Abstract
Applying calculus to the horizontal and vertical unit hyperbolas we derive the obtuse exterior and the acute angle of parallelism formulas for the arc length of a unit circle. When the analogous arc length of a hyperbola is figured, we further define metric distance with radian measure. Employing a non-conventional velocity of light formula, we explain several aspects of physics with a logical geometry. The hyperbolic geometry Lobacevskii angle of parallelism is mathematically measured to agree with the physical measure, without using the gravitational constant, but rather the electromagnetic spectrum and geodesics. Space is hereby measured with units of radians and time. Half-angle formulas for slopes and boosts pertain to spinor algebra. The new light velocity is justified when its derivative results in Einstein's mass–energy formula. Relativity is reformulated with these half-angle formulas.
Similar content being viewed by others
References
Asimov, I. (1965) Of Time, Space, and other things. New York: Discus of Avon.
Beltrami, E. (1868) Saggio di interpretazione della geometria non-Euclidean, Giornale di Mathematiche 6, 298(12).
Beyer, W.H. (1987) CRC Handbook of Mathematical Sciences. Boca Raton: CRC Press.
Bolyai, J. (1987) Appendix, the Theory of Space, North-Holland Math. Studies, vol. 138, Amsterdam.
Busemann, H. (1950) Non-Euclidean geometry, Math. Mag. 24(9), 19–34.
Chrystal, G. (1931) Algebra. II, Black, London.
Coxeter, H.S.M. (1961) Introduction to Geometry, 2nd ed. New York: John Wiley.
Coxeter, H.S.M. (1996) Non-Euclidean Geometry, 6th ed. Toronto: Univ. Toronto Press.
Coxeter, H.S.M. (1978) Parallel lines, Canad. Math. Bull. 21(4), 385–397.
Einstein, A. (1957) Relativity, the Special and the General Theory, 15th ed. London: Methuen.
Eskew, R.C. (1989) Relating a circle and a hyperbola, J. Undergrad. Math. 21(2), Guilford College, Greensboro, 49–54.
Gauss, C.F. (1899) Briefwechsel Zwischen Carl Friedrich Gauss und Wolfgang Bolyai, Leipzig.
Halliday, D., Resnick, R. and Walker, J. (1997) Fundamentals of Physics. New York: John Wiley.
Hawking, S.W. (1988) A brief history of time. Toronto: Bantam Books.
Iaglom, I.M. (1969) A Simple non-Euclidean Geometry and its Physical basis, Nauka, Moscow (Russian); English transl., 1979, New York: Springer-Verlag.
Kagan, V. (1957) N. Lobachevsky and His Contribution to Science. Moscow: Foreign Languages Publishing House.
Lobachevskii, N.I. (1840) Geometrical researches on the theory of parallels. Karzan, Berlin (Russian); English transl., 1891, Austin: Univ. Texas Press.
Lorentz, H.A. (1923) The Principle of Relativity. New York: Dover.
Martin, G.E. (1972) The Foundations of Geometry and the non-Euclidean Plane. New York: Intext Educational.
Millman, R.S. and Parker, G.D. (1981) Geometry, a Metric Approach with Models. New York: Springer-Verlag.
Misner, C.W., Thorne, K.S. and Wheeler, J.A. (1970) Gravitation. San Francisco: W.H. Freeman and Co.
Morgan, F. (1993) Riemannian Geometry, a Beginner's Guide. Boston: Jones and Bartlett.
Poincaré, H. (1921) Des fondements de la geometrie. Paris.
Saccheri, G. (1733) Euclides ab omni naevo vindicatus.
Schutz, B. (1985) A First Course in General Relativity. Cambridge: Cambridge University Press.
Taylor, E.F. and Wheeler, J.A. (1963) Spacetime Physics. San Francisco: W.H. Freeman.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Eskew, R.C. Time, gravity and the exterior angle of parallelism. Speculations in Science and Technology 21, 213–225 (1998). https://doi.org/10.1023/A:1005401228301
Issue Date:
DOI: https://doi.org/10.1023/A:1005401228301