The Geometry of Minkowski Spacetime: An Introduction to the Mathematics of the Special Theory of Relativity (Applied Mathematical Sciences) [Hardcover]

Gregory L. Naber (Author)

Book Description

ISBN-10: 0387978488 | ISBN-13: 978-0387978482 | Publication Date: July 23, 1992 | Edition: 1st Edition

This book offers a presentation of the special theory of relativity that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. It treats, in addition to the usual menu of topics one is accustomed to finding in introductions to special relativity, a wide variety of results of more contemporary origin. These include Zeeman’s characterization of the causal automorphisms of Minkowski spacetime, the Penrose theorem on the apparent shape of a relativistically moving sphere, a detailed introduction to the theory of spinors, a Petrov-type classification of electromagnetic fields in both tensor and spinor form, a topology for Minkowski spacetime whose homeomorphism group is essentially the Lorentz group, and a careful discussion of Dirac’s famous Scissors Problem and its relation to the notion of a two-valued representation of the Lorentz group. This second edition includes a new chapter on the de Sitter universe which is intended to serve two purposes. The first is to provide a gentle prologue to the steps one must take to move beyond special relativity and adapt to the presence of gravitational fields that cannot be considered negligible. The second is to understand some of the basic features of a model of the empty universe that differs markedly from Minkowski spacetime, but may be recommended by recent astronomical observations suggesting that the expansion of our own universe is accelerating rather than slowing down. The treatment presumes only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and, in two appendices, some elementary point-set topology. The first edition of the book received the 1993 CHOICE award for Outstanding Academic Title. Reviews of first edition: “… a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics.” (American Mathematical Society, 1993) “Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.” (CHOICE, 1993) “… his talent in choosing the most significant results and ordering them within the book can’t be denied. The reading of the book is, really, a pleasure.” (Dutch Mathematical Society, 1993)

Editorial Reviews

From the Back Cover

This book offers a presentation of the special theory of relativity that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. It treats, in addition to the usual menu of topics one is accustomed to finding in introductions to special relativity, a wide variety of results of more contemporary origin. These include Zeeman’s characterization of the causal automorphisms of Minkowski spacetime, the Penrose theorem on the apparent shape of a relativistically moving sphere, a detailed introduction to the theory of spinors, a Petrov-type classification of electromagnetic fields in both tensor and spinor form, a topology for Minkowski spacetime whose homeomorphism group is essentially the Lorentz group, and a careful discussion of Dirac’s famous Scissors Problem and its relation to the notion of a two-valued representation of the Lorentz group. This second edition includes a new chapter on the de Sitter universe which is intended to serve two purposes. The first is to provide a gentle prologue to the steps one must take to move beyond special relativity and adapt to the presence of gravitational fields that cannot be considered negligible. The second is to understand some of the basic features of a model of the empty universe that differs markedly from Minkowski spacetime, but may be recommended by recent astronomical observations suggesting that the expansion of our own universe is accelerating rather than slowing down. The treatment presumes only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and, in two appendices, some elementary point-set topology. The first edition of the book received the 1993 CHOICE award for Outstanding Academic Title. Reviews of first edition: “… a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics.” (American Mathematical Society, 1993) “Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.” (CHOICE, 1993) “… his talent in choosing the most significant results and ordering them within the book can’t be denied. The reading of the book is, really, a pleasure.” (Dutch Mathematical Society, 1993)

About the Author

Gregory L. Naber is a Professor in the Department of Mathematics at Drexel University in Philadelphia, PA.

Product Details

• Hardcover: 350 pages
• Publisher: Springer; 1st Edition edition (July 23, 1992)
• Language: English
• ISBN-10: 0387978488
• ISBN-13: 978-0387978482
• Product Dimensions: 9.2 x 6.1 x 0.8 inches
• Shipping Weight: 1.2 pounds
• Average Customer Review: 4.3 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #2,413,819 in Books (See Top 100 in Books)

More About the Author

Discover books, learn about writers, read author blogs, and more.

