We present a modified Schwarzschild solution for a model of evaporation of a black hole with information preservation. By drawing a direct analogy to the quantum pure accelerating mirror (dynamical Casimir effect of a 1D horizon), we derive a Schwarzschild metric with not only the usual Schwarzschild radius but an additional length scale related to the Planck length. The black hole has thermal particle production that leads to complete evaporation of the black hole, resulting in non-divergent entanglement entropy, Page curve turn-over, and an asymptotic quantum pure state with no information loss.
It is shown that the extremal Reissner-Nordström black hole, the non-extremal one with multiple scattering particles, and the Schwarzschild black hole with radial head-on particles are stable under the collision of the particles near the horizon, if the back-reaction effect and the effect generated by gravity of particles are involved. Moreover, the collision near Reissner-Nordström black holes with astrophysically typical mass can not generate the Planck-scale center-of-mass energy. However, the head-on collision near the typical primordial black hole could just occur at the Planck-energy sca…
Considering corrections to all orders in the Planck length on the quantum state density from a generalized uncertainty principle (GUP), we calculate the statistical entropy of the scalar field on the background of the Schwarzschild black hole without any cutoff. We obtain the entropy of the massive scalar field proportional to the horizon area.
We revisit the connection between trajectories of accelerated mirrors and spacetime metrics. We present the general (1+1)D effective metric that can be obtained with a fibre-optical analogue through the Kerr effect. Then we introduce a new connection between accelerated mirrors and the optical metric. In particular, we connect them for two specific trajectories: The first one is the black mirror that perfectly recreates the Schwarzchild spacetime. The second one is the Schwarzschild-Planck metric that is a regularized version of the Schwarzschild case. The regularization depends on a length sc…
A simple model is suggested by assuming an effective interaction at the crossing point of the two curves, the Schwarzschild radius and the Compton wave length, that is at the Planck length and Planck mass. It is argued that there would be a physical effect that may be measurable at, say, nucleon mass and size, while the Schwarzschild radius would remain unaffected for macroscopic lengths and masses.
A winding number analysis of Schwarzschild black hole stability in light of Planck-scale modified kinematics
Institute for Research in Fundamental Sciences (IPM)（基础科学研究院）；Damghan University（达姆甘大学）；Khazar University（克扎尔大学）；Fesenkov Astrophysical Institute（费森柯夫天体物理研究所）；Al-Farabi Kazakh National University（阿里-法拉比哈萨克国立大学）；Baku State University（巴库国立大学）
We propose a novel nonsingular black-hole spacetime representing a strong deformation of the Schwarzschild solution with mass $M$ by an additional hair $\ell$, which may be hierarchically larger than the Planck scale. Our black-hole model presents a de Sitter core and $\mathcal{O}(\ell^2/r^2)$ slow-decaying corrections to the Schwarzschild solution. Our black-hole solutions are thermodynamically preferred when $0.2 \lesssim \ell/GM \lesssim \, 0.3$ and are characterized by strong deviations in the orbits of test particles from the Schwarzschild case. In particular, we find corrections to the p…
The Black Hole Uncertainty Principle correspondence suggests that there could exist black holes with mass beneath the Planck scale but radius of order the Compton scale rather than Schwarzschild scale. We present a modified, self-dual Schwarzschild-like metric that reproduces desirable aspects of a variety of disparate models in the sub-Planckian limit, while remaining Schwarzschild in the large mass limit. The self-dual nature of this solution under $M \leftrightarrow M^{-1}$ naturally implies a Generalized Uncertainty Principle with the linear form $Δx \sim \frac{1}{Δp} + Δp$. We also demons…
Though popular presentations give the Schwarzschild singularity as a point it is known that it is spacelike and not timelike. Thus it has a "length" and is not a "point". In fact, its length must necessarily be infinite. It has been proved that the proper length of the Qadir-Wheeler suture model goes to infinity [1], while its proper volume shrinks to zero, and the asymptotic behaviour of the length and volume have been calculated. That model consists of two Friedmann sections connected by a Schwarzschild "suture". The question arises whether a similar analysis could provide the asymptotic beh…
Using the canonical quantum theory of spherically symmetric pure gravitational systems, we present a direct correspondence between the Friedmann-Robertson-Walker (FRW) cosmological model in the interior of a Schwarzschild black hole and the nth energy eigenstate of a linear harmonic oscillator. Such type of universe has a quantized mass of the order of the Planck mass and harmonic oscillator wave functions
We consider a spacetime foam model of the Schwarzschild horizon, where the horizon consists of Planck size black holes. According to our model the entropy of the Schwarzschild black hole is proportional to the area of its event horizon. It is possible to express geometrical arguments to the effect that the constant of proportionality is, in natural units, equal to one quarter.
