Abstract
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.
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References
Aazami A.B., Javaloyes M.A.: Penrose’s singularity theorem in a Finsler spacetime. Class. Quantum Grav. 33, 025003 (2016)
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)
Anderson J.L., Finkelstein D.: Cosmological constant and fundamental length. Am. J. Phys. 39, 901–904 (1971)
Asanov G.S.: Finsler geometry, relativity and gauge theories. D. Reidel Publishing Co, Dordrecht (1985)
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)
Beem J.K.: Indefinite Finsler spaces and timelike spaces. Can. J. Math. 22, 1035–1039 (1970)
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)
Benoist Y.: Convexes divisibles. C. R. Acad. Sci. Paris Sér. I Math. 332, 387–390 (2001)
Blaschke, W.: Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Band II., Affine Differentialgeometrie. J. Springer, Berlin (1923)
Bock R.D.: Local scale invariance and general relativity. Int. J. Theor. Phys. 42, 1835–1847 (2003)
Bombelli L., Lee J.-H., Meyer D., Sorkin R.D.: Space-time as a causal set. Phys. Rev. Lett. 59, 521–524 (1987)
Brickell F.: A new proof of Deicke’s theorem on homogeneous functions. Proc. Am. Math. Soc. 16, 190–191 (1965)
Calabi E.: Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens. Michigan Math. J. 5, 105–126 (1958)
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)
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)
Castro C.: Gravity in curved phase-spaces, Finsler geometry and two-times physics. Int. J. Mod. Phys. A. 27, 1250069 (2012)
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)
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)
Cheng S.-Y., Yau S.-T.: Complete affine hypersurfaces. I. The completeness of affine metrics. Comm. Pure Appl. Math. 39, 839–866 (1986)
Deicke A.: Über die Finsler–Räume mit A i = 0. Arch. Math. 4, 45–51 (1953)
Dillen F., Vrancken L.: Calabi-type composition of affine spheres. Diff. Geom. Appl. 4, 303–328 (1994)
Dixon W.G.: On the uniqueness of the Newtonian theory as a geometric theory of gravitation. Commun. Math. Phys. 45, 167–182 (1975)
Duval C., Burdet G., Künzle H.P., Perrin M.: Bargmann structures and Newton–Cartan theory. Phys. Rev. D 31, 1841–1853 (1985)
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)
Fox D.J.F.: Functions dividing their Hessian determinants and affine spheres. Asian J. Math. 20(3), 503–530 (2016)
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)
Geroch R.: A method for generating solutions of Einstein’s equations. J. Math. Phys. 12, 918–923 (1971)
Ghomi, M.: The problem of optimal smoothing for convex functions. Proc. Am. Math. Soc. 130, 2255–2259 (2002) (electronic)
Gigena S.: Integral invariants of convex cones. J. Diff. Geom. 13, 191–222 (1981)
Gigena S.: On a conjecture by E. Calabi. Geom. Dedicata 11, 387–396 (1981)
Godbillon C.: Géométrie différentielle et mécanique analytique. Hermann, Paris (1969)
Hartman P.: Ordinary differential equations. Wiley, New York (1964)
Hildebrand R.: Analytic formulas for complete hyperbolic affine spheres. Contrib. Algebra Geometr. 55, 497–520 (2014)
Hildebrand R.: Canonical barriers on convex cones. Math. Oper. Res. 39, 841–850 (2014)
Hildebrand R.: Centro-affine hypersurface immersions with parallel cubic form. Contrib. Algebra Geometr. 56, 593–640 (2015)
Hu Z., Li H., Vrancken L.: Locally strongly convex affine hypersurfaces with parallel cubic form. J. Differ. Geom. 87, 239–308 (2011)
Horváth J.I.: A geometrical model for the unified theory of physical fields. Phys. Rev. 80, 901 (1950)
Horváth J.I., Moór A.: Entwicklung einer einheitlichen feldtheorie begründet auf die finslersche geometrie. Z. Physik 131, 544–570 (1952)
Ikeda S.: On the theory of gravitational field in Finsler spaces. Lett. Nuovo Cimento 26, 277–281 (1979)
Ishikawa H.: Einstein equation in lifted Finsler spaces. Il Nuovo Cimento 56, 252–262 (1980)
Ishikawa H.: Note on Finslerian relativity. J. Math. Phys. 22, 995–1004 (1981)
Jian H., Wang X.-J.: Bernstein theorem and regularity for a class of Monge–Ampère equations. J. Differ. Geom. 93, 431–469 (2013)
Jo K.: Quasi-homogeneous domains and convex affine manifolds. Topol. Appl. 134, 123–146 (2003)
Jörgens K.: Über die Lösungen der Differentialgleichung \({rt-s^2=1}\). Math. Ann. 127, 130–134 (1954)
Knebelman M.S.: Conformal geometry of generalized metric spaces. Proc. N. A. S. 15, 376–379 (1929)
Kobayashi, S., Nomizu, K.: Foundations of differential geometry. vol. I of Interscience tracts in pure and applied mathematics. Interscience Publishers, New York (1963)
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)
Künzle H.P.: Covariant Newtonian limit of Lorentz space-times. Gen. Rel. Grav. 7, 445–457 (1976)
Lämmerzahl C., Perlick V., Hasse W.: Observable effects in a class of spherically symmetric static Finsler spacetimes. Phys. Rev. D. 86, 104042 (2012)
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)
Li A.-M.: Calabi conjecture on hyperbolic affine hyperspheres. II. Math. Ann. 293, 485–493 (1992)
