By J. N. Islam

This e-book presents a concise advent to the mathematical elements of the foundation, constitution and evolution of the universe. The publication starts with a short assessment of observational and theoretical cosmology, in addition to a quick advent of basic relativity. It then is going directly to talk about Friedmann versions, the Hubble consistent and deceleration parameter, singularities, the early universe, inflation, quantum cosmology and the far-off way forward for the universe. This re-creation incorporates a rigorous derivation of the Robertson-Walker metric. It additionally discusses the bounds to the parameter house via a number of theoretical and observational constraints, and provides a brand new inflationary answer for a 6th measure strength. This publication is acceptable as a textbook for complicated undergraduates and starting graduate scholars. it is going to even be of curiosity to cosmologists, astrophysicists, utilized mathematicians and mathematical physicists.

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7) it follows that the ﬁrst derivatives of the metric tensor also vanish at this point. This is one form of the equivalence principle, according to which the gravitational ﬁeld can be ‘transformed away’ at any point by choosing a suitable frame of reference. 11) where x0 ϭct, t being the time and (x1, x2, x3) being Cartesian coordinates. 12) is the Riemann tensor deﬁned by where R ϭ⌫ Ϫ⌫ ϩ⌫ ⌫␣ Ϫ⌫ ⌫␣ . 14c) and satisﬁes the Bianchi identity: R ϩR ϩR ϭ0. 15) The Ricci tensor R is deﬁned by R ϭ R ϭR .

43) and so is a Killing vector. Suppose we have only n linearly independent Killing vectors (i), i ϭ1, 2, . . , n and no more. Then the commutator of any two of these is a Killing vector and so must be a linear combination of some or all of the n Killing vectors with constant coeﬃcients since there are no other solutions of Killing’s equation. Thus we have the result (i); ( j) Ϫ ( j); (i) ϭ n ͚a kϭ1 ij (k) , k i, jϭ1, . , n. 50) In coordinate independent notation, we can write [ (i), ( j)]ϭ n ͚a kϭ1 ij (k) , k i, jϭ1, .

A space-like three-surface is a surface deﬁned by f(x0,x1,x2,x3)ϭ0 such that f, f, Ͼ0 when fϭ0. The unit normal vector to this surface is given by n ϭ( ␣f,␣ f,)Ϫ1/2 f,. Given a vector ﬁeld , one can deﬁne a set of curves ﬁlling all space such that the tangent vector to any curve of this set at any point coincides with the value of the vector ﬁeld at that point. This is done by solving the set of ﬁrst order diﬀerential equations. 30) where on the right hand side we have put x for all four components of the coordinates.