It is known that, in the noncommutative Schwarzschild black hole spacetime, the point-like object is replaced by the smeared object, whose mass density is described by a Gaussian distribution of minimal width $\sqrtθ$ with $θ$ the noncommutative parameter. The elimination of the point-like structures makes it quite different from the conventional Schwarzschild black hole. In this paper, we mainly investigate the area spectrum and entropy spectrum for the noncommutative Schwarzschild black hole with $0\leq θ\leq (\frac{M}{1.90412})^{2}$. By the use of the new physical interpretation of the quasinormal modes of black holes presented by Maggiore, we obtain the quantized area spectrum and entropy spectrum with the modified Hod's and Kunstatter's methods, respectively. The results show that: (1) The area spectrum and entropy spectrum are discrete. (2) The spectrum spacings are dependent on the parameter $\frac{M}{\sqrtθ}$. (3) The spacing of the area spectrum of the noncommutative Schwarzschild black hole is smaller than that of the conventional one. So does the spacing of the entropy spectrum. (4) The spectra from the two methods are consistent with each other.
A deformed Schwarzschild solution in noncommutative gauge theory of gravitation is obtained. The gauge potentials (tetrad fields) are determined up to the second order in the noncommutativity parameters $Θ^{μν}$. A deformed real metric is defined and its components are obtained. The noncommutativity correction to the red shift test of General Relativity is calculated and it is concluded that the correction is too small to have observable effects. Implications of such a deformed Schwarzschild metric are also mentioned.
We show the importance of the Voros product in defining a noncommutative Schwarzschild black hole. The corrected entropy/area-law is then computed in the tunneling formalism. Two types of corrections are considered; one, due to the effects of noncommutativity and the other, due to the effects of going beyond the semiclassical approximation. The leading correction to the semiclassical entropy/area-law is logarithmic and its coefficient involves the noncommutative parameter.
We apply the generalization of the Parikh-Wilczek method to the tunneling of massive particles from noncommutative inspired Schwarzschild black holes. By deriving the equation of radial motion of the tunneling particle directly, we calculate the emission rate which is shown to be dependent on the noncommutative parameter besides the energy and mass of the tunneling particle. After equating the emission rate to the Boltzmann factor, we obtain the modified Hawking temperature which relates to the noncommutativity and recovers the standard Hawking temperature in the commutative limit. We also dis…
Using a graphical analysis, we show that for the horizon radius $r_h\gtrsim 4.8\sqrtθ$, the standard semiclassical Bekenstein-Hawking area law for noncommutative Schwarzschild black hole exactly holds for all orders of $θ$. We also give the corrections to the area law to get the exact nature of the Bekenstein-Hawking entropy when $r_h<4.8\sqrtθ$ till the extremal point $r_h=3.0\sqrtθ$.
Based on the perturbation expansion, we compute the noncommutative corrections to the minimal surface areas of the pure AdS spacetime and Schwarzschild-AdS black hole, where the noncommutative background is suitably constructed in terms of the Poincaré coordinate system. In particular, we find a reasonable tetrad with subtlety, which not only matches the metrics of the pure AdS spacetime and Schwarzschild-AdS black hole in the commutative case, but also makes the corrections real rather than complex in the noncommutative case. For the pure AdS spacetime, the nocommutative effect is only a loga…
In this paper we study through tunneling formalism, the effect of noncommutativity to Hawking radiation and the entropy of the noncommutative Schwarzschild black hole. In our model we have considered the noncommutativity implemented via the Lorentzian distribution. We obtain non-commutative corrections to the Hawking temperature using the Hamilton-Jacobi method and the Wentzel-Kramers-Brillouin (WKB) approximation. In addition, we found corrections of the logarithmic and other types due to noncommutativity and quantum corrections from the generalized uncertainty principle (GUP) for the entropy…
We investigate the behavior of a noncommutative radiating Schwarzschild black hole. It is shown that coordinate noncommutativity cures usual problems encountered in the description of the terminal phase of black hole evaporation. More in detail, we find that: the evaporation end-point is a zero temperature extremal black hole even in the case of electrically neutral, non-rotating, objects; there exists a finite maximum temperature that the black hole can reach before cooling down to absolute zero; there is no curvature singularity at the origin, rather we obtain a regular DeSitter core at shor…
In this contribution we present a noncanonical phase-space noncommutative (NC) extension of a Kantowski Sachs (KS) cosmological model to describe the interior of a Schwarzschild black hole (BH). We evaluate the thermodynamical quantities inside this NC Schwarzschild BH and compare with the well known quantities. We find that for a NCBH the temperature and entropy have the same mass dependence as the Hawking quantities for a Schwarzschild BH.
