Abstract
The holographic principle sets an upper bound on the total (Boltzmann) entropy content of the Universe at around ( being Boltzmann’s constant). In this work we point out the existence of a remarkable duality between nonrelativistic quantum mechanics on the one hand, and Newtonian cosmology on the other. Specifically, nonrelativistic quantum mechanics has a quantum probability fluid that exactly mimics the behaviour of the cosmological fluid, the latter considered in the Newtonian approximation. One proves that the equations governing the cosmological fluid (the Euler equation and the continuity equation) become the very equations that govern the quantum probability fluid after applying the Madelung transformation to the Schroedinger wavefunction. Under the assumption that gravitational equipotential surfaces can be identified with isoentropic surfaces, this model allows for a simple computation of the gravitational entropy of a Newtonian Universe. In a first approximation, we model the cosmological fluid as the quantum probability fluid of free Schroedinger waves. We find that this model Universe saturates the holographic bound. As a second approximation, we include the Hubble expansion of the galaxies. The corresponding Schroedinger waves lead to a value of the entropy lying three orders of magnitude below the holographic bound. Current work on a fully relativistic extension of our present model can be expected to yield results in even better agreement with empirical estimates of the entropy of the Universe.
Highlights
There is a widespread belief that the continuum description of spacetime as provided by general relativity must necessarily break down at very short length scales and/or very high curvatures
Suffice it to say that whatever the atoms of spacetime may turn out to be, at the moment there exists a large body of well-established knowledge concerning the thermodynamics of spacetime
Nonrelativistic quantum mechanics has a quantum probability fluid that exactly mimics the behaviour of the cosmological fluid—the latter considered in the Newtonian approximation
Summary
There is a widespread belief that the continuum description of spacetime as provided by general relativity must necessarily break down at very short length scales and/or very high curvatures. Even if the precise nature of the latter is unknown as of yet, one can still make progress following a thermodynamical approach: one ignores large amounts of detailed knowledge (e.g., the precise motions followed by the atoms of a gas) while concentrating only on a few coarse-grained averages (e.g., the overall pressure exerted by the atoms of a gas on the container walls) This way of approaching the problem has come to be called the emergent approach. The total potential at any point within the cosmological fluid is the sum of two harmonic potentials; Hubble’s constant H0 is the frequency of this total harmonic potential In this way, the Newtonian space R3 is foliated by a continuous succession of concentric spheres with growing radii. A stable vacuum state is guaranteed to exist
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