DOI: 10.4236/jamp.2018.68140 1643 Journal of Applied Mathematics and Physics In this paper we reviewed the free quantum electrodynamics in static spheri- cally symmetric spacetime of arbitrary dimensions in a modified Feynman gauge [4]. Using the physical modes functions, we calculate the response rate of a static charge outside the d -dimensional **Schwarzschild** **black** **hole** in the Unruh va- cuum [5]. Limited to four-dimensional **Schwarzschild** **black** **hole**, the response rate is consistent with the result in Ref. [4].

In studying linear perturbations of a **Schwarzschild** **black** **hole** we are able to study its static space-time proper- ties and the emission of gravitation radiation. The gravitational radiation emitted by a **Schwarzschild** **black** **hole** carries information about its mass (as well as spin and charge for rotating and/or charged **black** holes). Also by studying the perturbations of a **Schwarzschild** **black** **hole** it is possible to make conclusions about the stability of the Einstein equations [1]. Because of the challenges of studying the gravitational radiation analytically, people have developed numerical techniques [2] to solve the field equations by evolving the metric. Different ap- proaches are used in numerical relativity to tackle these problems in standard coordinates, the most approach being the ADM formalism [3] [4] which is based on the split of spacetime into space and time. However, the natural formalism based on the fact that gravitational radiation travels at the speed of light and uses null coordi- nates, is called Bondi-Sachs formalism [5] [6]. Important numerical studies involving **black** **hole**-**black** **hole**, **black** **hole**-neutron star, and neutron star-neutron star binaries have been done [7]-[12] in this direction.

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case of a point particle emitting gravitational radiation, in geodesic circular motion around a **Schwarzschild** **black** **hole**. We also obtain numerical results for the emitted power of gravitational radiation. In Sec. IV we com- pare these results to an analogous case in flat spacetime, namely the radiation emitted by a particle orbiting a Newtonian massive object. We conclude this paper with some remarks in Sec. V. We present the derivation of the normalization factor of one type of the modes, the scalar- type modes, in Appendix A. Throughout this paper we use the metric signature − + ++ and natural units such that G = c = ~ = 1.

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radiated by single electron hydrogen atom during its radial fall in **Schwarzschild** **black** **hole**, in terms of frequency , eq. (23) is considered, that is, when radiated energy follows the condition x = Rs. For calculating the results, the value τ =1 and the mass mo= 9.1 x 10 -31 Kg as the standard value of electron has been taken. Fig. 1 shows the frequency υ range of this one electron H atom when it fall into **Schwarzschild** **black** **hole** of mass taken in solar masses M O .In deriving the energy spectrum Planck’s constant h has been also introduced for relation υ = ξ h . The energy spectrum from the calculations is calculated to fall in gamma region of electromagnetic spectrum of range 10 Hz and 23 24

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 δ-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 equa- tion 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.

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have considered the application of this interchange of roles – a symmetry which is particular to the case of a uniformly accelerated particle. In the case of radial motion, outside the horizon and motion along the direction of homogeneity inside the horizon, one find that the equations of motion on both sides of the horizon are of the same form. In the case of circular motion on a photon sphere and “circular” motion on a photon sphere analogue, outside and inside the horizon, respectively, the equations of motion are found to be different. In both cases, with either the same or different equations of motion, one gets the very different solutions. Outside the horizon one reproduces more or less well-known outcomes; inside the horizon the outcomes are found to be rather unexpected. Outside the horizon a radially accelerated particle departs with ever increasing speed (if the acceleration is larger than some critical value); the speed of a test particle uniformly accelerated along the homogeneity axis inside the horizon, whose equation of motion is the same as the one outside the horizon, initially increases but then decreases to zero when approaching the ultimate singularity. The radial acceleration during circular motion of the test particle on the photon sphere is independent of its speed; the speed of a test particle following accelerated motion along a circle belonging to the photon sphere analogue (of ever decreasing radius) increases to the speed of light when approaching the ultimate singularity. Having in mind an (intimate) symmetry (7) between the exterior and interior of the **Schwarzschild** **black** **hole** leading in the simplest case to the same equation of motion, one may wonder how it arises that the properties of the solutions are so different. The answer is: it is because 𝑟 and 𝑡, spatial (radial) and temporal coordinates, respectively, outside horizon interchange their roles, becoming temporal and spatial (in the direction of homogeneity) coordinates inside the horizon. Such an exchange of roles has a deeper consequence: outside the horizon spacetime is spherically symmetric and static and inside the horizon it is no longer static, it is dynamically changing (ie an “anisotropic cosmology”, see [11]) but homogeneous along one of its spatial directions.

