The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity.
2nd ed.

*(English)*Zbl 1241.83005
Applied Mathematical Sciences 92. Berlin: Springer (ISBN 978-1-4419-7837-0/hbk; 978-1-4419-7838-7/ebook). xvi, 324 p. (2012).

This is the second edition of the masterpiece by G. L. Naber [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: “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.” Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book “…a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics” and from the Dutch Mathematical Society (1993) which praised the author by writing “…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”.

According to the author’s intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between A. Einstein and H. Minkowski [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.

In the Introduction the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by H. A. Lorentz, A. Einstein, H. Minkowski and H. Weyl [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the “event world” through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by E. C. Zeeman [J. Math. Phys. 5, 490–493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.

The basic geometrical information about this model is supplied in Chapter 1 with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by A. A. Robb [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. Chapter 1 also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman’s theorem on the causal automorphism based on some results by G. Salmon [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by N. H. Kuiper [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of Chapter 1 shows that the fractional linear transformation of the “celestial sphere” has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by L. V. Ahlfors [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose’s theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. Chapter 1 ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by Herbert E. Ives and G. R. Stilwell in 1938.

Chapter 2 describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by I. N. Herstein [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from S. Lang [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. Chapter 2 culminates with variable fields and introduces the Maxwell’s (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading S. Parrott [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].

Chapter 3 is a detailed exposition of the algebraic Theory of Spinors devised by E. Cartan [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by O. Laporte and G. E. Uhlenbeck [Phys. Rev., II. Ser. 37, 1380–1397 (1931; Zbl 0002.09001)], developed by O. Veblen [Science, New York 80, 415–419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111–127 (1937; Zbl 0018.32604)] and by W. T. Payne [Am. J. Phys. 20, 253–262 (1952; Zbl 0046.43705)] and by W. L. Bade and H. Jehle [Rev. Mod. Phys. 25, 714–728 (1953; Zbl 0051.20705)] and further refined by E. D. Bolker [Am. Math. Mon. 80, 977–984 (1973; Zbl 0287.55001)]. Some results from I. M. Gel’fand, R. A. Minlos and Z. Ya. Shapiro [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). Chapter 3 offers also a “Petrov-type” classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.

Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by W. de Sitter [Bull. Astron. Inst. Netherlands 7, 97–105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205–216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597–601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new Chapter 4 explores some features of the “de Sitter Universe”, a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like S. Akcay and R. A. Matzner [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], Ion I. Cot{\b a}escu and C. Crucean [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], M. Faizal [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], D. Bini, G. Esposito and A. Geralico [Gen. Relativ. Gravitation 44, No. 2, 467–490 (2012; Zbl 1235.83039)], D.-Y. Jia, R.-H. Yue and S.-M. Huang [Commun. Theor. Phys. 55, No. 1, 75–79 (2011; Zbl 1223.83030)].

Appendix A investigates the “path topology” for \(M\) by S. W. Hawking, A. R. King and P. J. McCarthy [J. Math. Phys. 17, 174–181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the “fine topology” by E. C. Zeeman [Topology 6, 161–170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by M. H. Freedman and F. Luo [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [G. L. Naber, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27–123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1–83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt S. Willard [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.

In Appendix B the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac’s famous “Scissors Problem” and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is M. J. Greenberg [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from R. P. Feynman, R. B. Leighton and M. Sands [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from A. M. R. Magnon [J. Math. Phys. 28, 1364–1369 (1987; Zbl 0638.53067)] and further from M. Spivak [A comprehensive introduction to differential geometry. Vol. 1–5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].

The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader.

According to the author’s intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between A. Einstein and H. Minkowski [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.

In the Introduction the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by H. A. Lorentz, A. Einstein, H. Minkowski and H. Weyl [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the “event world” through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by E. C. Zeeman [J. Math. Phys. 5, 490–493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.

The basic geometrical information about this model is supplied in Chapter 1 with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by A. A. Robb [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. Chapter 1 also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman’s theorem on the causal automorphism based on some results by G. Salmon [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by N. H. Kuiper [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of Chapter 1 shows that the fractional linear transformation of the “celestial sphere” has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by L. V. Ahlfors [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose’s theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. Chapter 1 ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by Herbert E. Ives and G. R. Stilwell in 1938.

Chapter 2 describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by I. N. Herstein [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from S. Lang [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. Chapter 2 culminates with variable fields and introduces the Maxwell’s (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading S. Parrott [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].

Chapter 3 is a detailed exposition of the algebraic Theory of Spinors devised by E. Cartan [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by O. Laporte and G. E. Uhlenbeck [Phys. Rev., II. Ser. 37, 1380–1397 (1931; Zbl 0002.09001)], developed by O. Veblen [Science, New York 80, 415–419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111–127 (1937; Zbl 0018.32604)] and by W. T. Payne [Am. J. Phys. 20, 253–262 (1952; Zbl 0046.43705)] and by W. L. Bade and H. Jehle [Rev. Mod. Phys. 25, 714–728 (1953; Zbl 0051.20705)] and further refined by E. D. Bolker [Am. Math. Mon. 80, 977–984 (1973; Zbl 0287.55001)]. Some results from I. M. Gel’fand, R. A. Minlos and Z. Ya. Shapiro [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). Chapter 3 offers also a “Petrov-type” classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.

Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by W. de Sitter [Bull. Astron. Inst. Netherlands 7, 97–105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205–216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597–601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new Chapter 4 explores some features of the “de Sitter Universe”, a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like S. Akcay and R. A. Matzner [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], Ion I. Cot{\b a}escu and C. Crucean [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], M. Faizal [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], D. Bini, G. Esposito and A. Geralico [Gen. Relativ. Gravitation 44, No. 2, 467–490 (2012; Zbl 1235.83039)], D.-Y. Jia, R.-H. Yue and S.-M. Huang [Commun. Theor. Phys. 55, No. 1, 75–79 (2011; Zbl 1223.83030)].

Appendix A investigates the “path topology” for \(M\) by S. W. Hawking, A. R. King and P. J. McCarthy [J. Math. Phys. 17, 174–181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the “fine topology” by E. C. Zeeman [Topology 6, 161–170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by M. H. Freedman and F. Luo [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [G. L. Naber, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27–123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1–83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt S. Willard [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.

In Appendix B the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac’s famous “Scissors Problem” and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is M. J. Greenberg [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from R. P. Feynman, R. B. Leighton and M. Sands [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from A. M. R. Magnon [J. Math. Phys. 28, 1364–1369 (1987; Zbl 0638.53067)] and further from M. Spivak [A comprehensive introduction to differential geometry. Vol. 1–5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].

The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader.

Reviewer: Enzo Bonacci (Latina)

##### MSC:

83-02 | Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory |

83A05 | Special relativity |

83C60 | Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism |

53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |

53Z05 | Applications of differential geometry to physics |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

83C10 | Equations of motion in general relativity and gravitational theory |

22E10 | General properties and structure of complex Lie groups |

83F05 | Relativistic cosmology |

78A25 | Electromagnetic theory (general) |