An introduction to mathematical cosmology by Jamal Nazrul Islam

By Jamal Nazrul Islam

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

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If we put ␦Iϭ0, we will obtain Einstein’s vacuum field equations: R␮␯ ϭ 0. 86a) where R* ϭ ␮␯(⌫␭ ␭ ␮␯,␭ Ϫ ⌫ ␮␭,␯), Qϭ ␮␯(⌫␭ ⌫␴ Ϫ ⌫␴ ⌫␭ ). ␮␯ ␭␴ ␮␭ ␯␴ We first remove the second derivatives of the these occur in the expression R*. 86b) ␮␯) ⌫␴ . 87) TLFeBOOK 29 Some special topics in general relativity We can use the divergence theorem to convert the first two integrals into surface integrals over Ѩ⍀, and so they will not contribute to the variation ␦I since the ␦ ␮␯ vanish on Ѩ⍀. 7) one can show that ( ␮␯) ,␴ ϭ(Ϫ ␯␤⌫␮ Ϫ ␤␴ ␮␣⌫␯ ϩ ␣␴ ␮␯⌫␳ ) ␴␳ .

4) one can readily show that ␦ ␮␯ ϭϪ ␮␣ ␯␤␦ ␣␤. 68) one can obtain the following relation: ␦ ϭ 21 ␣␤␦ ␣␤, so that ␦( ␮␯ ) ϭϪ( ␮␣ ␯␤ ) ␦ 1 ␮␯ ␣␤ Ϫ2 ␣␤. 22)). 26) can also be obtained by a variation principle, if it is not a null geodesic. 99) A which represents the ‘length’ (in the case of time-like curves this is the proper time) from the point A to the point B of the curve. 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␯.

This set of curves is referred to as the congruence of curves generated by the given vector field. In general there is a unique member of this congruence passing through any given point. A particular member of the congruence is sometimes referred to as an orbit. Consider now the vector field given by (␨ 0,␨ 1,␨ 2,␨ 3)ϭ(1,0,0,0). 30) we see that the congruence of this vector field is the set of curves given by (x0 ϭ ␭,x1 ϭconstant, x2 ϭconstant, x3 ϭconstant). 31) This vector field is also referred to as the vector field Ѩ/Ѩx0.

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