No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
The full text of this article is not currently available.
Spacetime and observer space symmetries in the language of Cartan geometry
É. Cartan, Exposés de Géométrie V, Actualités scientifiques et industrielles Vol. 194 (Hermann, Paris, 1935).
R. W. Sharpe, Differential Geometry (Springer, New York, 1997).
A. Einstein, S. Preuss. Akad. Wiss. 1923, 32.
A. Einstein, S. Preuss. Akad. Wiss. 1923, 137.
A. S. Eddington, The Mathematical Theory of Relativity (University Press, Cambridge, 1924).
É. Cartan, C. R. Acad. Sci. Paris 174, 593 (1922).
É. Cartan, Ann. Sci. Ecole Norm. Sup. 40, 325 (1923).
M. Blagojević, Gravitation and Gauge Symmetries (IOP, Bristol, 2002).
M. Blagojević and F. W. Hehl, Gauge Theories of Gravitation: A Reader with Commentaries (Imperial College Press, London, 2013).
, in Encyclopedia of Mathematical Physics
, edited by J.-P. Francoise
, G. L. Naber
, and S. T. Tsou
), Vol. 2, pp. 189
; e-print arXiv:gr-qc/0606062
A. Einstein, S. Preuss. Akad. Wiss. 1915, 844.
S. Carroll, Spacetime and Geometry–An Introduction to General Relativity (Addison Wesley, San Francisco, 2004).
A. Einstein, S. Preuss. Akad. Wiss. 1928, 217.
R. Aldrovandi and J. G. Pereira, Teleparallel Gravity: An Introduction (Springer, Dordrecht, 2013).
D. Bao, S. S. Chern, and Z. Shen, An Introduction to Riemann-Finsler Geometry (Springer, New York, 2000).
I. Bucataru and R. Miron, Finsler-Lagrange Geometry (Editura Academiei Romane, Bucharest, 2007).
S. I. Vacaru
, in Clifford and Riemann Finsler Structures in Geometric Mechanics and Gravity
(Geometry Balkan Press
); e-print arXiv:hep-th/0211068
C. Pfeifer, DESY-THESIS-2013-049.
H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers, and E. Herlt, Exact Solutions of Einstein’s Field Equations (University Press, Cambridge, 2003).
, in Mathematical Structures of the Universe
, edited by M. Heller
, M. Eckstein
, and S. Szybka
(Copernicus Center Press
), pp. 13
; e-print arXiv:1403.4005
M. Ostrogradski, Mem. Ac. St. Petersburg VI4, 385 (1850).
D. J. Saunders, The Geometry of Jet Bundles (University Press, Cambridge, 1989).
G. Giachetta, L. Mangiarotti, and G. Sardanashvily, Advanced Classical Field Theory (World Scientific, Singapore, 2009).
A. Čap and J. Slovák, Parabolic Geometries I (American Mathematical Society, Providence, 2009).
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Wiley, New York, London, 1963).
I. Kolář, P. Michor, and J. Slovák, Natural Operations in Differential Geometry (Springer, Berlin, 1993).
K. Yano and S. Ishihara, Tangent and Cotangent Bundles, Pure and Applied Mathematics Vol. 1 (Marcel Dekker, New York, 1973).
Note that there exist different conventions in the literature regarding the order of the lower two indices. Here we choose the second lower index to be the “derivative” index.
Note that Ref. 45 also discusses symmetries of Cartan spaces. However, these are different geometric structures than the Cartan geometries we reviewed in Section II C.
Article metrics loading...
We introduce a definition of symmetry generating vector fields on manifolds which are equipped with a first-order reductive Cartan geometry. We apply this definition to a number of physically motivated examples and show that our newly introduced notion of symmetry agrees with the usual notions of symmetry of affine, Riemann-Cartan, Riemannian, and Weizenböck geometries, which are conventionally used as spacetime models. Further, we discuss the case of Cartan geometries which can be used to model observer space instead of spacetime. We show which vector fields on an observer space can be interpreted as symmetry generators of an underlying spacetime manifold, and may hence be called “spatio-temporal.” We finally apply this construction to Finsler spacetimes and show that symmetry generating vector fields on a Finsler spacetime are indeed in a one-to-one correspondence with spatio-temporal vector fields on its observer space.
Full text loading...
Most read this month