Spin and mass of the nearest supermassive black hole

Quasi-periodic oscillations (QPOs) of the hot plasma spots or clumps orbiting an accreting black hole contain information on the black hole mass and spin. The promising observational signatures for the measurement of black hole mass and spin are the latitudinal oscillation frequency of the bright spots in the accretion flow and the frequency of black hole event horizon rotation. Both of these frequencies are independent of the accretion model and defined completely by the properties of the black hole gravitational field. Interpretation of the known QPO data by dint of a signal modulation from the hot spots in the accreting plasma reveals the Kerr metric rotation parameter, \(a=0.65\pm 0.05\) , and mass, \(M=(4.2\pm 0.2)10^6M_\odot \) , of the supermassive black hole in the Galactic center. At the same time, the observed 11.5 min QPO period is identified with a period of the black hole event horizon rotation, and, respectively, the 19 min period is identified with a latitudinal oscillation period of hot spots in the accretion flow. The described approach is applicable to black holes with a low accretion rate, when accreting plasma is transparent up to the event horizon region.     

Horizon structure of rotating Bardeen black hole and particle acceleration

We investigate the horizon structure and ergosphere in a rotating Bardeen regular black hole, which has an additional parameter (g) due to the magnetic charge, apart from the mass (M) and the rotation parameter (a). Interestingly, for each value of the parameter g, there exists a critical rotation parameter (\(a=a_{E}\)), which corresponds to an extremal black hole with degenerate horizons, while for \(a<a_{E}\) it describes a non-extremal black hole with two horizons, and no black hole for \(a>a_{E}\). We find that the extremal value \(a_E\) is also influenced by the parameter g, and so is the ergosphere. While the value of \(a_E\) remarkably decreases when compared with the Kerr black hole, the ergosphere becomes thicker with the increase in g. We also study the collision of two equal mass particles near the horizon of this black hole, and explicitly show the effect of the parameter g. The center-of-mass energy (\(E_\mathrm{CM}\)) not only depend on the rotation parameter a, but also on the parameter g. It is demonstrated that the \(E_\mathrm{CM}\) could be arbitrarily high in the extremal cases when one of the colliding particles has a critical angular momentum, thereby suggesting that the rotating Bardeen regular black hole can act as a particle accelerator.     

Komar energy and Smarr formula for noncommutative inspired Schwarzschild black hole

We calculate the Komar energy E for a noncommutative inspired Schwarzschild black hole. A deformation from the conventional identity E = 2ST _H is found in the next to leading order computation in the noncommutative parameter θ (i.e. \({\mathcal{O}(\sqrt{\theta}e^{-M^2/\theta})}\)) which is also consistent with the fact that the area law now breaks down. This deformation yields a nonvanishing Komar energy at the extremal point T _H  = 0 of these black holes. We then work out the Smarr formula, clearly elaborating the differences from the standard result M = 2ST _H , where the mass (M) of the black hole is identified with the asymptotic limit of the Komar energy. Similar conclusions are also shown to hold for a deSitter–Schwarzschild geometry.     

Horizon thermodynamics in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) theory

We investigate whether the new horizon first law proposed recently still work in f(R) theory. We identify the entropy and the energy of black hole as quantities proportional to the corresponding value of integration, supported by the fact that the new horizon first law holds true as a consequence of equations of motion in f(R) theories. The formulas for the entropy and energy of black hole found here are in agreement with the results obtained in literatures. For applications, some nontrivial black hole solutions in f(R) theories have been considered, the entropies and the energies of black holes in these models are firstly computed, which may be useful for future researches.     

Quasi-normal Modes of Rerssner-Nordström Black Hole

The minimum interval of event horizon area of Rerssner-Nordström black hole was calculated via using the loop quantum gravity theory. Based on the first law of black hole thermodynamics, the real part of quasi-normal modes frequency of the black hole was calculated. The expression of asymptotically quasi-normal mode frequency of Rerssner-Nordström black hole was deduced strictly. By analyzing the value of the minimum spin j _{m i n} , the two families of quasi-normal mode spectra of the charged black hole were obtained for j _{m i n} = 1/2 and j _{m i n} = 1 respectively. Our conclusion is in complete agreement with the analytical results of Hod. Our results provide the theoretical basis for the source of the real part of the quasi-normal mode frequency of the black hole.     

