Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2010-10-21
New J.Phys.12:115001,2010
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
Scientific paper
10.1088/1367-2630/12/11/115001
There are competing schools of thought about the question of whether spacetime is fundamentally either 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 and discrete information, which is of key importance in information theory, is established by Shannon sampling theory: of any bandlimited 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 bandlimit. 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 we here also consider the generalization to curved spacetimes, i.e., to Lorentzian manifolds.
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