We study modifications of the Schwarzschild solution within the noncommutative gauge theory of gravity. In the present analysis, the deformed solutions are obtained by solving the field equations perturbatively, up to the second order in the noncommutativity parameter $Θ$, for both exterior and interior solutions of the equations of motion for $e_μ^a \left(x\right)$. Remarkably, we find that this new noncommutive solution is analogous to the Reissner-Nordström solution in the ordinary spacetime, in which the square of electric charge is replaced by the square of the noncommutativity parameter,…
The importance of the Voros product in defining a noncommutative Schwarzschild black hole is shown. The entropy is then computed and the area law is shown to hold upto order $\frac{1}{\sqrtθ}e^{-M^2/θ}$. The leading correction to the entropy (computed in the tunneling formalism) is shown to be logarithmic. The Komar energy $E$ for these black holes is then obtained and a deformation from the conventional identity $E=2ST_H$ is found at the order $\sqrtθe^{-M^2/θ}$. This deformation leads to a nonvanishing Komar energy at the extremal point $T_{H}=0$ of these black holes. Finally, the Smarr form…
We study the effects of noncommutative spaces on the horizon, the area spectrum and Hawking temperature of a Schwarzschild black hole. The results show deviations from the usual horizon, area spectrum and the Hawking temperature. The deviations depend on the parameter of space-space noncommutativity.
A quantum Schwarzschild spacetime and a quantum Schwarzschild-de Sitter spacetime with cosmological constant $Λ$ are constructed within the framework of a noncommutative Riemannian geometry developed in an earlier publication. The metrics and curvatures of the quantum Schwarzschild spacetime and the quantum Schwarzschild-de Sitter spacetime are computed. It is shown that up to the second order in the deformation parameter, the quantum spacetimes are solutions of a noncommutative Einstein equation which is proposed in this paper.
This work aims to implement the idea of noncommutativity in the subject of black holes. Its principal contents deal with a study of Schwarzschild black holes in a D-dimensional noncommutative space. Various aspects related to the non commutative extension are discussed and some non trivial results are derived.
In this paper we study noncommutative black holes. We use a diffeomorphism between the Schwarzschild black hole and the Kantowski-Sachs cosmological model, which is generalized to noncommutative minisuperspace. Through the use of the Feynman-Hibbs procedure we are able to study the thermodynamics of the black hole, in particular, we calculate the Hawking's temperature and entropy for the noncommutative Schwarzschild black hole.
In this paper we investigate the problem of ordinary baryonic matter accretion onto the noncommutative geometry inspired Schwarzschild black hole. The fundamental equations governing the spherically symmetric steady state matter accretion are deduced. These equations are seen to be modified due to the presence of noncommutativity. The matter accretion rate is computed and is found to increase rapidly with the increase in strength of the noncommutative parameter. The sonic radius reduces while the sound speed at the sonic point increases with the increase in the strength of noncommutativity. Th…
The event horizon of Schwarzschild black hole is obtained in noncommutative spaces up to the second order of perturbative calculations. Because this type of black hole is non-rotating, to the first order there is no any effect on the event horizon due to the noncommutativity of space. A lower limit for the noncommutativity parameter is also obtained. As a result, the event horizon in noncommutative spaces is less than the event horizon in commutative spaces.
