Search

It is shown how a noncommutative spacetime leads to an area, mass and entropy quantization condition which allows to derive the Schwarzschild black hole entropy A 4G, the logarithmic corrections, and further corrections, from the discrete mass transitions taking place among different mass states in D = 4. The higher-dimensional generalization of the results in D = 4 follows. The discretization of the entropy-mass relation S = S(M) leads to an entropy quantization of the form S = S(Mn) = n, such that one may always assign n “bits” to the discrete entropy, and in doing so, make contact with quantum information. The physical applications of mass quantization, like the counting of states contributing to the black hole entropy, black hole evaporation, and the direct connection to the black holes-string correspondence [G. Horowitz and J. Polchinski, A correspondence principle for black holes and strings, Phys. Rev. D 55 (1997) 6189.] via the asymptotic behavior of the number of partitions of integers, follows. To conclude, it is shown how the recent large N Matrix model (fuzzy sphere) of C.-S. Chu [A matrix model proposal for QG and the QM of black holes, preprint, arXiv:2406.01466] leads to very similar results for the black hole entropy as the physical model described in this work which is based on the discrete mass transitions originating from the noncommutativity of the spacetime coordinates. [ABSTRACT FROM AUTHOR]

A brief review of the essentials of asymptotic safety and the renormalization group (RG) improvement of the Schwarzschild black hole that removes the r = 0 singularity is presented. It is followed with a RG improvement of the Kantowski-Sachs metric associated with a Schwarzschild black hole interior such that there is no singularity at t = 0 due to the running Newtonian coupling G(t) vanishing at t = 0. Two temporal horizons at [t.sub.-] [equivalent] [t.sub.P] and [t.sub.+] [equivalent] [t.sub.H] are found. For times below the Planck scale t < [t.sub.P], and above the Hubble time t > [t.sub.H], the components of the Kantowski-Sachs metric exhibit a key sign change, so the roles of the spatial z and temporal t coordinates are exchanged, and one recovers a repulsive inflationary de Sitter-like core around z = 0, and a Schwarzschild-like metric in the exterior region z > [R.sub.H] = 2[G.sub.o]M. The inclusion of a running cosmological constant [LAMBDA](t) follows. We proceed with the study of a dilaton-gravity (scalar-tensor theory) system within the context of Weyl's geometry that permits singling out the expression for the classical potential V([phi]) = [kappa][[phi].sup.4], instead of being introduced by hand, and find a family of metric solutions that are conformally equivalent to the (anti) de Sitter metric. To conclude, an ansatz for the truncated effective average action of ordinary dilaton gravity in Riemannian geometry is introduced, and a RG-improved cosmology based on the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric is explored where instead of recurring to the cutoff identification k = k(t) = [xi]H(t), based on the Hubble function H(t), with [xi] a positive constant, one now has k = k(t) = [xi][phi](t), when [phi] is a positive-definite dilaton scalar field that is monotonically decreasing with time. Key words: Weyl geometry, cosmology, asymptotic safety, dilaton-gravity, Kantowski-Sachs, Schwarzschild metric. Dans cette breve revue, nous presentons les elements essentiels de l'amelioration de la securite asymptotique et du groupe de renormalisation (GR/RG) pour le trou noir de Schwarzschild, qui permettent de lever la singularite a r =0. C'est suivi par l'amelioration GR sur la metrique de Kantowski-Sachs qui est associee a l'interieur du trou noir de Schwarzschild, ce qui permet qu'il n'y ait plus de singularite a t = 0, consequence de la variation (running) de la constante de couplage de Newton (qui tend a zero a t = 0). Nous trouvons deux horizons temporels, a [t.sub.-] [equivalent] [t.sub.P] et a [t.sub.+] [equivalent] [t.sub.H]. Pour des temps sous l'echelle de Planck, t < [t.sub.P] et pour des temps au-dessus du temps de Hubble, t > [t.sub.H], les composantes de la metrique de Kantowski-Sachs montrent un changement de signe significatif, de telle sorte que les coordonnees d'espace z et de temps t sont interchangees et qu'on retrouve alors un coeur de type de Sitter autour de z = 0 et une metrique de type Schwarzschild dans la region exterieure z > [R.sub.H] = 2[G.sub.o]M. S'ensuit l'introduction d'une constante cosmologique variable K(t). Nous poursuivons avec l'etude d'un systeme gravite- dilaton dans la geometrie de Weyl, qui permet d'identifier un potentiel classique V([phi]) = [kappa][[phi].sub.4], au lieu d'avoir a l'introduire a la main et nous trouvons une famille de solutions metriques qui ont une equivalence conforme a la metrique (anti) de Sitter. Pour conclure, nous introduisons un ansatz pour l'action moyenne effective tronquee de la gravite-dilation ordinaire dans une geometrie de Riemann et nous explorons une cosmologie avec amelioration GR, basee sur la metrique FLRW ou, au lieu de recourir a l'identification d'un seuil de coupure k = k(t) = [phi]H(t) qui est basee sur la constante de Hubble H(t) avec [xi] une constante positive, nous avons a la place k = k(t) = [xi][phi](t), ou [phi](t) est une fonction de champ scalaire definie positive qui decroit de facon monotone avec le temps. [Traduit par la Redaction] Mots-cles : geometrie de Weyl, securite asymptotique, Kantowski-Sachs, cosmologie, gravite-dilaton., 1. Introduction The problem of dark energy and the solution to the cosmological constant problem is one of the most challenging problems facing cosmology today. There are a vast number [...]