We investigate the thermodynamics of Schwarzschild-Tangherlini black hole in the context of the generalized uncertainty principle. The corrections to the Hawking temperature, entropy and the heat capacity are obtained via the modified Hamilton-Jacobi equation. These modifications show that the GUP changes the evolution of Schwarzschild-Tangherlini black hole. Specially, the GUP effect becomes susceptible when the radius or mass of black hole approach to the order of Planck scale, it stops radiating and leads to black hole remnant. Meanwhile, the Planck scale remnant can be confirmed through th…
In three spatial dimensions, the Compton wavelength $(R_C \propto M^{-1}$) and Schwarzschild radius $(R_S \propto M$) are dual under the transformation $M \rightarrow M_{P}^2/M$, where $M_{P}$ is the Planck mass. This suggests that there is a fundamental link -- termed the Black Hole Uncertainty Principle or Compton-Schwarzschild correspondence -- between elementary particles in the $M < M_{P}$ regime and black holes in the $M > M_{P}$ regime. In the presence of $n$ extra dimensions, compactified on some scale $R_E$, one expects $R_S \propto M^{1/(1+n)}$ for $R < R_E$, which breaks this dualit…
We consider scalar field entanglement entropy generated with black hole of (sub)planck mass scale thus implying the unitary evolution of gravity. The dependence on the dimension of the Hilbert space for degrees of freedom located behind the horizon is taken into account. The obtained results contain polylogarithmic terms.
In three spatial dimensions, the Compton wavelength $(R_C \propto M^{-1}$) and Schwarzschild radius $(R_S \propto M$) are dual under the transformation $M \rightarrow M_{P}^2/M$, where $M_{P}$ is the Planck mass. This suggests that there could be a fundamental link -- termed the Black Hole Uncertainty Principle or Compton-Schwarzschild correspondence -- between elementary particles with $M < M_{P}$ and black holes in the $M > M_{P}$ regime. In the presence of $n$ extra dimensions, compactified on some scale $R_E$ exceeding the Planck length $R_P$, one expects $R_S \propto M^{1/(1+n)}$ for $R_P…
Based on the perspective that continuum gravitational physics is an emergent quantum gravitational phenomenon, and that spacetime thermodynamic is the natural langauge in which it can be described, we derive a modified Schwarzschild-de Sitter geometry in static coordinates from quantum gravity inspired logarithmic corrections to the entropy-area law. The resulting quantum corrections to classical geometry can be encoded in explicitly radius dependent black hole and cosmological constant parameters. By considering the pure black hole case we study linear perturbations on this modified backgroun…
Greybody factors in black hole physics modify the naive Planckian spectrum that is predicted for Hawking radiation when working in the limit of geometrical optics. We consider the Schwarzschild geometry in (3+1) dimensions, and analyze the Regge-Wheeler equation for arbitrary particle spin S and wave-mode angular momentum L, deriving rigourous bounds on the greybody factors as a function of S, L, wave frequency (omega), and the black hole mass, m.
By considering the back-reaction of the spacetime through the spherically symmetric quantum fluctuations of the background metric, Kazakov and Solodukhin removed the singularity of the Schwarzschild black hole. This regular Schwarzschild black hole has a spherical central region with a radius of the order of the Planck length. On the other hand, due to the positively accelerating expansion of the Universe, it seems that there exists a universal repulsive force known as dark energy. In the framework of quantum field theories, the quintessence field is a candidate model for investigating and mod…
Mass thresholds, lifetimes, entropy and heat capacity for micro black holes close to their late Schwarzschild phase are computed using two different generalized uncertainty principles, in the framework of models with extra spatial dimensions. Emissions of both photons and gravitons (in the bulk) are taken into account. Results are discussed and compared.