Quintessence Reissner Nordström Anti de Sitter Black Holes and Joule Thomson Effect

In this work we investigate corrections of the quintessence regime of the dark energy on the Joule-Thomson (JT) effect of the Reissner Nordström anti de Sitter (RNAdS) black hole. The quintessence dark energy has equation of state as p _q = ωρ _q in which \(-1<\omega <-\frac {1}{3}\). Our calculations are restricted to ansatz: ω = ??1 (the cosmological constant regime) and \(\omega =-\frac {2}{3}\) (quintessence dark energy). To study the JT expansion of the AdS gas under the constant black hole mass, we calculate inversion temperature T _i of the quintessence RNAdS black hole where its cooling phase is changed to heating phase at a particular (inverse) pressure P _i . Position of the inverse point {T _i , P _i } is determined by crossing the inverse curves with the corresponding Gibbons-Hawking temperature on the T-P plan. We determine position of the inverse point versus different numerical values of the mass M and the charge Q of the quintessence AdS RN black hole. The cooling-heating phase transition (JT effect) is happened for M > Q in which the causal singularity is still covered by the horizon. Our calculations show sensitivity of the inverse point {T _i , P _i } position on the T-P plan to existence of the quintessence dark energy just for large numerical values of the AdS RN black holes charge Q. In other words the quintessence dark energy dose not affect on position of the inverse point when the AdS RN black hole takes on small charges.     

Effective information loss outside the horizon

If a system falls through a black hole horizon, then its information is lost to an observer at infinity. But we argue that the accessible information is lost before the horizon is crossed. The temperature of the hole limits information carrying signals from a system that has fallen too close to the horizon. Extremal holes have T = 0, but there is a minimum energy required to emit a quantum in the short proper time left before the horizon is crossed. If we attempt to bring the system back to infinity for observation, then acceleration radiation destroys the information. All three considerations give a critical distance from the horizon \({d\sim \sqrt\frac{r_H}{\Delta E}}\), where r _H is the horizon radius and ΔE is the energy scale characterizing the system. For systems in string theory where we pack information as densely as possible, this acceleration constraint is found to have a geometric interpretation. These estimates suggest that in theories of gravity we should measure information not as a quantity contained inside a given system, but in terms of how much of that information can be reliably accessed by another observer.     

The accretion of dark energy onto a black hole

The stationary, spherically symmetric accretion of dark energy onto a Schwarzschild black hole is considered in terms of relativistic hydrodynamics. The approximation of an ideal fluid is used to model the dark energy. General expressions are derived for the accretion rate of an ideal fluid with an arbitrary equation of state p = p(ρ) onto a black hole. The black hole mass was found to decrease for the accretion of phantom energy. The accretion process is studied in detail for two dark energy models that admit an analytical solution: a model with a linear equation of state, p = α(ρ ? ρ₀), and a Chaplygin gas. For one of the special cases of a linear equation of state, an analytical expression is derived for the accretion rate of dark energy onto a moving and rotating black hole. The masses of all black holes are shown to approach zero in cosmological models with phantom energy in which the Big Rip scenario is realized.     

Vaidya spacetime in the diagonal coordinates

We have analyzed the transformation from initial coordinates (v, r) of the Vaidya metric with light coordinate v to the most physical diagonal coordinates (t, r). An exact solution has been obtained for the corresponding metric tensor in the case of a linear dependence of the mass function of the Vaidya metric on light coordinate v. In the diagonal coordinates, a narrow region (with a width proportional to the mass growth rate of a black hole) has been detected near the visibility horizon of the Vaidya accreting black hole, in which the metric differs qualitatively from the Schwarzschild metric and cannot be represented as a small perturbation. It has been shown that, in this case, a single set of diagonal coordinates (t, r) is insufficient to cover the entire range of initial coordinates (v, r) outside the visibility horizon; at least three sets of diagonal coordinates are required, the domains of which are separated by singular surfaces on which the metric components have singularities (either g ₀₀ = 0 or g ₀₀ = ∞). The energy–momentum tensor diverges on these surfaces; however, the tidal forces turn out to be finite, which follows from an analysis of the deviation equations for geodesics. Therefore, these singular surfaces are exclusively coordinate singularities that can be referred to as false fire-walls because there are no physical singularities on them. We have also considered the transformation from the initial coordinates to other diagonal coordinates (η, y), in which the solution is obtained in explicit form, and there is no energy–momentum tensor divergence.     