Gauss-Bonnet invariant in a **Schwarzschild** background. The setup corresponds to the decoupling limit of theory (1). We focused on initial data for which the scalar van- ishes initially and its derivatives either vanish or are given by a Gaussian shell. In all cases, the scalar eventually re- laxes to the known static configuration of Eq. (8). This is the configuration of the known **black** **hole** solution ob- tained in Refs. [9, 21] (and also in Ref. [24] in a different setup) by working perturbatively in the coupling λ.

change the results, at least for stable orbits. 13 A more realistic scenario was pre- sented in Ref. 14, where the scalar source was replaced by an electric charge and the emission of photons by this charge was analyzed, together with the absorption of the electromagnetic radiation by the **black** **hole**. The case for gravitational radia- tion, where the source is now a particle in a geodesic circular orbit, was considered in Ref.15. It has been shown that there is an enhancement of the high multipole modes for unstable orbits, which would be characterized as synchrotron radiation. However, the low multipoles, especially the quadrupole one, still have relevant con- tributions to the emitted power, even for unstable orbits. Thus, the gravitational radiation is not as concentrated in the plane of orbit as in the scalar radiation case. In this context, the purpose of this paper is to present the analysis of the emitted radiation by a source in circular orbit around a **Schwarzschild** **black** **hole**, in a unified framework, treating the radiation (scalar, electromagnetic, or gravitational) as a quantum field propagating in the curved background. A perturbative approach is taken to compute the one-particle-emission amplitudes at tree level. We then compute the emitted power of radiation for each case. Although equivalent to the classical methods, the use of the QFT framework provides a very simple conceptual point of view as well as tools for possible extensions.

In this paper only one basic assumption has been made: if we try to describe **black** holes, their behavior should be understood in the same lan- guage as the one we use for particles; **black** holes should be treated just like atoms. They must be quantum forms of matter, moving in accordance with Schrödinger equations just like everything else. In particular, Rosen’s quantization approach to the gravitational collapse is applied in the simple case of a pressureless “star of dust” by finding the gravitational potential, the Schrödinger equation and the solution for the collapse’s energy levels. By applying the constraints for a **Schwarzschild** **black** **hole** (BH) and by using the concept of BH effective state, previously introduced by one of the authors (CC), the BH quantum gravitational potential, Schrödinger equation and the BH energy spectrum are found. Remarkably, such an energy spectrum is in agreement (in its absolute value) with the one which was conjectured by Bekenstein in 1974 and consistent with other ones in the literature. This approach also permits to find an interesting quantum representation of the **Schwarzschild** BH ground state at the Planck scale. Moreover, two fundamental issues about **black** **hole** quantum physics are addressed by this model: the area quantization and the singularity res- olution. As regards the former, a result similar to the one obtained by Bekenstein, but with a different coefficient, has been found. About the latter, it is shown that the traditional classical singularity in the core of the **Schwarzschild** BH is replaced, in a full quantum treatment, by

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In the fourth chapter we give a brief introduction into the required tools of functional calculus which we need in order to explain the Schr¨ odinger representation of quantum field theory (first on Minkowski space-time, afterwards on curved spaces). After the preliminaries we state the definition of quantum completeness, i.e. completeness where the only adequate description is in terms of quantum field theory. Our criterion is due to the ground-states of the Schr¨ odinger representation but we will show that this is so far sufficient. Then we apply the criterion to the **Schwarzschild** space-time, and afterwards to Kasner space- time. Moreover we analyse non-Gaussian deformations of the ground-state wave-functionals such as excitation with respect to the ground state and self-interaction of the quantum probes. For the latter we give an argument why they cannot change the result, whatsoever. In the last two sections of Chapter 4 we calculate the energy density which is in full accordance with the result of quantum completeness and show that charges are conserved inside **Schwarzschild** **black** **hole**. In the end we will draw a link to the **black** **hole** final- state proposed by Horowitz and Maldacena. The link between Heisenberg and Schr¨ odinger representation is presented in the appendix.

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Rutvij H. Jhaveri [7] proposed a protocol MR-AODV which is modification of R-AODV. MR-AODV establishes the secure route for data transmission by detecting **black** **hole** and gray **hole** nodes during route discovery phase. As soon as malicious node is detected MR-AODV updates the routing table with malicious node entry and discards RREP. MR- AODV does not forward on reverse path and also it does not require any flag. Thus RREP indicating shortest fresher path will be chosen for data transmission by the source node. MR- AODV reduces overhead by not forwarding RREP after detection of misbehavior.

When there is an accretion disk, circling matter will touch the surface of **black** **hole**, where the vibration movements along a longitude convert the circling matter into lay- ers of spheres wrapping around the **black** **hole**. The newly formed spheres twist up- wards. When these particles reach a pole, they are accelerated by out-going **black** **hole** energy at the poles. The particles are continuously pulled into the new sphere from the rotating disk and ejected at the top of pole. This dynamic process forms a **black** **hole** jet Figure 7.

