Communications in Mathematical Physics

, Volume 350, Issue 2, pp 749–801 | Cite as

Affine Sphere Relativity

  • E. MinguzziEmail author


We investigate spacetimes whose light cones could be anisotropic. We prove the equivalence of the structures: (a) Lorentz–Finsler manifold for which the mean Cartan torsion vanishes, (b) Lorentz–Finsler manifold for which the indicatrix (observer space) at each point is a convex hyperbolic affine sphere centered on the zero section, and (c) pair given by a spacetime volume and a sharp convex cone distribution. The equivalence suggests to describe (affine sphere) spacetimes with this structure, so that no algebraic-metrical concept enters the definition. As a result, this work shows how the metric features of spacetime emerge from elementary concepts such as measure and order. Non-relativistic spacetimes are obtained replacing proper spheres with improper spheres, so the distinction does not call for group theoretical elements. In physical terms, in affine sphere spacetimes the light cone distribution and the spacetime measure determine the motion of massive and massless particles (hence the dispersion relation). Furthermore, it is shown that, more generally, for Lorentz–Finsler theories non-differentiable at the cone, the lightlike geodesics and the transport of the particle momentum over them are well defined, though the curve parametrization could be undefined. Causality theory is also well behaved. Several results for affine sphere spacetimes are presented. Some results in Finsler geometry, for instance in the characterization of Randers spaces, are also included.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aazami A.B., Javaloyes M.A.: Penrose’s singularity theorem in a Finsler spacetime. Class. Quantum Grav. 33, 025003 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Álvarez Paiva, J.C., Thompson, A. C.: Volumes on normed and Finsler spaces. In: A sampler of Riemann–Finsler geometry, vol. 50, pp. 1–48. Cambridge Univ. Press, Cambridge. Math. Sci. Res. Inst. Publ. (2004)Google Scholar
  3. 3.
    Anderson J.L., Finkelstein D.: Cosmological constant and fundamental length. Am. J. Phys. 39, 901–904 (1971)ADSCrossRefGoogle Scholar
  4. 4.
    Asanov G.S.: Finsler geometry, relativity and gauge theories. D. Reidel Publishing Co, Dordrecht (1985)CrossRefzbMATHGoogle Scholar
  5. 5.
    Basilakos S., Kouretsis A.P., Saridakis E.N., Stavrinos P.: Resembling dark energy and modified gravity with Finsler-Randers cosmology. Phys. Rev. D. 88, 123510 (2013)ADSCrossRefGoogle Scholar
  6. 6.
    Beem J.K.: Indefinite Finsler spaces and timelike spaces. Can. J. Math. 22, 1035–1039 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Beem, J.K.: On the indicatrix and isotropy group in Finsler spaces with Lorentz signature. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 54(8), 385–392 (1974) (1973)Google Scholar
  8. 8.
    Benoist Y.: Convexes divisibles. C. R. Acad. Sci. Paris Sér. I Math. 332, 387–390 (2001)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Blaschke, W.: Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Band II., Affine Differentialgeometrie. J. Springer, Berlin (1923)Google Scholar
  10. 10.
    Bock R.D.: Local scale invariance and general relativity. Int. J. Theor. Phys. 42, 1835–1847 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bombelli L., Lee J.-H., Meyer D., Sorkin R.D.: Space-time as a causal set. Phys. Rev. Lett. 59, 521–524 (1987)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Brickell F.: A new proof of Deicke’s theorem on homogeneous functions. Proc. Am. Math. Soc. 16, 190–191 (1965)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Calabi E.: Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens. Michigan Math. J. 5, 105–126 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Calabi, E.: Complete affine hyperspheres. I. In: Symposia Mathematica, Vol. X (Convegno di Geometria Differenziale, INDAM, Rome, 1971), pp. 19–38. Academic Press, London (1972)Google Scholar
  15. 15.
    Cartan E.: Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie). Ann. Sci. École Norm. Sup. (3) 40, 325–412 (1923)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Castro C.: Gravity in curved phase-spaces, Finsler geometry and two-times physics. Int. J. Mod. Phys. A. 27, 1250069 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cheng S.-Y., Yau S.-T.: On the regularity of the Monge-Ampère equation \({{\rm det}(\partial ^{2}u/\partial x_{i} \partial x_{j})=F(x,u)}\). Comm. Pure Appl. Math. 30, 41–68 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cheng, S.Y., Yau, S.-T.: The real Monge-Ampère equation and affine flat structures. In: Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3 (Beijing, 1980), pp. 339–370. Science Press, Beijing (1982)Google Scholar
  19. 19.
