By Jamal Nazrul Islam

This e-book is a concise creation to the mathematical features of the starting place, constitution and evolution of the universe. The booklet starts off with a short review of observational cosmology and basic relativity, and is going directly to speak about Friedmann versions, the Hubble consistent, types with a cosmological consistent, singularities, the early universe, inflation and quantum cosmology. This publication is rounded off with a bankruptcy at the far away way forward for the universe. The e-book is written as a textbook for complex undergraduates and starting graduate scholars. it's going to even be of curiosity to cosmologists, astrophysicists, astronomers, utilized mathematicians and mathematical physicists.

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As before, we denote by the determinant of the covariant tensor ␣ considered as a matrix. 58) where Јϭdet( Ј ). 59) TLFeBOOK Some special topics in general relativity 23 1 where in the second equation we have introduced the notation ϭ(Ϫ )2, 1 Јϭ (Ϫ Ј)2, since this quantity occurs in various contexts (the symbol is to be read as ‘curly ’). Consider now a scalar ﬁeld quantity which remains invariant under a coordinate transformation. If we call it S, then Sϭ SЈ; S could be AB, for example, where A is a covariant vector and B a contravariant one.

Consider now the transformation given by xЈϭ xϩ aЈϪa, yЈϭyϩbЈϪb, zЈϭz ϩcЈϪc. 16) This transformation takes the point P to the point PЈ, because when (x, y, z)ϭ (a, b, c), we get (xЈ, yЈ, zЈ)ϭ(aЈ, bЈ, cЈ). 15) has in the old coordinates. 16) represents an isometry of the metric, which is not just inﬁnitesimal but a ﬁnite or a global isometry. 15) represents a homogeneous space. In terms of Killing vectors, it is easily veriﬁed that the vectors given by ϭ(0, 1, 0, 0), ϭ(0, 0, 1, 0) and ϭ(0, 0, 0, 1) are all Killing vectors, as are any linear combinations of these with arbitrary constant coeﬃcients.

8) where the ␥ij are functions of (x , x , x ) only. Consider the three-space given by 1 2 3 dЈ2 ϭ␥ijdx i dx j. 9) We assume this three-space to be homogeneous and isotropic. According to a theorem of diﬀerential geometry, this must be a space of constant curvature (see, for example, Eisenhart (1926) or Weinberg (1972)). In such a space the Riemann tensor can be constructed from the metric (and not its derivatives) and constant tensors only. 10) where k is a constant. 10) if the ␥ij are chosen to be given by the following metric (Weinberg 1972, Chapter 13): dЈ2 ϭ(1ϩ 14krЈ2)Ϫ2[(dx1)2 ϩ(dx2)2 ϩ(dx3)2], rЈ2 ϭ(x1)2 ϩ(x2)2 ϩ(x3)2.