The vacuum activity generated by the curvature of the Schwarzschild black-hole geometry close to the event horizon is studied for the case of a massless, conformal scalar field. The associated approximation to the unknown, exact propagator in the Hartle-Hawking vacuum state for small values of the radial coordinate above $ r = 2M$ results in an analytic expression which manifestly features its dependence on the background space-time geometry. This approximation to the Hartle-Hawking scalar propagator on the Schwarzschild black-hole geometry is, for that matter, distinct from all other. It is shown that the stated approximation is valid for physical distances which range from the event horizon to values which are orders of magnitude above the scale within which quantum and backreaction effects are comparatively pronounced. An expression is obtained for the renormalised $ <ϕ^2(x)>$ in the Hartle-Hawking vacuum state which reproduces the established results on the event horizon and in that segment of the exterior geometry within which the approximation is valid. In contrast to previous results the stated expression has the superior feature of being entirely analytic. The effect of the manifold's causal structure to scalar propagation is also studied.
We show that the hidden conformal symmetry of the Schwarzschild black hole is realized from the AdS$_2$ sector of the AdS$_2\times S^2$, but not from the Rindler spacetime which is the genuine near-horizon geometry of the Schwarzschild black hole. This implies that purely imaginary quasinormal frequencies obtained using the hidden conformal symmetry is not suitable for describing the largely damped modes around the Schwarzschild black hole.
The asymptotic conformal Yano--Killing tensor proposed in J. Jezierski, On the relation between metric and spin-2 formulation of linearized Einstein theory [GRG, in print (1994)] is analyzed for Schwarzschild metric and tensor equations defining this object are given. The result shows that the Schwarzschild metric (and other metrics which are asymptotically ``Schwarzschildean'' up to O(1/r^2) at spatial infinity) is among the metrics fullfilling stronger asymptotic conditions and supertranslations ambiguities disappear. It is also clear from the result that 14 asymptotic gravitational charges …
The analytic expression obtained in the preceding project for the massless conformal scalar propagator in the Hartle-Hawking vacuum state for small values of the Schwarzschild radial coordinate above $ r = 2M$ is analytically extended into the interior of the Schwarzschild black hole. The result of the analytical extension coincides with the exact propagator for a small range of values of the Schwarzschild radial coordinate below $ r = 2M$ and is an analytic expression which manifestly features its dependence on the background space-time geometry. This feature as well as the absence of any ass…
We show that the scalar wave equation at low frequencies in the Schwarzschild geometry enjoys a hidden SL(2,R) invariance, which is not inherited from an underlying symmetry of the spacetime itself. Contrary to what happens for Kerr black holes, the vector fields generating the SL(2,R) are globally defined. Furthermore, it turns out that under an SU(2,1) Kinnersley transformation, which maps the Schwarzschild solution into the near horizon limit AdS_2 x S^2 of the extremal Reissner-Nordstr"om black hole (with the same entropy), the Schwarzschild hidden symmetry generators become exactly the is…
We show that the Schwarzschild-(A)dS black hole mechanics possesses a hidden symmetry under the three-dimensional Poincaré group. This symmetry shows up after having gauge-fixed the diffeomorphism invariance in the symmetry-reduced homogeneous Einstein-$Λ$ model and stands as a physical symmetry of the system. It dictates the geometry both in the black hole interior and exterior regions, as well as beyond the cosmological horizon in the Schwarzschild-dS case. It follows that one can associate a set of non-trivial conserved charges to the Schwarzschild-(A)dS black hole which act in each causall…
We study the excitations of a massive Schwarzschild black hole of mass M resulting from the capture of infalling matter described by a massless scalar field. The near-horizon dynamics of this system is governed by a Hamiltonian which is related to the Virasoro algebra and admits a one-parameter family of self-adjoint extensions described by a parameter z \in R . The density of states of the black hole can be expressed equivalently in terms of z or M, leading to a consistent relation between these two parameters. The corresponding black hole entropy is obtained as S = S(0) - 3/2 log S(0) + C, w…
Near-horizon conformal structure of a massive Schwarzschild black hole of mass M is analyzed using a scalar field as a simple probe of the background geometry. The near-horizon dynamics is governed by an operator which is related to the Virasoro algebra and admits a one-parameter family of self-adjoint extensions described by a real parameter z. When z satisfies a suitable contraint, the corresponding wavefunctions exhibit scaling behaviour in a band-like region near the horizon of the black hole. This formalism is consistent with the Bekenstein-Hawking entropy formula and naturally produces t…