∆ + ⋅ ∆ + = which we state would be due our construction a necessary condition for a complete quantum gravity analysis of gravitons being emitted from a Kerr-Newman **black** **hole**. We state that these two points have to be determined and investigated, and also that an optimal value of d , for dimen- sions for a problem, involving Kerr Newman **black** holes would have to be as- certained in future research. Finally, we refer the reader to references [66]-[72] or additional ideas which may be used in future projects. Note also that Valev wrote [73]

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of the Anti deSitter (Ads) **black** holes in a canonical ensemble is quite assured where the exploration of quantum gravity and several super conformal gauge theories can be done in a wider perspective via its critical thermodynamic analysis [ 1998; Chaturvedi, 2017]. Mostly due to the occurrence of a phase transition within the dual thermal field theory and also the confession of the gauge duality of the **black** holes via Ads/CFT correspondence, **black** **hole** thermodynamics had acquired attention in the framework of many researches [

malicious node and could send route reply even if it had no fresh enough route to the destination to make a **black** **hole** attack. They proposed a solution that the source node would send another route request to the next hop of the intermediate node to verify the authenticity of the route from the intermediate node to the destination node. If the route exists, the intermediate node is trusted; otherwise, the reply message from the intermediate node is discarded. Sanjay Ramaswamy et al. [18] proposed a technique for identifying multiple **black** **hole** nodes in WSN. They are initially suggesting solution for cooperative **black** **hole** attack in ad-hoc network. Author in some extent modified AODV protocol by introducing data routing information table (DRI) and cross checking of routing table data where, each entry of the mobile node is maintained. They are depending on the trustworthy nodes to transmit the packets. Source sends The Route request (RREQ) to every node and it send packet to the node from where it gets the RREP. The intermediate node should send NHN and the DRI entry to the table. The source mobile node (SN) check own DRI table whether intermediate node (IN) node is trustworthy or not. In ad hoc network, source node sending the supplementary request to next hop node (NHN) for IN (intermediate node). If SN uses IN to send the packet, then it is considered as trustworthy node otherwise not. Cross checking is done on the intermediate nodes and this is one- time procedure. The spending of cross checking is more and it can be making economical by letting mobile nodes sharing their trusted nodes record list with each other.

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However in case the SN gets an RREP for the RIP, then it means that, there is a **black** **hole** in that route. In this case the SN initiates the process of **Black** **Hole** detection. The SN at the beginning notifies the neighbours of the node from which it got the RREP to RIP, to enter in to promiscuous form, to make sure they pay attention not simply to the actual packet bound to them, but likewise to the packet bound to the defined Destination node. Now the SN sends a small number of artificial data packets to the destination, while the neighbouring nodes start off keeping track of the packet flow. These kinds of neighboring nodes further send out the monitor message to the next hop of the artificial data packet & so on. At a point when the monitoring nodes finds out that the artificial data packet loss is way more than the standard anticipated loss in a network, it informs the SN about this particular Intermediate Node(IN). This time with regards to the critical information received by the various monitoring nodes, the SN detects the location of the **Black** **Hole**.

Since the reduced action (19) does not contain any infor- mation that the spacetime represents a **black** **hole**, canon- ical quantization of this system does not provide quantum mechanics of the **black** **hole** given by Eq. (3). For this pur- pose, one therefore needs another canonical transforma- tion from the Kuchaˇr action (19). This is what Louko and Mäkelä did for the four-dimensional spherically symmet- ric spacetime without a cosmological constant [1]. In [6], we generalized their result in the case of general k with Λ ( ≤ 0) in arbitrary dimensions. We summarize our result in this section.

Einstein gravity describes **black** holes. In string theory, **black** holes have been obtained in a low energy limit of string effective action 12–15 . Quantum effects to Einstein gravity can also be addressed non-perturbatively using s-duality in superstring theories (10 dimensional). The type IIA- superstring theory in a strong coupling limit is known to incorporate an extra spatial dimensionon S 1 and has been identified with the non-perturbative M-theory in eleven dimensions. Generically M- theory has been identified with the stringy vacua in various dimensions 16, 17 . In a low energy limit M-theory is known to describe an eleven dimensional supergravity. However a complete non- perturbative formulation of M-theory is still unknown.

In Einstein theory of gravity, energy can also be carried by vibrating waves of space and time, which travel at the speed of light. In the same way that **black** **hole** are made just to space and time, gravitational waves are also “pure” space and time. They interact very weakly with matter and penetrate anything without losing strength. While this makes them powerful probes of extreme condition. It also make them hard to detect **black** **hole** of steller mass are expected to for when very massive stars collapse at the end of their life cycle. After a **black** **hole** formed, it can contineous to grow by absorbing other stars and merging with other **black** **hole**.