The stability of the Schwarzschild black hole is restudied in the Painlevé coordinates. Using the Painlevé time coordinate to define the initial time, we reconsider the odd perturbation and find that the Sch...The stability of the Schwarzschild black hole is restudied in the Painlevé coordinates. Using the Painlevé time coordinate to define the initial time, we reconsider the odd perturbation and find that the Schwarzschild black hole in the Painlevé coordinates is unstable. The Painlevé metric in this paper corresponds to the white-hole-connected region of the Schwarzschild black hole （r 〉 2m） and the odd perturbation may be regarded as the angular perturbation. Therefore, the white-hole-connected region of the Schwarzschild black hole is unstable with respect to the rotating perturbation.展开更多
In our previous work [Chin. Phys. Lett. 35（2018） 010410], the quasinormal modes of massless scalar field perturbation in a noncommutative-geometry-inspired Schwarzschild black hole spacetime are studied using the th...In our previous work [Chin. Phys. Lett. 35（2018） 010410], the quasinormal modes of massless scalar field perturbation in a noncommutative-geometry-inspired Schwarzschild black hole spacetime are studied using the third-order Wentzel–Kramers–Brillouin approximative approach. In this study, we extend the work to the cases of gravitational, electromagnetic and massless Dirac perturbations. The result further confirms that the noncommutative parameter plays an important role for the quasinormal frequencies.展开更多
The quasinormal modes（QNMs） of massless scalar field perturbation in a noncommutative-geometry-inspired Schwarzschild black hole spacetime are studied using the third-order Wentzel-Kramers-Brillouin approximative ap...The quasinormal modes（QNMs） of massless scalar field perturbation in a noncommutative-geometry-inspired Schwarzschild black hole spacetime are studied using the third-order Wentzel-Kramers-Brillouin approximative approach. The result shows that the noncommutative parameter plays an important role for the quasinormal（QNM） frequencies.展开更多
Energy and thermodynamics are investigated in the Schwarzschild black hole spacetime when considering corrections due to quantum vacuum fluctuations. The Einstein and M？ller prescriptions are used to derive the expre...Energy and thermodynamics are investigated in the Schwarzschild black hole spacetime when considering corrections due to quantum vacuum fluctuations. The Einstein and M？ller prescriptions are used to derive the expressions of the energy in the background. The temperature and heat capacity are also derived. The results show that due to the quantum fluctuations in the background of the Schwarzschild black hole, all the energies increase and the Einstein energy differs from M？ller＇s one. Moreover, when increasing the quantum correction factor aa, the difference between Einstein and M？ller energies, the Unruh–Verlinde temperature as well as the heat capacity of the black hole increases while the Hawking temperature remains unchanged.展开更多
In present paper, by using the quantization of an electromagnetic field in the background of static spherically symmetric d-dimensional spacetime in the Boulware vacuum, we calculated the response rate of a static cha...In present paper, by using the quantization of an electromagnetic field in the background of static spherically symmetric d-dimensional spacetime in the Boulware vacuum, we calculated the response rate of a static charge outside d-dimensional Schwarzschild black hole in the low-frequency regime, which can be expressed as the summation of hypergeometric functions.展开更多
We model analytically a relativistic problem consisting of two quasi-particles each with mass m in close orbit around a static Schwarzschild black hole with mass M = 1 situated at the center of mass of the system. The...We model analytically a relativistic problem consisting of two quasi-particles each with mass m in close orbit around a static Schwarzschild black hole with mass M = 1 situated at the center of mass of the system. The angular momentum l of the system is taken to be 2. We model the mass density of the orbiting particles as a δ-function and we assume that there are no deformations. To model the system, we apply the second-order differential equation obtained elsewhere for a dynamic thin matter shell on a Schwarzschild background. As it is the case in this paper, the framework on which the equation was obtained is Bodi-Sachs. The only change in the equation is that now the quasi-normal mode parameter represents the particle’s orbital frequency from which we are able to analytically compute the gravitational radiation emitted by the system at null infinity. We note that in a real astrophysical scenario the dynamics of the particles paths will be very dynamic and complicated and that the analytical methods used here will have to be developed further to accommodate that.展开更多
We analytically model a relativistic problem consisting of a point-particle with mass m in close orbit around a stationary Schwarzschild black hole with mass M = 1 using the null-cone formalism when l = 2. We use the ...We analytically model a relativistic problem consisting of a point-particle with mass m in close orbit around a stationary Schwarzschild black hole with mass M = 1 using the null-cone formalism when l = 2. We use the -function to model the matter density of the particle. To model the whole problem, we apply the second order differential equation obtained elsewhere for a dynamic thin matter shell around a Schwarzschild black hole. The only thing that changes on the equation is the quasi-normal mode parameter which now represent the orbital frequency of the particle. We compare our results with that of the standard 5.5 PN formalism and found that there is a direct proportionality factor that relates the two results, i.e. the two formalisms.展开更多
