The role of the Rankine–Hugoniot circular boundary conditions within has been attributed a characterization in the analysis of the sonic behaviour of accretion objects in Astrophysical blackhole spacetimes in General Relativity; within the same framework, the definition of cusps on the accretion objects is found to be determined after the same shock properties. The ongoing blackhole-imaging experiments and the projected ones are a suitable phenomenological setting for the validation of the correspondence of the blackhole images with respect to the location of the cusps. The cusps are here described analytically from their geometrical origin; the numerical methods are then compared.

I point out that the schemes are now ready for the modeling of jets within the analysis of the accretion-object cusps.

It is also my intention to delineate the incompleteness of the further non-geometrical methods.

1. Rankine–Hugoniot Circular Boundary Conditions for the Solutions of the Einstein Field Equations

The general relativistic Rankine–Hugoniot (RH) boundary conditions [1-3] were implemented after the work of Taub [4] and were explained with reference to [5], after which the phenomenon is derived, according to which the inequality holds that the velocity of the shock wave [6] is always smaller than that of light in vacuum, i.e., when the shocks are sufficiently strong. The position of RH boundary conditions in GR is discussed in [5], after which the proof is granted that the circular-symmetry accretion object remains defining a static spacetime, i.e., the absence of gravitational torque is ensured. As a result, the application of the RH boundary conditions to the geometrical part of the Einstein Field Equations (EFEs) allows one to describe the presence of one cusp or of more than one cusp in the accretion object of blackholes after the analysis of the sonic behaviour of the macroscopic matter of which the accretion flow consists.

2. Blackhole Figures and Cusps of the Accretion Objects

The role of the geometrical aspects of the blackhole accretion discs in the definition of the shadow of the blackholes is underlined in the recent work of Wang [7]. Ibidem, the Schwarzschild spacetime is taken as a useful instance for the sake to investigate blackhole-imaging methodologies which follow the geometrical aspects of the geometrically thick accretion discs.

Within the study of the presence of cusps of geometrical origin, on accretion objects, the description is provided with of the different blackhole optical manifestations; the geometrical aspects of the position of the RH boundary conditions on the solution of the EFEs is represented: in this manner, the attempts to ascribe the features of the cusp(s) to hypothetical non-geometrical phenomena is proven up-to-nowadays ruinous, i.e., such as the attempt to relate the geometry of the cusp(s) as ’dark-matter spikes’ is probed incomplete [8], or even mathematically inconsistent [9], i.e., as explained in [10].

2.1. The Analytical Geometrical Approach to Blackhole Imaging

The consistency of the approach towards the geometry of the disc is probed in [7] after the application of the developped methodologies to all the manifestations of the consequences of the geometrical structures, i.e., the direct image, the lensing ring, and the photon ring within the optical regimes which define the emission regions of the optically thin discs, of the optically thick discs, and of partially optically-thick discs.

In particular, the method followed is one of prescribing the measure of the inclination angle and that of the inner radius of a chosen set of one-dimensional luminous segments; the explanation of the characterization of the diverse disc configurations is provided below.

One of the purposes of the study from [7] is to adapt a theoretical paradigm to shape the future data-analysis techniques dedicated to high-resolution experiments of blackhole observations. According to the work of Gralla et al. [11], the structure of the accretion process is indicated as discriminant regarding the formation of the photon ring; unassociatedly, the analysis of Narayan et al. [12] is aimed at outlining the role of the geometry as surpassing that of the quality of the macroscopic matter chosen as solution of the EFEs when the edge of the shadow is observed—it is now possible to attribute prominence to the presence of a cus in the accretion object in correspondence of the geometrical manifestations of the black hole figures, differently form [12], according to which a Bondi accretion should be considered.

From [7], the optically thin discs are didascalized to exhibit the consequences of the flux which is theoretically expected to be diverging at the critical value of the impact parameter i the case the inner edge of the disc emitting region should approach the photon ring: in accordance with these new findings, the diverging value of the flux is tamed after the presence of a cusp (also sharp) close to the critical parameter, which is paired with the presence of the photon ring.

From ibidem, the several roles of the inner radius are further exemplified, i.e., with respect to the orbital speed of the emitting material.

From ibidem, the location of the inner radius is explored to define the lensing ring.

The focus on the angular boundaries of the disc is established accordingly; i.e., the edges of the emitting regions are explicated to possess different brightness. The importance of the inner edge is further understood to determine the characterization of the lensing ring; in contrast, the outer edge is depicted as being influential on the region of the direct emission.

3. General Relativistic Hydrodynamics Techniques

General-Relativistic hydrodynamics (GRHD) is a powerful tool to describe the accretion flow onto blackholes in the pertinent spacetimes.

The natural angular momentum distribution in thick discs is surveyed in [13]. In the work of GU et al. [14], the Astrophysical configurations of Rankine–Hugoniot shocks are schematized for viscous discs. The parameters studied in ibidem are the specific angular momentum of the accreting blackhole and the energy. The comparison with the inviscid discs is proposed according to the shock formation; more in detail, a continuity in the modifications of the parameter spaces from the inviscid case to the viscous one is postulated.

The dissipative flows are considered in [15]; the accretion rate and the angular momentum are taken as initial-value parameter sets for the study of the variation in the energy dissipation at the shock.