TeV signatures of compact UHECR accelerators

We numerically study particle acceleration by the electric field induced near the horizon of a rotating supermassive (M ～ 10⁹–10¹⁰M_⊙) black hole embedded in the magnetic field B. We find that acceleration of protons to the energy E ～ 10²⁰ eV is possible only at extreme values of M and B. We also find that the acceleration is very inefficient and is accompanied by a broad-band MeV-TeV radiation whose total power exceeds the total power emitted in ultrahigh energy cosmic rays (UHECRs) at least by a factor of 1000. This implies that if O(10) nearby quasar remnants were sources of proton events with an energy E > 10²⁰ eV, then each quasar remnant would, e.g., overshine the Crab Nebula by more than two orders of magnitude in the TeV energy band. Recent TeV observations exclude this possibility. A model in which O(100) sources are situated at 100–1000 Mpc is not ruled out and can be experimentally tested by present TeV γ-ray telescopes. Such a model can explain the observed UHECR flux at moderate energies E ≈ (4–5) × 10¹⁹ eV.     

Boundedness of Massless Scalar Waves on Reissner-Nordström Interior Backgrounds

We consider solutions of the scalar wave equation \({\Box_g\phi=0}\), without symmetry, on fixed subextremal Reissner-Nordström backgrounds \({(\mathcal{M}, g)}\) with nonvanishing charge. Previously, it has been shown that for ? arising from sufficiently regular data on a two ended Cauchy hypersurface, the solution and its derivatives decay suitably fast on the event horizon \({\mathcal{H}^+}\). Using this, we show here that ? is in fact uniformly bounded, \({|\phi| \leq C}\), in the black hole interior up to and including the bifurcate Cauchy horizon \({\mathcal{C}\mathcal{H}^+}\), to which ? in fact extends continuously. The proof depends on novel weighted energy estimates in the black hole interior which, in combination with commutation by angular momentum operators and application of Sobolev embedding, yield uniform pointwise estimates. In a forthcoming companion paper we will extend the result to subextremal Kerr backgrounds with nonvanishing rotation.     

Solar system constraints on asymptotically flat IR modified Hořava gravity through light deflection

In this paper, we study the motion of photons around a Kehagias–Sfetsos (KS) black hole and obtain constraints on IR modified Ho?ava gravity without cosmological constant (~Λ _W ). An analytic formula for the light deflection angle is obtained. For a propagating photon, the deflection angle δφ increases with large values of the Ho?ava gravity parameter ω. Under the UV limit \({\omega \longrightarrow \infty}\), deflection angle reduces to the result of usual Schwarzschild case, 4GM/R. It is also found that with increasing scale of astronomical observation system the Ho?ava–Lifshitz gravity should satisfy |ω M ²| > 1.1725 × 10^?16 with 12% precision for Earth system, |ω M ²| > 8.27649 × 10^?17 with 17% precision for Jupiter system and |ω M ²| > 8.27650 × 10^?15 with 0.17% precision for solar system.     

<Emphasis Type="Italic">P</Emphasis>–<Emphasis Type="Italic">v</Emphasis> criticality in the extended phase space of a noncommutative geometry inspired Reissner–Nordström black hole in AdS space-time

The P–v criticality and phase transition in the extended phase space of a noncommutative geometry inspired Reissner–Nordström (RN) black hole in Anti-de Sitter (AdS) space-time are studied, where the cosmological constant appears as a dynamical pressure and its conjugate quantity is thermodynamic volume of the black hole. It is found that the P–v criticality and the small black hole/large black hole phase transition appear for the noncommutative RN-AdS black hole. Numerical calculations indicate that the noncommutative parameter affects the phase transition as well as the critical temperature, horizon radius, pressure and ratio. The critical ratio is no longer universal, which is different from the result in the van de Waals liquid–gas system. The nature of phase transition at the critical point is also discussed. Especially, for the noncommutative geometry inspired RN-AdS black hole, a new thermodynamic quantity \(\varPsi \) conjugate to the noncommutative parameter \(\theta \) has to be defined further, which is required for consistency of both the first law of thermodynamics and the corresponding Smarr relation.     