We study the thermodynamics of spherically symmetric, neutral and non-rotating black holes in conformal (Weyl) gravity. To this end, we apply different methods: (i) the evaluation of the specific heat; (ii) the study of the entropy concavity; (iii) the geometrical approach to thermodynamics known as \textit{thermodynamic geometry}; (iv) the Poincaré method that relates equilibrium and out-of-equilibrium thermodynamics. We show that the thermodynamic geometry approach can be applied to conformal gravity too, because all the key thermodynamic variables are insensitive to Weyl scaling. The first …
We study thermodynamic quantities and phase transitions of a spherically symmetric Schwarzschild black hole by taking into account the back reaction through the conformal anomaly of matter fields, and show that there exists an additional phase transition to the conventional Hawking-Page phase transition. The small black hole is more probable than the hot flat space above a second critical temperature, while it is less probable than the hot flat space in the classical Schwarzschild black hole. However, the unstable small black hole eventually should decay into the stable large black hole becaus…
The conformal gravity remarkably boosts our prehension of gravity theories. We find a series of dynamical solutions in the $W^2$-conformal gravity, including generalized Schwarzschild-Friedmann-Robertson-Walker (GSFRW), charged generalized Schwarzschild-Friedmann-Robertson-Walker (CGSFRW), especially rotating Friedmann-Robertson-Walker (RFRW), charged rotating Friedmann-Robertson-Walker (CRFRW), and a dynamical cylindrically symmetric solutions. The RFRW, CRFRW and the dynamical cylindrically symmetric solutions are never found in the Einstein gravity and modified gravities. The GSFRW and CGSF…
The propagation of electromagnetic plane waves with negative phase velocity (NPV) is considered in Schwarzschild-(anti-)de Sitter spacetime. It is demonstrated that NPV propagation occurs in Schwarzschild-de Sitter spacetime at lower values of the cosmological constant than is the case for de Sitter spacetime. Furthermore, we report that neither is NPV propagation observed in Schwarzschild-anti-de Sitter spacetime, nor is it possible outside the event horizon of a Schwarzschild blackhole.
Conformal invariance can ameliorate or eliminate the singularities residing in the black holes, and may still exist in the strong gravity regimes close to these black holes. In this paper, we try to probe this conformal invariance by looking into the wave absorption and scattering by the nonsingular static spherical black holes. The partial and total absorption cross section, as well as the differential scattering cross section, are presented for black holes with different choices of conformal parameters. Although the photon trajectories are unchanged from the Schwarzschild case since the spac…
Vacuum spherically symmetric Einstein gravity in $N\ge 4$ dimensions can be cast in a two-dimensional conformal nonlinear sigma model form by first integrating on the $(N-2)$-dimensional (hyper)sphere and then performing a canonical transformation. The conformal sigma model is described by two fields which are related to the Arnowitt-Deser-Misner mass and to the radius of the $(N-2)$-dimensional (hyper)sphere, respectively. By quantizing perturbatively the theory we estimate the quantum corrections to the ADM mass of a black hole.
In this paper, we first investigate the thermal stability of black holes in conformal Weyl gravity with a comparison with the Schwarzschild black holes. Then, we consider a minimally coupled massive scalar perturbation and calculate the quasinormal modes in asymptotically dS spacetime by employing the sixth order WKB approximation and asymptotic iteration method. The deviations from those of the Schwarzschild-dS solutions are obtained and the possibility of the presence of quasi-resonance modes for Weyl black hole solutions is investigated. Finally, we consider a massless scalar perturbation i…
It is believed that in the near future, gravitational wave detections will become a promising tool not only to test gravity theories, but also to probe extremely curved spacetime regions in our universe, such as the surroundings of black holes. In this paper, we investigate the quasinormal modes (QNMs) of the axial gravitational perturbations of a class of non-singular black holes conformally related to the Schwarzschild black hole. These non-singular black holes can be regarded as the vacuum solution of a family of conformal gravity theories which are invariant under conformal transformations…
We show that the causal properties of asymptotically flat spacetimes depend on their dimensionality: while the time-like future of any point in the past conformal infinity $\mathcal{I}^-$ contains the whole of the future conformal infinity $\mathcal{I}^+$ in $(2+1)$ and $(3+1)$ dimensional Schwarzschild spacetimes, this property (which we call the Penrose property) does not hold for $(d+1)$ dimensional Schwarzschild if $d>3$. We also show that the Penrose property holds for the Kerr solution in $(3+1)$ dimensions, and discuss the connection with scattering theory in the presence of positive ma…
Based on the hidden conformal symmetry, some authors have proposed a Harrison metric for the Schwarzschild black hole. We give a procedure which can generate a family of Harrison metrics starting from a general set of SL(2, R) vector fields. By analogy with the subtracted geometry of the Kerr black hole, we find a new Harrison metric for the Schwarzschild case. Its conformal generators are also investigated using the Killing equations in the near-horizon limit.
