This is a republication of a book that was originally published in 1945. While the mathematical approaches to Relativity Theory have changed since then, this book still provides one of the most accessible treatments of the subject. There are, however, two very quirky features of this book of which any prospective reader should be made aware. (I mention these quirky features so that a prospective reader will look beyond them in order to see a very interesting book.) Professor Lieber had the idea that the book would be more easily read by putting each phrase on a separate line, making the book look like poetry. I found just the opposite. Furthermore, when I first saw this book I put it back thinking that is was some sort of cute attempt to discuss Relativity Theory in verse. This incorrect view was unfortunately supported by the numerous drawings that the author's husband contributed. For the most part, I found the drawings innocuous and if you looked at them carefully they did have some relevance to the text. Putting one phrase per line greatly reduced the amount of text per page. Had a conventional layout been used, and the drawings eliminated, the book would have been less than 150 pages long. The upside to this arrangement is that it did provide plenty of white space upon which I could write numerous notes.
This book aims at providing the mathematics of both the Special and General Theories of Relativity, in a manner that is reasonably accessible to someone with at least some mathematical background. However, this is not the book if you are looking for explainations using people on trains firing guns, twins in space ships or people in falling elevators. There is almost none of that sort of thing in this book. If that is what your are looking for I would recommend a book like Martin Gardner's "Relativity Simply Explained" or Wolfson's "Simply Einstein". The book being reviewed here focuses on the mathematics, not on general explanations.
The first third of the book is concerned with Special Relativity, and the treatment in this book is very similar to that provided in many other books, especially Einstein's "Relativity, The Special and General Theory", which was written for a general audience. (If you buy a copy of Einstein's book be sure to get the 15th and final edition, which contains all of the appendices and corrections.) Some of the text of Professor Lieber's book comes (with the proper attribution) from Einstein's book. I liked Einstein's treatment better (it had some of the moving train analogies as well as the math discussed here) and if you are just interested in Special Relativity I recommend it over the book being reviewed here.
The final two thirds of the book being reviewed here are devoted to the mathematics of the General Theory of Relativity, and if you are interested in this subject you should be interested in this book. The General Theory of Relativity is one of those subjects that is so daunting that every other treatment for a general audience eschews providing any mathematics at all and in-depth treatments are frighteningly complex. In my opinion, Professor Lieber did the near impossible in that she made the subject intelligible. I cannot say that after reading this book I can solve problems in General Relativity, but at least I now know what kind of mathematics Einstein utilized and I now have a general idea how General Relativity problems are attacked. General relativity replaces the idea of a gravitational force by the idea of the curvature of space and that this curvature is described by tensor calculus, by means of a curvature tensor. Solve the curvature tensor and you solve the gravitational problem. As illustrations, Professor Lieber uses the calculation of the phrihelion of Mercury, the bending of light by the Sun and the shift in spectra lines due to the curvature of space caused by a large mass (such as that of a star). The phrihelion solution cleared up a problem that had been unexplained for about 50 years when Einstein provided a solution, and was unexplainable by Newtonian mechanics. The confirmation of his prediction of the bending of light made Einstein world famous and the spectral line shift was something that had heretofore not been suspected, but when measured was further confirmation of General Relativity.
In addition to the original author, there are two editors who have done much more that just edit the book. They have provided 21 pages of notes that solve some of the problems that are left for the reader, provide the text from books that are referenced but are no longer in print or available, and provide modern references. The editors fill in some of the gaps that are left for the reader to search out from other books. This is a set of editor's notes that are essential and should not be skipped.
As I have noted the mathematics of General Relativity is that of tensor calculus. Professor Lieber defines contravariant and covariant tensors and how they are manipulated, but does not provide anything like a complete discussion of tensor calculus. For the most part, Professor Lieber defines the relevant tenors and how they are to be algebraically manipulated. Tensor mathematics utilizes a very compressed notation that allows hundreds of equations to be expressed by a single one, but one that utilizes subscripts, superscripts and many different symbols and Greek letters that define complex mathematical operations. The result is a very complex looking expression, but one that I think someone who is adept at algebraic manipulation may be able to follow. However, without the necessary physics and mathematical background they may not really understand why the mathematical operations that the equation describes are being performed. For someone with the right background this may not be too much of an impediment.
An important question to be considered is how much background mathematics does one require in order to get anything out of this book? I have been exposed to some tensor mathematics, but only the simple tensors used to relate vector qualities such as stress and strain. I had never been exposed to contravariant and covariant tensors. I found the discussions to be relatively clear, but the resulting algebraic manipulation of the tensors was daunting to say the least. I doubt that someone who had not studied college level mathematics would get very much from this book, but at least one other reviewer seems to have found otherwise.
The major problem that I had with the book was that there were many places where results were just stated and not properly explained. The editors cleared some of this up, but there were still many places where I wanted to know why a particular operation was being performed. I guess one cannot expect more from a book that effectively devotes only about 100 pages to the mathematics of the General Theory of Relativity. Anyone looking at a text on this subject will see that many, many, more pages of very, very, complex text are required to give a more detailed presentation, which is usually taught as a graduate level course. I am giving the book four rather than five stars because this lack of exposition and because of a concern that in spite of the efforts of the author and editors, the book may be over the head of a reader without the necessary background and I do not want to mislead prospective readers into thinking otherwise. However, with the right background, you should be able to understand what sort of mathematics is used and where to go to learn more (especially from the modern references provided by the editors). For me, that was sufficient and made the book well worth the time I spent with it.