    Cheng S.-Y., Yau S.-T.: Complete affine hypersurfaces. I. The completeness of affine metrics. Comm. Pure Appl. Math. 39, 839–866 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Deicke A.: Über die Finsler–Räume mit A i = 0. Arch. Math. 4, 45–51 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Dillen F., Vrancken L.: Calabi-type composition of affine spheres. Diff. Geom. Appl. 4, 303–328 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Dixon W.G.: On the uniqueness of the Newtonian theory as a geometric theory of gravitation. Commun. Math. Phys. 45, 167–182 (1975)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Duval C., Burdet G., Künzle H.P., Perrin M.: Bargmann structures and Newton–Cartan theory. Phys. Rev. D 31, 1841–1853 (1985)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Duval, C., Gibbons, G.W., Horvathy, P.A., Zhang, P.M.: Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time. Class. Quantum Grav. 31 (2014)Google Scholar
  25. 25.
    Fox D.J.F.: Functions dividing their Hessian determinants and affine spheres. Asian J. Math. 20(3), 503–530 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Fox D.J.F.: A Schwarz lemma for Kähler affine metrics and the canonical potential of a proper convex cone. Annali di Matematica 194, 1–42 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Geroch R.: A method for generating solutions of Einstein’s equations. J. Math. Phys. 12, 918–923 (1971)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ghomi, M.: The problem of optimal smoothing for convex functions. Proc. Am. Math. Soc. 130, 2255–2259 (2002) (electronic)Google Scholar
  29. 29.
    Gigena S.: Integral invariants of convex cones. J. Diff. Geom. 13, 191–222 (1981)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Gigena S.: On a conjecture by E. Calabi. Geom. Dedicata 11, 387–396 (1981)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Godbillon C.: Géométrie différentielle et mécanique analytique. Hermann, Paris (1969)zbMATHGoogle Scholar
  32. 32.
    Hartman P.: Ordinary differential equations. Wiley, New York (1964)zbMATHGoogle Scholar
  33. 33.
    Hildebrand R.: Analytic formulas for complete hyperbolic affine spheres. Contrib. Algebra Geometr. 55, 497–520 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Hildebrand R.: Canonical barriers on convex cones. Math. Oper. Res. 39, 841–850 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Hildebrand R.: Centro-affine hypersurface immersions with parallel cubic form. Contrib. Algebra Geometr. 56, 593–640 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Hu Z., Li H., Vrancken L.: Locally strongly convex affine hypersurfaces with parallel cubic form. J. Differ. Geom. 87, 239–308 (2011)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Horváth J.I.: A geometrical model for the unified theory of physical fields. Phys. Rev. 80, 901 (1950)ADSCrossRefzbMATHGoogle Scholar
  38. 38.
    Horváth J.I., Moór A.: Entwicklung einer einheitlichen feldtheorie begründet auf die finslersche geometrie. Z. Physik 131, 544–570 (1952)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Ikeda S.: On the theory of gravitational field in Finsler spaces. Lett. Nuovo Cimento 26, 277–281 (1979)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Ishikawa H.: Einstein equation in lifted Finsler spaces. Il Nuovo Cimento 56, 252–262 (1980)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Ishikawa H.: Note on Finslerian relativity. J. Math. Phys. 22, 995–1004 (1981)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Jian H., Wang X.-J.: Bernstein theorem and regularity for a class of Monge–Ampère equations. J. Differ. Geom. 93, 431–469 (2013)zbMATHGoogle Scholar
  43. 43.
    Jo K.: Quasi-homogeneous domains and convex affine manifolds. Topol. Appl. 134, 123–146 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Jörgens K.: Über die Lösungen der Differentialgleichung \({rt-s^2=1}\). Math. Ann. 127, 130–134 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Knebelman M.S.: Conformal geometry of generalized metric spaces. Proc. N. A. S. 15, 376–379 (1929)ADSCrossRefzbMATHGoogle Scholar
  46. 46.
    Kobayashi, S., Nomizu, K.: Foundations of differential geometry. vol. I of Interscience tracts in pure and applied mathematics. Interscience Publishers, New York (1963)Google Scholar
  47. 47.
    Künzle H.P.: Galilei and Lorentz structures on space-time: comparison of the correspondig geometry and physics. Ann. Inst. H. Poincaré Phys. Theor. 17, 337–362 (1972)Google Scholar
  48. 48.
    Künzle H.P.: Covariant Newtonian limit of Lorentz space-times. Gen. Rel. Grav. 7, 445–457 (1976)ADSMathSciNetzbMATHGoogle Scholar
  49. 49.
    Lämmerzahl C., Perlick V., Hasse W.: Observable effects in a class of spherically symmetric static Finsler spacetimes. Phys. Rev. D. 86, 104042 (2012)ADSCrossRefGoogle Scholar
  50. 50.