We construct a kind of thermal potential and then put the black hole thermodynamic system in it. In this regard, some thermodynamic properties of the black hole are related to the geometric characteristics of the thermal potential. Driven by the intrinsic thermodynamic fluctuations, the behavior of the black hole in the thermal potential is stochastic. With the help of solving the Fokker-Planck equation analytically, we obtain the discrete energy spectrum of Schwarzschild and Banados-Teitelboim-Zanelli (BTZ) black holes in the thermal potential. For Schwarzschild black hole, the energy spectru…
We discuss quantum black holes in asymptotically safe quantum gravity with a scale identification based on the Kretschmann scalar. After comparing this scenario with other scale identifications, we investigate in detail the Kerr-(A)dS and Schwarzschild-(A)dS space-times. The global structure of these geometries is studied as well as the central curvature singularity and test particle trajectories. The existence of a Planck-sized, extremal, zero temperature black hole remnant guarantees a stable endpoint of the evaporation process via Hawking radiation.
Quantum gravity has exciting peculiarities on the Planck scale.The effect of generalized uncertainty principle (GUP) to the entangled scalar/fermion particles' tunneling from a Schwarzschild black hole immersed in an electromagnetic Universe is investigated by the help of semi-classical tunneling method. The quantum corrected Hawking temperature of this black hole with an external parameter "a" modifies the Hawking temperature for the entangled particles.
It was recently claimed that the Planck length is not a limit to the precision by which we can measure distances, but that instead it is merely the Planck volume that limits the precision by which we can measure volumes. Here, we investigate this claim and show that the argument does not support the conclusion.
In this paper we suggest that the Planck length $l_{pl}$ and the Cosmological Constant scale $r_Λ=\frac{1}{\sqrtΛ}$ could in principle be dual each other if we take seriously the so-called q-Bargmann Fock space representation as has been previously suggested by Kempf and others and if additionally we introduce $l_{pl}$ as an ultraviolet cut-off and $r_Λ=\frac{1}{\sqrtΛ}$ as an infrared one. As a consequence, it is possible to demonstrate that a Generalized Uncertainty Principle (GUP) given by $ΔX ΔP\geq \frac{\hbar}{2}+\frac{l_{pl}^2}{2\hbar}(ΔP)^2+\frac{\hbar}{2r_Λ^2}(ΔX)^2$, can reproduce ap…
We study two different ways to analyze the Hawking evaporation of a Schwarzschild-de Sitter black hole. The first one uses the standard approach of surface gravity evaluated at the possible horizons. The second method derives its results via the Generalized Uncertainty Principle (GUP) which offers a yet different method to look at the problem. In the case of a Schwarzschild black hole it is known that this methods affirms the existence of a black hole remnant (minimal mass $M_{\rm min}$) of the order of Planck mass $m_{\rm pl}$ and a corresponding maximal temperature $T_{\rm max}$ also of the …
We calculate the statistical entropy of a scalar field on the background of (2+1)-dimensional de Sitter space without an artificial cutoff considering corrections to all orders in the Planck length from a generalized uncertainty principle (GUP) on the quantum state density. The desired entropy proportional to the horizon perimeter is obtained.
We study the effects, to all orders in the Planck length from a generalized uncertainty principle (GUP), on the statistical entropy of massive scalar bulk fields in the Randall-Sundrum black brane world. We show that the Bekenstein-Hawking area law is not preserved, and contains small corrections terms proportional to the black hole inverse area.
The formulae for Planck length, Hawking temperature and Unruh-Davies temperature are derived by using only laws of classical physics together with the Heisenberg principle. Besides, it is shown how the Hawking relation can be deduced from the Unruh relation by means of the principle of equivalence; the deep link between Hawking effect and Unruh effect is in this way clarified.
We study the statistical entropy of a scalar field on the (1+1)-dimensional Maxwell-dilaton background without an artificial cutoff considering corrections to all orders in the Planck length from a generalized uncertainty principle (GUP) on the quantum state density. In contrast to the previous results of the higher dimensional cases having adjustable parameter, we obtain an unadjustable entropy due to the independence of the minimal length while this entropy is proportional to the Bekenstein-Hawking entropy.
