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From Pythagoras To Einstein: The Hyperbolic Pythagorean Theorem

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Abstract

A new form of the Hyperbolic Pythagorean Theorem, which has a striking intuitive appeal and offers a strong contrast to its standard form, is presented. It expresses the square of the hyperbolic length of the hypotenuse of a hyperbolic right-angled triangle as the “Einstein sum” of the squares of the hyperbolic lengths of the other two sides, Fig. 1, thus completing the long path from Pythagoras to Einstein. Following the pioneering work of Varičak it is well known that relativistic velocities are governed by hyperbolic geometry in the same way that prerelativistic velocities are governed by Euclidean geometry. Unlike prerelativistic velocity composition, given by the ordinary vector addition, the composition of relativistic velocities, given by the Einstein addition, is neither commutative nor associative due to the presence of Thomas precession. Following the discovery of the mathematical regularity that Thomas precession stores, it is now possible to extend Thomas precession by abstraction, (i) allowing hyperbolic geometry to be studied by means of analogies that it shares with Euclidean geometry; and, similarly (ii) allowing velocities and accelerations in relativistic mechanics to be studied by means of analogies that they share with velocities and accelerations in classical mechanics. The abstract Thomas precession, called the Thomas gyration, gives rise to gyrovector space theory in which the prefix gyro is used extensively in terms like gyrogroups and gyrovector spaces, gyroassociative and gyrocommutative laws, gyroautomorphisms, gyrotranslations, etc. We demonstrate the superiority of our gyrovector space formalism in capturing analogies by deriving the Hyperbolic Pythagorean Theorem in a form fully analogous to its Euclidean counterpart, thus contrasting it with the standard form in which the Hyperbolic Pythagorean Theorem is known in the literature. The hyperbolic metric, which supports the Hyperbolic Pythagorean Theorem, has a dual metric. We show that the dual metric does not support a Pythagorean theorem but, instead, it supports the π-Theorem according to which the sum of the three dual angles of a hyperbolic triangle is π.

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REFERENCES

  1. John Stillwell, Mathematics and Its History (Springer, New York, 1989).

    Google Scholar 

  2. Elisha Scott Loomis, The Pythagorean Proposition (Reston, Va., National Council of Teachers of Mathematics, 1968).

    Google Scholar 

  3. Abraham A. Ungar, “Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics,” Found. Phys. 27, 881-951 (1997).

    Google Scholar 

  4. Edward C. Wallace and Stephen F. West, Roads to Geometry, 2nd edn. (Prentice Hall, New Jersey, 1998) pp. 362-363. Wallace and West present a numerical counterexample to demonstrate the invalidity of the hyperbolic Pythagorean theorem as they view it. Their numerical example, however, if correctly interpreted, should take the form ∥(-0.6, 0)∥2 ⊕ ∥(0, 0.5)∥2= ∥(-0.6, 0)⊖ (0, 0.5)∥2, resulting in a valid hyperbolic Pythagorean equality, in accordance with our Hyperbolic Pythagorean Theorem 4.3. Here ⊕ is the Möbius addition, Eq. (2.4) (or, equivalently, Eq. ( 2.3) in the complex representation of the unit disk), and ⊖ is the Möbius subtraction, Eq. (2.10). When applied to parallel vectors ( particularly, vectors of dimension 1) the Möbius addition is identical with the Einstein velocity addition. The lesson is clear, hyperbolic geometry should be revised: (I) The standard metric in a hyperbolic space should be the natural one, Eq. (5.9), allowing the presentation of the elegant Hyperbolic Pythagorean Theorem; and ( II) a dual metric, Eq. (5.9), should be introduced, turning hyperbolic geometries into bimetric spaces, allowing the presentation of the elegant πTheorem. The resulting bimetric hyperbolic geometry captures geometric properties that have seemingly been lost in the transition from Euclidean to hyperbolic geometry. Thus, for instance, the seemingly lost hyperbolic parallelism and the πTheorem in fact reappear in hyperbolic geometry, but under its dual metric.

    Google Scholar 

  5. I. M. Yaglom, Felix Klein and Sophus Lie, Evolution of the Idea of Symmetry in the Nineteenth Century (Birkhauser, Boston, 1988). The idea that groups need not be the only algebraic structure that measures symmetry is found in Alan Weinstein, “Groupoids: unifying internal and external symmetry,” Not. AMS 43, 744-751 (1996).

