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The Mass of Graviton and Its Relation to the Number of Information According to the Holographic Principle

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The Mass of Graviton and Its Relation to the Number of Information According to the Holographic Principle Hindawi Publishing Corporation International Scholarly Research Notices Volume 2014, Article ID 718251, 8 pages http://dx.doi.org/10.1155/2014/718251 Research Article The Mass of Graviton and Its Relation to the Number of Information according to the Holographic Principle Ioannis Haranas1 and Ioannis Gkigkitzis2 1 Department of Physics and Astronomy, York University, 4700 Keele Street, Toronto, ON, Canada M3J 1P3 2 Departments of Mathematics and Biomedical Physics, East Carolina University, 124 Austin Building, East Fifth Street, Greenville, NC 27858-4353, USA Correspondence should be addressed to Ioannis Haranas; [email protected] Received 2 June 2014; Accepted 27 July 2014; Published 29 October 2014 Academic Editor: Sergi Gallego Copyright © 2014 I. Haranas and I. Gkigkitzis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate the relation of the mass of the graviton to the number of information in a flat universe. As a result we find that ∝1/√ the mass of the graviton scales as gr . Furthermore, we find that the number of gravitons contained inside the observable ∝ horizon is directly proportional to the number of information ;thatis, gr . Similarly, the total mass of gravitons that exist ∝ √ in the universe is proportional to the number of information ;thatis, gr .Inanefforttoestablisharelationbetween the graviton mass and the basic parameters of the universe, we find that the mass of the graviton is simply twice the Hubble mass as it is defined by Gerstein et al. (2003), times the square root of the quantity −1/2,where is the deceleration parameter of the universe. In relation to the geometry of the universe we find that the mass of the graviton varies according to the relation ∝ √ gr sc, and therefore gr obviously controls the geometry of the space time through a deviation of the geodesic spheres from the spheres of Euclidean metric. 1. Introduction and quantum field theory predict the existence of graviton particles. In quantum field theory graviton is the elementary In Einstein’s theory of general relativity, linearization of the particle that mediates the gravitational force and is expected field equations demonstrates that small perturbations of the to be massless, and that is because the gravitational force metric obey a wave equation [1]. These small disturbances, itself has an infinite range. Furthermore, graviton must bea referred to as gravitational waves, travel at the speed of light. spin-2 boson, which results from the fact that the source of However, some other gravity theories predict a dispersive gravitation is the stress-energy tensor itself. Additionally, it propagation (see [2] for references). The most commonly can be shown that any massless spin-2 field could give rise considered form of dispersion supposes that the waves obey to a force that is indistinguishable from gravitation because a a Klein-Gordon-type equation that is given below: massless spin-2 field must couple to the stress-energy tensor 2 in the same way that the gravitational field does3 [ ]. Therefore, 2 2 1 2 ( −∇ −( ) )=0. (1) this result suggests that if a massless spin-2 particle is dis- 2 2 ℏ covered, it must be the graviton [1]. Thus graviton detection remains vital in the validation of the theories and also in Physically, the dispersive term is ascribed to the quantum of the research that strives to unify quantum mechanics with gravitation having a nonzero rest mass ,orequivalentlya general relativity. Although physicists normally speak as if noninfinite Compton wavelength =ℎ/,where is the bosons mediating the gravitational force exist, the extremely potential function of the gravitational field. weak character of the gravitational interaction makes the Moreover, several of today’s theories like string theory, detection of graviton an extremely hard issue. Recently, superstringtheory,M-theoryandloopquantumgravity, Dyson has suggested that “the detection of a single graviton 2 International Scholarly Research Notices mayinfactberuledoutintherealuniverse”[4]. If researchers .Similarly,inBousso[10] the author argues that the answer Dyson’s question in an affirmative way, concluding total observable entropy is bounded by the inverse of the that graviton detection is impossible, this will immediately cosmological constant. This holds for all space-times with a raise issues in relation to the necessity of gravity quanti- positive cosmological constant, including cosmologies dom- zation. However, attempts to extend the standard model inated by ordinary matter and recollapsing universes [10]. with gravitons have run into serious theoretical difficulties Boussos’ conjecture is largely influenced by Banks’ idea of at high energies (processes with energies close to or above cosmological constant [11]. Moreover, in Mongan [12], the the Planck scale) because of infinities arising due to quantum author examines a vacuum-dominated Friedmann universe effects (in other words, gravitation is nonrenormalizable). asymptotic to a de Sitter space, with a cosmological event Since classical general relativity and quantum mechanics are horizon that its area in Planck’s units determines the maxi- incompatible at such energies, from a theoretical point of mum amount of information that will ever be available to any view, the present situation is not tenable. Some proposed observer. Moreover, in a recent paper by Mureika and Mann models of quantum gravity [5] attempt to address these issues, [13], the authors examine the idea of information transfer but these are speculative theories. between a test particle and the holographic screen in entropic There are only a few conceivable sources of graviton gravity. The transfer respects both the uncertainty principle production, like black hole decay, spontaneous emission andcausality,andalowerlimitonthenumberofinformation of gravitons from neutral hydrogen, bremsstrahlung from bits in the universe relative to the mass may be derived. The electron-electron collisions in stellar interiors, and conserva- corresponding limits indicate that particles travelling with tions of photons to gravitons by interstellar magnetic fields. the speed of light—photon and/or graviton—have a nonzero −68 Here we will only briefly touch upon the graviton production mass ≥10 kg. Their result is found to be in excellent by black hole decay, which appears to be the most promising agreement with the current experimental mass bound of mechanism. Black holes of mass possess a Hawking photon and graviton, suggesting that entropic gravity might temperature that is equal to betheresultofarecentlocalsymmetrythatissoftlybroken [13]. In particular cosmological holography postulates that ℏ3 = , all the information content in our universe is encoded on its BH (2) 8 cosmological horizon. This proposal has been put forward by Smoot [14]andinshortstatesthatallpossiblepastandfuture where is the Boltzmann constant, is the gravitational histories of our universe are encoded on its apparent horizon constant, ℏ is Planck’s constant, and is speed of light. A 19 and by that making a connection. Moreover, the authors primordial black hole with mass ≤10 kg can in principle ≥10 proceed by asking how much the universe as a whole, mass, evaporate gravitons of energy eV or higher and, with and information included can tell us about its parts or the no observational constraints on the mass of the primordial lightest possible mass of the elementary particles. The authors black holes applied [6],canconstitutemostoftheuniverse’s further claim that in entropic gravity there is a lower limit to dark matter. On the other hand, primordial black holes with ≤1012 thenumberofbitsofholographicscreenwhichprovidesthe mass kg that emit particles with energy higher information transfer between the test particle and the screen, ≥108 eVwouldhavealreadylongbeenevaporated. which obeys causality as well as the uncertainty principle. Recently, Finn and Sutton [7] have examined the binary Similarly, in Haranas and Gkigkitzis [15], the authors pulsar, PSR B1913 + 16, of which the observed decay rate coin- examine the Bekenstein bound of information number and cides with that expected from relativity to approximately 0.3% its relation to cosmological parameters in a universe, where [7]. A nonzero graviton mass would upset this remarkable in Gkigkitzis et al. [16] the authors use a recent result for the agreement altering the predicted orbital decay, implying an number of information derived from Landauer’s principle; upper limit on the graviton mass. The authors have obtained they obtain an expression for the cosmological constant Λ. a crude estimate on this bound using dimensional analysis. Finally, in Haranas and Gkigkitzis [17], the authors investigate For a system with characteristic frequency one expects the thenumberofinformation as related to the minimum effects of a graviton mass to appear at second order in / quantum and gravitational masses in a vacuum-dominated [7]. For gravitational waves at twice the orbital frequency of 2 universe.Inthiscontributionweseektoinvestigateand PSR B1913 + 16, requiring (/) < 0.003 implies an upper understand the physics behind any possible relations that −20 2 limitoforder10 eV/c . This is comparable to the best limit mightresultfromtherelationofthegravitonmasstothe −21 2 from solar system observations, < 0.44 × 10 eV/c . surface area of
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