By considering a deformation of the Schwarzschild metric in the presence of a minimal measurable length which still respects the equivalence principle, we study corrections to the standard general relativistic predictions for some astrophysical phenomena such as stability of circular orbits of black hole accretion disks, redshift of black hole accretion disks, gravitational tidal forces and the geodetic drift rate. We use the \emph{Gravity Probe B} data to see robustness of our results. Our analysis shows also that the relevant deformation parameter $\varepsilon$ which has a geometric origin, …
Motivated by quantum mechanical corrections to the Newtonian potential, which can be translated into an $\hbar$-correction to the $g_{00}$ component of the Schwarzschild metric, we construct a quantum mechanically corrected metric assuming $-g_{00}=g^{rr}$. We show how the Bekenstein black hole entropy $S$ receives its logarithmic contribution provided the quantum mechanical corrections to the metric are negative. In this case the standard horizon at the Schwarzschild radius $r_S$ increases by small terms proportional to $\hbar$ and a remnant of the order of Planck mass emerges. We contrast th…
We investigate the effects to all orders in the Planck length from a generalized un- certainty principle (GUP) on black holes thermodynamics. We calculate the corrected Hawking temperature, entropy, and examine in details the Hawking evaporation process. As a result, the evaporation process is accelerated and the evaporation end-point is a zero entropy, zero heat capacity and finite non zero temperature black hole remnant (BHR). In particular we obtain a drastic reduction of the decay time, in comparison with the results obtained in the Hawking semi classical picture and with the GUP to leadin…
We consider a Schwarzschild type solution in the discrete Regge calculus formulation of general relativity quantized within the path integral approach. Earlier, we found a mechanism of a loose fixation of the background scale of Regge lengths. This elementary length scale is defined by the Planck scale and some free parameter of such a quantum extension of the theory. Besides, Regge action was reduced to an expansion over metric variations between the tetrahedra and, in the main approximation, is a finite-difference form of the Hilbert-Einstein action. Using for the Schwarzschild problem a pri…
We investigate the effects to all orders in the Planck length, from a generalized uncertainty principle, on the thermodynamic parameters of radiating Schwarzschild black holes in a scenario with large extra dimesions. We show that black holes in this framework are hotter, decay faster and are less classical objects. Particularly, we show that the final stage of the radiation process is a black hole remnant with zero entropy, zero heat capacity and non zero finite temperature. We finally compare our results with the ones obtained in the standard Hawking picture and with the generalized uncertai…
We discuss the opportunity that the singularity inside a Schwarzschild black hole could be replaced by a regular bounce, described as a regular minimum of the spherical radius (instead of zero) and a regular maximum of the longitudinal scale (instead of infinity) in the corresponding Kantowski-Sachs metric. Such a metric in a vicinity of the bounce is shown to be a solution to the Einstein equations with the stress-energy tensor representing vacuum polarization of quantum matter fields, described by a combination of curvature-quadratic terms in the effective action. The indefinite parameters o…
If a physical significance should be attributed to the cosmological large number relationship obtained from Sciama's formulation of Mach's Principle, then a number of interesting physical conclusions may be drawn. The Planck length is naturally obtained as the amplitude of waves in a medium whose properties are implied by the relationship. The relativistic internal energy associated with a rest mass is explicitly related to the gravitational potential energy of the Universe, and consistency with the Einstein photon energy is demonstrated. Broader cosmological consequences of this formulation a…
We discuss the properties of the previously constructed model of a Schwarzschild black hole interior where the singularity is replaced by a regular bounce, ultimately leading to a white hole. The model is semiclassical in nature and uses as a source of gravity the effective stress-energy tensor (SET) corresponding to vacuum polarization of quantum fields, and a minimum spherical radius is a few orders of magnitude larger than the Planck length, so that the effects of quantum gravity should be still negligible. We estimate the other quantum contributions to the effective SET, caused by a nontri…
In this paper we present a suitably adjusted Higgs-like mechanism producing black holes at, and beyond, the Planck energy. Planckian objects are difficult to classify either as "particles", or as "black holes", since the Compton wavelength and the Schwarzschild radius are comparable. Due to this unavoidable ambiguity, we consider more appropriate a quantum field theoretical (QFT) approach rather than General Relativity (GR), which is known to break down as the Planck scale is approached. A posteriori, a connection between the two description can be established for masses large enough with resp…
This author's recent proposal of interferometric tests of Planck-scale-related properties of space-time is here revisited from a strictly phenomenological viewpoint. The results announced previously are rederived using elementary dimensional considerations. The dimensional analysis is then extended to the other two classes of experiments (observations of neutral kaons at particle accelerators and observations of the gamma rays we detect from distant astrophysical sources) which have been recently considered as opportunities to explore "foamy" properties of space-time. The emerging picture sugg…
The Wheeler equation, for electromagnetic disturbances in a gravitational field, was found by Fiziev to have exact solutions both above and below the event horizon, in the form of waves propagating both inwardly and outwardly. This observation can be interpreted and verified from the optical point of view, entirely on the basis of the Schwarzschild metric for length contraction and time dilation, in order to derive a differential version of Snell's law for the Schwarzschild black hole. It reveals interesting physics, including the correct amount of light deflection by the Sun, internal and ext…
Planck introduced his famous units of mass, length and time a hundred years ago. The many interesting facets of the Planck mass and length are explored. The Planck mass ubiquitously occurs in astrophysics, cosmology, quantum gravity, string theory, etc. Current aspects of its implications for unification of fundamental interactions, energy dependence of coupling constants, dark energy, etc. are discussed.