    Google Scholar 

  6. Abraham A. Ungar, Thomas rotation and the parametrization of the Lorentz transformation group, Found. Phys. Lett. 1, 57-89 (1988).

    Google Scholar 

  7. Abraham A. Ungar, “Thomas precession and its associated grouplike structure,” Am. J. Phys. 59, 824-834 (1991).

    Google Scholar 

  8. For NASA GP-B space gyroscope experiment to test general relativity; see footnote 37 inRef. 3, and URL http://www.einstein.stanford.edu.

  9. Kurt Otto Friedrichs, From Pythagoras to Einstein (Random House, New York, 1966).

    Google Scholar 

  10. Cornelius Lanczos, Space through the Ages: The Evolution of Geometrical Ideas from Pythagoras to Hilbert and Einstein (Academic Press, New York, 1970).

    Google Scholar 

  11. Werner Krammer and Helmuth K. Urbantke, “K-loops, gyrogroups, and symmetric spaces,” preprint.

  12. Harold E. Wolf, Introduction to Non-Euclidean Geometry, p. 63 for the term “hyperbolic geometry” and p. 144 for the traditional hyperbolic Pythagorean theorem (The Dryden Press, New York, 1945). A nice derivation of the traditional hyperbolic Pythagorean theorem is found in David C. Kay, College Geometry( Holt, Reinhart & Winston, New York, 1969), p. 317.

    Google Scholar 

  13. A. Einstein, “Zur Elektrodynamik Bewegter Körper (On the Electrodynamics of Moving Bodies),” Ann. Phys. (Leipzig ) 17, 891-921 (1905). For English translation see H. M. Schwartz, Am. J. Phys. 45, 18 (1977); and H. A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl, The Principle of Relativity( trans. W. Perrett and G. B. Jeffrey, Dover, New York, 1952), pp. 37-65.

    Google Scholar 

  14. Vladimir Varičak, “Anwendung der Lobatschefkijschen Geometrie in der Relativtheorie,” Phys. Z. 11, 93-96 and 287-293 (1910); and Vladimir Varičak, “Über die nichteuklidische Interpretation der Relativitatstheorie,” Jahresbericht der Deutschen Math. Verinigung 21, 103-127 (1912). The connections between special relativity and hyperbolic geometry were discussed by Pauli and by Fock. (i) W. Pauli, Theory of Relativity(Pergamon, New York, 1958), based on Pauli's 1921 Mathematical Encyclopediaarticle. In a famous footnote on p. 74, having just referred to Varičak in the main text, Pauli points out that Varičak had not observed that Klein's projective point of view very much clarifies matters; (ii) V. A. Fock, Theory of Space, Time, and Gravitation(Pergamon, New York, 1959). Recent studies of the hyperbolic structure of relativistic velocity spaces are available in the literature; see, for instance, D. K. Sen, “3-Dimensional hyperbolic geometry and relativity,” in A. Coley, C. Dyer, and T. Tupper ( eds.), Proceedings of the2nd Canadian Conference on General Relativity and Relativistic Astrophysics(World Scientific, 1988), p. 264-266; and Lars-Erik Lundberg, “Quantum theory, hyperbolic geometry, and relativity,” Rev. Math. Phys. 6, 39-49 (1994).

    Google Scholar 

  15. Abraham A. Ungar, “The abstract Lorentz transformation group,” Am. J. Phys. 60, 815-828 (1992).

    Google Scholar 

  16. Abraham A. Ungar, “Axiomatic approach to the nonassociative group of relativistic velocities,” Found. Phys. Lett. 2, 199-203 (1989). See also Abraham A. Ungar, “The relativistic noncommutative nonassociative group of velocities and the Thomas rotation,” Results Math. 16, 168-179 (1989). In this paper the author introduced the term “K-loop” into the literature to honor Karzel's work on near domains, where several gyrogroup identities are found in the study of their additive substructure. Later, the author realized in Ref. 28 that several gyrogroup identities are also found in Kikkawa's work (including Kikkawa's Ref. 13 in Ref. 28). The author, therefore, dedicates the term “K-loop” to both Karzel and Kikkawa. In the present author's “gyroterminology” K-loops are called gyrocommutative gyrogroups, indicating that nongyrocommutative gyrogroups do exist as well.(25) The term K-loop is now established in the study of incidence geometry, while coincidentally, the same K-loop structure is known in algebra as a Bruck loop. We however employ our gyroterminology, rather than using the old term Bruck loop or its equivalent newer term, K-loop, in order to (i) emphasize the capability of gyrogroups and gyrovector spaces to capture analogies that are shared with groups and vector spaces; and to (ii) record the historical fact that the gyrogroup structure first emerged in the special theory of relativity following the exposition of the mathematical regularity that the Thomas gyration stores. The Thomas gyration is obtained from the Thomas precession of the special theory of relativity by abstraction, giving rise to the prefix gyrothat we extensively use in describing the resulting novel analogies.

