An introduction to mathematical cosmology - download pdf or read online

By J. N. Islam

ISBN-10: 0511018495

ISBN-13: 9780511018497

This e-book offers a concise creation to the mathematical points of the starting place, constitution and evolution of the universe. The publication starts off with a short review of observational and theoretical cosmology, in addition to a brief advent of common 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 away way forward for the universe. This re-creation includes a rigorous derivation of the Robertson-Walker metric. It additionally discusses the boundaries to the parameter area via a variety of theoretical and observational constraints, and provides a brand new inflationary answer for a 6th measure capability. This e-book is appropriate 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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Let each point of the curve with coordinate x␮ be moved to x␮ ϩdx␮. If dx␮ is an element along the curve, we have ds2 ϭ ␮␯dx ␮dx␯. Taking variations of both sides, we get 2ds␦(ds) ϭ dx␮dx␯ ␮␦(dx␭). 100) the following expression for ␦(ds): ␦(ds) ϭ Ά dx␯ ␭ ␦x ϩ ds ds dx 1 2 ␮␯,␭ ␮ ␮␭ · dx␮ d␦x␭ ds. ds ds Therefore, ␦ Ύ B ds ϭ A Ύ B ΎΆ · B ␦(ds)ϭ A 1 ␮ ␯ ␭ 2 ␮␯,␭u u ␦x ϩ d (␦x␭) ds. 101) ds ␮ ␮␭u A We carry out partial integration with respect to s and use the fact that ␦x␭ ϭ0 at A and B, to get Ύ ds ϭ Ύ Ά B ␦SAB ϭ ␦ B A 1 ␮ ␯ 2 ␮␯,␭u u Ϫ A d ( ds ␮) ␮␭u ·␦ x␭ds.

40) gives a solution to Killing’s equation. 39) is satisfied and the metric is stationary. A similar result can be established for any of the other three coordinates. We now derive a property of Killing vectors which we will use later. 38). We define the commutator of these two Killing vectors as the vector ␨ ␮ given by ␨ ␮ ϭ ␰ (1)␮;␭␰ (2)␭ Ϫ ␰ (2)␮;␭␰ (1)␭. 42) In coordinate independent notation the commutator of ␰ (1) and ␰ (2) is written as [␰ (1), ␰ (2)]. 42) can be replaced by ordinary derivatives.

Consider a curve x␮(␭), where x␮ are suitably differentiable functions of the real parameter ␭, varying over some interval of the real line. It is readily verified that dx␮/d␭ transforms as a contravariant vector. This is the tangent vector to the curve x␮(␭). For an arbitrary vector field Y␮ its covariant derivative along the curve (defined along the curve) is Y␮;␯(dx␯/d␭). The vector field Y␮ is said to be parallelly transported along the curve if Y␮;␯ dx␯ dx␯ dx␯ ϭY␮,␯ ϩ⌫␮␯␴Y␴ d␭ d␭ d␭ ϭ dY␮ dx␯ ϩ⌫␮␯␴Y␴ ϭ0.

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An introduction to mathematical cosmology by J. N. Islam

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