It was argued in a number of papers that the gravitational potential calculated by using the modified QFT that follows from the Planck-length deformed uncertainty relation implies the existence of black-hole remnants of the order of the Planck-mass. Usually this sort of QFTs are endowed with two specific features, the modified dispersion relation, which is universal, and the concept of minimum length, which, however, is not universal. While the emergence of the minimum-length most readily leads to the idea of the black hole remnants, here we examine the behaviour of the potential that follows …
Black holes play an important role in linking microphysics with macrophysics, with those of the Planck mass ($M_P \sim10^{-5}$g) featuring in any theory of quantum gravity. In particular, the Compton-Schwarzschild correspondence posits a smooth transition between the Compton wavelength ($R_C \propto 1/M$) below the Planck mass and the Schwarzschild radius ($R_{\rm S} \propto M$) above it. The duality between $R_{\rm C}$ and $R_{\rm S}$ implies a form of the Generalized Uncertainty Principle (GUP) and suggests that elementary particles may be sub-Planckian black holes. The simplest possibility …
A basic arbitrariness in the determination of the topology of a manifold at the Planck length is discussed. An explicit example is given of a `smooth' change in topology from the 2-sphere to the 2-torus through a sequence of noncommuting geometries. Applications are considered to the theory of D-branes within the context of the proposed $M$(atrix) theory.
Based on the Euclidean approach, we consider the effects of quantum gravity and mass-less matter on the thermodynamic properties of Schwarzschild black hole. The techniques of effective field theory are utilized to analytically construct the partition function at the one-loop level. Using the non-local heat kernel formalism, the partition function is expressed as a curvature expansion. We extensively discuss the effect of the corrections on the thermodynamic stability. The one-loop free energy shows, remarkably, that a large number of gauge fields is able to render Schwarzschild black hole the…
The interior of a Schwarzschild black hole is investigated at the level of phenomenological dynamics with the discreteness corrections of loop quantum geometry implemented in two different improved quantization schemes. In one scheme, the classical black hole singularity is resolved by the quantum bounce, which bridges the black hole interior with a white hole interior. In the other scheme, the classical singularity is resolved and the event horizon is also diffused by the quantum bounce. Jumping over the quantum bounce, the black hole gives birth to a baby black hole with a much smaller mass.…
Inspired by the recent conjecture that black holes are condensates of gravitons, we investigate a simple model for the black hole degrees of freedom that is consistent both from the point of view of Quantum mechanics and of General Relativity. Since the two perspectives should "converge" into a unified picture for small, Planck size, objects, we expect our construction is a useful step for understanding the physics of microscopic, quantum black holes. In particular, we show that a harmonically trapped condensate gives rise to two horizons, whereas the extremal case (corresponding to a remnant …
We show why and how Compton wavelength and Schwarzschild radius should be combined into one single new length scale, which we call the Compton-Schwarzschild length. Doing so offers a resolution of the black hole information loss paradox, and suggests Planck mass remnant black holes as candidates for dark matter. It also compels us to introduce torsion, and identify the Dirac field with a complex torsion field. Dirac equation, and Einstein equations, are shown to be mutually dual limiting cases of an underlying gravitation theory which involves the Compton-Schwarzschild length scale, and includ…
Using the mass-smeared scheme of black holes, we study the thermodynamics of black holes. Two interesting models are considered. One is the self-regular Schwarzschild-AdS black hole whose mass density is given by the analogue to probability densities of quantum hydrogen atoms. The other model is the same black hole but whose mass density is chosen to be a rational fractional function of radial coordinates. Both mass densities are in fact analytic expressions of the $δ$-function. We analyze the phase structures of the two models by investigating the heat capacity at constant pressure and the Gi…
