A vacuum solution of modified Einstein equations based on fractional calculus

Abstract

In this work, we construct a modified version of the Einstein field equations for a vacuum and spherically symmetric spacetime in terms of the Riemann–Liouville fractional derivative. The main difference between our approach and other works is that we ensure that both the classical differential equations and the classical solutions are exactly recovered in the limit when the fractional parameter is turned off. We assume that the fractional equations are valid inside and near the horizon radius and match the classical solution at the horizon. Our approach resembles the Herrera–Witten strategy (Adv High Energy Phys 2018:3839103, 2018, https://doi.org/10.1155/2018/3839103, arXiv:1806.07143 [gr-qc]), where the authors constructed an alternative black hole solution by assuming that inside the horizon the spacetime is hyperbolically symmetric and matches the classical spherically symmetric exterior solution at one point at the horizon. We obtain that, depending on the value of the fractional parameter, the solutions can be interpreted as a regular black hole or a gravastar. As a final step, we compute the fractional curvature scalars and show that the solution is regular everywhere inside the horizon.

Fractional cosmic strings

“…On the contrary, LSM heavily relies on GR: the switch from partial to fractional derivative is performed on the Euler-Lagrange equations obtained from the usual Einstein-Hilbert action. This is by far the oldest and the most manageable method, and as such it has been used to extend Friedman-Robertson-Walker cosmology [33] and the Schwarzschild exterior solution [15]. ISM is an in-between approach: in the case of the fractional action-like variational approach, the switch from partial to fractional derivative is applied to the Einstein-Hilbert action before this latter is extremized (see for example equations ( 20) and ( 33) in [31]).…”

“…In [14], the authors used a fractional action principle (based on the Caputo fractional derivative) built from a FLRW ansatz to derive the fractional Friedman equations: they set stringent constraints on the fractional parameter to fit to observational data. In another remarkable work [15], the authors determined a vacuum solution for a static and spherically-symmetric time-space based on the fractional Riemann-Liouville derivative: while the exterior Schwarzschild's metric is retrieved outside the horizon, the interior solutions are not only regular at r = 0, but depending on the fractional parameter, they can match with gravastar models. More recently, the authors of [16] have proposed a fractional anisotropy function relating both the tangential and radial pressure of a spherically symmetric fluid based on the Grünwald-Letnikov fractional derivative, which is a discrete version of the Riemann-Liouville fractional derivative.…”

Geometrized Vacuum Physics. Part I. Algebra of Stignatures

“…We choose from the sixteen 4-bases shown in Figure 7, one 4-basis, for example, e 5 (e0 (5) ,e1 (5) ,e2 (5) ,e3 (5) ) and letʹs call it ʺbaseʺ (see Figure 8). An affine 4-dimensional space, the directions of the axes of which are given by the 4-basis e 5 (e0 (5) ,e1 (5) ,e2 (5) ,e3 (5) ) with the conventional signature {+ + + +}.…”

“…We conditionally accept that the directions of all unit vectors of the ʺbaseʺ are positive (see Figure 8) ei (5) (e0 (5) ,e1 (5) ,e2 (5) ,e3 (5) ) = (+1, +1,+ 1, +1)  {+ + + +}.…”

“…In modern physics, there are [1][2][3][4][5][6][7][8]: technical vacuum (discharged gas); physical vacuum (the lowest energy state of a set of scalar, vector, tensor and spinor quantum fields); Einstein vacuum (generally a curved 4-dimensional space-time continuum surrounding neutral or charged physical bodies); ideal vacuum (emptiness, i.e. 3-dimensional space, in which any curvature and particles are completely absent).…”