We revisit the nonlinear Klein–Gordon-like equation that was proposed by us which captures how quantum mechanical probability densities curve spacetime, and find an exact solution that may appear to be “trivial” but with important physical implications related to the physics of frozen stars and with Mach’s principle. The nonlinear Klein–Gordon-like equation is essentially the static spherically symmetric relativistic analog of the Newton–Schrödinger equation. We finalize by studying the higher-dimensional generalizations of the nonlinear Klein–Gordon-like equation and examine the relativistic Bohm–Poisson equation as yet another equation capturing the interplay between quantum mechanical probability densities and gravity. [ABSTRACT FROM AUTHOR]

Starting with a brief description of Born's reciprocal relativity theory (BRRT), based on a maximal proper force, maximal speed of light, and inertial and non-inertial observers, we derive the exact thermal relativistic corrections to the Schwarzschild, Reissner-Nordstrom, and Kerr-Newman black hole entropies and provide a detailed analysis of the many novel applications and consequences to the physics of black holes, quantum gravity, minimal area, minimal mass, Yang-Mills mass gap, information paradox, arrow of time, dark matter, and dark energy. We finish by outlining our proposal towards a space-time-matter unification program where matter can be converted into spacetime quanta and vice versa. Key words: thermal relativity, gravity, black holes, entropy, Born reciprocity, phase spaces, maximal acceleration, dark matter, Finsler geometry. Debutant avec une breve description du theoreme de relativite reciproque de Born (TRRB/BRRT), base sur une force propre maximale, sur la vitesse maximale de la lumiere et sur les observateurs inertiels et non inertiels, nous derivons les corrections thermiques exactes aux entropies des trous noirs de Schwarzchild, de Reissner-Nordstrom et de Kerr-Newman et fournissons une analyse detaillee du grand nombre de nouvelles applications et consequences a la physique des trous noirs, de la gravite quantique, de la surface minimale, de la masse minimale, du gap de masse de Yang-Mills, du paradoxe d'information, de la fleche du temps, de la matiere noire et de l'energie sombre. Nous concluons en soulignons notre proposition vers un programme d'unification espace-temps-matiere, ou la matiere peut etre transformee en quanta d'espace-temps et vice-versa. [Traduit par la Redaction] Mots-cles : relativite thermique, gravite, trous noirs, principe de reciprocite de Born, entropie, espaces de phase, acceleration maximale, matiere noire, geometrie de Finsler., 1. Introduction The deep origins of the connection between black holes and thermodynamics is still a mystery (to our knowledge). We shall argue that the principle of thermal relativity holds [...]

