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From Classical to Discrete Gravity through Exponential Non-Standard Lagrangians in General Relativity

College of Mathematics and Information Science, Neijiang Normal University, Neijiang, Sichuan 641112, China
Academic Editor: Palle E.T. Jorgensen
Mathematics 2015, 3(3), 727-745; https://doi.org/10.3390/math3030727
Received: 24 May 2015 / Revised: 30 July 2015 / Accepted: 4 August 2015 / Published: 14 August 2015
(This article belongs to the Special Issue Mathematical physics)
Recently, non-standard Lagrangians have gained a growing importance in theoretical physics and in the theory of non-linear differential equations. However, their formulations and implications in general relativity are still in their infancies despite some advances in contemporary cosmology. The main aim of this paper is to fill the gap. Though non-standard Lagrangians may be defined by a multitude form, in this paper, we considered the exponential type. One basic feature of exponential non-standard Lagrangians concerns the modified Euler-Lagrange equation obtained from the standard variational analysis. Accordingly, when applied to spacetime geometries, one unsurprisingly expects modified geodesic equations. However, when taking into account the time-like paths parameterization constraint, remarkably, it was observed that mutually discrete gravity and discrete spacetime emerge in the theory. Two different independent cases were obtained: A geometrical manifold with new spacetime coordinates augmented by a metric signature change and a geometrical manifold characterized by a discretized spacetime metric. Both cases give raise to Einstein’s field equations yet the gravity is discretized and originated from “spacetime discreteness”. A number of mathematical and physical implications of these results were discussed though this paper and perspectives are given accordingly. View Full-Text
Keywords: non-standard Lagrangians; general relativity; modified geodesic equations; discrete gravity; discrete spacetime; quantized cosmological constant non-standard Lagrangians; general relativity; modified geodesic equations; discrete gravity; discrete spacetime; quantized cosmological constant
MDPI and ACS Style

El-Nabulsi, R.A. From Classical to Discrete Gravity through Exponential Non-Standard Lagrangians in General Relativity. Mathematics 2015, 3, 727-745.

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