Approximate Symmetries in General Relativity

The problem of finding an appropriate geometrical/physical index for measuring a degree of inhomogeneity for a given space-time manifold is posed. Interrelations with the problem of understanding the gravitational/informational entropy are pointed out. An approach based on the notion of approximate symmetry is proposed. A number of related results on definitions of approximate symmetries known from literature are briefly reviewed with emphasis on their geometrical/physical content. A definition of a Killing-like symmetry is given and a classification theorem for all possible averaged space-times acquiring Killing-like symmetries upon averaging out a space-time with a homothetic Killing symmetry is proved. E-mail address: [email protected] 1 ENTROPY, INFORMATION, INHOMOGENEITY 1 1 Entropy, Information, Inhomogeneity In modern gravitational physics there exist a number of proposed definitions of gravitational entropy. Amongst them one can mention the black hole entropy by Bekenstein [1] and Hawking [2] which relates the gravitational entropy with the black hole surface area, the space-time entropy by Penrose [3] based on the idea of employing the Weyl tensor for ‘measuring’ the pure gravitational content of a given space-time, Hu’s cosmological entropy [4] due to particle production in anisotropically expanding Universe, BlanderbergerMukhanov-Prokopec’s non-equilibrium entropy [5] for a classical stochastic field applied for density perturbations in an inflationary Universe, the intrinsic entropy by Smolin [6] measuring the irreversibility inherent in conversion any form of matter into gravitational radiation. Despite relative consensus of opinions regarding the usefulness and physical adequacy in applying the known definitions of gravitational entropy for the corresponding domains of gravitational phenomena, there is still much controversy in understanding (formulating) the underlying foundations of classical and/or quantum gravitational physics that are generally expected to bring about a generic definition of the notion of gravitational entropy, it being expected to be of geometrical nature. Two aspects here are of the primary importance: (i) missing links and interrelations between different definitions; (ii) absence, in most cases, of clear geometrical interpretations of the proposed notions of entropy. The challenging problem of gravitational entropy can be considered in broader context as that of finding relations between the informational and gravitational entropy (see a discussion in [1], [7], [8] and references therein). The manifold’s entropy here is to be understood as a kind of geometrical entropy measuring the information encoded in a spacetime manifold and being related with manifold’s inhomogeneity. In such approach the key point (assumption) is that an evolving gravitational system tends to a final symmetric (relatively) homogeneous state with subsequently increasing the system’s entropy, or, its physical homogeneity which is reflected, in general, in the homogeneity of the space-time. Indeed, a ‘structureless’ smooth, highly symmetric homogeneous manifold (i.e. when the system is disordered and requires less information to describe it) may be considered to possess maximum entropy, while a lumpy inhomogeneous, highly structured manifold (i.e. when the system is ordered and requires a great deal of information to describe it) does have a smaller value of entropy. In such context the entropy of a system measures one’s uncertainty or lack of information about the actual internal configuration of the system. Given a set of system’s states determined by probabilities pn, the system entropy is defined due to Shannon’s formula [10] as