In the work of Das et al. [16], the methodology is implemented for the study of the axisymmetric flows onto Kerr blackholes being the spin of the blackhole part of the linearized expression of the location of the stationary critical points within the analysis of the transonic behaviour of the accretion object.

The experimental validation of the newly found role of the blackhole spin of the model should be observed within black hole-shadow images.

The study of galactic-centre black holes, i.e., such as Sgr A*, is performed after considering all the macroscopic matter orbiting the supermassive blackhole [17-20]; the methodology is traced back to [21].

3.1. The Analytical Studies

The case analyzed in [22] is one of a model of axially symmetric, stationary, rotating blackhole spacetime in a pseudo-Newtonian potential.

Ibidem, entropy is taken as varying at the shock locations only, the scheme remaining isoentropic, with adiabatic index γ = 4/3. Isothermal shocks are also studied.

The wind solutions are scrutinized.

The velocities of small acoustic perturbations are studied.

From ibidem, the variation in the energy as a function of the location of the critical points is plotted after the choice of a set of angular momenta.

In the further analysis of [23], the locations of the sonic points and those of the standing shock waves are investigated analytically with the aim of confirming the existence of global shocks in centrifugal pressure-dominated flows, where the gravitational-pressure term is neglected.

The analytical methods developed ibidem comprehend severe approximations. The standing shocks in the regions near the cusps are investigated and compared with the results of [22] as far as the location of the cusp is concerned. The numerical studies and the analytical ones are commented as not being in agreement about the determination of the qualities of the cusp when the specific energy is calculated as a function of the angular momentum.

Within the framework of the investigation of the shocks within the unit compression ratio, the variation in the energy, as a function of the specific angular momentum i, was researched again to identify the location of the cusp as that coinciding with the boundary of the disc. The possible mismatch between the approximated analytical methods and the numerical ones is once more outlined.

The analysis of [23] is targeted at extrapolating the spectral properties of the shock oscillations and their frequencies. The strategy to relate the theoretically found approximated results with the quasi-periodic oscillations (QPOs) of hard X-rays is adopted; more in detail, the QPOs are put ibidem in connection with the radiative-transfer phenomena of the Comptonization (innermost) region of the accretion disc.

3.2. The Numerical Investigations

In the case analyzed in [24], the GRHD system is implemented numerically in order to study the sub-Keplerian advection-dominated accretion objects which exhibit a cusp in black hole spacetimes.

Ibidem, the two instances are studied, i.e., the Schwarzschild spacetime and the Kerr one.

In the case the black hole is non-rotating, the initial conditions on the specific angular momentum and on the location of the inner edge of the accretion object are chosen for the accretion material not to fill the Roche lobe: the location of the cusp is calculated numerically accordingly.

In the case of the Kerr spacetime, the initial conditions on the specific angular momentum of the accretion objects, as well as that on the location of the inner edge, are set; the location of the cusp is calculated not to coincide with the inner edge of the disc.

The relevance of the position of the RH boundary conditions is therefore conceived in the study of the initial condition of the flow parameters, i.e., the specific angular momentum of the accretion object and its specific energy. The choice of the flow parameters defines the radial velocity of the accretion object, according to which the presence of (two) X-type critical sonic points is made possible. The wind solution is demonstrated to be involved with the shock jump. The radial variation in the Mach numbers can thus be studied numerically also in the case of the Kerr spacetimes.

4. Conclusion

Theoretical and experimental viewpoints are developed about the geometrical properties of the cusp(s), which are to qualify the different types of blackhole accretion objects.

The analytical theory referenced descended from the imposition of the Rankine–Hugoniot boundary conditions on the EFEs for the macroscopic-matter content (solution of the EFEs) of the blackhole accretion objects: their sonic behaviour is this way directly related to the arising of the cusp structures.

The presence of cusps modifies the spacetimes—the sought-after evidence in diverse blackhole figures is, in turn, testified in the present blackhole-imaging experiments and is also here advocated to be applied to future data analysis of high-resolution imaging.

The derivation of blackhole accretion objects cusps from non-geometrical theoretical endeavours has proven inadequate to describe the spacetimes of GR as ones of global properties; moreover, these theoretical attempts have also proven not to be reduced to any description of the solution of the EFE's nor to any Newtonian limit.

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Biography

Prof. Dr. Orchidea Maria Lecian graduated and defended her PhD Thesis at Sapienza University of Rome and ICRA, Italy. She was postdoctoral Fellow at IHES, Bures s/Yvette, and at Sapienza University of Rome. She was invited in intensive-research programmes, such as at the Max Plank Institute- Potsdam, at The Fields Institute for Research in Mathematical Sciences- Toronto, and at Milan State University. She received the SAIA-NS’P International Researcher’s Fellowship and was appointed Erasmus Lecturer at Comenius University-Bratislava . She was Visiting Professor at Kursk State University after the Programme Education in Russia. She was Assistant Professor and has been Associate Professor at Sapienza University of Rome. She has been serving several International Journals with editorial positions. She has participated in several National Conferences and International ones. She is member of several research consortia. She is author of research papers, review papers, conference-proceeding papers, encyclopedia entries, six books and several book chapters.