Return to Modern Relativity

Each section presumes that the reader has worked through the previous sections and chapters. It is also assume the reader has an understanding of mathematics through calculus and partial differential equations. The relevant tensor calculus is presented throughout as needed.

Every author has his own conventions, his own way of doing general relativity. When mixing equations from this book with equations from other authors of relativity, make sure to account for the differences in definitions and conventions as needed.

This book will use the following conventions. The space-time signature will be (+ - - -). Where the word mass or the letter m is used unqualified it will be defined as invariant as discussed in the section on the definition of mass. The Ricci tensor will be the contraction over the Riemann tensor's first and third indices. Except for the section on Kaluza-Klein theories, Greek superscripts and subscripts will be indices running 0,1,2,3 where 0 will represent the time index. Indices i and j will only be spatial indices running 1,2,3. A comma will represent a partial derivative. In other words F^l,_r will mean ¶F^l/¶x^r. A semicolon will represent the partial covariant derivative. The meaning of a covariant derivative is discussed in its section below.

The Einstein summation convention will be used in the following way. Unless otherwise stated, whenever a product of quantities or a single quantity appears where one index is high and the other is low summation over the repeated index will be implied. Also recall, that for Greek indices they run 0,1,2,3 , but the indices i and j will only run 1,2,3. For instance the expression

R^l_mln

means

R⁰_m0n + R¹_m1n + R²_m2n + R³_m3n

For another example

P_iPⁱ = P₁P¹ + P₂P² + P₃P³

Unless otherwise stated an index that is repeated but both are high, or both are low, then summation is not implied. For instance T^mm will simply be the mm element of T^mn. There is no summation here.