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)
Li A.-M., Xu R.: A cubic form differential inequality with applications to affine kähler–ricci flat manifolds. Res. Math. 54, 329–340 (2009)
Li A.-M., Xu R.: A rigidity theorem for an affine Kähler–Ricci flat graph. Res. Math. 56, 141–164 (2009)
Li, X., Chang, Z.: Exact solution of vacuum field equation in Finsler spacetime. Phys. Rev. D. 90, 064049. arXiv:1401.6363v1 (2014)
Lin F.H., Wang L.: A class of fully nonlinear elliptic equations with singularity at the boundary. J. Geom. Anal. 8, 583–598 (1998)
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)
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)
Loftin J.C.: Riemannian metrics on locally projectively flat manifolds. Am. J. Math. 124, 595–609 (2002)
Matsumoto M.: On c-reducible Finsler spaces. Tensor 24, 29–37 (1972)
Matsumoto M.: On the indicatrices of a Finsler space. Period. Math. Hung. 8, 187–191 (1977)
Matsumoto M., Hōjō S.: A conclusive theorem on c-reducible Finsler spaces. Tensor 32, 225–230 (1978)
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
Minguzzi, E.: Convex neighborhoods for Lipschitz connections and sprays. Monatsh. Math. 177, 569–625 (2015). arXiv:1308.6675
Minguzzi, E.: Light cones in Finsler spacetime. Commun. Math. Phys. 334, 1529–1551 (2015). arXiv:1403.7060
Minguzzi, E.: Raychaudhuri equation and singularity theorems in Finsler spacetimes. Class. Quantum Grav. 32, 185008 (2015). arXiv:1502.02313
Minguzzi, E.: How many futures on Finsler spacetime? J. Phys. Conf. Ser. 626, 012029 (2015). arXiv:1502.02313
Minguzzi, E.: A divergence theorem for pseudo-Finsler spaces (2015). arXiv:1508.06053
Minguzzi, E.: Affine sphere spacetimes which satisfy the relativity principle. Phys. Rev. D. (2016) in press)
Minguzzi, E.: An equivalence of Finslerian relativistic theories. Rep. Math. Phys. 77, 45–55 (2016). arXiv:1412.4228
Miron R.: On the Finslerian theory of relativity. Tensor 44, 63–81 (1987)
Miron R., Rosca R., Anastasiei M., Buchner K.: New aspects of Lagrangian relativity. Found. Phys. Lett. 5, 141–171 (1992)
Mo, L., Xiaohuan, Huang: On characterizations of Randers norms in Minkowski space. Int. J. Math. 21 (2010)
Nomizu K., Sasaki T.: Affine differential geometry. Cambridge University Press, Cambridge (1994)
Perlick V.: Fermat principle in Finsler spacetimes. Gen. Relat. Gravit. 38, 365–380 (2006)
Pfeifer C., Wohlfarth M.N.R.: Finsler geometric extension of Einstein gravity. Phys. Rev. D. 85, 064009 (2012)
Pimenov, R. I.: Axiomatics of generally relativistic and Finsler space-times by means of causality. Sibirsk. Mat. Zh. 29, 133–143, 218 (1988)
Pogorelov A.V.: On the improper convex affine hyperspheres. Geom. Dedicata 1, 33–46 (1972)
Randers G.: On an asymmetric metric in the four-space of general relativity. Phys. Rev. D. 59, 195–199 (1941)
Rutz S.F.: A Finsler generalisation of Einstein’s vacuum field equations. Gen. Relat. Gravit. 25, 1139–1158 (1993)
Sasaki T.: Hyperbolic affine hyperspheres. Nagoya Math. J. 77, 107–123 (1980)
Simon U.: Zur Relativgeometrie: Symmetrische Zusammenhänge auf Hyperflächen. Math. Z. 106, 36–46 (1968)
Stavrinos P.C.: Gravitational and cosmological considerations based on the Finsler and Lagrange metric structures. Nonlinear Anal. 71, e1380–e1392 (2009)
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)
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)
Takano Y.: Gravitational field in Finsler spaces. Lettere al Nuovo Cimento 10, 747–750 (1974)
Takano Y.: Variation principle in Finsler spaces. Lettere al Nuovo Cimento 11, 486–490 (1974)
Teitelboim, M. H.C.: The cosmological constant and general covariance. Phys. Lett. B. 222 (1989)
Toupin R.A.: World invariant kinematics. Arch. Rational Mech. Anal. 1, 181–211 (1958)
Trautman A.: Sur la théorie newtonienne de la gravitation. C. R. Acad. Sci. Paris. 257, 617–620 (1963)
Trudinger N.S., Wang X.-J.: Affine complete locally convex hypersurfaces. Invent. Math. 150, 45–60 (2002)
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)
Vacaru, S.I.: Principles of Einstein–Finsler gravity and perspectives in modern cosmology. Int. J. Mod. Phys. D. 21, 1250072, 40 (2012)
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)
Vinberg È.B., Kac V.G.: Quasi-homogeneous cones. Mat. Zametki 1, 347–354 (1967)
Voicu N.: New considerations on Einstein equations in anisotropic spaces. AIP Conf. Proc. 1283, 249–257 (2010)
Wald R.M.: General Relativity. The University of Chicago Press, Chicago (1984)
Xu, R., Zhu, L.: A simple proof of a rigidity theorem for an affine Kähler–Ricci flat graph. Res. Math. (2015) (in press)
Yan M.: Extension of convex function. J. Convex Anal. 21, 965–987 (2014)
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Communicated by P. T. Chruściel
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Minguzzi, E. Affine Sphere Relativity. Commun. Math. Phys. 350, 749–801 (2017). https://doi.org/10.1007/s00220-016-2802-9
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DOI: https://doi.org/10.1007/s00220-016-2802-9