arXiv:gr-qc/0001007v2 6 Jan 2000 Is the cosmological singularity really unavoidable in general relativity? Israel Quiros * Departamento de Fisica. Universidad Central de Las Villas. Santa Clara. CP: 54830 Villa Clara. Cuba (February 7, 2008) The initial singularity problem in standard general relativity is treated on the light of a viewpoint asserting that this formulation of Einstein’s theory and its conformal formulations are physically equivalent. We show that flat Friedmann-Robertson-Walker universes and open dust-filled and radiation-filled universes are singularity free when described in terms of the formulation of general relativity conformal to the canonical one. 04.20.-q, 04.20.Dw, 04.70.Bw One of the most undesirable features of general relativity is the ocurrence of spacetime singularities where the laws of physics breakdown. The famous Hawking-Penrose singularity theorems predict that, if ordinary matter obeys some reasonable energy conditions, then the ocurrence of spacetime singularities in general relativity is inevitable [1]. In particular, the initial singularity is a problem in standard general relativity given by the action: S = 1 16π  d 4 x  −ˆ g( ˆ R + 16π ˆ L matter ) (1) where ˆ R is the Ricci scalar of the metric ˆ g ab and ˆ L matter is the Lagrangian for ordinary matter. Recent developments of string theory suggest that the initial singularity can be resolved by including new degrees of freedom like p-brane [2]. However, the low-energy limit of string theory is usually linked with Brans-Dicke gravity so, the new degree of freedom is treated, precisely, in the frame of this theory. In particular, in reference [3], it has been argued that the gas of solitonic p-brane treated as a perfect-fluid-type matter in Brans-Dicke theory can resolve the initial singularity without any explicit solution. In this letter we shall treat the initial singularity problem in the light of a viewpoint, first presented in reference [4] for general relativity in empty space, according to which the usual formulation of Einstein’s theory given by the action (1) and its conformal formulation given by: S = 1 16π  d 4 x √ −g[φR + 3 2 φ -1 (∇φ) 2 + 16πL matter ] (2) are physically equivalent. In eq.(2) R is the Ricci scalar given in terms of the metric g ab that is conformal to ˆ g ab : ˆ g ab = φg ab (3) φ is some scalar function given on the manifold and L matter = φ 2 ˆ L matter is the Lagrangian for ordinary matter nonminimally coupled to the scalar φ. In what follows we shall call the formulation of general relativity due to (1) as Einstein frame general relativity while, its conformal formulation due to (2), we call as Jordan frame general relativity, a terminology usual in scalar-tensor gravity. The physical equivalence of Einstein frame general relativity and Jordan frame one is supported by the fact that physical experiment is not sensitive to the transformation (3), that can be interpreted as a particular units transfor- mation [5]. Actually, measurements of dimensional quantities represent ratios with respect to standard units so the measurables of the theory are always dimensionless numbers 1 and, then, are unchanged under the units transformation (3). In particular, the dimensionless gravitational coupling constant Gm 2 (¯ h = c = 1), where m is the rest mass of some particle and G is the dimensional gravitational constant, is kept unchanged under the transformation (3) [5]. Then, the fact that Gm 2 is a constant in general relativity, is a conformal invariant feature of this theory. In the formulation due to (1) both, G and m, are constant over the manifold, while in the conformal formulation due to (2), both G and m are variable in spacetime: G ∼ φ -1 and m ∼ √ φ. * israel@uclv.etecsa.cu 1 For a readable discussion on the dimensionless nature of measurements see section II of reference [6] 1