    Laugwitz, D.: Geometrical methods in the differential geometry of Finsler spaces. In: Geometria del calcolo delle variazioni, pp. 173–226. Springer, Heidelberg, Fondazione C.I.M.E., Florence, vol. 23 of C.I.M.E. Summer Sch. (2011) (Reprint of the 1961 original)Google Scholar
  51. 51.
    Li A.-M.: Calabi conjecture on hyperbolic affine hyperspheres. II. Math. Ann. 293, 485–493 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Li, A.M., Simon, U., Zhao, G.S.: Global affine differential geometry of hypersurfaces. Vol. 11 of de Gruyter Expositions in Mathematics. Walter de Gruyter & Co., Berlin (1993)Google Scholar
  53. 53.
    Li A.-M., Xu R.: A cubic form differential inequality with applications to affine kähler–ricci flat manifolds. Res. Math. 54, 329–340 (2009)CrossRefzbMATHGoogle Scholar
  54. 54.
    Li A.-M., Xu R.: A rigidity theorem for an affine Kähler–Ricci flat graph. Res. Math. 56, 141–164 (2009)CrossRefzbMATHGoogle Scholar
  55. 55.
    Li, X., Chang, Z.: Exact solution of vacuum field equation in Finsler spacetime. Phys. Rev. D. 90, 064049. arXiv:1401.6363v1 (2014)
  56. 56.
    Lin F.H., Wang L.: A class of fully nonlinear elliptic equations with singularity at the boundary. J. Geom. Anal. 8, 583–598 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Loewner, C., Nirenberg, L.: Partial differential equations invariant under conformal or projective transformations. In: Contributions to analysis (a collection of papers dedicated to Lipman Bers), pp. 245–272. Academic Press, New York (1974)Google Scholar
  58. 58.
    Loftin, J.: Survey on affine spheres. In: Handbook of geometric analysis, No. 2., pp. 161–191. Int. Press, Somerville, MA, vol. 13 of Adv. Lect. Math. (ALM) (2010)Google Scholar
  59. 59.
    Loftin J.C.: Riemannian metrics on locally projectively flat manifolds. Am. J. Math. 124, 595–609 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Matsumoto M.: On c-reducible Finsler spaces. Tensor 24, 29–37 (1972)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Matsumoto M.: On the indicatrices of a Finsler space. Period. Math. Hung. 8, 187–191 (1977)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Matsumoto M., Hōjō S.: A conclusive theorem on c-reducible Finsler spaces. Tensor 32, 225–230 (1978)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Minguzzi, E.: The connections of pseudo-Finsler spaces. Int. J. Geom. Meth. Mod. Phys. 11, 1460025 (2014). Erratum ibid 12 (2015) 1592001. arXiv:1405.0645
  64. 64.
    Minguzzi, E.: Convex neighborhoods for Lipschitz connections and sprays. Monatsh. Math. 177, 569–625 (2015). arXiv:1308.6675
  65. 65.
    Minguzzi, E.: Light cones in Finsler spacetime. Commun. Math. Phys. 334, 1529–1551 (2015). arXiv:1403.7060
  66. 66.
    Minguzzi, E.: Raychaudhuri equation and singularity theorems in Finsler spacetimes. Class. Quantum Grav. 32, 185008 (2015). arXiv:1502.02313
  67. 67.
    Minguzzi, E.: How many futures on Finsler spacetime? J. Phys. Conf. Ser. 626, 012029 (2015). arXiv:1502.02313
  68. 68.
    Minguzzi, E.: A divergence theorem for pseudo-Finsler spaces (2015). arXiv:1508.06053
  69. 69.
    Minguzzi, E.: Affine sphere spacetimes which satisfy the relativity principle. Phys. Rev. D. (2016) in press)Google Scholar
  70. 70.
    Minguzzi, E.: An equivalence of Finslerian relativistic theories. Rep. Math. Phys. 77, 45–55 (2016). arXiv:1412.4228
  71. 71.
    Miron R.: On the Finslerian theory of relativity. Tensor 44, 63–81 (1987)MathSciNetzbMATHGoogle Scholar
  72. 72.
    Miron R., Rosca R., Anastasiei M., Buchner K.: New aspects of Lagrangian relativity. Found. Phys. Lett. 5, 141–171 (1992)MathSciNetCrossRefGoogle Scholar
  73. 73.
    Mo, L., Xiaohuan, Huang: On characterizations of Randers norms in Minkowski space. Int. J. Math. 21 (2010)Google Scholar
  74. 74.
