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Matching of Stephani and de Sitter solutions on the hypersurface of constant time

The spherically symmetric solution for perfect fluid with homogeneous density and inhomogeneous pressure has been considered. This solution is known as Stephani solution. The matching of this solution and de Sitter solution has been done on a hypersurface of constant time. The matching has been done for general case and for particular cases (flat, closed, open universe). An equality of the densities and a bound of the pressures have been shown on the matching hypersurface. Also, restrictions on some arbitrary functions have been found.


Introduction
The most general class of non-static, perfect fluid solutions of Einstein's equations that are conformally flat is known as the "Stephani Universe" [1][2][3][4][5]. The spherically symmetric Stephany solution has been examined in numerous papers. A comprehensive review is presented in [5]. There are many papers devoted to applying this solution as star models, as generalization of the FLRW, as cosmological model [5][6][7]. In our opinion, this solution is attractive for cosmological model for many reasons. Firstly, it is shear-free and inhomogeneous. The absence of a shear makes it simple for the cosmological purpose. The assumption of homogeneity is just a first approximation introduced to simplify Einstein's equations. So far this assumption has worked well, but future and modern observations can not be precise without taking into account inhomogeneity. And due to the fact, that modern and future observation dates become more and more precisely and that the smallest deviations from the standard model can be detected with high level of accuracy soon, makes inhomogeneous models actual. Secondly, Stephani solution has general form in contradistinction to the FLRW solution, where three solutions (flat, open, closed), non-transforming into each other, exist. Thirdly, the spatial curvature of this solution depends on time only via an arbitrary function, this fact is discussed in [3,8].
Physical interpretation of the Stephani Universe is obscure. It is due to the many arbitrary functions and peculiar inhomogeneity -inhomogeneity is contained in the pressure (depend on time and spatial coordinates), but density is homogeneous (depend on time only). It needs matching in order to determine some arbitrary functions. May be the main reason to use this solution in cosmological modeling is the fact that it is generalization of the FLRW solution, and, in our opinion, studying more general solution is promising. This solution generalizes not only the FLRW but the de Sitter solution too [3]. In this connection, the idea to examine the Stephani solution on the de Sitter background looks reasonable.
In the first part of the artice the solution for perfect fluid with inhomogeneous pressure (the Stephani solution) has been reobtained with mass-function method [9][10][11][12]. Some properties of it are discussed, in particular, we discuss a sense of the arbitrary functions and transformation to the FLRW and to the de Sitter solutions.
In the second part of the article the matching of the de Sitter and the Stephani solutions on the hypersurface of constant time is done. Lichnerowicz -Darmois conditions were used. Some consequences of matching are discussed.

Mass-function method
Mass-function method essentially simplifies the appearance of the Einstein's equation in contrast to the standard one, it makes them easier for work. The mass-function was introduced in [9] and discussed in [10][11][12]. As shown in [9], the mass-function is invariant and in our consideration may be determined as full energy limited by some hypersurface of constant time and coordinates. For spherically symmetric metric: .
Einstein's equations with mass-function have following form:

Obtaining Stephani solution
This solutions was first found by Stephani [13] as a special example of a space-time embeddable in a flat five-dimensional space, and later reobtained by Krasinski [2]. We reobtained this solution with mass-function method.
Expressing the mass-function from the first equation of the set and the substituting it into the third one gives: from this equation expression for Φ is obtained: 2 2 ln , where ( ) t ψ ψ = -arbitrary function of integration. The fourth equation of the set (5) gives us the expression for Ω: where ( ) k k R = -arbitrary function (prime is used for convenience). From (1) and (3) the metric is obtained: With (7) and (8) the mass-function is obtained: The expression (10) with the first equation of (1.5) gives us: it is integrated in elementary functions providing the expressing for and ( ) t η is arbitrary function of integration.
Depending on the sign of ( ) t ζ the expression (12) is: The existence of this solution shows that the distinction between the closed and open Universe is not required by Einstein's theory of gravitation as such, but is due to the very strong symmetry assumptions, which are put into the models right from the beginning. From (5) it is also possible to obtain equation that links density and pressure:

The FLRW and the de Sitter solutions as special cases of the Stephani solution
Stephani solution, as mentioned above, is a generalization of the FLRW solution and the de Sitter one.
When ( ) 0 t ζ = , the Stephani solution is transformed to the flat FLRW solution. If ( ) 0 t ζ ≠ transformation occurs in such a way: A density of the de Sitter solution is The function ( ) k R may be chosen arbitrary, because it leads to a transition to another coordinate system, only.
The part (9) is just transformation from R to k . The coordinate transformation may be chosen in such way, that spatial part of the solution (9) is conformal to one of the three homogeneous and isotropic spaces: Chosen ( ) t η is also referring to the coordinate transformation. Thus (9) takes the form: The analysis of invariants of the spatial curvature tensor of the metric (9) showed that invariants depend on the arbitrary function ( ) t ζ , only. The scalar curvature tensor and the Kretschmann scalar, for example, are:

Lichnerowicz -Darmois conditions
The Lichnerowicz-Darmois matching conditions [14] are two metrics    From the equality of the first fundamental forms, two matching conditions can be obtained: X a e t λ τ λ a e X λ τ λ Right-hand sides of these equations are equal, from this equality we obtain: Equality of right-hand sides of (53) and (54) with (52) gives us: The two possibilities exist in this case: the first is 2 1, X ′ = (56) or the second one reads But (57) The equality of the right-hand sides of (66) and (67) gives: Two possibilities exist for satisfy this equation: 1) 2 1, X ′ =

X X tX
The last set of equation is an incompatible system, so, we conclude from (69) This system of equation is incompatible, so, we conclude from (82) and (78), that Stephani's time and radial coordinate. Matching was done both for special cases (flat, open, closed) and for the general case, that not concretize type of curvature. From the matching conditions the equality of densities on the matching hypersurfase has been obtained. From (15) a bound of the pressure exists. Also it was obtained, that de Sitter radial coordinate distinguishes from Stephani one on some shift and de Sitter time is a some arbitrary function depending on Stephani's time.