Born's reciprocal relativity theory (BRRT), based on a maximal proper-force, maximal speed of light, and inertial and non-inertial observers, is re-examined in full detail. Relativity of locality and chronology are natural consequences of this theory, even in flat phase space. The advantage of BRRT is that Lorentz invariance is preserved and there is no need to introduce Hopf algebraic deformations of the Poincare algebra, de Sitter algebra, nor non- commutative space-times. After a detailed study of the notion of generalized force, momentum, and mass in phase space, we explain that what one may interpret as 'dark matter' in galaxies, for example, is just an effect of observing ordinary galactic matter in different accelerating frames of reference than ours. Explicit calculations are provided that explain these novel relativistic effects due to the accelerated expansion of the Universe, and which may generate the present-day density parameter value [[OMEGA].sub.DM] ~ 0.25 of dark matter. The physical origins behind the numerical coincidences in black hole cosmology are also explored. We finalize with a rigorous study of the curved geometry of (co)tangent bundles (phase space) within the formalism of Finsler geometry, and provide a short discussion on Hamilton spaces.Key words: Born reciprocity, phase spaces, maximal acceleration, gravity, dark matter, Finsler geometry.Nous reexaminons en detail la theorie de la relativite reciproque de Born (TRRB) basee sur une force propre maximale, une vitesse de la lumiere maximale et des observateurs inertiels et non inertiels. La relativite de localite et la chronologie sont des consequences naturelles de cette theorie, meme dans un espace plat. L'avantage de la TRRB est que l'invariance de Lorentz est preservee et il n'est pas necessaire d'introduire les deformations algebriques de Hopf de l'algebre de Poincare, d'algebre de de Sitter ni d'espace-temps non commutatifs. Apres une etude detaillee de la notion de force generalisee, d'impulsion et de masse dans l'espace de phase, nous expliquons que nous pouvons interpreter la matiere noire dans les galaxies par exemple, comme resultant de l'observation de matiere ordinaire dans des referentiels accelerant differemment du notre. Nous presentons des calculs explicites qui expliquent ces nouveaux effets relativistes dus a l'acceleration de l'expansion de l'univers et capables de generer la valeur d'aujourd'hui du parametre de densite [[OMEGA].sub.DM] ~ 0.25 de matiere noire. Nous explorons aussi les origines physiques de la coincidence numerique en cosmologie du trou noir. Nous terminons avec une etude rigoureuse de la geometrie courbe du fibre (co)tangent (espace de phase) a l'interieur du formalisme de la geometrie de Finsler et fournissons une breve discussion sur les espaces de Hamilton. [Traduit par la Redaction]Mots-cles: reciprocite de Born, gravite, geometrie de Finsler, matiere noire, espace de phase., 1. Born's reciprocal relativity in phase space and maximal proper forceMost of the work devoted to quantum gravity has been focused on the geometry of space-time rather than phase space [...]

Ref.: Can. J. Phys. 99(6): 455-465 (2020). doi:10.1139/cjp-2020-0348. It has come to the publisher's attention that the reference list in the published version of this paper erroneously omitted several references [...]

We construct the novel Clifford-Yang algebra which is an extension of the Yang algebra in noncommutative phase spaces. The Clifford-Yang algebra allows us to write down the commutators of the n o n c o m m u t a t i v e polyvector-valued coordinates and momenta which are compatible with the Jacobi identities, the Weyl–Heisenberg algebra, and paves the way for a formulation of quantum mechanics (QM) in noncommutative Clifford spaces. We continue with a detailed study of the isotropic 3D quantum oscillator in noncommutative spaces and find the energy eigenvalues and eigenfunctions. These findings differ considerably from the ordinary quantum oscillator in commutative spaces. We find that QM in noncommutative spaces leads to very different solutions, eigenvalues, and uncertainty relations than ordinary QM in commutative spaces. The generalization of QM to noncommutative Clifford (phase) spaces is attained via the Clifford-Yang algebra. The operators are now given by the generalized angular momentum operators involving polyvector coordinates and momenta. The eigenfunctions (wave functions) are now more complicated functions of the polyvector coordinates. We conclude with some important remarks. [ABSTRACT FROM AUTHOR]

Using simple arguments derived from the Boolean hypercube configuration, the structure of natural spaces, and the recursive exponential generation of the set of natural numbers, a linear classification of the natural numbers is presented. The definition of a pseudolinear Collatz operator, the description of the set of powers of 2, and the construction of the natural numbers via this power set might heuristically prove the Collatz conjecture from an empirical point of view. [ABSTRACT FROM AUTHOR]