Essay Abstract

From the Physics point of view, time is now best described through General Relativity, as part of space-time which is a dynamical object encoding gravity. Time possesses also some intrinsic irreversibility due to thermodynamics, quantum mechanical effects... This irreversibility can look puzzling since time-like loops (and hence time machines) can appear in General Relativity (for example in the Godel universe, a solution of Einstein's equations). We take this apparent discrepancy as a warning bell pointing to us that time as we understand it, might not be fundamental and that whatever theory, lying beyond General Relativity, may not include time as we know it as a fundamental structure. We propose therefore, following the philosophy of analog models of gravity, that time and gravity might not be fundamental per se, but only emergent features. We illustrate our proposal using a toy-model where we show how the Lorentzian signature and Nordstrom gravity (a diffeomorphisms invariant scalar gravity theory) can emerge from a timeless non-dynamical space.

Author Bio

F. Girelli has done his PhD in Marseille (France). He went to the Perimeter Institute (Canada) and SISSA (Italy) for some postdocs. He is now postdoc at the University of Sydney. S. Liberati has done his PhD in SISSA. He did a postdoc at the University of Maryland (USA) before becoming assistant professor at SISSA. L. Sindoni is currently finishing his PhD at SISSA.

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Dear Florian,

I have read your interesting work. I have a doubt concerning the boundary conditions in the evolution problems which arise in your model. For the fields psi I would expect boundary conditions like those for the Laplace equation, namely some condition in the boudary of a closed domain of (4-dimensional) euclidean space. On the contrary for the perturbations phi I would expect different boundary conditions because the evolution equations are hyperbolic (thus the boundary conditions would have to be given on a Cauchy hypersurface). Now, psi'=psi+phi is just another psi type solution, thus it must satisfy the first type of boudary conditions that I mentioned above, but then I would expect phi to be constrained on the boundary of a closed set in 4 dimensional space and not on a open hypersurface. There seems to be a conflict between the evolution problems. How is it solved? Many thanks, Ettore

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dear Ettore, thanks for your question.

Let us restate it: at first order, the field Psi coincides with the perturbations phi, and in this sense, both of them must satisfy hyperbolic boundary conditions, if the Minkowski metric has emerged. This might seem contradictory with the fact that the field Psi a priori is living in a euclidian space and therefore should have some elliptic boundary conditions.

There is in fact no contradiction:

If one considers the full theory, that is we have nonlinear kinetic terms, there is a mismatch between signature of the metric and hyperbolicity-ellipticity, the latter being determined by a matrix involving the metric AND the derivatives of the fields. More precisely, the signature of the metric tensor determines the hyperbolicity-ellipticity only for canonical kinetic terms, for which the matrix in front of the second derivatives in the PDE is just the metric tensor. For nonlinear kinetic terms (our case), this is not true.

As a consequence, when dealing with the full theory, the chosen boundary conditions can be elliptic or hyperbolic, but when restricted to the (linear) operator constructed at *first* order (so that everything becomes linear), they can definitely be interpreted as hyperbolic boundary conditions, and there is no mismatch.

Hope this helps!

Florian, Stefano, Lorenzo

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Thanks for the clear explanation. Ettore

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Dear Dr. Girelli, Dr. Liberati and Dr. Sidoni,

You raise a counterargument against the fundamental nature of time: the discrepancy between the irreversibility and the possibility of closed timelike curves. Starting from here, and from condensed matter inspired emergent gravity, you construct an interesting toy model, which shows a possible mechanism for the emergence of Lorentz and diffeomorphism symmetries (and time). I think that this idea worth to be explored.

Congratulations,

Cristi Stoica

Flowing with a Frozen River

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