We emphasize the importance of the Voros product in defining noncommutative inspired black holes. The computation of entropy for both the noncommutative inspired Schwarzschild and Reissner-Nordström black holes show that the area law holds upto order $\frac{1}{\sqrtθ}e^{-M^2/θ}$. The leading correction to the entropy (computed in the tunneling formalism) is shown to be logarithmic. The Komar energy $E$ for these black holes is then obtained and a deviation from the standard identity $E=2ST_H$ is found at the order $\sqrtθe^{-M^2/θ}$. This deviation leads to a nonvanishing Komar energy at the e…
We calculate the Komar energy $E$ for a noncommutative 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. $\mathcal{O}(\sqrtθe^{-M^2/θ})$) 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 identifie…
We present a toy model of fuzzy Schwarzschild space slice (as a noncommutative manifold) which quantum mean values and quantum quasi-coherent states (states minimizing the quantum uncertainties) have properties close to the classical slice of the $(r,θ)$ Schwarzschild coordinates (the so-called Flamm's paraboloid). This fuzzy Schwarzschild slice is built as a deformation of the noncommutative plane. Quantum time observables are introduced to add a time quantization in the model. We study the structure of the quasi-coherent state of the fuzzy Schwarzschild slice with respect to the quasi-cohere…
We find a new rotating black hole in three-dimensional anti-de Sitter space using an anisotropic perfect fluid inspired by the noncommutative black hole. We deduce the thermodynamical quantities of this black hole and compare them with those of a rotating BTZ solution.
We investigate the effects of space noncommutativity and the generalized uncertainty principle on the thermodynamics of a radiating Schwarzschild black hole. We show that evaporation process is in such a way that black hole reaches to a maximum temperature before its final stage of evolution and then cools down to a nonsingular remnant with zero temperature and entropy. We compare our results with more reliable results of string theory. This comparison Shows that GUP and space noncommutativity are similar concepts at least from view point of black hole thermodynamics.
Using a noncommutative inspired Einstein-Euler-Heisenberg black hole, we analyse the associated graviton and light rings that together with the event horizon, provide the necessary components to build its shadow. An analysis of the angular radius of the noncommutative inspired Einstein-Euler-Heisenberg black hole shows that for observers located in its vicinity, it becomes smaller when compared with the commutative case; this effect is more noticeable as noncommutativity increases. The existence of marginally stable bound orbits and the critical angle for the escaping of photons of a noncommut…
A phase-space noncommutativity in the context of a Kantowski-Sachs cosmological model is considered to study the interior of a Schwarzschild black hole. Due to the divergence of the probability of finding the black hole at the singularity from a canonical noncommutativity, one considers a non-canonical noncommutativity. It is shown that this more involved type of noncommutativity removes the problem of the singularity in a Schwarzschild black hole.
In this paper we propose a generalization to the Schwarzschild metric and define noncommutative SUSY black holes. We introduce the noncommutative deformation to the minisuperspace variables and derive the noncommutative supersymmetric (SUSY) Wheeler-DeWitt (WDW) equation for the Schwarzschild black hole. We calculate the metric and find that the singularities are not removed.
Recently, a new perspective of gravitational-thermodynamic duality as an entropic force arising from alterations in the information connected to the positions of material bodies is found. In this paper, we generalize some aspects of this model in the presence of noncommutative Schwarzschild black hole by applying the method of coordinate coherent states describing smeared structures. We implement two different distributions: (a) Gaussian and (b) Lorentzian. Both mass distributions prepare the similar quantitative aspects for the entropic force. Our study shows, the entropic force on the smalle…
We find a new charged black hole in three-dimensional anti-de Sitter space using an anisotropic perfect fluid inspired by the noncommutative black hole as the source of matter and a gaussian distribution of electric charge. We deduce the thermodynamical quantities of this black hole and compare them with those of a charged BTZ solution.
We emphasize the importance of the Voros product in defining a noncommutative inspired Reissner-Nordström black hole. The entropy of this black hole is then computed in the tunneling approach and is shown to obey the area law at the next to leading order in the noncommutative parameter $θ$. Modifications to entropy/area law is then obtained by going beyond the semi-classical approximation. The leading correction to the semiclassical entropy/area law is found to be logarithmic and its coefficient involves the noncommutative parameter $θ$.
