Abstract
We develop an approach to the theory of nonholonomic relativistic stochastic processes in curved spaces. The Itô and Stratonovich calculus are formulated for spaces with conventional horizontal (holonomic) and vertical (nonholonomic) splitting defined by nonlinear connection structures. Geometric models of the relativistic diffusion theory are elaborated for nonholonomic (pseudo) Riemannian manifolds and phase velocity spaces. Applying the anholonomic deformation method, the field equations in Einstein’s gravity and various modifications are formally integrated in general forms, with generic off-diagonal metrics depending on some classes of generating and integration functions. Choosing random generating functions we can construct various classes of stochastic Einstein manifolds. We show how stochastic gravitational interactions with mixed holonomic/nonholonomic and random variables can be modelled in explicit form and study their main geometric and stochastic properties. Finally, the conditions when non-random classical gravitational processes transform into stochastic ones and inversely are analyzed.
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References
C. Chevalier, F. Debbasch, J. Math. Phys. 49, 043303 (2008).
J. Dunkel, P. Hänggi, Phys. Rev. E 72, 036106 (2005).
Z. Haba, Phys. Rev. E 79, 021128 (2009).
J. Franchi, Y. Le Jan, Curvature Diffusions in General Relativity, arXiv:1003.3849.
I. Bailleul, A Probabilistic View on Singularities and Spacetime Boundary, arXiv:1009.4865.
J. Herrmann, Phys. Rev. E 80, 051110 (2009).
J. Herrmann, Diffusion in the general theory of relativity, arXiv:1003.3753.
S. Vacaru, Locally Anisotropic Stochastic Processes in Fiber Bundles, in Proceeding of the Workshop “Global Analysis, Differential Geometry and Lie Algebras”, December 16–18, 1995, Thessaloniki, Greece, edited by G. Tsagas (Geometry Balkan Press, Bucharest, 1997) pp. 123-140, arXiv:gr-qc/9604014.
S. Vacaru, Interactions, Strings and Isotopies in Higher Order Anisotropic Superspaces (Hadronic Press, Palm Harbor, 1998) p. 450, math-ph/0112065.
S. Vacaru, Ann. Phys. (N.Y.) 290, 83 (2001).
S. Vacaru, Ann. Phys. (Leipzig) 9, 175 (2000).
A. Einstein, Investigations on the Theory of Brownian Motion (Dover, New York, 1956), Reprint of the 1st English edition (1926).
S. Vacaru, P. Stavrinos, E. Gaburov, D. Gonţa, Clifford and Riemann-Finsler Structures in Geometric Mechanics and Gravity, Selected Works, Differential Geometry --- Dynamical Systems, Monograph 7 (Geometry Balkan Press, 2006) www.mathem.pub.ro/dgds/mono/va-t.pdf and arXiv:gr-qc/0508023.
S. Vacaru, Int. J. Geom. Meth. Mod. Phys. 4, 1285 (2007).
S. Vacaru, Int. J. Geom. Meth. Mod. Phys. 5, 473 (2008).
S. Vacaru, Int. J. Geom. Meth. Mod. Phys. 8, 9 (2011) arXiv:0909.3949v1 [gr-qc] and 1106.4660 [physics.gen-ph].
S. Vacaru, Int. J. Theor. Phys. 49, 884 (2010).
S. Vacaru, Int. J. Theor. Phys. 48, 1973 (2009).
S. Vacaru, J. Math. Phys. 46, 042503 (2005).
S. Vacaru, Class. Quantum. Grav. 27, 105003 (2010).
S. Vacaru, Diffusion and Self-Organized Criticality in Ricci Flow Evolution of Einstein and Finsler Spaces, arXiv:1010.2021.
D.S. Lemons, An Introduction to Stochastic Processes in Physics (The John Hopkins University Press, Baltimore, 2002).
K.D. Elworthy, Stochastic Differential Equations on Manifolds, in London Math. Soc. Lecture Notes, Vol. 79 (Cambridge University Press, 1982).
N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes (Noth Holand Publishing Company, Amsterdam, 1981) (Russian translation (Nauka, Moscow, 1986)).
M. Emery, Stochastic Calculus on Manifolds (Springer-Verlag, Berlin, Heidelberg, 1989).
B.L. Hu, E. Verdaguer, Living Rev. Relativity 11, 3 (2008) http://www.livingreviews.org/lrr-2008-3.
M. Christensen, J. Comput. Phys. 201, 421 (2004).
P.H. Damgard, H. Hüffel, Phys. Rep. 152, 227 (1987).
M. Namiki, Stochastic Quantization (Springer, Hedielberg, 1992).
R. Dijkgraaf, D. Orlando, S. Reffert, Relating Field Theories via Stochastic Quantization, arXiv:0903.0732.
F.-W. Shu, Y.-S. Wu, Stochastic Quantization of Hořava Gravity, arXiv:0906.1645.
L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields, 2nd and 3rd editions, Vol. 2 (Pergamon, London, 1962 and 1967) the 4th edition does not contain the imaginary unity for pseudo-Euclidean metrics.
C. Møller, Theory of Relativity, 2nd edition (Oxford University Press, 1972).
D. Rapoport, Int. J. Theor. Phys. 30, 287 (1991).
D. Rapoport, Int. J. Theor. Phys. 35, 287 (1996).
S.I. Vacaru, Stochastic Calculus on Generalized Lagrange Spaces, in The Program of the Iaşi Academic Days, October 6-9, 1994 (Academia Româna, Filiala Iaşi, 1994) p. 30.
S. Vacaru, Izv. Acad. Nauk Respubliky Moldova, Fiz. Tekh. 3, 13 (1996).
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Vacaru, S.I. Nonholonomic relativistic diffusion and exact solutions for stochastic Einstein spaces. Eur. Phys. J. Plus 127, 32 (2012). https://doi.org/10.1140/epjp/i2012-12032-0
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DOI: https://doi.org/10.1140/epjp/i2012-12032-0