4.1. Mixmaster evolution

 

Here, x, s (jc, y, z) and the coordinate ranges are 0 jc 4tt, 0 y tr, 0 z 2tt. In this frame,

 

The three-spaces are closed and have finite volume given by

 

where ' is d/dT and the new time coordinate r is related to the synchronous coordinate time. /. b\

If we take the structure constants to be of type IX then the space metric in (3.7) can be diagonalised^ with the basis vectors chosen as:

 

 

The field equations (3.9)—(3.11) give three evolution equations for the scale factors, мисс R R R 3= 0 we have [31] the cyclic equations,

 

 

Typically ahc~t and r~ In t so the cosmological singularity at t = 0 lies at r = The addition of equations (4.4}-(4.6) also gives a first integral, tThis is not the most general case. When matter is present with 7^*0 components in the enerjs> momentum tenvt J"* the ivn non-diagonal models. However, the new leatutes that this introduces do not affect the results of this section m am sismx.sm t.^tuor o.i discuss the more general case in section 5.m

Belinskii et al. [31] have shown how the evolution described by (4.4H4.8) can be analysed quan­titatively. Alternatively, a qualitative picture can be developed using the Hamiltonian techniques of Misner [36]. Note that the solution to the system (4.4H4.8) will contain three arbitrary constants when all scale transformation freedom has been used.

Equations, (4.4H4.6) describe the motion of a particle within a closed time-dependent potential. The terms on the right-hand side of these equations specify the potential. When the right-hand sides of (4.4)_(4.6) are set equal to zero the field equations reduce exactly to those for the Bianchi I or Kasner Universe described by (3.14-15).

To establish the Mixmaster behaviour as f->0 (r-*-°°) suppose it begins to evolve with a>b>c then (4.4-4.8) solve as

 

 

they give the amplitude of the relative scales at the maximum of a(t). That such maxima occur can be seen from (4.9). The solution (4.9)-(4.11) describes a half-period oscillation of a\t) and b2(t). Far from the turning points of a2(t) and b\t) the solution is described by Kasner behaviour. The model then evolves through a sequence of oscillations in a2(t) and b2(t) during which the function c(t) falls monotonically (t->-=°). By matching the corresponding solutions of the form (4.9)-(4.12), recurrence relations can be established to link the parameters describing these small oscillations

The Kasner approximation to (4.9H4 U) away from the turning points of a{t) and b(t) is described by the parameter u which corresponds to that introduced in (2.16)

A graph of the long-term behaviour on approach to the singularity is given in fig. 2.

A series of small oscillations from one Kasner behaviour to auoïher occurs until the value of the constant uH falls below unity. In this case the invariance properties of pt(u) in the Kasner model. (3.17) show the next cycle of oscillations commences with [30,31,45]

 

The following remarks are in order:

The evolution towards the singularity proceeds through an infinite number of oscillations of the scale factors o(t\ bit) and c(t) on any open interval of time t F (0T T). Physically speaking one is following the evolution of a ball of gravitational wave energy as it collapses to zero volume. The collapse follows a series of cycles during which two of the scale factors ("radii") execute small oscillations \\ htlst the third collapses monotonically. The change of behaviour indicating the onset of a new cycle is the attainment of a local minimum by the monotonically falling function. During the new cycle the monotonic scale of the old cycle executes small oscillations whilst the scale factor it replaced now undertakes monotonic behaviour until the next cycle commences.

During any interval of time in which none of a, b or c have local maxima or minima the expansion is described to a good approximation by the Kasner model (3.14,3.15). II one thinks of (4.4H46) as a Hamiitonian description then the left-hand side of these equations represent the kinetic terms whilst the right-hand side describes the potential. Whilst the motion is far from the potential walls the right-hand sides are negligible and the Kasner solution obtains. After a momentary collision with the potential wall the model is perturbed into a different Kasner model. The law of reflection is encoded in the constant u introduced in (3.16) If the Kasner behaviour at the commencement erf the first cycle is encoded by Mo > 1 then that after the first small oscillation is given by (w«-1), that after the second by (u<,-2) and so on. Clearly each cycle contains [u0] small oscillations where [• * •] denotes the integer part (for example, [7r] = 3). The u value describing the Kasner approximation at the commencement of the next cycle of small oscillations is given by where, (4.18),

 

 

These recurrence relations describe the Mixmaster oscillation over many cycles of evolution as t-»-*. As the col;apse occurs the amplitude of both the large and the small oscillations increases steadily even though the overall volume decreases.