In this study,we investigate the thermodynamic characteristics of the Rindler–Schwarzschild black hole solution.Our analysis encompasses the examination of energy emission,Gibbs free energy,and thermal fluctuations.W...In this study,we investigate the thermodynamic characteristics of the Rindler–Schwarzschild black hole solution.Our analysis encompasses the examination of energy emission,Gibbs free energy,and thermal fluctuations.We calculate various quantities such as the Hawking temperature,geometric mass,and heat capacity to assess the local and global thermodynamic stability.The temperature of the black hole is determined using the first law of thermodynamics,while the energy emission rate is evaluated as well.By computing the Gibbs free energy,we explore the phase transition behavior exhibited by Rindler–Schwarzschild black hole,specifically examining the swallowing tails.Moreover,we derive the corrected entropy to investigate the influence of thermal fluctuations on small and large black holes.Notably,we compare the impact of correction terms on the thermodynamic system by comparing the results obtained for large black holes and small black holes.展开更多
In this study,we observe that,in the presence of the string cloud parameter a and the quintessence parameterγ,with the equation of state parameter,ω_(q) = -2/3the radius of the shadow of the Schwarzschild black hole...In this study,we observe that,in the presence of the string cloud parameter a and the quintessence parameterγ,with the equation of state parameter,ω_(q) = -2/3the radius of the shadow of the Schwarzschild black hole increases as compared with that in the pure Schwarzschild black hole case.The existence of both quintessential dark energy and the cloud of strings increases the shadow size;hence,the strength of the gravitational field around the Schwarzschild black hole increases.Using the data collected by the Event Horizon Telescope(EHT)collaboration for M87*and Sgr A*,we obtain upper bounds on the values of a andγ.Further,we see the effects of a andγon the rate of emission energy for the Schwarzschild black hole.We notice that the rate of emission energy is higher in the presence of clouds of strings and quintessence.Moreover,we study the weak deflection angle using the Gauss-Bonnet theorem.We show the influence of a andγon the weak deflection angle.We notice that both a andγincrease the deflection angleα.展开更多
We find exact solutions to the Klein-Gordon equation in the vicinity of Schwarzschild black holes.For particles with a zero angular momentum,the convergence range of the solution is r < 4M.One of the solutions desc...We find exact solutions to the Klein-Gordon equation in the vicinity of Schwarzschild black holes.For particles with a zero angular momentum,the convergence range of the solution is r < 4M.One of the solutions describes an exponential enhancement of the density of particles in the vicinity of Schwarzschild black holes,which might be the mechanism of gamma-ray bursts.展开更多
The harmonic metric for Schwarzschild black hole with a uniform velocity is presented. In the limit of weak field and low velocity, this metric reduces to the post-Newtonian approximation for one moving point mass. As...The harmonic metric for Schwarzschild black hole with a uniform velocity is presented. In the limit of weak field and low velocity, this metric reduces to the post-Newtonian approximation for one moving point mass. As an application, we derive the dynamics of particle and photon in the weak-field limit for the moving Schwarzschild black hole with an arbitrary velocity. It is found that the relativistic motion of gravitational source can induce an additional centripetal force on the test particle, which may be comparable to or even larger than the conventional Newtonian gravitational force.展开更多
Applying Parikh＇s quantum tunnelling method, this paper has studied the quantum tunnelling radiation of Schwarzschild de Sitter black hole with a global monopole. The result shows that the tunnelling rates at the eve...Applying Parikh＇s quantum tunnelling method, this paper has studied the quantum tunnelling radiation of Schwarzschild de Sitter black hole with a global monopole. The result shows that the tunnelling rates at the event horizon and the cosmological horizon are related to Bekenstein-Hawking entropy and the derived radiation spectrum is not precisely thermal when considering energy conservation and self-gravitation interaction.展开更多
The quasinormal modes of the Schwarzschild black hole surrounded by the quintessence in Rastall gravity are studied using the sixth-order Wentzel-Kramers-Brillouin approximative approach. The effect of the Rastall par...The quasinormal modes of the Schwarzschild black hole surrounded by the quintessence in Rastall gravity are studied using the sixth-order Wentzel-Kramers-Brillouin approximative approach. The effect of the Rastall parameter on the quasinormal modes of gravitational, electromagnetic and massless scalar perturbations is explored. Compared to the case of Einstein gravity, it is found that, when η < 0, the gravitational field, electromagnetic field as well as massless scalar field damp more rapidly and have larger real frequency of oscillation in Rastall gravity, while when η > 0, the gravitational field, electromagnetic field as well as massless scalar field damp more slowly and have smaller real frequency of oscillation in Rastall gravity. It is also found that the gravitational field, electromagnetic field as well as massless scalar field damp more and more slowly and the real frequency of oscillation for the gravitational perturbation, electromagnetic perturbation as well as massless scalar perturbation becomes smaller and smaller as the Rastall parameter η increases.Compared among the quasinormal frequencies of gravitational, electromagnetic and massless scalar perturbations, I find that, for fixed η,(l, n), ε and Nq, the oscillation damps most slowly for the gravitational perturbation, mediate for the electromagnetic perturbation and most rapidly for the massless scalar perturbation, and the real frequency of oscillation is the smallest for the gravitational perturbation, mediate for the electromagnetic perturbation and the largest for the massless scalar perturbation in Rastall gravity.展开更多
