Download e-book for kindle: An introduction to mathematical cosmology by Jamal Nazrul Islam
By Jamal Nazrul Islam
This publication is a concise advent to the mathematical features of the starting place, constitution and evolution of the universe. The publication starts off with a short assessment of observational cosmology and basic relativity, and is going directly to talk about Friedmann versions, the Hubble consistent, types with a cosmological consistent, singularities, the early universe, inflation and quantum cosmology. This e-book is rounded off with a bankruptcy at the far-off way forward for the universe. The booklet is written as a textbook for complex 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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Additional resources for An introduction to mathematical cosmology
5 Some further topics In this section we consider some additional topics. First we consider the action principle in the presence of matter. 23). We will deal with the simpler situation in which the pressure p vanishes and the material particles move along geodesics. We have in mind a distribution of matter in which the velocity varies from one element to a neighbouring one continuously. The worldlines of material particles ﬁll up all space-time or a portion of it, a typical worldline being denoted by z(s), in which the points along the worldline are distinguished by values of a parameter s which measures the interval along the line.
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.
The latter are used in formulating an action principle from which ﬁeld equations can be derived in a convenient manner. We shall use this principle to obtain the ﬁeld equations with a scalar (Higgs) ﬁeld in connection with inﬂationary cosmologies. Consider a transformation from coordinates x to xЈ. 52) TLFeBOOK 22 Introduction to general relativity where J is the Jacobian of the transformation given by Έ Έ ѨxЈ0 Ѩx0 ѨxЈ1 0 1 2 3 Ѩx0 Ѩ(xЈ ,xЈ ,xЈ ,xЈ ) Jϭ ϵ ѨxЈ2 Ѩ(x0,x1,x2,x3 ) Ѩx0 ѨxЈ3 Ѩx0 ѨxЈ0 Ѩx1 ѨxЈ1 Ѩx1 ѨxЈ2 Ѩx1 ѨxЈ3 Ѩx1 ѨxЈ0 Ѩx2 ѨxЈ1 Ѩx2 ѨxЈ2 Ѩx2 ѨxЈ3 Ѩx2 ѨxЈ0 Ѩx3 ѨxЈ1 Ѩx3 .
An introduction to mathematical cosmology by Jamal Nazrul Islam