A review on algebraic extensions in general relativity

For a rigorous approach to reconcile principles of general relativity (GR) and quantum mechanics (QM), we suggest applying the generalized relativistic noncommutative Heisenberg algebra on spacetime coordinates and momenta of a test particle on curved Riemann geometry manifold, which in turn is extended to an eight-dimensional manifold. The natural generalization of the four-dimensional Riemann geometry is the Finsler geometry, in which the quadratic restriction on the length measure is relaxed. With the minimum measurable length derived from the relativistic generalized four-dimensional uncertainty principle, the quantum-induced corrections to the fundamental tensor could be determined in the relativistic regime. Accordingly, the affine connections could be revisited. With the quantum-induced revisiting Riemann curvature tensor and its contractions, the Ricci curvature tensor, and then the Ricci scalar, we have been able to construct a quantum-induced revision of the Einstein tensor, in which besides quantum-induced corrections, additional geometric structures emerge. On the surface of the 2-sphere, we compared the quantum-induced revisiting and non-revisiting Einstein tensor and concluded that the difference between both versions strongly depends on the minimum measurable length and the local geodesics of the test particle through the additional curvature. K E Y W O R D Salternatives to general relativity, general relativity in relativistic quantum regime, generalized noncommutative Heisenberg algebra, reconciling principles of quantum mechanics with general relativity

{dx}^{\mu }

and the fundamental tensor on four‐ and eight‐dimensional manifolds, noncommutation relations between 1‐form

{dx}^{\mu }

Section: Discussionmentioning

confidence: 99%

“…• for the curved spacetime another fully quantizable algebraic or geometrical or phenomenological model should be utilized Hess (2021); Vasconcellos (2017), and…”

Section: Discussionmentioning

confidence: 99%

On quantum‐induced revisiting Einstein tensor in the relativistic regime

Tawfik

2022

“…To summarize, a proper full quantization of

{\tilde{g}}_{\mu \nu}

is likely conditioned to.expressing

{\tilde{g}}_{\mu \nu}

as a quantum operator or suggesting noncommutation relations for the coordinate basis vectors or imposing probability distribution functions or choosing another model for the spacetime quantization Hess (2021), Vasconcellos (2017) or introducing full quatization to the local geodesic

\mid {\ddot{x}}^2\mid

characterizing the additional curvature. …”

Section: Discussionmentioning

confidence: 99%

On possible quantization of the fundamental tensor in the relativistic regime

Tawfik

2022

Inspired by the quantum gravity prescriptions, for instance, string theory, doubly-special relativity, and quantum loop gravity, we apply the generalized relativistic noncommutative Heisenberg algebra on the eight-dimensional Finsler manifold, the natural generalization of the four-dimensional Riemann manifold. The Finsler manifold is conjectured to relax the quadratic restriction on the length measure, especially in the relativistic regime. The additional curvature coupled with the Finsler manifold is assumed to mimic the quantization on the Riemann manifold. The second ingredient we suggest is the existence of a minimum measurable length, whose value is determined from the relativistic generalized uncertainty principle, and then applying in to the coordinates of the Finsler manifold. The quantum-induced corrections to the fundamental tensor are determined under extreme conditions of a relativistic energy. Whether this proposal is adequate for a full quantization, the various limitations of the proposed theory are elaborated and discussed, especially the limitations of the curved spacetime model assumed for the line element measure. We have conservatively formulated our conclusion that the quantization of the fundamental tensor might be conditioned to an adequate quantization of the line element measure, at least the one on the eight-dimensional Finsler manifold. K E Y W O R D Salternatives to general relativity, general relativity in relativistic quantum regime, generalized noncommutative Heisenberg algebra, reconciling principles of quantum mechanics with general relativity

“…Another prediction concerns the possible light emission from the surface of the dark star. The redshift

z

at the surface depends on the azimuthal angle

\Theta

and is lowest at

\Theta =0

, that is, near the poles (see Figure 3) (Hess 2021). Although

z

is very large at the orbital plane, it decreases toward the poles, also depending on the rotational Kerr parameter

a

.…”

Section: Introductionmentioning

confidence: 99%

The pseudo‐complex General Relativity and effects of a minimal length on the structure of black holes

Maghlaoui

Hess

2022

After resuming some main predictions of the pseudo‐complex General Relativity (pcGR), the effects of a minimal length, on the structure near the event horizon of a Schwarzschild and Kerr black hole, are investigated within the pcGR. It is shown that for small mass black holes there are strong effects, for example, avoiding the accumulation of mass, which might be important in the formation of primordial black holes during the Big Bang. The pcGR adds to the metric a contribution characterized by a parameter α$$ \alpha $$, which has a critical value at which barely an event horizon exists. For macroscopic black holes, these effects vanished for below and above this critical value, but at the critical value effects still persist up to m=10−13$$ m={10}^{-13} $$ cm, though quickly disappearing. For very large mass black holes the minimal length can be safely neglected. The limit between a small and a macroscopic black hole, in units of length, is between 10−15$$ {10}^{-15} $$ cm and 10−13$$ {10}^{-13} $$ cm.