PDF Ebook Spacetime and Singularities: An Introduction (London Mathematical Society Student Texts), by Gregory L. Naber
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Spacetime and Singularities: An Introduction (London Mathematical Society Student Texts), by Gregory L. Naber
PDF Ebook Spacetime and Singularities: An Introduction (London Mathematical Society Student Texts), by Gregory L. Naber
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Naber provides an elementary introduction to the geometrical methods and notions used in special and general relativity. Particular emphasis is placed on the ideas concerned with the structure of space-time and that play a role in the Penrose-Hawking singularity theorems. The author's primary purpose is to give a rigorous proof of the simplest of these theorems, by the one that is representative of the whole. He provides exercises and examples at the end of each chapter. No previous exposure either to relativity theory of differential geometry is required of the reader, as necessary concepts are developed when needed, though some restrictions ae imposed on the types of space considered.
- Sales Rank: #3165445 in Books
- Brand: Brand: Cambridge University Press
- Published on: 1989-03-31
- Original language: English
- Number of items: 1
- Dimensions: 8.98" h x .43" w x 5.98" l, .53 pounds
- Binding: Paperback
- 192 pages
- Used Book in Good Condition
Review
"...fills a need to introduce the singularity theorems, their concepts and techniques to seniors or beginning graduate students." Physics in Canada
Most helpful customer reviews
0 of 0 people found the following review helpful.
Pedagogic Prelude To Singularity Theorems
By G. A. Schoenagel
Not an easy text to review. Some have noted its brevity as to the downside:
"...sacrifices in the cause of brevity..." review, Journal of General Relativity and Gravitation, Volume 22, Number 8,1990,
Yet, another, in praise of its virtues:
"...quite successful of an attempt," review in Astrophysics and Space Science,Volume 172,Number 1,1990.
My assessment falls between those two extremes. After all of roughly 170 pages of text, ultimate success of this monograph
falls upon the shoulders of the author---who writes in the Preface: " ...we wish to provide brief, reasonably self-contained and
elementary introduction to the ideas required for a rigorous understanding of the simplest of these Hawking Singularity Theorems."
(That is, Theorems as presented in Hawking's 1966 paper, The Occurrence of Singularities in Cosmology ).
Therefore, in order to ascertain how well this book succeeds, one turns to Hawking's paper after studying this book.
And, upon perusal of said paper, I must conclude that Naber has definitely succeeded in his stated goals.
In fact, ignoring perusal of Hawking's 1966 paper would make review of Naber's book much more difficult.
Study Naber, then study Hawking, after which a student is prepared for their own assessment of the book's efficacy.
Without proper mathematical compass this text will be unassailable to even the ablest student. However, more than
mere computational facility is required; a dose of mathematical maturity will serve one well. Multivariable Calculus,
at more than a computational plateau, should prove adequate. (Elementary point-set topology,here, ends chapter one).
Also, a nodding acquaintance with Taylor and Wheeler's Spacetime Physics can provide physical understanding.
Now, as to the contents of this text (slim text such as it is! ):
Exercises (interwoven, easier) and Problems (end of chapter, harder) are essential to the entire textual material.
Not to be eschewed. Especially, the First Chapter: Each exercise serves a specific purpose for material presented later in the book.
( Note: typo--page 74-- for the solution to Problem 2-H, part six, Chapter Two: the fourth-component should be squared;
not the second-component, as herein written.).
Page 130: "...it is often very difficult to decide whether a given Singularity is 'real' or only due to an unfortunate choice of
coordinates." There follows an exercise (3.8.1) in just such an occurrence. Then, a nice remark concerning geodesic
incompleteness. The first three chapters are very much in the manner of preparation for the final chapter-- the pinnacle.
Chapters One,Two and Three provide survey of, and discussion pertaining to : Minkowski, Special Relativity, Gravity.
Along this route is encountered (first): four-geometry, geodesics, parametrization, causality relations, and more.
Second: Energy-Momentum, Electromagnetic field, and concepts from relativistic mechanics.
Third: Gravity ("...the operative word is local..."); Page 87 gives brief (!) discussion of deSitter spacetime, immediately
followed by an exercise explicating such. After this, Einstein-deSitter Spacetime ( "....simplest cosmological model ").
This third chapter is rather remarkable in its coverage--or, its brevity--depending upon one's outlook. As noted, page 95:
"...henceforth we leave it as a standing exercise to verify that definitions which appear to depend on the choice of coordinate
system, in fact, do not." Topics such as exponential map and star-shaped region are a bit rushed. On to page 108, where we
"....record our first indication of the influence of topology on physics...". Stable Causality and Global Hyperbolicity are prelude
to Einstein's Field Equations, and here, we read, page 124: "....in order to gain some sense of the enormous complexity of
these equations, we recommend writing out the left-hand side of the Einstein Equations entirely in terms of the metric components
and their derivatives." This is excellent advice ! Also, the Problem Set pertaining to Conformal Equivalence is apt (Page 139).
Finally: the fourth chapter, Hawking's Theorem, an exposition in twenty-five pages. Not easy-going, the hardest chapter by far.
We are in a position to state Hawking's Theorem: " Let M be a time oriented, stable causal, globally hyperbolic, spacetime which
satisfies two conditions ( Ricci and Cauchy conditions ), then, M is timelike geodesically incomplete." Hard work ensues.
A pedagogic Problem--of great utility--near end of the book: It delineates the difference between Riemann and Lorentz geometries.
Quite an accomplishment, all told. As prelude to: Hawking (1966) and, then Hawking and Ellis (1973) this text is unique.
And, --- again---from the review of 1990, Journal of General Relativity and Gravitation :
"....short, elementary, and for the most part, highly readable."
My assessment: A highly recommended companion.
Requires effort, but ever so worth the effort expended !
16 of 16 people found the following review helpful.
A Stimulating and Interesting Book
By Andy Gregory
This book is concerned primarily with a geometrical and in places, a topological approach to spacetime, leading to a full proof of one of Hawking's singularity theorems.The first part introduces the geometry of Minkowski Spacetime as.. 'a 4-dimensional ral vector space on which is defined a nondegenerate symmetric bilinear form of index one'.Some mathematical maturity is required to attempt this book on one's own.Chapter two develops relativistic mechanics in quite an abstract way (certainly for a first encounter) and chapter three develops spacetimes from the point of view of maps between manifolds.This chapter ends with a statement of one of Hawking's theorems. Chapter four sets out a full rigorous proof. There are no hints/partial solutions for the exercises although there are some 'examples'. The first three chapters were enjoyable and I managed to do quite a lot of the exercises and problems.As someone who works entirely independently at this kind of thing for 'fun',I found chapter four very hard going.Having no-one to ask when stuck made it a bit frustrating.The book was very stimulating though and encouraged me to research other sources for similar material to fill in gaps in my mathematical knowledge.
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