Mathematical Physics Seminar

University of Memphis

Summer 2011

Dwiggins

The following notes were taken from Chapters 8, 9, 10 of the book,Principles of Physical Cosmology, by Professor P. J. E. Peebles,published by Princeton University Press in 1993.

These notes come from the opening section of the middle part of the book,which is divided into three parts:

Part I = The Development of Physical Cosmology

Part II = General Relativity and Cosmology

Part III = Topics in Modern Cosmology

General Covariance.

Coordinates for space-time events have no absolute significance.

They are merely labels relative to a chosen coordinate system.

Scalars are measurable (observable) quantities which have values independent of the coordinate system. Note that scalar values do of course depend on the chosen scale — for example, where is the origin (zero) on this scale? What defines positive/negative?

Vectors and Tensors have components which depend on the given coordinate system.If the coordinate system changes then the components change according to specific rules.These rules are specified in order to make certain space-time quantities invariant,An observable associated with a vector or tensor is a scalar quantity, calculated in terms of the coordinate system, such that when the coordinate system changes the observable maintains the same value. (One example would be energy.)

Note: In order to make the concepts and notation of tensor analysis consistent, tensors of various order are defined, with scalars considered as zero-order tensors. First-order tensors are called vectors, of which there are two types: covariant (think of these as column vectors) and contravariant (row vectors). Second-order tensors are classified as covariant, contravariant, or mixed. The first two types behave like vectors with different sections blocked out, and mixed second-order tensors are represented as matrices. Second-order tensors are also called dyads or dyadics.

As just described, the answer to the question, “What is a tensor?” involves the study of what happens when coordinate systems are changed, i.e. how to physical quantities transform under general coordinate transformations. We turn now to introducing the notation used in this type of analysis.