BKL Singularity - Generalized Kasner Solution

Generalized Kasner Solution

Further generalization of solutions depended on some solution classes found previously. The Friedmann solution, for example, is a special case of a solution class that contains three physically arbitrary coordinate functions. In this class the space is anisotropic; however, its compression when approaching the singularity has "quasi-isotropic" character: the linear distances in all directions diminish as the same power of time. Like the fully homogeneous and isotropic case, this class of solutions exist only for a matter-filled space.

Much more general solutions are obtained by a generalization of an exact particular solution derived by Edward Kasner for a field in vacuum, in which the space is homogeneous and has Euclidean metric that depends on time according to the Kasner metric

(eq. 2)

(see ). Here, p₁, p₂, p₃ are any 3 numbers that are related by

(eq. 3)

Because of these relationships, only 1 of the 3 numbers is independent. All 3 numbers are never the same; 2 numbers are the same only in the sets of values and (0, 0, 1). In all other cases the numbers are different, one number is negative and the other two are positive. If the numbers are arranged in increasing order, p₁ < p₂ < p₃, they change in the ranges

(eq. 4)

The numbers p₁, p₂, p₃ can be written parametrically as

(eq. 5)

All different values of p₁, p₂, p₃ ordered as above are obtained by changing the value of the parameter u in the range u ≥ 1. The values u < 1 are brought into this range according to

(eq. 6)

Figure 1 is a plot of p₁, p₂, p₃ with an argument 1/u. The numbers p₁(u) and p₃(u) are monotonously increasing while p₂(u) is monotonously decreasing function of the parameter u.

In the generalized solution, the form corresponding to eq. 2 applies only to the asymptotic metric (the metric close to the singularity t = 0), respectively, to the major terms of its series expansion by powers of t. In the synchronous reference frame it is written in the form of eq. 1 with a space distance element

(eq. 7)

where

(eq. 8)

The three-dimensional vectors l, m, n define the directions at which space distance changes with time by the power laws eq. 8. These vectors, as well as the numbers p_l, p_m, p_n which, as before, are related by eq. 3, are functions of the space coordinates. The powers p_l, p_m, p_n are not arranged in increasing order, reserving the symbols p₁, p₂, p₃ for the numbers in eq. 5 that remain arranged in increasing order. The determinant of the metric of eq. 7 is

(eq. 9)

where v = l. It is convenient to introduce the following quantitities

(eq. 10)

The space metric in eq. 7 is anisotropic because the powers of t in eq. 8 cannot have the same values. On approaching the singularity at t = 0, the linear distances in each space element decrease in two directions and increase in the third direction. The volume of the element decreases in proportion to t.

The Einstein equations in vacuum in synchronous reference frame are

(eq. 11)

(eq. 12)

(eq. 13)

where is the 3-dimensional tensor, and P_αβ is the 3-dimensional Ricci tensor, which is expressed by the 3-dimensional metric tensor γ_αβ in the same way as R_ik is expressed by g_ik; P_αβ contains only the space (but not the time) derivatives of γ_αβ.

The Kasner metric is introduced in the Einstein equations by substituting the respective metric tensor γ_αβ from eq. 7 without defining a priori the dependence of a, b, c from t:

where the dot above a symbol designates differentiation with respect to time. The Einstein equation eq. 11 takes the form

(eq. 14)

All its terms are to a second order for the large (at t → 0) quantity 1/t. In the Einstein equations eq. 12, terms of such order appear only from terms that are time-differentiated. If the components of P_αβ do not include terms of order higher than 2, then

(eq. 15)

where indices l, m, n designate tensor components in the directions l, m, n. These equations together with eq. 14 give the expressions eq. 8 with powers that satisfy eq. 3.

However, the presence of 1 negative power among the 3 powers p_l, p_m, p_n results in appearance of terms from P_αβ with an order greater than t−2. If the negative power is p_l (p_l = p₁ < 0), then P_αβ contains the coordinate function λ and eq. 12 become

(eq. 16)

Here, the second terms are of order t−2(p_m + p_n − p_l) whereby p_m + p_n − p_l = 1 + 2 |p_l| > 1. To remove these terms and restore the metric eq. 7, it is necessary to impose on the coordinate functions the condition λ = 0.

The remaining 3 Einstein equations eq. 13 contain only first order time derivatives of the metric tensor. They give 3 time-independent relations that must be imposed as necessary conditions on the coordinate functions in eq. 7. This, together with the condition λ = 0, makes 4 conditions. These conditions bind 10 different coordinate functions: 3 components of each of the vectors l, m, n, and one function in the powers of t (any one of the functions p_l, p_m, p_n, which are bound by the conditions eq. 3). When calculating the number of physically arbitrary functions, it must be taken into account that the synchronous system used here allows time-independent arbitrary transformations of the 3 space coordinates. Therefore, the final solution contains overall 10 − 4 − 3 = 3 physically arbitrary functions which is 1 less than what is needed for the general solution in vacuum.

The degree of generality reached at this point is not lessened by introducing matter; matter is written into the metric eq. 7 and contributes 4 new coordinate functions necessary to describe the initial distribution of its density and the 3 components of its velocity. This makes possible to determine matter evolution merely from the laws of its movement in an a priori given gravitational field. These movement laws are the hydrodynamic equations

(eq. 17)

(eq. 18)

where ui is the 4-dimensional velocity, ε and σ are the densities of energy and entropy of matter. For the ultrarelativistic equation of state p = ε/3 the entropy σ ~ ε1/4. The major terms in eq. 17 and eq. 18 are those that contain time derivatives. From eq. 17 and the space components of eq. 18 one has

resulting in

(eq. 19)

where 'const' are time-independent quantities. Additionally, from the identity u_iui = 1 one has (because all covariant components of u_α are to the same order)

where u_n is the velocity component along the direction of n that is connected with the highest (positive) power of t (supposing that p_n = p₃). From the above relations, it follows that

(eq. 20)

or

(eq. 21)

The above equations can be used to confirm that the components of the matter stress-energy-momentum tensor standing in the right hand side of the equations

are, indeed, to a lower order by 1/t than the major terms in their left hand sides. In the equations the presence of matter results only in the change of relations imposed on their constituent coordinate functions.

The fact that ε becomes infinite by the law eq. 21 confirms that in the solution to eq. 7 one deals with a physical singularity at any values of the powers p₁, p₂, p₃ excepting only (0, 0, 1). For these last values, the singularity is non-physical and can be removed by a change of reference frame.

The fictional singularity corresponding to the powers (0, 0, 1) arises as a result of time line coordinates crossing over some 2-dimensional "focal surface". As pointed out in, a synchronous reference frame can always be chosen in such way that this inevitable time line crossing occurs exactly on such surface (instead of a 3-dimensional caustic surface). Therefore, a solution with such simultaneous for the whole space fictional singularity must exist with a full set of arbitrary functions needed for the general solution. Close to the point t = 0 it allows a regular expansion by whole powers of t.