Abstract
There are competing schools of thought about the question of whether spacetime is fundamentally continuous or discrete. Here, we consider the possibility that spacetime could be simultaneously continuous and discrete, in the same mathematical way that information can be simultaneously continuous and discrete. The equivalence of continuous information and discrete information, which is of key importance in signal processing, is established by the Shannon sampling theory: for any band-limited signal, it suffices to record discrete samples to be able to perfectly reconstruct it everywhere, if the samples are taken at a rate of at least twice the band limit. It is known that physical fields on generic curved spaces obey a sampling theorem if they possess an ultraviolet cutoff. Most recently, methods of spectral geometry have been employed to show that also the very shape of a curved space (i.e. of a Riemannian manifold) can be discretely sampled and then reconstructed up to the cutoff scale. Here, we develop these results further and also consider the generalization to curved spacetimes, i.e. to Lorentzian manifolds.
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GENERAL SCIENTIFIC SUMMARY Introduction and background. To unify quantum theory and general relativity is difficult in large part because of a basic dilemma: general relativity indicates that spacetime should be a continuum, but quantum field theory indicates that spacetime should be discrete, with a spacing at the Planck length of 10−35 m or larger. Correspondingly, there are candidate quantum gravity theories in which it is assumed that spacetime is continuous, while there are competing theories in which spacetime is assumed to be discrete. In the former theories it is difficult to ensure the absence of infinities, while in the latter theories it tends to be difficult to ensure that a continuum emerges.
Main results. Here, aiming to resolve this dilemma, we present results that show that spacetime could be simultaneously continuous and discrete, in the same way that information can be. The transformation rules between continuous and discrete representations of information are described in Shannon's sampling theory, which is in ubiquitous use in all signal processing. In this paper, the underlying mathematics has been generalized and applied to physical fields and spacetime itself. One of the key results is the finding that the local curvature of spacetime can be understood to be a local variation in its density of degrees of freedom, i.e., in the local information carrying capacity of spacetime.
Wider implications. The new information-theoretic concept of a spacetime that is both continuous and discrete could be very helpful for describing spacetime both general relativistically and quantum theoretically.