    Google Scholar 

  17. Abraham A. Ungar, “Weakly associative groups,” Results Math. 17, 149-168 (1990).

    Google Scholar 

  18. Abraham A. Ungar, “Quasidirect product groups and the Lorentz transformation group,” in T. M. Rassias (ed.), Constantin Caratheodory: An International Tribute, Vol. II (World Scientific, New Jersey, 1991), pp. 1378-1392.

    Google Scholar 

  19. A. A. Ungar (1994) ArticleTitleThe abstract complex Lorentz transformation group with real metric I: Special relativity formalism to deal with the holomorphic automorphism group of the unit ball in any complex Hilbert space J. Math. Phys. 35 1408–1425

    Google Scholar 

  20. The existence of “rigid motions” is an assumption about the curvature of space; seeRichard L. Faber, Foundations of Euclidean and Non-Euclidean Geometry ( Marcel Dekker,New York, 1983), p. 109.

  21. Helmuth K. Urbantke, “Physical holonomy, Thomas precession, and Clifford algebra;” Sec. Ill, Am. J. Phys. 58, 747-750 (1990).

    Google Scholar 

  22. Marvin Jay Greenberg, Euclidean and Non-Euclidean Geometries: Development and History, 2nd ed. (Freeman, San Francisco, 1980), p. 326.

    Google Scholar 

  23. Henri Poincaré, Science and Hypothesis (Dover, New York, 1952), p. 50.

    Google Scholar 

  24. Marvin J. Greenberg, Euclidean and Non-Euclidean Geometries: Development and History, 2nd ed. (Freeman, San Francisco, 1980); and Arlan Ramsay and Robert D. Richtmyer, Introduction to Hyperbolic Geometry(Springer, New York, 1995).

    Google Scholar 

  25. Tuval Foguel and Abraham A. Ungar, “Transversals, gyrotransversals, and gyrogroups,” preprint.

  26. A. M. Polyakov, Gauge Fields and Strings (Harwood Academic, New York, 1987), p. 183.

    Google Scholar 

  27. Michael Henle, Modern Geometries the Analytic Approach (Prentice Hall, New Jersey, 1997), p. 77.

    Google Scholar 

  28. Abraham A. Ungar, “The holomorphic automorphism group of the complex disk,” Aequat. Math. 47, 240-254 (1994).

    Google Scholar 

  29. Lars V. Ahlfors, “An extension of Schwartz's lemma,” Trans. Am. Math. Soc. 43, 359-364 (1938).

    Google Scholar 

  30. Steven G. Krantz, Complex Analysis: The Geometric Viewpoint (Carus Mathematical Monographs, 23, Math. Assoc. of Amer., Washington, DC, 1990), pp. 52 and 58.

    Google Scholar 

  31. Abraham A. Ungar, “Extension of the unit disk gyrogroup into the unit ball of any real inner product space,” J. Math. Anal. Appl. 202, 1040-1057 (1996).

    Google Scholar 

  32. Abraham A. Ungar, “Midpoints in gyrogroups,” Found. Phys. 26, 1277-1328 (1996).

    Google Scholar 

  33. See, for instance, Ian N. Sneddon, ed., Encyclopedic Dictionary of Mathematics for Engineers and Applied Scientists, (Pergamon, New York, 1976), p. 320.