The exact metric of a Schwarzschild black hole in the true radiation gauge was recently reported. In this work, we base on this gravity and calculate the gravitational deflection of relativistic massive particles up t...The exact metric of a Schwarzschild black hole in the true radiation gauge was recently reported. In this work, we base on this gravity and calculate the gravitational deflection of relativistic massive particles up to the fourth post-Minkowskian order. It is found that the result is consistent with the previous formulations for both the case of dropping the fourth-order contribution and the case of light deflection. Our result might be helpful for future high-accuracy observations.展开更多
The surface gravity of Schwarzschild black hole can be quantized from the test particle moving around different energy states analog to the Bohr's atomic model. We have quantized the Hawking temperature and entrop...The surface gravity of Schwarzschild black hole can be quantized from the test particle moving around different energy states analog to the Bohr's atomic model. We have quantized the Hawking temperature and entropy of Schwarzschild black hole from quantization of surface gravity. We also have shown that the change of entropy reduces to zero when the boundary shrinks to very small size.展开更多
A new simpler mathematic method is proposed to study fermions tunneling from black holes. According to this method, by using semiclassical approximation theory, it simplifies the Dirac equation of curved spacetime and...A new simpler mathematic method is proposed to study fermions tunneling from black holes. According to this method, by using semiclassical approximation theory, it simplifies the Dirac equation of curved spacetime and then the relationship of the gamma matrix and the component of contravariant metric is considered in order to transform the set of difficult quantum equations into a simple equation. Finally, the fermion tunneling and Hawking radiation of black holes are obtained. The method is very effective and simple, and we will take the Schwarzschild black hole with global monopole and the higher-dimensional Reissner-Nordstrom de Sitter black hole as two examples to show the fact.展开更多
We provide a simple way for calculating the entropy of a Schwarzschild black hole from the entropy of its Hawking radiation. To this end, we show that if a thermodynamic system loses its energy only through the black ...We provide a simple way for calculating the entropy of a Schwarzschild black hole from the entropy of its Hawking radiation. To this end, we show that if a thermodynamic system loses its energy only through the black body radiation, its loss of entropy is always 3/4 of the entropy of the emitted radiation. This proposition enables us to relate the entropy of an evaporating black hole to the entropy of its Hawking radiation. Explicitly, by calculating the entropy of the Hawking radiation emitted in the full period of evaporation of the black hole, we find the Bekenstein-Hawking entropy of the initial black hole.展开更多
The black hole is a region in space where things may fall into it but nothing can come out. We present a study of the physics of a black hole using a quantum field theory frame based on the WZW model in a suitable mat...The black hole is a region in space where things may fall into it but nothing can come out. We present a study of the physics of a black hole using a quantum field theory frame based on the WZW model in a suitable mathematical frame. Based on the Schwarzschild metric, we show the different regions of our universe with the present singularities. Then we introduce the calculation of a black hole mass using the perturbation theory. We further discuss Hawking radiation and its quantum mechanical implications. At some limits, the space time can represent a black hole with a singularity hidden by the horizon.展开更多
Null geodesics for massless particles in Schwarzschild spacetime are obtained by direct integration of the trajectory equation in spatial coordinates without transformation to the inverse space. The results are classi...Null geodesics for massless particles in Schwarzschild spacetime are obtained by direct integration of the trajectory equation in spatial coordinates without transformation to the inverse space. The results are classified following Chandrasekhar depending on the ratio of the impact parameter of the trajectory to its critical value. In the subcritical and supercritical cases the geodesics are expressed in terms of elliptic integrals of the first kind. Some results are formally different from the classical ones, but in fact equivalent to them, being at the same time more compact and descriptive.展开更多
基金Project supported by the National Natural Science Foundation of China （Grant Nos 10475013, 10373003 and 10375087）, the National Basic Research Program （Grant No 2004CB318000） and the Post-Doctor Foundation of China.