We study the asymptotic quasi-normal modes for the scalar perturbation of the non-commutative geometry inspired Schwarzschild black hole in (3+1) dimensions. We have considered $M\geq M_0$, which effectively correspond to a single horizon Schwarzschild black hole with correction due to non-commutativity. We have shown that for this situation the real part of the asymptotic quasi-normal frequency is proportional to $\ln (3)$. The effect of non-commutativity of spacetime on quasi-normal frequency arises through the constant of proportionality, which is Hawking temperature $T_H(θ)$. We also consi…
Black holes in noncommutative geometry background are considered to be quantized in accordance with the holographic principle. Incomplete gamma function involving the effective black hole mass is replaced by a discrete sum. The mass spectrum as well as the temperature of black holes is presented. The spectra are discrete ones but the shapes are very consistent with the known continuous results.
We consider noncommutative gravity on a space with canonical noncommutativity that is based on the commutative MacDowell-Mansouri action. Gravity is treated as gauge theory of the noncommutative $SO(1,3)_\star$ group and the Seiberg-Witten (SW) map is used to express noncommutative fields in terms of the corresponding commutative fields. In the commutative limit the noncommutative action reduces to the Einstein-Hilbert action plus the cosmological term and the topological Gauss-Bonnet term. After the SW expansion in the noncommutative parameter the first order correction to the action, as expe…
Using the noncommutative geometry inspired Vaidya metric obtained in terms of coordinate coherent states and also utilizing the generalized uncertainty principle (GUP), we show that the nonthermal nature of the Hawking spectrum leads to Planck-scale nonthermal correlations between emitted modes of evaporation. Our analysis thus exhibits that owing to self-gravitational effects plus noncommutativity and GUP influences, information can emerge in the form of Planck-scale correlated emissions from the black hole.
We find a new, non-commutative geometry inspired, solution of the coupled Einstein-Maxwell field equations describing a variety of charged, self-gravitating objects, including extremal and non-extremal black holes. The metric smoothly interpolates between deSitter geometry, at short distance, and Reissner-Nordstroem geometry far away from the origin. Contrary to the ordinary Reissner-Nordstroem spacetime there is no curvature singularity in the origin neither "naked" nor shielded by horizons. We investigate both the Hawking process and pair creation in this new scenario.
We derive noncommutative Einstein equations for abelian twists and their solutions in consistently symmetry reduced sectors, corresponding to twisted FRW cosmology and Schwarzschild black holes. While some of these solutions must be rejected as models for physical spacetimes because they contradict observations, we find also solutions that can be made compatible with low energy phenomenology, while exhibiting strong noncommutativity at very short distances and early times.
We discuss the thermodynamics of the noncommutative inspired Kerr black hole by means of a reformulated Hamilton-Jacobi method and a dimensional reduction technique. In order to investigate the effect of the angular momentum of the tunneling particle, we calculate the wave function to the first order of the WKB ansatz. Then, using a density matrix technique we derive the radiation spectrum from which the radiation temperature can be read out. Our results show that the radiation of this noncommutative inspired black hole corresponds to a modified temperature which involves the effect of noncomm…
We investigated regular black holes with fuzzy sources in three and four dimensions. The density distributions of such fuzzy sources are inspired by noncommutative geometry and given by Gaussian or generalized Gaussian functions. We utilized mass functions to give a physical interpretation of the horizon formation condition for the black holes. In particular, we investigated three-dimensional BTZ-like black holes and four-dimensional Schwarzschild-like black holes in detail, and found that the number of horizons is related to the spacetime dimensions, and the existence of a void in the vicinit…
In this review article we discuss some of the applications of noncommutative geometry in physics that are of recent interest, such as noncommutative many-body systems, noncommutative extension of Special Theory of Relativity kinematics, twisted gauge theories and noncommutative gravity.