    Nomizu K., Sasaki T.: Affine differential geometry. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  75. 75.
    Perlick V.: Fermat principle in Finsler spacetimes. Gen. Relat. Gravit. 38, 365–380 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    Pfeifer C., Wohlfarth M.N.R.: Finsler geometric extension of Einstein gravity. Phys. Rev. D. 85, 064009 (2012)ADSCrossRefGoogle Scholar
  77. 77.
    Pimenov, R. I.: Axiomatics of generally relativistic and Finsler space-times by means of causality. Sibirsk. Mat. Zh. 29, 133–143, 218 (1988)Google Scholar
  78. 78.
    Pogorelov A.V.: On the improper convex affine hyperspheres. Geom. Dedicata 1, 33–46 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    Randers G.: On an asymmetric metric in the four-space of general relativity. Phys. Rev. D. 59, 195–199 (1941)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  80. 80.
    Rutz S.F.: A Finsler generalisation of Einstein’s vacuum field equations. Gen. Relat. Gravit. 25, 1139–1158 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  81. 81.
    Sasaki T.: Hyperbolic affine hyperspheres. Nagoya Math. J. 77, 107–123 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  82. 82.
    Simon U.: Zur Relativgeometrie: Symmetrische Zusammenhänge auf Hyperflächen. Math. Z. 106, 36–46 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  83. 83.
    Stavrinos P.C.: Gravitational and cosmological considerations based on the Finsler and Lagrange metric structures. Nonlinear Anal. 71, e1380–e1392 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  84. 84.
    Stavrinos P.C., Kouretsis A.P., Stathakopoulos M.: Friedman-like Robertson–Walker model in generalized metric space-time with weak anisotropy. Gen. Relat. Gravit. 40, 1403–1425 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  85. 85.
    Storer T.P.: Generalized relativity: a unified field theory based on free geodesic connections in Finsler space. Internat. J. Theoret. Phys. 39, 1351–1374 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  86. 86.
    Takano Y.: Gravitational field in Finsler spaces. Lettere al Nuovo Cimento 10, 747–750 (1974)MathSciNetCrossRefGoogle Scholar
  87. 87.
    Takano Y.: Variation principle in Finsler spaces. Lettere al Nuovo Cimento 11, 486–490 (1974)MathSciNetCrossRefGoogle Scholar
  88. 88.
    Teitelboim, M. H.C.: The cosmological constant and general covariance. Phys. Lett. B. 222 (1989)Google Scholar
  89. 89.
    Toupin R.A.: World invariant kinematics. Arch. Rational Mech. Anal. 1, 181–211 (1958)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  90. 90.
    Trautman A.: Sur la théorie newtonienne de la gravitation. C. R. Acad. Sci. Paris. 257, 617–620 (1963)MathSciNetzbMATHGoogle Scholar
  91. 91.
    Trudinger N.S., Wang X.-J.: Affine complete locally convex hypersurfaces. Invent. Math. 150, 45–60 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  92. 92.
    Trudinger, N.S., Wang, X.-J.: The Monge–Ampère equation and its geometric applications. In: Handbook of geometric analysis. No. 1, pp. 467–524. Int. Press, Somerville, MA, vol. 7 of Adv. Lect. Math. (ALM) (2008)Google Scholar
  93. 93.
    Vacaru, S.I.: Principles of Einstein–Finsler gravity and perspectives in modern cosmology. Int. J. Mod. Phys. D. 21, 1250072, 40 (2012)Google Scholar
  94. 94.
    Vinberg, È.B.: The theory of convex homogeneous cones. Trudy Moskov. Mat. Obšč. 12, 303–358 (1963). [Trans. Mosc. Math. Soc. 12, 340–403 (1963)Google Scholar
  95. 95.
    Vinberg È.B., Kac V.G.: Quasi-homogeneous cones. Mat. Zametki 1, 347–354 (1967)MathSciNetzbMATHGoogle Scholar
  96. 96.
    Voicu N.: New considerations on Einstein equations in anisotropic spaces. AIP Conf. Proc. 1283, 249–257 (2010)ADSCrossRefGoogle Scholar
  97. 97.
    Wald R.M.: General Relativity. The University of Chicago Press, Chicago (1984)CrossRefzbMATHGoogle Scholar
  98. 98.
    Xu, R., Zhu, L.: A simple proof of a rigidity theorem for an affine Kähler–Ricci flat graph. Res. Math. (2015) (in press)Google Scholar
  99. 99.
    Yan M.: Extension of convex function. J. Convex Anal. 21, 965–987 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Dipartimento di Matematica e Informatica “U. Dini”Università degli Studi di FirenzeFirenzeItaly

Personalised recommendations