文摘The stability of the Schwarzschild black hole is restudied in the Painlevé coordinates. Using the Painlevé time coordinate to define the initial time, we reconsider the odd perturbation and find that the Schwarzschild black hole in the Painlevé coordinates is unstable. The Painlevé metric in this paper corresponds to the white-hole-connected region of the Schwarzschild black hole （r 〉 2m） and the odd perturbation may be regarded as the angular perturbation. Therefore, the white-hole-connected region of the Schwarzschild black hole is unstable with respect to the rotating perturbation.
基金Supported by the Natural Science Foundation of Education Department of Shannxi Province under Grant No 15JK1077the Doctorial Scientific Research Starting Fund of Shannxi University of Science and Technology under Grant No BJ12-02
文摘In our previous work [Chin. Phys. Lett. 35（2018） 010410], the quasinormal modes of massless scalar field perturbation in a noncommutative-geometry-inspired Schwarzschild black hole spacetime are studied using the third-order Wentzel–Kramers–Brillouin approximative approach. In this study, we extend the work to the cases of gravitational, electromagnetic and massless Dirac perturbations. The result further confirms that the noncommutative parameter plays an important role for the quasinormal frequencies.
基金Supported by the Natural Science Foundation of Education Department of Shannxi Province under Grant No 15JK1077the Doctorial Scientific Research Starting Fund of Shannxi University of Science and Technology under Grant No BJ12-02
文摘The quasinormal modes（QNMs） of massless scalar field perturbation in a noncommutative-geometry-inspired Schwarzschild black hole spacetime are studied using the third-order Wentzel-Kramers-Brillouin approximative approach. The result shows that the noncommutative parameter plays an important role for the quasinormal（QNM） frequencies.
文摘Energy and thermodynamics are investigated in the Schwarzschild black hole spacetime when considering corrections due to quantum vacuum fluctuations. The Einstein and M？ller prescriptions are used to derive the expressions of the energy in the background. The temperature and heat capacity are also derived. The results show that due to the quantum fluctuations in the background of the Schwarzschild black hole, all the energies increase and the Einstein energy differs from M？ller＇s one. Moreover, when increasing the quantum correction factor aa, the difference between Einstein and M？ller energies, the Unruh–Verlinde temperature as well as the heat capacity of the black hole increases while the Hawking temperature remains unchanged.
文摘In present paper, by using the quantization of an electromagnetic field in the background of static spherically symmetric d-dimensional spacetime in the Boulware vacuum, we calculated the response rate of a static charge outside d-dimensional Schwarzschild black hole in the low-frequency regime, which can be expressed as the summation of hypergeometric functions.
文摘We model analytically a relativistic problem consisting of two quasi-particles each with mass m in close orbit around a static Schwarzschild black hole with mass M = 1 situated at the center of mass of the system. The angular momentum l of the system is taken to be 2. We model the mass density of the orbiting particles as a δ-function and we assume that there are no deformations. To model the system, we apply the second-order differential equation obtained elsewhere for a dynamic thin matter shell on a Schwarzschild background. As it is the case in this paper, the framework on which the equation was obtained is Bodi-Sachs. The only change in the equation is that now the quasi-normal mode parameter represents the particle’s orbital frequency from which we are able to analytically compute the gravitational radiation emitted by the system at null infinity. We note that in a real astrophysical scenario the dynamics of the particles paths will be very dynamic and complicated and that the analytical methods used here will have to be developed further to accommodate that.
