Universe mixmaster

The universe mixmaster is a solution of the equations of the General relativity of Einstein studied by Charles Misner in 1969. It describes the temporal evolution of an empty matter universe, homogeneous but anisotropic, from which the Rate growth differs in the three directions from space. This model corresponds to type IX of the Classification of Bianchi. It particularly is interested in the evolution of the universe in the vicinity of a gravitational Singularité like the Big Bang.

Discussion

The model is similar to the universe closed of Friedmann-Lemaître-Robertson-Walker, in whom the “space sections” are positively curved and are topologically of the 3 - Sphère S

S^3

. Cepandant, in the universe of FLRW, the

S^3

can only extend or contract: the only dynamic parameter is the total dimension of

S^3

, and parameterized by the Scale factor

a (T)

. In the universe mixmaster, the

S^3

can extend or contract, but also twist da anisotropic manner. Its evolution is described by the scale factor

a (T)

as well as by the two parameters of form

\ beta_ \ pm (T)

. The values of the parameters of form describe the distortions of the

S^3

which preserve its volume and which also maintain a scalar constant of the curve of Ricci. Consequently, like the 3 parameters

a, \ beta_ \ pm

aussement of the different values, the homogeneity is preserved, but not the isotropy.

The dynamics of the universe mixmaster appears extremely rich. Misner in particular showed that in the vicinity of the gravitational singularity, the universe extends in two and only two directions, and contracts along the third. This behavior is usually called Singularité BKL, of the name of V.A. Belinsky, I. Mr. Khalatnikov and E. Mr. Lifschitz, which studied it in detail in 1970. It was conjectured that it was generic with a phenomenon of gravitational Effondrement. It presents a certain number of characteristics of a chaotic phenomenon , in particular by the fact that the direction in which the universe contracts erratic exchange of way during time, and that the various contraction or growth rates jump of a value to another in a quasi random way.

Metric

The metric one studied by Misner (slightly modified for its notation) is given by:

$\ text {D} s^2 = - \ text {D} t^2 + \ sum_ {k=1} ^3 {L (T)}_k^2 \sigma_k \otimes \sigma_k$ .

where the $\ sigma_k$ , regarded a long time as differential forms, are defined by:

$\ sigma_1 = \ sin \ psi \ text {D} \ theta - \ cos \ psi \ sin \ theta \ text {D} \ phi$ ,

\ sigma_2 = \ cos \ psi \ text {D} \ theta + \ sin \ psi \ sin \ theta \ text {D} \ phi

\ sigma_3 = - \ text {D} \ psi - \ cos \ theta \ text {D} \ phi

In term of coordinates $(\ theta, \ psi, \ phi)$ , those satisfy:

$\ text {D} \ sigma_i = \ frac {1} {2} \ epsilon_ {ijk} \ sigma_j \ wedge \ sigma_k$ .

where $\ text {D}$ is the external Dérivée and $\ wedge$ , the produced wedge of the differential forms. This relation points out the relations of commutation of the Algèbre of Dregs of '' KNOWN (2) ''. The group of variety KNOWN (2) is the 3-sphere $S^3$ and in truth the $\ sigma_k$ describes the metric one of a $S^3$ which can twist in an anisotropic way thanks to the presence of $L_k (T)$ .

Then, Misner defines the parameter $\ Omega (T)$ and $R (T)$ which measures the volume of the space sections, as well as the “parameters of form” $\ beta_k$ by:

$R (T) = e^ {- \ Omega (T)} = (L_1 (T) L_2 (T) L_3 (T))^ {1/3}, \ quad L_k = R (T) e^ {\ beta_k}, \ quad \ sum_ {k=1} ^3 \ beta_k (T) = 0$ .

As there is a codition on the three $\ beta_k$ , there decrait to be only two function S free, which Misner chooses as being $\ beta_ \ pm$ , definite like:

$\ beta_+ = \ beta_1 + \ beta_2 = - \ beta_3, \ quad \ beta_- = \ frac {\ beta_1 - \ beta_2} {\ sqrt {3}}$ .

The evolution of the Universe is then described by finding $\ beta_ \ pm$ like a function of $\ Omega$ .

Applications to the Cosmology

He hoped that chaos barraterait and would level the Universe in his first moments. Also, during the periods in which a direction was static (i.e.: energy of the expansion to the contraction), the horizon of Hubble

H^ {- 1}

is infinite in this direction. He suggested by there that the Problème of the horizon could be solved. As the directions of the expansion and the contraction vary, the problem of the horizon would be solved in all the directions.

While there was an interesting example of gravitational chaos, it is largely recognized that the cosmological problems that the universe mixmaster tries to solve are more elegantly solved by the cosmic Inflation. The study of metric of Misner is also known under the name of metric of Bianchi of the type

See too

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