Narrow Results

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 $\frac{A}{4G}$, 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 $S=S(M_n)=n$, 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. D55 (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.

The idea of applying the gauge principle to formulate the general theory of relativity started with Utiyama in 1956. In this article, various applications of the gauge principle applied to different aspects of the gravitational interactions, are reviewed.

The main result of the paper is a new representation of the Weyl Lagrangian (massless Dirac Lagrangian). As the dynamical variable, we use the coframe, i.e. an orthonormal tetrad of covector fields. We write down a simple Lagrangian — wedge product of axial torsion with a lightlike element of the coframe — and show that variation of the resulting action with respect to the coframe produces the Weyl equation. The advantage of our approach is that it does not require the use of spinors, Pauli matrices or covariant differentiation. The only geometric concepts we use are those of a metric, differential form, wedge product and exterior derivative. Our result assigns a variational meaning to the tetrad representation of the Weyl equation suggested by J. B. Griffiths and R. A. Newing.

A gauge theory of gravity based on a nonlinear realization (NLR) of the local Conform-Affine (CA) group of symmetry transformations is presented. The coframe fields and gauge connections of the theory are obtained. The tetrads and Lorentz group metric are used to induce a spacetime metric. The inhomogenously transforming (under the Lorentz group) connection coefficients serve as gravitational gauge potentials used to define covariant derivatives accommodating minimal coupling of matter and gauge fields. On the other hand, the tensor valued connection forms serve as auxiliary dynamical fields associated with the dilation, special conformal and deformational (shear) degrees of freedom inherent in the bundle manifold. The bundle curvature of the theory is determined. Boundary topological invariants are constructed. They serve as a prototype (source free) gravitational Lagrangian. The Bianchi identities, covariant field equations and gauge currents are obtained.

A compact, self-contained approach to gravitation, based on the local Poincaré gauge invariance, is proposed. Starting from the general invariance principle, we discuss the global and the local Poincaré invariance developing the spinor, vector and tetrad formalisms. These tools allow to construct the curvature, torsion and metric tensors by the Fock–Ivanenko covariant derivative. The resulting Einstein–Cartan theory describes a space endowed with non-vanishing curvature and torsion while the gravitational field equations are similar to the Yang–Mills equations of motion with the torsion tensor playing the role of the Yang–Mills field strength.

It has been argued that gravity acts dissipatively on quantum-mechanical systems, inducing thermal fluctuations that become indistinguishable from quantum fluctuations. This has led some authors to demand that some form of time irreversibility be incorporated into the formalism of quantum mechanics. As a tool toward this goal, we propose a thermodynamical approach to quantum mechanics, based on Onsager's classical theory of irreversible processes and Prigogine's nonunitary transformation theory. An entropy operator replaces the Hamiltonian as the generator of evolution. The canonically conjugate variable corresponding to the entropy is a dimensionless evolution parameter. Contrary to the Hamiltonian, the entropy operator is not a conserved Noether charge. Our construction succeeds in implementing gravitationally-induced irreversibility in the quantum theory.

We give a geometrical description of gravitational theories from the viewpoint of symmetries and affine structure. We show how gravity, considered as a gauge theory, can be consistently achieved by the nonlinear realization of the conformal-affine group in an indirect manner: due to the partial isomorphism between CA(3, 1) and the centrally extended Sp( 8), we perform a nonlinear realization of the centrally extended (CE)Sp( 8) in its semi-simple version. In particular, starting from the bundle structure of gravity, we derive the conformal-affine Lie algebra and then, by the nonlinear realization, we define the coset field transformations, the Cartan forms and the inverse Higgs constraints. Finally, we discuss the geometrical Lagrangians where all the information on matter fields and their interactions can be contained.

A Clifford-gravity-based model is exploited to build a generalized action (beyond the current ones used in the literature) and arrive at relevant numerical results which are consistent with the presently-observed de Sitter accelerating expansion of the universe driven by a very small vacuum energy density ρ_obs ~ 10^-120(M_P)⁴ (M_P is the Planck mass) and provide promising dark energy/matter candidates in terms of the 16 scalars corresponding to the degrees of freedom associated with a Cl(3, 1)-algebra-valued scalar field Φ in four dimensions.

We consider the most general torsional completion of gravitation together with electrodynamics for the Dirac spinorial material fields, and we show that consistency arguments constrain torsion to be completely antisymmetric and the dynamics to be parity-invariant and described by actions that are either least-order derivative or renormalizable.