Customer Reviews

Most Helpful Customer Reviews

17 of 17 people found the following review helpful:

5.0 out of 5 stars Special Relativity for the graduate student., January 18, 2007

By 

Rehan Dost (Canada) - See all my reviews
(REAL NAME)   

This review is from: The Geometry of Minkowski Spacetime: An Introduction to the Mathematics of the Special Theory of Relativity (Paperback)

This book is NOT for the pop science buff or the novice with little understanding of Special Relativity.

This book is designed for graduate level students in mathematics or physics who want a deeper understanding of Minkowski space. It presupposes a solid foundation in SR.

Having said this, the book is phenomenal. It brings out startling relationship between mathematics and physics explaining esoteric phenomena in SR.

For example:

1) The author shows how Lorentz transformations can be realized as fractional linear transformations of the Riemann sphere. By doing so we can use the full power of complex analysis to derive far reaching results. One property of such tranforms is that they map circles to circles thus explaining why an observer at rest who sees a circle ( say lit by lights ) will also see a circle, NOT ellipse from length contraction, when he moves relative to the circle.

2) Using a simple example ( scissors, chair and rubber band ) the author shows how a 360 degree rotation may not leave a system in the same state requiring the need of a new mathematical object ( spinor ) to describe this transformation.

3) The author clearly develops the mathematics of spinors. In fact this is the best introduction to spinors I have read. He develops the notion of spin vectors and realizes spinors as multi-linear functionals with inputs as spin vectors, their duals, their conjugate, and the conjugate duals. He then lays out the transformation properties of the spinors and shows that certain spinors have exactly the transformation properties needed to model particles with spin.

4) There is a great section on the relationship of SL (2,C) to the lorentz group. The author shows how Minkowski space can be represented by certain combinations of 2x2 complex matrices and shows how SL ( 2,C) can then operate on these. This operation is actually equivalent to a lorentz transformation thus giving a mapping between the two groups. He then shows that we can easily analyze SL (2,C) by breaking it down into irreducible representations ( which are known ) and that to each of these representations there exist a unique representation of the Lorentz group ( provided certain conditions are imposed ). If that condition is not met the representation leads to the all familiar 2-valued representation of the Lorentz group one hears so much about. Thus by studying SL ( 2,C ) which we know alot about we can represent the Lorentz group which is generally harder to study but of the most relevance in physics.

The books is filled with such insights and I would recommend it to anyone who wishes to understand particle physics or relativity.

13 of 13 people found the following review helpful:

4.0 out of 5 stars Fascinating but not for the general reader, June 28, 2006

By 

Serious Inquirer (Jacksonville, AL USA) - See all my reviews

Amazon Verified Purchase

This review is from: The Geometry of Minkowski Spacetime: An Introduction to the Mathematics of the Special Theory of Relativity (Paperback)

Starting with a quick overview of certain structures from linear algebra (bilinear forms) the book moves to discussing Minkowski spacetime. Unfortunately for many, the text is highly esoteric without even a single descriptive section that doesn't make use of some fairly advanced mathematics.

The level of mathematical maturity required is comparable to a fourth year mathematics major at any decent university. The relationship between the mathematics involved and the special theory of relativity is fully explained.

A solid introduction to special relativity for the earnest mathematician.

8 of 10 people found the following review helpful:

4.0 out of 5 stars Very good on electromagtisim in Relativity Theory, December 29, 2007

By 

R. Clow - See all my reviews
(REAL NAME)   

Amazon Verified Purchase

This review is from: The Geometry of Minkowski Spacetime: An Introduction to the Mathematics of the Special Theory of Relativity (Paperback)

Better as a textbook than an after dinner reading book,
because of the numerous exercises which contain many
of the key points. More worked out examples would
make it more informative to read (without pencil and
paper at hand) as on buses or airplanes.