The functional mechanics in General relativity

Functional reformulation of classical mechanics was proposed by I. V. Volovich to solve the irreversibility problem of macroscopic dynamics arising in the justification of thermodynamics. Description of the microscopic state of the system by distribution function in the functional mechanics allows to include directly in the equations of dynamics the finite precision of the measurements. This report discusses the derivation of the basic equation of the functional mechanics (special form of the Liouville equation) for a material point on the $(d+1)$-dimensional space-time manifold in General relativity. The conditions of normalization of the distribution function and the formulation of the Cauchy problem, which significantly depends on the choice of the reference system, are specified. The relationship between the probability densities in different noninertial frames of reference is established by example of two-dimensional Rindler space (classical analogue of Unruh effect). The penetration under the event horizon of a Schwarzschild black hole distribution function corresponding to the solution of the Liouville equation for a freely falling particle is described, which may allow to advance in the resolution of the paradox formation of black holes.