文摘We analytically model a relativistic problem consisting of a point-particle with mass m in close orbit around a stationary Schwarzschild black hole with mass M = 1 using the null-cone formalism when l = 2. We use the -function to model the matter density of the particle. To model the whole problem, we apply the second order differential equation obtained elsewhere for a dynamic thin matter shell around a Schwarzschild black hole. The only thing that changes on the equation is the quasi-normal mode parameter which now represent the orbital frequency of the particle. We compare our results with that of the standard 5.5 PN formalism and found that there is a direct proportionality factor that relates the two results, i.e. the two formalisms.
基金funded by the National Natural Science Foundation of China 11975145Scientific and Technological Research Institution of Turkey(TUBITAK)the Sponsoring Consortium for Open Access Publishing in Particle Physics(or SCOAP3)for their support。
文摘In this study,we investigate the thermodynamic characteristics of the Rindler–Schwarzschild black hole solution.Our analysis encompasses the examination of energy emission,Gibbs free energy,and thermal fluctuations.We calculate various quantities such as the Hawking temperature,geometric mass,and heat capacity to assess the local and global thermodynamic stability.The temperature of the black hole is determined using the first law of thermodynamics,while the energy emission rate is evaluated as well.By computing the Gibbs free energy,we explore the phase transition behavior exhibited by Rindler–Schwarzschild black hole,specifically examining the swallowing tails.Moreover,we derive the corrected entropy to investigate the influence of thermal fluctuations on small and large black holes.Notably,we compare the impact of correction terms on the thermodynamic system by comparing the results obtained for large black holes and small black holes.
基金the Grant No.ZC304022919 to support his Postdoctoral Fellowship at Zhejiang Normal Universitythe support of Inha University in Tashkent and research work has been supported by the Visitor Research Fellowship at Zhejiang Normal Universitysupported by Research Grant FZ-20200929344,F-FA-2021-510 and F-FA-2021-432 of the Uzbekistan Ministry for Innovative Development。
文摘In this study,we observe that,in the presence of the string cloud parameter a and the quintessence parameterγ,with the equation of state parameter,ω_(q) = -2/3the radius of the shadow of the Schwarzschild black hole increases as compared with that in the pure Schwarzschild black hole case.The existence of both quintessential dark energy and the cloud of strings increases the shadow size;hence,the strength of the gravitational field around the Schwarzschild black hole increases.Using the data collected by the Event Horizon Telescope(EHT)collaboration for M87*and Sgr A*,we obtain upper bounds on the values of a andγ.Further,we see the effects of a andγon the rate of emission energy for the Schwarzschild black hole.We notice that the rate of emission energy is higher in the presence of clouds of strings and quintessence.Moreover,we study the weak deflection angle using the Gauss-Bonnet theorem.We show the influence of a andγon the weak deflection angle.We notice that both a andγincrease the deflection angleα.
基金supported by the National Natural Science Foundation of China (Grant No. 11073007)
文摘We find exact solutions to the Klein-Gordon equation in the vicinity of Schwarzschild black holes.For particles with a zero angular momentum,the convergence range of the solution is r < 4M.One of the solutions describes an exponential enhancement of the density of particles in the vicinity of Schwarzschild black holes,which might be the mechanism of gamma-ray bursts.
基金Supported by the Program for New Century Excellent Talents in University under Grant No.NCET-10-0702the National Basic Research Program of China(973 Program)under Grant No.2013CB328904the Ph.D.Programs Foundation of Ministry of Education of China under Grant No.20110184110016
文摘The harmonic metric for Schwarzschild black hole with a uniform velocity is presented. In the limit of weak field and low velocity, this metric reduces to the post-Newtonian approximation for one moving point mass. As an application, we derive the dynamics of particle and photon in the weak-field limit for the moving Schwarzschild black hole with an arbitrary velocity. It is found that the relativistic motion of gravitational source can induce an additional centripetal force on the test particle, which may be comparable to or even larger than the conventional Newtonian gravitational force.
基金Proiect suooorted by the National Natural Science Foundation of China （Grant No 10347008）.
文摘Applying Parikh＇s quantum tunnelling method, this paper has studied the quantum tunnelling radiation of Schwarzschild de Sitter black hole with a global monopole. The result shows that the tunnelling rates at the event horizon and the cosmological horizon are related to Bekenstein-Hawking entropy and the derived radiation spectrum is not precisely thermal when considering energy conservation and self-gravitation interaction.
