On cosmological constant of generalized Robertson-Walker space-times

Abstract

We study Einstein's equation in (m + n)D and (1 + n)D warped spaces (M ,ḡ) and classify all such spaces satisfying Einstein equationsḠ = −Λḡ. We show that the warping function not only can determine the cosmological constantΛ but also it can determine the cosmological constant Λ appearing in the induced Einstein equations G = −Λh on (M 2 , h). Moreover, we discuss on the origin of the 4D cosmological constant as an emergent effect of higher dimensional warped spaces.

On Einstein equations with cosmological constant in braneworld models

“…In Space-Time-Matter (STM) theory, the reduction of Einstein equations in warped product metrics was studied and the following results were obtained. [20] -…”

“…The 5-dimensional brane world theories like RS [2], [3] and DGP models [4], may be considered as warped product spacetimes. We have already obtained, in the context of warped product spacetimes, the Einstein equations with cosmological constant in arbitrary (m + n)D and (1 + n)D multidimensional spacetime with multiply-warped product metric ( M , ḡ) [19], [20]. In this paper, motivated by the previous works, we consider arbitrary multidimensional brane world metrics of RS and DGP models with a cosmological constant, as warped product spacetimes.…”

Classification of Einstein equations with cosmological constant in warped product space-time

“…The warped product spaces were widely studied in the context of general relativity theory to construct new metrics with interesting geometrical and physical properties. In this regard, some solutions of Einstein equations, like generalized Friedmann-Robertson-Walker metric, generalized Schwarzschild black hole metric, generalized Reissner-Nordstrom black hole metric, generalized (2 + 1)-dimensional Banados-Teitelboim-Zanelli (BTZ) black hole metric, generalized (2+1)-dimensional de Sitter black hole metric, generalized standard static metric, 5-dimensional Randall-Sundrum and Dvali-Gabadadze-Porrati metrics, generalized twisted product structure and special base conformal warped product structure were shown by the authors to be expressed in terms of multi warped products in Lorentzian geometry [2]- [5].…”

“…In recent years, a large amount of attention has been paid on the modification of Einstein equations [7], specially in models with space-time dimensions other than four [24]. In this line of activity, we have previously obtained the Einstein's equation in (m + n)D and (1 + n)D multidimensional space-time with multiply-warped product metric ( M , ḡ) [27], [28]. Specially, we have discussed on the origin of 4D cosmological constant as an emergent effect of higher dimensional warped spaces.…”

Multiply-warped product metrics and reduction of Einstein equations

“…Then, we show that the Einstein equationsḠ AB = −Λḡ AB on the manifold (M ,ḡ) with a cosmological constantΛ are reducible to the Einstein equations G αβ = −Λ 1 g αβ and G ij = −Λ 2 h ij on the submanifolds (M 1 , g) and (M 2 , h), with cosmological constants Λ 1 and Λ 2 respectively, such thatΛ, Λ 1 and Λ 2 are given in terms of the geometric features f 1 , f 2 and n 1 , n 2 . By using [3,4], we consider some black hole solutions as typical examples and transform their metrics into the multiply warped product form of generalized Friedmann-Robertson-Walker metricM = K × f1 M 1 × f2 M 2 having warp functions f 1 and f 2 . Then, we derive the corresponding Einstein equationsḠ AB = −Λḡ AB , and the reduced Einstein equations G αβ = −Λ 1 g αβ and G ij = −Λ 2 h ij for each black hole solution.…”