Abstract
We introduce the notion of relative volume entropy for two spacetimes with preferred compact spacelike foliations. This is accomplished by applying the notion of Kullback-Leibler divergence to the volume elements induced on spacelike slices. The resulting quantity gives a lower bound on the number of bits which are necessary to describe one metric given the other. For illustration, we study some examples, in particular gravitational waves, and conclude that the relative volume entropy is a suitable device for quantitative comparison of the inhomogeneity of two spacetimes.
Highlights
How much information is required to describe one spacetime in terms of another? More detailed manifestations of this question are: Just how much more complex is a solution of the Einstein equation in the presence of matter than a vacuum solution? How “complicated” is a gravitational wave? In this paper, we provide a possible answer to these questions in terms of the relative entropy of the volume elements associated with the metric tensors of two given spacetimes
We have introduced the relative entropy of volume, an information theoretical measure in classical gravity, measuring part of the information content of one solution of the Einstein equation relative to another
In a technical relative entropy direction, it would be interesting to extend the theory beyond the case of compact spacelike slices and metrics with a preferred time direction
Summary
We provide a possible answer to these questions in terms of the relative entropy of the volume elements associated with the metric tensors of two given spacetimes. We propose to compute the relative entropy of two continuous probability distributions associated with our metrics, namely the normalized volume densities in spacelike slices. For this method to make sense without further complications, we restrict ourselves to the following situation. This certainly is a very satisfying result: The more particle sources are in the energy-momentum tensor, the more information is necessary to describe spacetime over a vacuum background. We proceed to work out some examples, in order to illustrate how the relative volume entropy behaves for different classes of spacetimes
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