I will review a new mean field theory for the general relativistic gravitational field. It will be shown that it makes both mathematical and physical sense to associate to a statistical ensemble of space-times sharing a common topology a single averaged space-time where the metric is simply the mean value of the metrics of the various space-times constituting the ensemble.
Several consequences of this theory will be explored. I will present statistical ensembles of black holes representing a single black hole observed with finite precision and I will show that, in many physically interesting cases, the averaged space-time still describes a black hole; the parameters and thermodynamical properties of this averaged black hole are generally different from those of the unaveraged one, and I will spend some time discussing these differences, stressing the implications of our results for theoretical physics and observational astrophysics as well.
The new mean field theory predicts some very peculiar effects which may be particularly relevant in the cosmological context, and these will also be adressed. For example, given an arbitrary statistical ensemble of space-times, the stress-energy tensor of the corresponding averaged spacetime is generally different from the mean value of the stress-energy tensors of the various space-times in the ensemble. Similarly, the value taken in the averaged space-time by the conserved 4-current associated to an arbitrary gauge symmetry is also generally different from the mean value of the corresponding 4-currents in the various space-times of the ensemble. Thus, a space-time with vanishing stress-energy tensor and Maxwell current may appear as non-vacuum and electrically charged when observed with finite precision.
I will finally give some very rough order of magnitude estimates which do support the usual assumption that most ‘real’ trajectories can be well approximated, even on the cosmological scale, by geodesics of a standard homogeneous isotropic model.