Together with collaborators, we introduced a noncommutative Riemannian geometry over Moyal algebras and systematically developed it for noncommutative spaces embedded in higher dimensions in the last few years. The theory was applied to construct a noncommutative version of general relativity, which is expected to capture some essential structural features of spacetime at the Planck scale. Examples of noncommutative spacetimes were investigated in detail. These include quantisations of plane-fronted gravitational waves, quantum Schwarzschild spacetime and Schwarzschild-de Sitter spacetime, and…
We study the formation of the (noncommutative) Schwarzschild black hole from collapsing shell {of the} generalized matters containing polytropic and Chaplygin gas. We show that this collapsing shell depending on various parameters forms either a black hole or a naked singular shell with the help of the pressure.Furthermore, by considering the smeared gravitational sources, we investigate the noncommutative black holes formation. Though this mild noncommutative correction of matters cannot ultimately resolve the emergence of the naked singularity, we show that in some parameter region the colla…
Recently Verlinde proposed that gravity can be described as an emergent phenomena arising from changes in the information associated with the positions of material bodies. By using noncommutative geometry as a way to describe the microscopic, microstructure of quantum spacetime, we derive modified Friedmann equation in this setup and study the entropic force modifications to the inflationary dynamics of early universe.
We calculate leading long-distance noncommutative corrections to the classical Schwarzschild black hole which is sourced by a massive noncommutative scalar field. The energy-momentum tensor is taken up to ${\cal O}(\ell^4)$ in noncommutative parameter, and is treated in semiclassical (tree level) approximation. These noncommutative corrections can dominate classical post-post-Newtonian corrections providing $\ell > 1/M_P$, however, they are still too small to be observable in present-day experiments.
Recently, a new noncommutative geometry inspired solution of the coupled Einstein-Maxwell field equations including black holes in 4-dimension is found. In this paper, we generalize some aspects of this model to the Reissner-Nordström (RN) like geometries with large extra dimensions. We discuss Hawking radiation process based on noncommutative inspired solutions. In this framework, existence of black hole remnant and possibility of its detection in LHC are investigated.
In this paper we investigate quasinormal modes (QNM) for a scalar field around a noncommutative Schwarzschild black hole. We verify the effect of noncommutativity on quasinormal frequencies by applying two procedures widely used in the literature. The first is the Wentzel-Kramers-Brillouin (WKB) approximation up to sixth order. In the second case we use the continuous fraction method developed by Leaver. Besides, we also show that due to noncommutativity, the shadow radius is reduced when we increase the noncommutative parameter. In addition, we find that the shadow radius is nonzero even at t…
When embedding models of noncommutative geometry inspired black holes into the peridium of large extra dimensions, it is natural to relate the noncommutativity scale to the higher-dimensional Planck scale. If the Planck scale is of the order of a TeV, noncommutative geometry inspired black holes could become accessible to experiments. In this paper, we present a detailed phenomenological study of the production and decay of these black holes at the Large Hadron Collider (LHC). Noncommutative inspired black holes are relatively cold and can be well described by the microcanonical ensemble durin…
We search for all Poisson brackets for the BTZ black hole which are consistent with the geometry of the commutative solution and are of lowest order in the embedding coordinates. For arbitrary values for the angular momentum we obtain two two-parameter families of contact structures. We obtain the symplectic leaves, which characterize the irreducible representations of the noncommutative theory. The requirement that they be invariant under the action of the isometry group restricts to $R\times S^1$ symplectic leaves, where $R$ is associated with the Schwarzschild time. Quantization may then le…
We show at one-loop and first order in the noncommutativity parameters that in any noncommutative GUT inspired theory the total contribution to the fermionic four point functions coming only from the interaction between fermions and gauge bosons, though not UV finite by power counting, is UV finite at the end of the day. We also show that this is at odds with the general case for noncommutative gauge theories --chiral or otherwise-- defined by means of Seiberg-Witten maps that are the same --barring the gauge group representation-- for left-handed spinors as for right-handed spinors. We believ…
Thermodynamic metric usually works only for those black holes with more than one conserved charge, therefore the Schwarzschild black hole was excluded. In this letter, we compute and compare different versions of offshell thermodynamic metric for the Schwarzschild-like black hole by introducing a new degree of freedom. This new degree of freedom could be the running Newton constant, a cutoff scale for regular black hole, a noncommutative deformation, or the deformed parameter in the nonextensive Tsallis-Renyi entropy. The onshell metric of the deformed Schwarzschild solution would correspond t…
In order to find a noncommutative analog of Schwarzschild or Schhwarzschild-de Sitter blackhole we investigate spherically symmetric spaces generated by four noncommutative coordinates in the frame formalism. We present two solutions which however do not posess the prescribed commutative limit. Our analysis indicates that the appropriate noncommutative space might be found as a subspace of a higher-dimensional space.