基金Supported by the National Nature Science Foundation of Education Department of Shannxi Province under Grant No.15JK1077the Doctorial Scientific Research Starting Fund of Shannxi University of Science and Technology under Grant No.BJ12-02
文摘The quasinormal modes of the Schwarzschild black hole surrounded by the quintessence in Rastall gravity are studied using the sixth-order Wentzel-Kramers-Brillouin approximative approach. The effect of the Rastall parameter on the quasinormal modes of gravitational, electromagnetic and massless scalar perturbations is explored. Compared to the case of Einstein gravity, it is found that, when η < 0, the gravitational field, electromagnetic field as well as massless scalar field damp more rapidly and have larger real frequency of oscillation in Rastall gravity, while when η > 0, the gravitational field, electromagnetic field as well as massless scalar field damp more slowly and have smaller real frequency of oscillation in Rastall gravity. It is also found that the gravitational field, electromagnetic field as well as massless scalar field damp more and more slowly and the real frequency of oscillation for the gravitational perturbation, electromagnetic perturbation as well as massless scalar perturbation becomes smaller and smaller as the Rastall parameter η increases.Compared among the quasinormal frequencies of gravitational, electromagnetic and massless scalar perturbations, I find that, for fixed η,(l, n), ε and Nq, the oscillation damps most slowly for the gravitational perturbation, mediate for the electromagnetic perturbation and most rapidly for the massless scalar perturbation, and the real frequency of oscillation is the smallest for the gravitational perturbation, mediate for the electromagnetic perturbation and the largest for the massless scalar perturbation in Rastall gravity.
基金Supported by the National Natural Science Foundation of China under Grant No.11647314the Research Foundation of Education Department of Hunan Province under Grant No.18C0427
文摘The exact metric of a Schwarzschild black hole in the true radiation gauge was recently reported. In this work, we base on this gravity and calculate the gravitational deflection of relativistic massive particles up to the fourth post-Minkowskian order. It is found that the result is consistent with the previous formulations for both the case of dropping the fourth-order contribution and the case of light deflection. Our result might be helpful for future high-accuracy observations.
文摘The surface gravity of Schwarzschild black hole can be quantized from the test particle moving around different energy states analog to the Bohr's atomic model. We have quantized the Hawking temperature and entropy of Schwarzschild black hole from quantization of surface gravity. We also have shown that the change of entropy reduces to zero when the boundary shrinks to very small size.
基金supported by the National Natural Science Foundation of China(Grant Nos.10773008 and 11075224)the Chongqing University Postgraduates Science and Innovation Fund,China(Grant No.200811B1A0100299)
文摘A new simpler mathematic method is proposed to study fermions tunneling from black holes. According to this method, by using semiclassical approximation theory, it simplifies the Dirac equation of curved spacetime and then the relationship of the gamma matrix and the component of contravariant metric is considered in order to transform the set of difficult quantum equations into a simple equation. Finally, the fermion tunneling and Hawking radiation of black holes are obtained. The method is very effective and simple, and we will take the Schwarzschild black hole with global monopole and the higher-dimensional Reissner-Nordstrom de Sitter black hole as two examples to show the fact.
文摘We provide a simple way for calculating the entropy of a Schwarzschild black hole from the entropy of its Hawking radiation. To this end, we show that if a thermodynamic system loses its energy only through the black body radiation, its loss of entropy is always 3/4 of the entropy of the emitted radiation. This proposition enables us to relate the entropy of an evaporating black hole to the entropy of its Hawking radiation. Explicitly, by calculating the entropy of the Hawking radiation emitted in the full period of evaporation of the black hole, we find the Bekenstein-Hawking entropy of the initial black hole.
文摘The black hole is a region in space where things may fall into it but nothing can come out. We present a study of the physics of a black hole using a quantum field theory frame based on the WZW model in a suitable mathematical frame. Based on the Schwarzschild metric, we show the different regions of our universe with the present singularities. Then we introduce the calculation of a black hole mass using the perturbation theory. We further discuss Hawking radiation and its quantum mechanical implications. At some limits, the space time can represent a black hole with a singularity hidden by the horizon.
文摘Null geodesics for massless particles in Schwarzschild spacetime are obtained by direct integration of the trajectory equation in spatial coordinates without transformation to the inverse space. The results are classified following Chandrasekhar depending on the ratio of the impact parameter of the trajectory to its critical value. In the subcritical and supercritical cases the geodesics are expressed in terms of elliptic integrals of the first kind. Some results are formally different from the classical ones, but in fact equivalent to them, being at the same time more compact and descriptive.