    Google Scholar 

  34. See Sec. 7 in Abraham A. Ungar, “The holomorphic automorphism group of the complex disk,” Aequat Math. 47, 240-254 (1994); and, for instance, H. S. Bear, “Distance-decreasing functions on the hyperbolic plane,” Mich. Math. J. 39, 271-279 (1992). Detailed explanation why the distance function 2 tanh-1 ∥x⊝ y∥ is preferred in the literature over our Poincarédistance functions ∥x⊝y∥, as well as a presentation of the traditional hyperbolic Pythagorean theorem, may be found in Hans Schwerdtfeger, Geometry of Complex Numbers, Circle Geometry, Möbius Transformations, Non-Euclidean Geometry(Dover, New York, 1979), pp. 141, 146-147, 149-151. See also Michael Henle, Modern Geometries: The Analytic Approach(Prentice Hall, New Jersey, 1997), p. 93; and Christian Grosche, “The path integral on the Poincarédisk, the Poincaréupper half plane, and on the hyperbolic strip,” Fortschr. Phys. 38, 531-569 (1990).

    Google Scholar 

  35. Lars V. Ahlfors, Conformal Invariants Topics in Geometric Function Theory (McGraw-Hill, New York, 1973), p. 12.

    Google Scholar 

  36. It is customary in the literature to select s= 1 without loss of generality. However, there are circumstances when the retention of sas a free positive parameter is clearly useful; see, e.g., Olli Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, translated from the German by K. W. Lucas, 2nd edn. (Springer, Berlin, New York, 1973, p. 66.

  37. See Eq. (3.3) in Lars V. Ahlfors, “Old and new in Möbius groups,” Ann. Acad. Sci. Fenn. Ser. A I Math. 9, 93-105 (1984); and p. 25 in Lars V. Ahlfors, Möbius Transformations in Several Dimensions, Lecture Notes (University of Minnesota, Minneapolis, 1981).

    Google Scholar 

  38. Olli Lehto, Univalent Functions and Teichmuller Spaces (Springer, Berlin, New York, 1987), pp. 105-106.

    Google Scholar 

  39. Rafael Artzy, Geometry: An Algebraic Approach (Wissenschaftsverlag, Mannheim, 1992).

    Google Scholar 

  40. Michael K. Kinyon, private communication.

  41. Edward Andalafte and Raymond Freese, “Weak homogeneity of metric Pythagorean orthogonality,” J. Geom. 56, 3-8 (1996), and references therein. In particular, see R. C. James, “Orthogonality in normed linear spaces,” Duke Math. J. 12, 291-302 (1945). In that paper the Pythagorean orthogonality of vectors xand yin a normed linear space is defined by the equation ∥x+ y∥−2= ∥x∥ 1+ ∥y∥ 2. This relation is said to be homogeneous if Pythagorean orthogonality between xand yimplies Pythagorean orthogonality between xand r yfor all r∈ {ie1321-1}. The homogeneity of Pythagorean orthogonality is important since it characterizes inner product spaces. In full analogy, Eq. (4.2) indicates that hyperbolic-Pythagorean orthogonality in gyrovector spaces is homogeneous, that is, if aand bare hyperbolic-Pythagorean orthogonal in a gyrovector space, then so are aand r⊙bfor all r∈ {ie1321-2}.

    Google Scholar 

  42. Michael K. Kinyon and Abraham A. Ungar, “The complex unit disk,” preprint.

  43. See Exercise 10 in Saul Stahl, The PoincaréHalf-Plane: A Gateway to Modern Geometry (Jones and Bartlett, Boston, 1993), p. 205.

    Google Scholar 

  44. The curvature Kof an arbitrary Riemann metric Edx 2+ 2Fdx dy+ Gdy 2 is defined on p. 201 in Saul Stahl, The Poincaré Half-Plane: A Gateway to Modern Geometry (Jones and Bartlett, Boston, 1993).

    Google Scholar 

  45. For an English translation of Riemann's inaugural address, see D. E. Smith, A SourceBook in Mathematics ( McGraw Hill, New York, 1929) , pp. 272–286.

  46. Sigurdur Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Academic Press, New York, 1978).

    Google Scholar 

  47. Yaakov Friedman and Abraham A. Ungar, “Gyrosemidirect product structure of bounded symmetric domains,” Res. Math. 26, 28-38 (1994).

    Google Scholar 

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Ungar, A.A. From Pythagoras To Einstein: The Hyperbolic Pythagorean Theorem. Foundations of Physics 28, 1283–1321 (1998). https://doi.org/10.1023/A:1018874826277

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