We present an exhaustive analysis of scalar, electromagnetic and gravitational perturbations in the background of a Schwarzchild-de Sitter spacetime. The field propagation is considered by means of a semi-analytical (WKB) approach and two numerical schemes: the characteristic and general initial value integrations. The results are compared near the extreme regime, and a unifying picture is established for the dynamics of different spin fields. Although some of the results just confirm usual expectations, a few surprises turn out to appear, as the dependence on the non-characteristic initial co…
The characteristic features of $<T_{μν}>$ in the Boulware, Unruh and Hartle-Hawking states for a conformal massless scalar field propagating in the Schwarzschild space-time are obtained by means of effective actions deduced by the trace anomaly. The actions are made local by the introduction of auxiliary fields and boundary conditions are carefully imposed on them in order to select the different quantum states.
It is well-known that the $(1+1)$ dimensional Schwarzschild and spatially flat FLRW spacetimes are conformally flat. This work examines entanglement harvesting from the conformal field vacuums in these spacetimes between two Unruh-DeWitt detectors, moving along outgoing null trajectories. In $(1+1)$ dimensional Schwarzschild spacetime, we considered the Boulware and Unruh vacuums for our investigations. In this analysis, one observes that while entanglement harvesting is possible in $(1+1)$ dimensional Schwarzschild and $(1+3)$ dimensional de Sitter spacetimes, it is not possible in the $(1+1)…
Light propagation and quasinormal modes of a topologically charged Schwarzschild-Klinkhamer wormhole
Universidade Federal do Espírito Santo（圣埃斯皮里图联邦大学）；Instituto Federal de Educação Ciência e Tecnologia do Maranhão（马拉尼昂联邦教育、科学和技术学院）；University of Valladolid UVA（巴利亚多利德大学）；Universidade Estadual do Maranhão（马拉尼昂州立大学）；Universidade Federal do Pará（帕拉联邦大学）；Universidade Federal da Paraíba（帕拉伊巴联邦大学）；Universidade Federal de Campina Grande（坎皮纳格兰德联邦大学）
The statistical entropy of a Schwarzschild black string in five dimensions is obtained by counting the black string states which form a representation of the near-horizon conformal symmetry with a central charge. The statistical entropy of the string agrees with its Bekenstein-Hawking entropy as well as that of the Schwarzschild black hole in four dimensions. The choice of the string length which gives the Virasoro algebra also reproduces the precise value of the Bekenstein-Hawking entropy and lies inside the stability bound of the string.
