Classical and quantum gravity with fractional operators

Calcagni, Gianluca

doi:10.1088/1361-6382/ac1081

Cited by 27 publications

(23 citation statements)

References 80 publications

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“…Other avenues to explore in the future may come across other multi-fractional theories, those with fractional operators [13,44]. A phenomenological Newtonian model with fractional Laplacian seems able to fit galaxy rotation curves [19,20] but it has not been embedded in a covariant theory.…”

Section: Discussionmentioning

confidence: 99%

Gravitational potential and galaxy rotation curves in multi-fractional spacetimes

Calcagni,

Varieschi

2021

Preprint

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Multi-fractional theories with integer-order derivatives are simple models of gravitational and matter fields living in spacetimes with variable Hausdorff and spectral dimension, originally proposed as descriptions of spacetimes arising in quantum gravity. In this paper, we pose the question of whether they can serve as an alternative to dark matter. We give a preliminary positive answer. We find the Poisson equation and the Newtonian potential of the multi-fractional theories with integer-order derivatives starting from their covariant modified Einstein's equations. We show that neither the theory T v with weighted derivatives in the extreme fractional limit nor the theory T q with q-derivatives fit the rotation curve of any of the galaxies NGC7814, NGC6503 and NGC3741 in the SPARC catalogue. The rotation curve of all three galaxies can be explained by purely geometric effects in the theory T v with weighted derivatives with small fractional corrections when the fractional exponent takes the special value α = 4/3, but only at large radii. B Multipole expansion of inverse-power potentials 36 C Useful formulae 38 C.1 Angular integrals of Gegenbauer polynomials 38 C.2 Radial integrals 39 D Galaxy potential in the theory T v with exponential matter density 40 D.1 Calculation of eqs. (5.46) and (5.47) 40 D.2 Calculation of eqs. (5.49c) and (5.49d) 41 E Thick-disk data-based galaxy potential and gravitational field in T v 42 E.1 Extreme fractional limit 42 E.1.1 α = 2/3 42 E.1.2 α = 2/3 45 E.1.3 Comparison with data 47 E.2 Small fractional corrections 47 E.2.1 α = 4/3 47 E.2.2 α = 4/3 54

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Section: Discussionmentioning

confidence: 99%

Gravitational potential and galaxy rotation curves in multi-fractional spacetimes

Calcagni,

Varieschi

2021

Preprint

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“…This procedure is practically equivalent to the one used for scalar-tensor theories of gravity (see Ref. [52] for a general overview) and it has been used extensively by Calcagni in the context of multi-scale spacetimes and fractional gravity theories [7,8,10,11,15,17,18]. In the following subsections we will review these techniques and adapt them to our particular case.…”

Section: Relativistic Equations For Spaces With Non-integer Dimensionmentioning

confidence: 99%

“…Different multi-scale theories were then developed by Calcagni, with reference to the possible derivative operators ∂ being used: theory T 1 with ordinary derivatives, theory T v with weighted derivatives, and theory T q with q-derivatives (see [11,17] for full details). These models were then used in connection with the most general measure derived from first principles [13] and then applied to quantum field theories, quantum gravity, and cosmology [11,[15][16][17][18].…”

Section: Rfdg Field Equationsmentioning

confidence: 99%

“…In paper III, the use of a variable dimension D(r) as a function of the field point was also discussed and justified in terms of other similar existing studies. In all these three papers, we pointed out that NFDG is only loosely based on the methods of fractional calculus and fractional mechanics (see [6] and references therein), but is not a fractional theory in the sense used by other gravitational models [7][8][9][10][11][12][13][14][15][16][17][18][19][20]. NFDG field equations are of integer order, therefore local, as opposed to non-local field equations based on fractional differential operators.…”

Section: Introductionmentioning

confidence: 99%

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Relativistic Fractional-Dimension Gravity

Varieschi

2021

Universe

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This paper presents a relativistic version of Newtonian Fractional-Dimension Gravity (NFDG), an alternative gravitational model recently introduced and based on the theory of fractional-dimension spaces. This extended version—Relativistic Fractional-Dimension Gravity (RFDG)—is based on other existing theories in the literature and might be useful for astrophysical and cosmological applications. In particular, in this work, we review the mathematical theory for spaces with non-integer dimensions and its connections with the non-relativistic NFDG. The Euler–Lagrange equations for scalar fields can also be extended to spaces with fractional dimensions, by adding an appropriate weight factor, and then can be used to generalize the Laplacian operator for rectangular, spherical, and cylindrical coordinates. In addition, the same weight factor can be added to the standard Hilbert action in order to obtain the field equations, following methods used for scalar-tensor models of gravity, multi-scale spacetimes, and fractional gravity theories. We then apply the field equations to standard cosmology and to the Friedmann-Lemaître-Robertson-Walker metric. Using a suitable weight vtt, depending on the synchronous time t and on a single time-dimension parameter αt, we extend the Friedmann equations to the RFDG case. This allows for the computation of the scale factor at for different values of the fractional time-dimension αt and the comparison with standard cosmology results. Future additional work on the subject, including studies of the cosmological late-time acceleration, type Ia supernovae data, and related dark energy theory will be needed to establish this model as a relativistic alternative theory of gravity.

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“…The class of models discussed here does not encompass all possible nonlocal ghostfree theories of gravitation, since it leaves out the cases with non-entire form factors, such as theories with a minimal length [54], with a minimal proper time in Schwinger parametrization [55,56] or with fractional operators [57]. On the other hand, the present work has been motivated by the need to clarify the topic of scattering amplitudes in nonlocal gravity [4-6, 8, 9, 58, 59] but it actually has a very general applicability and validity.…”

Section: Jhep10(2021)169mentioning

confidence: 99%

Tree-level scattering amplitudes in nonlocal field theories

Modesto

Calcagni

2021

J. High Energ. Phys.

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We prove in two ways that, for a special class of nonlocal field theories consistent with linear and non-linear stability at the classical level, and with unitarity and super-renormalizability or finiteness at the quantum level, the n-point tree-level scattering amplitudes are the same as those of the underlying local theory. In particular, the n-point amplitudes of nonlocal gravity, with or without coupling to matter, are the same as for Einstein’s general relativity.

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Classical and quantum gravity with fractional operators

Cited by 27 publications

References 80 publications

Gravitational potential and galaxy rotation curves in multi-fractional spacetimes

Gravitational potential and galaxy rotation curves in multi-fractional spacetimes

Relativistic Fractional-Dimension Gravity

Tree-level scattering amplitudes in nonlocal field theories

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