© 2013

Einstein in Matrix Form

Exact Derivation of the Theory of Special and General Relativity without Tensors

  • Derives the fundamental equations of Einstein's theory of special and general relativity using matrix calculus, without the help of tensors

  • Provides necessary mathematical tools in a user-friendly way, either directly in the text or in the appendices

  • Appendices contain an introduction to classical dynamics as a refresher of known fundamental physics

  • Rehearses vector and matrix calculus, differential geometry, and some special solutions of general relativity in the appendices


Part of the Graduate Texts in Physics book series (GTP)

Table of contents

  1. Front Matter
    Pages I-XIV
  2. Günter Ludyk
    Pages 1-50
  3. Günter Ludyk
    Pages 51-107
  4. Günter Ludyk
    Pages 109-143
  5. Back Matter
    Pages 145-194

About this book


This book is an introduction to the theories of Special and General Relativity. The target audience are physicists, engineers and applied scientists who are looking for an understandable introduction to the topic - without too much new mathematics. The fundamental equations of Einsteins theory of Special and General Relativity are derived using matrix calculus, without the help of tensors. This feature makes the book special and a valuable tool for scientists and engineers with no experience in the field of tensor calculus. In part I the foundations of Special Relativity are developed, part II describes the structure and principle of General Relativity. Part III explains the Schwarzschild solution of spherical body gravity and examines the "Black Hole" phenomenon. Any necessary mathematical tools are user friendly provided, either directly in the text or in the appendices.


Black Holes Galilei Transformation Lorentz Transformation Matrix Calculus Motion in a Gravity Field Relativistic Electrodynamics Riemannian Geometry Schwarzschild Solution Theory of General Relativity Theory of Special Relativity

Authors and affiliations

  1. 1.Physics and Electrical EngineeringUniversity of BremenBremenGermany

About the authors

After receiving his PhD in 1967, Günter Ludyk habilitated and has been appointed „Scientific Advisor and Professor“ (associate professor) of the Technical University of Berlin in 1970. In  1971 he has been a visiting professor at the Technical University of Graz/Austrial. Since 1972 he is a Full Professor at the Physics/Electrical Engineering Faculty of the University of Bremen. His area of research includes the theory of dynamical systems and the application of interval mathematics to generate high-precision results. He published various books on these topics both in German and English, e. g.  "Time-Variant Discrete-Time-Systems” in 1981 and "Stability of Time-Variant Discrete-Time Systems” in 1985.

Bibliographic information


From the book reviews:

“Addressing physicists, applied scientists and engineers with no previous knowledge of tensor calculus, Ludyk … presents in a well-written manner a rather easy introduction to special as well as general relativity by solely using matrix calculus and elementary differential geometry. … It is more than certain that Ludyk’s treatment will make it possible for a larger number of students to get acquainted with special and general relativity theory at an introductory, undergraduate level.” (Theophanes Grammenos, Mathematical Reviews, April, 2014)

“The main audience for this book can be … those who have experience with practical calculations by use of matrices and vectors, but who have not the time to become familiar with tensor calculus. … this book presents essentially all ingredients what one expects from a book aimed to introduce special and general relativity … .” (Hans-Jürgen Schmidt, zbMATH, Vol. 1272, 2013)