NST430 Lecture Notes on Advanced Cosmology 1 of 4 (Cambridge)

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Lecture notes on Cosmology (ns-tp430m) by Tomislav Prokopec Part I: An introduction to the Einstein theory of gravitation Einstein’s theory of gravitation is a geometric theory, in the sense that gravitational forces exerted by masses are mediated by a nontrivial structure of space and time. In particular, in the presence of matter, physical distances between bodies change, and time lapses at a different rate. All information about the effects of a matter distribution on space and time are elegantly encoded in a symmetric metric tensor gµν with a Lorentzian signature, which means that a local Minkowski metric around the observer has the signature (the signs of the diagonal elements), sign[gµν ] = (+, −, −, −). (A completely equivalent sign convention, which is often used, is (−, +, +, +).) Once given, the metric tensor completely determines the Lorentzian manifold, which in turn provides a representation of gravitational interactions. In fact, the metric tensor gµν = gµν (~x, t) provides a complete information on how to measure physical distances and time lapses between space-time points. Furthermore, it specifies fully the dynamics of gravitating bodies, and thus it is of a fundamental importance for the Einstein relativistic theory of gravitation. Before we begin discussing the metric tensor, we shall now briefly consider the metric structure of the special theory of relativity.

A.

Special theory of relativity

Special relativity is an important special case of general theory of gravitation, where distances are determined by the Minkowski metric, which can be written in terms of an infinitesimal line element as follows

ds2 = ηµν dxµ dxν ,

ηµν

1 0

0

0

   0 −1 0 0    =  ≡ diag(1, −1, −1, −1) .  0 0 −1 0    0 0 0 −1

(1)

Here and throughout these lectures, we are using Einstein’s summation convention over the repeated P3 P3 µ ν µ i indices, e.g. ηµν dxµ dxν ≡ µ=0 ν=0 ηµν dx dx . Here x = (ct, x ) (i = 1, 2, 3) is a four-vector denoting a coordinate position of a point in space and time. The line element ds denotes an invariant

distance between two infinitesimally displaced points in space and time, and it is invariant under Lorentz transformations. A Lorentz transformation is any matrix belonging to real orthogonal matrices in 3+1 dimensional space and time. When taken together, they built up the orthogonal group O(1, 3), or


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