Andrea Giusti

Fractional relaxation with time-varying coefficient

From the point of view of the general theory of the hyper-Bessel operators, we consider a particular operator that is suitable to generalize the standard process of relaxation by taking into account both memory effects of power law type and time variability of the characteristic coefficient. According to our analysis, the solutions are still expressed in terms of functions of the Mittag-Leffler type as in case of fractional relaxation with constant coefficient but exhibit a further stretching in the time argument due to the presence of Erdélyi-Kober fractional integrals in our operator. We present solutions, both singular and regular in the time origin, that are locally integrable and completely monotone functions in order to be consistent with the physical phenomena described by non-negative relaxation spectral distributions.

Matter and gravitons in the gravitational collapse

Abstract We consider the effects of gravitons in the collapse of baryonic matter that forms a black hole. We first note that the effective number of (soft off-shell) gravitons that account for the (negative) Newtonian potential energy generated by the baryons is conserved and always in agreement with Bekensteins area law of black holes. Moreover, their (positive) interaction energy reproduces the expected post-Newtonian correction and becomes of the order of the total ADM mass of the system when the size of the collapsing object approaches its gravitational radius. This result supports a scenario in which the gravitational collapse of regular baryonic matter produces a corpuscular black hole without central singularity, in which both gravitons and baryons are marginally bound and form a Bose–Einstein condensate at the critical point. The Hawking emission of baryons and gravitons is then described by the quantum depletion of the condensate and we show the two energy fluxes are comparable, albeit negligibly small on astrophysical scales.

Non-singular rotating black hole with a time delay in the center

As proposed by Bambi and Modesto, rotating non-singular black holes can be constructed via the Newman–Janis algorithm. Here we show that if one starts with a modified Hayward black hole with a time delay in the centre, the algorithm succeeds in producing a rotating metric, but curvature divergences reappear. To preserve finiteness, the time delay must be introduced directly at the level of the non-singular rotating metric. This is possible thanks to the deformation of the inner stationarity limit surface caused by the regularisation, and in more than one way. We outline three different possibilities, distinguished by the angular velocity of the event horizon. Along the way, we provide additional results on the Bambi–Modesto rotating Hayward metric, such as the structure of the regularisation occurring at the centre, the behaviour of the quantum gravity scale alike an electric charge in decreasing the angular momentum of the extremal black hole configuration, or details on the deformation of the ergosphere.

Prabhakar-like fractional viscoelasticity

Abstract The aim of this paper is to present a linear viscoelastic model based on Prabhakar fractional operators. In particular, we propose a modification of the classical fractional Maxwell model, in which we replace the Caputo derivative with the Prabhakar one. Furthermore, we also discuss how to recover a formal equivalence between the new model and the known classical models of linear viscoelasticity by means of a suitable choice of the parameters in the Prabhakar derivative. Moreover, we also underline an interesting connection between the theory of Prabhakar fractional integrals and the recently introduced Caputo–Fabrizio differential operator.

On infinite order differential operators in fractional viscoelasticity

Abstract In this paper we discuss some general properties of viscoelastic models defined in terms of constitutive equations involving infinitely many derivatives (of integer and fractional order). In particular, we consider as a working example the recently developed Bessel models of linear viscoelasticity that, for short times, behave like fractional Maxwell bodies of order 1/2.

On infinite series concerning zeros of Bessel functions of the first kind

Abstract.A relevant result independently obtained by Rayleigh and Sneddon on an identity on series involving the zeros of Bessel functions of the first kind is derived by an alternative method based on Laplace transforms. Our method leads to a Bernstein function of time, expressed by Dirichlet series, that allows us to recover the Rayleigh-Sneddon sum. We also consider another method arriving at the same result based on a relevant formula by Calogero. Moreover, we also provide an electrical example in which this sum results to be extremely useful in order to recover the analytical expression for the response of the system to a certain external input.

On the propagation of transient waves in a viscoelastic Bessel medium

In this paper, we discuss the uniaxial propagation of transient waves within a semi-infinite viscoelastic Bessel medium. First, we provide the analytic expression for the response function of the material as we approach the wave front. To do so, we take profit of a revisited version of the so called Buchen–Mainardi algorithm. Secondly, we provide an analytic expression for the long-time behavior of the response function of the material. This result is obtained by means of the Tauberian theorems for the Laplace transform. Finally, we relate the obtained results to a peculiar model for fluid-filled elastic tubes.

Quantum corpuscular corrections to the Newtonian potential

We study an effective quantum description of the static gravitational potential for spherically symmetric systems up to the first post-Newtonian order. We start by obtaining a Lagrangian for the gravitational potential coupled to a static matter source from the weak field expansion of the Einstein-Hilbert action. By analyzing a few classical solutions of the resulting field equation, we show that our construction leads to the expected post-Newtonian expressions. Next, we show that one can reproduce the classical Newtonian results very accurately by employing a coherent quantum state, and modifications to include the first post-Newtonian corrections are considered. Our findings establish a connection between the corpuscular model of black holes and post-Newtonian gravity, and set the stage for further investigations of these quantum models.

Horizon quantum mechanics of rotating black holes

The horizon quantum mechanics is an approach that was previously introduced in order to analyze the gravitational radius of spherically symmetric systems and compute the probability that a given quantum state is a black hole. In this work, we first extend the formalism to general space-times with asymptotic (ADM) mass and angular momentum. We then apply the extended horizon quantum mechanics to a harmonic model of rotating corpuscular black holes. We find that simple configurations of this model naturally suppress the appearance of the inner horizon and seem to disfavor extremal (macroscopic) geometries.

On transient waves in linear viscoelasticity

Abstract The aim of this paper is to present a comprehensive review of method of the wave-front expansion, also known in the literature as the Buchen–Mainardi algorithm. In particular, many applications of this technique to the fundamental models of both ordinary and fractional linear viscoelasticity are thoroughly presented and discussed.