After a cursory introduction of the basic ideas behind Born’s Reciprocal Relativity theory, the geometry of the cotangent bundle of spacetime is studied via the introduction of nonlinear connections associated with certain nonholonomic modifications of Riemann–Cartan gravity within the context of Finsler geometry. A novel gauge theory of gravity in the $8D$ cotangent bundle $T^{\ast }M$ of spacetime is explicitly constructed and based on the gauge group $SO(6,2){\times }_s{R}^8$ which acts on the tangent space to the cotangent bundle $T_{(x,p)}{T}^{\ast }M$ at each point $(x,p)$. Several gravitational actions involving curvature and torsion tensors and associated with the geometry of curved phase-spaces are presented. We conclude with a brief discussion of the field equations, the geometrization of matter, quantum field theory (QFT) in accelerated frames, T-duality, double field theory, and generalized geometry.

The equations for gravitational plane waves produced by a typical binary system as a solution of linear approximation of Einstein equations are derived. The dynamics of the corresponding gravitational field is analyzed in a four-dimensional space-time manifold, endowed with a metric and taking into account torsion. In this context, the geometrical reason of the existence of torsion due to the presence of gravitational waves (GW) is highlighted and the geodesic deviation is obtained taking into account both curvature and torsion. In a laser interferometer gravitational detector, the delay time between the arrivals of the two laser beams traveling back and forth along the two arms in presence of gravitational waves is interpreted from this point of view. This delay is calculated for the NS–NS binary pulsar (1913 + 16) in two specific orientations with respect to the experimental device, corresponding to different polarizations of gravitational waves. In the specific case of this example, it is shown that the results obtained in the context of the standard general relativity (GR) and in the framework of teleparallel gravity are equivalent.

The premetric formalism is an alternative representation of a classical field theory in which the field equations are formulated without the spacetime metric. Only the constitutive relations between the basic field variables can involve the metric of the underlying manifold. In this paper, we present a brief pedagogical review of the premetric formalism in mechanics, electromagnetism, and gravity.

Starting with the study of the geometry on the cotangent bundle (phase space), it is shown that the maximal proper force condition, in the case of a uniformly accelerated observer of mass $m$ along the $x$ axis, leads to a minimum value of $x$ lying $inside$ the Rindler wedge and given by the black hole horizon radius $2Gm$. Whereas in the uniform circular motion case, we find that the maximal proper force condition implies that the radius of the circle cannot exceed the value of the horizon radius $2Gm$. A correspondence is found between the black hole horizon radius and a singularity in the curvature of momentum space. The fact that the geometry (metric) in phase spaces is observer-dependent (on the momentum of the massive particle/observer) indicates further that the matter stress energy tensor and vacuum energy in the underlying space-time may admit an interpretation in terms of the curvature in momentum spaces. Some final comments are made pertaining to the Asymptotic Safety program in gravity and why phase space geometry seems to be a proper arena for a space–time–matter unification.

In this paper, we will consider the birth and evolution of fields during formation of $N$-dimensional manifolds from joining point-like ones. We will show that at the beginning, only there are point-like manifolds which some strings are attached to them. By joining these manifolds, 1-dimensional manifolds are appeared and gravity, fermion, and gauge fields are emerged. By coupling these manifolds, higher dimensional manifolds are produced and higher orders of fermion, gauge fields and gravity are emerged. By decaying $N$-dimensional manifold, two child manifolds and a Chern–Simons one are born and anomaly is emerged. The Chern–Simons manifold connects two child manifolds and leads to the energy transmission from the bulk to manifolds and their expansion. We show that $F$-gravity can be emerged during the formation of $N$-dimensional manifold from point-like manifolds. This type of $F$-gravity includes both type of fermionic and bosonic gravity. $G$-fields and also $C$-fields which are produced by fermionic strings produce extra energy and change the gravity.

Recently, some authors have proposed a new model for the BIon which can be applied in both nanosystems and cosmology. A Bion is a configuration which has been constructed from two branes and a wormhole which connects them. In this paper, we consider the process of formation of a BIon by joining hexagonal accretion disks. These disks exchange gravitational waves with each other and produce some curvature. By increasing the number of connected disks and joining these waves, a wormhole emerges.

We present a general approach for the formulation of equations of motion for astrophysical compact objects in general relativistic theories. The objects are modeled as relativistic particles, which are assumed to be moving in a geometric background which in turn is asymptotically flat. The background can be affected by the compact object; so that our approach can be applied to binary systems; by concentrating on the main contributions coming from back reaction effects due to gravitational radiation.