Black hole (BH) solution in the conformal Weyl gravity is a generalization of the Schwarzschild spacetime which includes two additional constants appearing when integrating the third order differential equations for gravitational field. One constant looks like the effective cosmological constant providing the de Sitter asymptotic of the solution. The other constant allows one to describe flat rotation of galaxies without introducing of the dark matter. Here we show that the effective "dark matter" term in the metric function drastically changes the asymptotic behavior of the evolution of the w…
We study static massless minimally coupled scalar field created by a source in a static D-dimensional spacetime. We demonstrate that the corresponding equation for this field is invariant under a special transformation of the background metric. This transformation consists of the static conformal transformation of the spatial part of the metric accompanied by a properly chosen transformation of the red-shift factor. Both transformations are determined by one function of the spatial coordinates. We show that in a case of higher dimensional spherically symmetric black holes one can find such a b…
Exotic Spinor Deformation of a Schwarzschild Black Hole: Perturbations and Stability
Faculty of Engineering and Science of Guaratinguetá, UNESP（圣保罗州立大学瓜拉廷格塔工程与科学学院）；São Paulo State Department of Education (SEDUC-SP)（圣保罗州教育部）；Engineering School of Lorena - University of Sao Paulo (EEL-USP)（圣保罗大学洛雷达纳工程学院）
The construction of the conformal scalar propagator which has been obtained in the preceding two projects as an analytic function of the Schwarzschild black-hole space-time is completed with a boundary condition imposed by the physical context through contour integration in the exterior vicinity of the event horizon. It is shown that, as a consequence of the semi-classical character which the emitted quanta have in that exterior vicinity, the particle production by the Schwarzschild black hole which was formally established in the preceding project is identical to thermal Hawking radiation. By…
We determine the quasinormal frequencies for all gravitational perturbations of the d-dimensional Schwarzschild black hole, in the infinite damping limit. Using the potentials for gravitational perturbations derived recently by Ishibashi and Kodama, we show that in all cases the asymptotic real part of the frequency is proportional to the Hawking temperature with a coefficient of log 3. Via the correspondence principle, this leads directly to an equally spaced entropy spectrum. We comment on the possible implications for the spacing of eigenvalues of the Virasoro generator in the associated ne…
We revisit Carlip's approach to entropy counting. This analysis reemerged in a recently obtained Schwarzschild/CFT-correspondence as Sugawara-construction of a 2D stress-tensor. Here, for the example of a Schwarzschild black hole, we show how to single out diffeomorphisms forming in contrast to Carlip's analysis the full 2D local conformal algebra. We provide arguments, why their Hamiltonian generators are expected to be the symmetry generators of a possible conformal field theory describing the part of phase space responsible for black hole microstates. Then, we can infer central charges and …
We investigate the black holes in the new massive conformal gravity which is not invariant under conformal transformations because of the presence of the Einstein-Hilbert term. First, we show that the small Schwarzschild black hole is unstable against the $s$-mode of linearized Ricci tensor by solving the Lichnerowicz-Ricci tensor equation. This instability induces the appearance of the non-BBMB (Bocharova-Bronnikov-Melnikov-Bekenstein) black hole that has both Ricci tensor and conformal scalar hair.
The accretion of test fluids flowing onto a black hole is investigated. Particularly, by adopting a dynamical Hamiltonian approach, we are capable to find the critical points for various cases of black hole in conformal gravity. In these cases, we have analyzed the general solutions of accretion employing the isothermal equations of state. The steady state and spherically symmetric accretion of different test fluids onto the conformal gravity black hole has been considered. Further, we have classified these flows in the context of equations of state and the cases of conformal gravity black hol…
The exponential blueshift associated with the event horizon of a black hole makes conformal symmetry play a fundamental role in accounting for its thermal properties. Using a derivation based on two-point functions, we show that the spectrum of thermal radiation of scalar particles by Kerr (and Schwarzschild) black holes can be explicitly derived on the basis of a $2$-dimensional conformal symmetry arising in the wave equation near the horizon. This result reinforces the recently conjectured relation between Kerr geometry and a $2$-dimensional conformal field theory.
In the analysis of a binary black hole coalescence, it is necessary to include gravitational self-interactions in order to describe the transition of the gravitational wave signal from the merger to the ringdown stage. In this paper we study the phenomenology of the generation and propagation of nonlinearities in the ringdown of a Schwarzschild black hole, using second-order perturbation theory. Following earlier work, we show that the Green's function and its causal structure determines how both first-order and second-order perturbations are generated, and hence highlight that both of these s…
We study non conformal negative modes of Schwarzschild black holes in the five dimensional Einstein-Gauss-Bonnet gravity theory with a zero or negative cosmological constant, in the context of the semiclassical quantum gravity formulated by the path integral. We find an unique negative mode when the black hole has a negative heat capacity which is the same in Einstein theory. On the other hand, we still find negative modes inside a parameter region where a small black hole has a positive heat capacity. The number is one/two for the case of zero/negative cosmological constant respectively. In t…
We extend the derivation of the Hawking temperature of a Schwarzschild black hole via the Heisenberg uncertainty principle to the de Sitter and anti-de Sitter spacetimes. The thermodynamics of the Schwarzschild-(anti-)de Sitter black holes is obtained from the generalized uncertainty principle of string theory and non-commutative geometry. This may explain why the thermodynamics of (anti-)de Sitter-like black holes admits a holographic description in terms of a dual quantum conformal field theory, whereas the thermodynamics of Schwarzschild-like black holes does not.