Under two different metric ansatzes, the noncommutative geometry inspired black holes (NCBH) in the framework of Rastall gravity are derived and analyzed. We consider the fluid-type matter with the Gaussian-distribution smeared mass density. Taking a Schwarzschild-like metric ansatz, it is shown that the noncommutative geometry inspired Schwarzschild black hole (NCSBH) in Rastall gravity, unlike its counterpart in general relativity (GR), is not a regular black hole. It has at most one event horizon. After showing a finite maximal temperature, the black hole will leave behind a point-like mass…
We study the influence of the spacetime noncommutative parameter on the strong field gravitational lensing in the noncommutative Reissner-Nordström black-hole spacetime. Supposing that the gravitational field of the supermassive central object of the Galaxy is described by this metric, we estimate the numerical values of the coefficients and observables for strong gravitational lensing. Our results show that with the increase of the parameter $\sqrt{\vartheta}$, the observables $θ_{\infty}$ and $r_m$ decrease, while $s$ increases. Our results also show that i) if $\sqrt{\vartheta}$ is strong, …
One considers phase-space noncommutativity in the context of a Kantowski-Sachs cosmological model to study the interior of a Schwarzschild black hole. It is shown that the potential function of the corresponding quantum cosmology problem has a local minimum. One deduces the thermodynamics and show that the Hawking temperature and entropy exhibit an explicit dependence on the momentum noncommutativity regime and it is shown that the wave function vanishes in this limit.
We study the problem of a Schwarzschild-anti-deSitter black hole in a noncommutative geometry framework, thought to be an effective description of quantum-gravitational spacetime. As a first step we derive the noncommutative geometry inspired Schwarzschild-anti-deSitter solution. After studying the horizon structure, we find that the curvature singularity is smeared out by the noncommutative fluctuations. On the thermodynamics side, we show that the black hole temperature, instead of a divergent behavior at small scales, admits a maximum value. This fact implies an extension of the Hawking-Pag…
The Born-Infeld lagrangian for non-abelian gauge theory is adapted to the case of the generalized gauge fields arising in non-commutative matrix geometry. Basic properties of static and time dependent solutions of the scalar sector of this model are investigated.
We analyze the thermodynamics of the noncommutative high-dimensional Schwarzschild-Tangherlini AdS black hole with the non-Gaussian smeared matter distribution by regarding a noncommutative parameter as an independent thermodynamic variable named as the noncommutative pressure. In the new extended phase space that includes this noncommutative pressure and its conjugate variable, we reveal that the noncommutative pressure and the original thermodynamic pressure related to the negative cosmological constant make the opposite effects in the phase transition of the noncommutative black hole, i.e. …
We study a solution of the Einstein's equations generated by a self-gravitating, anisotropic, static, non-singular matter fluid. The resulting Schwarzschild like solution is regular and accounts for smearing effects of noncommutative fluctuations of the geometry. We call this solution regular Schwarzschild spacetime. In the presence of an Anti-deSitter cosmological term, the regularized metric offers an extension of the Hawking-Page transition into a van der Waals-like phase diagram. Specifically the regular Schwarzschild-Anti-deSitter geometry undergoes a first order small/large black hole tr…