Almost Commutative Riemannian Geometry: Wave Operators

Associated to any (pseudo)-Riemannian manifold M of dimension n is an n + 1-dimensional noncommutative differential structure (Ω¹, d) on the manifold, with the extra dimension encoding the classical Laplacian as a noncommutative ‘vector field’. We use the classical connection, Ricci tensor and Hodge Laplacian to construct (Ω², d) and a natural noncommutative torsion free connection \({(\nabla,\sigma)}\) on Ω¹. We show that its generalised braiding \({\sigma:\Omega^1\otimes\Omega^1\to \Omega^1\otimes\Omega^1}\) obeys the quantum Yang-Baxter or braid relations only when the original M is flat, i.e. their failure is governed by the Riemann curvature, and that σ ² = id only when M is Einstein. We show that if M has a conformal Killing vector field τ then the cross product algebra \({C(M)\rtimes_\tau\mathbb{R}}\) viewed as a noncommutative analogue of \({M\times\mathbb{R}}\) has a natural n + 2-dimensional calculus extending Ω¹ and a natural spacetime Laplacian now directly defined by the extra dimension. The case \({M=\mathbb{R}^3}\) recovers the Majid-Ruegg bicrossproduct flat spacetime model and the wave-operator used in its variable speed of light prediction, but now as an example of a general construction. As an application we construct the wave operator on a noncommutative Schwarzschild black hole and take a first look at its features. It appears that the infinite classical redshift/time dilation factor at the event horizon is made finite.     

The generalization of charged AdS black hole specific volume and number density

In this paper, by proposing a generalized specific volume, we restudy the P–V criticality of charged AdS black holes in the extended phase space. The results show that most of the previous conclusions can be generalized without change, but the ratio \({\tilde{\rho }}_c\) should be \(3 {\tilde{\alpha }}/16\) in general case. Further research on the thermodynamical phase transition of black hole leads us to a natural interpretation of our assumption, and more black hole properties can be generalized. Finally, we study the number density for charged AdS black hole in higher dimensions, the results show the necessity of our assumption.     

Structure of geodesics in the regular Hayward black hole space-time

The regular Hayward model describes a non-singular black hole space-time. By analyzing the behaviors of effective potential and solving the equation of orbital motion, we investigate the time-like and null geodesics in the regular Hayward black hole space-time. Through detailed analyses of corresponding effective potentials for massive particles and photons, all possible orbits are numerically simulated. The results show that there may exist four orbital types in the time-like geodesics structure: planetary orbits, circular orbits, escape orbits and absorbing orbits. In addition, when \(\ell \), a convenient encoding of the central energy density \(3/8\pi \ell ^{2}\), is 0.6M, and b is 3.9512M as a specific value of angular momentum, escape orbits exist only under \(b>3.9512M\). The precession direction is also associated with values of b. With \(b=3.70M\) the bound orbits shift clockwise but counter-clockwise with \(b=5.00M\) in the regular Hayward black hole space-time. We also find that the structure of null geodesics is simpler than that of time-like geodesics. There only exist three kinds of orbits (unstable circle orbits, escape orbits and absorbing orbits).     

Higher-dimensional black holes with Dirac–Born–Infeld (DBI) global defects

It is well-known that the exact solution of non-linear \(\sigma \) model coupled to gravity can be perceived as an exterior gravitational field of a global monopole. Here we study Einstein’s equations coupled to a non-linear \(\sigma \) model with Dirac–Born–Infeld (DBI) kinetic term in D dimensions. The solution describes a metric around a DBI global defects. When the core is smaller than its Schwarzschild radius it can be interpreted as a black hole having DBI scalar hair with deficit conical angle. The solutions exist for all D, but they can be expressed as polynomial functions in r only when D is even. We give conditions for the mass M and the scalar charge \(\eta \) in the extremal case. We also investigate the thermodynamic properties of the black holes in canonical ensemble. The monopole alter the stability differently in each dimensions. As the charge increases the black hole radiates more, in contrast to its counterpart with ordinary global defects where the Hawking temperature is minimum for critical \(\eta \). This behavior can also be observed for variation of DBI coupling, \(\beta \). As it gets stronger (\(\beta \ll 1\)) the temperature increases. By studying the heat capacity we can infer that there is no phase transition in asymptotically-flat spacetime. The AdS black holes, on the other hand, undergo a first-ordered phase transition in the Hawking–Page type. The increase of the DBI coupling renders the phase transition happen for larger radius.     

<Emphasis Type="Italic">f</Emphasis> (<Emphasis Type="Italic">R</Emphasis>) black holes

We study the f (R)-Maxwell black hole imposed by constant curvature and its all thermodynamic quantities, which may lead to the Reissner-Nordström-AdS black hole by redefining Newtonian constant and charge. Further, we obtain the f (R)-Yang-Mills black hole imposed by constant curvature, which is related to the Einstein-Yang-Mills black hole in AdS space. Since there is no analytic black hole solution in the presence of Yang-Mills field, we obtain asymptotic solutions. Then, we confirm the presence of these solutions in a numerical way.