Our premises are different from the traditional post-Newtonian and self-force approaches since we make strong use of the asymptotic properties of the self geometry of the particle; as is described below. In particular, we use a center of mass setting, and relativistic individual dynamical times for each object. We expect our model to complement the other approaches in different regimes.

After a brief review of the thermal relativistic corrections to the Schwarzschild black hole entropy, it is shown how the Stefan–Boltzman law furnishes large modifications to the evaporation times of Planck-size mini-black holes, and which might furnish important clues to the nature of dark matter and dark energy since one of the novel consequences of thermal relativity is that black holes do not completely evaporate but leave a Planck size remnant. Equating the expression for the modified entropy (due to thermal relativity corrections) with Wald’s entropy should, in principle, determine the functional form of the modified gravitational Lagrangian ${ℒ}(R_{abcd})$. We proceed to derive the generalized uncertainty relation which corresponds to the effective temperature $T_{\mathrm{\text{eff}}}=T_H{(1-\frac{T_H^2}{T_P^2})}^{-1/2}$ associated with thermal relativity and given in terms of the Hawking ($T_H$) and Planck ($T_P$) temperature, respectively. Such modified uncertainty relation agrees with the one provided by string theory up to first order in the expansion in powers of $\frac{{(\delta p)}^2}{M_P^2}$. Both lead to a minimal length (Planck size) uncertainty. Finally, an explicit analytical expression is found for the modifications to the purely thermal spectrum of Hawking radiation which could cast some light into the resolution of the black hole information paradox.

A geometrical construction for a global dynamical time for binary point-like particle systems, modeled by relativistic equations of motions, is presented. Thus, we provide a convenient tool for the calculation of the dynamics of recent models for the dynamics of black holes that use individual proper times. The construction is naturally based on the local Lorentzian geometry of the spacetime considered. Although in this presentation we are dealing with the Minkowskian spacetime, the construction is useful for gravitational models that have as a seed Minkowski spacetime. We present the discussion for a binary system, but the construction is obviously generalizable to multiple particle systems. The calculations are organized in terms of the order of the corresponding relativistic forces. In particular, we improve on the Darwin and Landau–Lifshitz approaches, for the case of electromagnetic systems. We discuss the possibility of a Lagrangian treatment of the retarded effects, depending on the nature of the relativistic forces. The higher-order contractions are based on a Runge–Kutta type procedure, which is used to calculate the quantities at the required retarded time, by increasing evaluations of the forces at intermediate times. We emphasize the difference between approximation orders in field equations and approximation orders in retarded effects in the equations of motion. We show this difference by applying our construction to the binary electromagnetic case.

Using the fact that neutrinos only participate in weak and gravitational interactions, we explore the possibility of having their masses emerged at the intersection between extended electroweak theory and theory of gravity. We describe how these two seemingly incompatible theories could be embedded in a lepton-number violating 5-dimensional Lagrangian ${{ℒ}}_5$. A peculiar feature of this approach is its ability to generate effective Majorana neutrino masses via the spontaneous symmetry breaking (SSB) of Grand Unified Theory (GUT), $G_{\mathrm{\text{GUT}}}\underrightarrow{\stackrel{}{M_X}}G(321)\underrightarrow{\stackrel{}{{\upsilon }_{\mathrm{\text{EW}}}}}G(1)$ and 4$\times 4$ symmetric matrix of gravitational couplings. Within the purview of this theoretical framework, we obtain values for the effective Majorana mass $M_{\mathrm{\text{eff}}}^{\mathrm{\text{mass}}}=3.7126$meV, and the Majorana neutrino masses $m_1=0.6672$meV, $m_2=14.4498$meV, $m_3=43.3494$meV, $m_4=0$meV and $\sum m=0.0585$eV. Our results are in good agreement with both experimental and cosmological data.

In this paper, we introduce a new geometric/topological approach to the emerging braneworld scenario in the context of D-branes using partially negative-dimensional product (PNDP) manifolds. The working hypothesis is based on the fact that the orientability of PNDP manifolds can be arbitrary, and starting from this, we propose that gravitational interaction can derive naturally from the non-orientability. According to this hypothesis, we show that topological defects can emerge from non-orientability and they can be identified as gravitational interaction at macroscopic level. In other words, the orientability of fundamental PNDPs can be related to the appearance of curvature at low-energy scales.