to depend on the choice of time variable [164, 19, 73]. Although MD itself is well-understood, its characterization as chaotic or not had been quite controversial .
LeBlanc et al  have shown (analytically and numerically) that MD can arise in Bianchi VI models with magnetic fields (see also ). In essence, the magnetic field provides the wall needed to close the potential in a way that yields the BKL map for u . A similar study of magnetic Bianchi I has been given by LeBlanc . Jantzen has discussed which vacuum and electomagnetic cosmologies could display MD .
To study the CL fractal and ergodic properties of Mixmaster evolution , one could take advantage of a new numerical algorithm due to Berger, Garfinkle, and Strasser . Symplectic methods are used to allow the known exact solution for a single Mixmaster bounce  to be used in the ODE solver. Standard ODE solvers  can take large time steps in the Kasner segments but must slow down at the bounce. The new algorithm patches together exactly solved bounces. Tens of orders of magnitude improvement in speed are obtained while the accuracy of machine precision is maintained. In , the new algorithm was used to distinguish Bianchi IX and magnetic Bianchi VI bounces. This required an improvement of the BKL map (for parameters other than u) to take into account details of the exponential potential.
There are also recent numerical studies of Mixmaster dynamics in other theories of gravity. For example, Carretero-Gonzalez et al  find evidence of chaotic behavior in Bianchi IX-Brans-Dicke solutions while Cotsakis et al  have shown that Bianchi IX models in 4th order gravity theories have stable non-chaotic solutions. Barrow and Levin find evidence of chaos in Bianchi IX Einstein-Yang-Mills (EYM) cosmologies . Their analysis may be applicable to the corresponding EYM black hole interior solutions.
Finally, we remark on a successful application of numerical Regge calculus in 3 + 1 dimensions. Gentle and Miller have been able to evolve the Kasner solution .
|Numerical Approaches to Spacetime Singularities
Beverly K. Berger
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