Download An Introduction to General Relativity and Cosmology by Jerzy Plebanski, Andrzej Krasinski PDF

By Jerzy Plebanski, Andrzej Krasinski

Normal relativity is a cornerstone of recent physics, and is of significant value in its purposes to cosmology. Plebanski and Krasinski are specialists within the box and supply an intensive advent to common relativity, guiding the reader via whole derivations of crucial effects. offering insurance from a special perspective, geometrical, actual and astrophysical houses of inhomogeneous cosmological types are all systematically and obviously offered, permitting the reader to keep on with and confirm all derivations. Many issues are integrated that aren't present in different textbooks.

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The collection of quantities obtained when each component of one tensor density (of type w k l ) is multiplied by each component of another tensor density (of type w k l ) is called a tensor product of the two densities, and is a tensor density of type w + w k + k l + l . For example, out of u and v one can form such tensor products as v v , v u , u u , v v v , v u v , v u v u v . The tensor product is denoted by ⊗, thus for example v u = v ⊗ u . 4. If a tensor density does not change its value when two indices (either both upper or both lower) are interchanged, then it is called symmetric with respect to this pair of indices.

A linear combination of two tensor densities of type w k l is a tensor density of the same type. ) 3. The collection of quantities obtained when each component of one tensor density (of type w k l ) is multiplied by each component of another tensor density (of type w k l ) is called a tensor product of the two densities, and is a tensor density of type w + w k + k l + l . For example, out of u and v one can form such tensor products as v v , v u , u u , v v v , v u v , v u v u v . The tensor product is denoted by ⊗, thus for example v u = v ⊗ u .

Note that only the symmetric part of the connection gives a nonzero contribution to the geodesic equation. 3 Exercises 1. Consider a vector on a Euclidean plane being transported parallely along a straight line. Find how its components change when they are given in polar coordinates. 2. Do the same for a vector in a 3-dimensional Euclidean space when its components are given in spherical coordinates. From the result, read out the connection coefficients of the Euclidean space in spherical coordinates.

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