Building on a pair of earlier papers, I investigate the various point-wise and averaged energy conditions for the quantum stress-energy tensor corresponding to a conformally-coupled massless scalar field in the in the (1+1)-dimensional Schwarzschild spacetime. Because the stress-energy tensors are analytically known, I can get exact results for the Hartle--Hawking, Boulware, and Unruh vacua. This exactly solvable model serves as a useful sanity check on my (3+1)-dimensional investigations wherein I had to resort to a mixture of analytic approximations and numerical techniques. Key results in (…
We analyze the classical stability of Schwarzschild black hole in massive conformal gravity which was recently proposed for another massive gravity model. This model in the Jordan frame is conformally equivalent to the Einstein-Weyl gravity in the Einstein frame. The coupled linearized Einstein equation is decomposed into the traceless and trace equation when one chooses $6m^2φ=δR$. Solving the traceless equation exhibits unstable modes featuring the Gregory-Laflamme $s$-mode instability of five-dimensional black string, while we find no unstable modes when solving the trace equation. It is sh…
We study the three dimensional Einstein gravity conformally coupled to a scalar field. Solutions of this theory are geometries with vanishing scalar curvature. We consider solutions with a constant scalar field which corresponds to an infinite Newton's constant. There is a class of solutions with possible curvature singularities which asymptotic symmetries are given by two copies of the Virasoro algebra. We argue that the central charge of the corresponding CFT is infinite. Furthermore, we construct a family of Schwarzschild solutions which can be conformally mapped to the Martinez-Zanelli sol…
In this paper, we consider a massless field, with spin j, in interaction with a Schwarzschild black hole in four dimensions, focusing mainly our study on the s-wave scattering. First, using a Fourier analysis, we show that one can have a simple and natural description of the Physics near the event horizon without using any conformal field approaches. Then, within the same "scattering picture", we derive analytically the imaginary part of the highly damped quasinormal complex frequencies and, as a natural consequence of our analysis, we show that thermal effects and in particular Hawking radiat…
We compute the spectra and total fluxes of quantum mechanically produced particles crossing the black hole and cosmological horizons in Schwarzschild de Sitter (SdS). Particle states are defined with respect to well-behaved, Kruskal coordinates near the horizons, and as a consequence we find that these spectra are generally non-thermal. The non-thermal Bogoliubov coefficient for a vacuum fluctuation near the black hole horizon to produce a particle that crosses the cosmological horizon is shown to equal to the convolution of two thermal coefficients, one at the cosmological temperature and one…
The trace anomaly for free propagation in the context of a conformally invariant scalar field theory defined on a curved manifold of positive constant curvature with boundary is evaluated through use of an asymptotic heat kernel expansion. In addition to their direct physical significance the results are also of relevance to the holographic principle and to Quantum Cosmology.
Motivated by the universality of Hawking radiation and that of the anomaly cancellation technique as well as that of the effective action method, we investigate the Hawking radiation of a Schwarzschild black hole in the isotropic coordinates via the cancellation of gravitational anomaly. After performing a dimensional reduction from the four-dimensional isotropic Schwarzschild metric, we show that this reduction procedure will, in general, result in two classes of two-dimensional effective metrics: the conformal equivalent and the inequivalent ones. For the physically equivalent class, the two…
We study the quasi-normal modes (QNM) of electromagnetic and gravitational perturbations of a Schwarzschild black hole in an asymptotically Anti-de Sitter (AdS) spacetime. Some of the electromagnetic modes do not oscillate, they only decay, since they have pure imaginary frequencies. The gravitational modes show peculiar features: the odd and even gravitational perturbations no longer have the same characteristic quasinormal frequencies. There is a special mode for odd perturbations whose behavior differs completely from the usual one in scalar and electromagnetic perturbation in an AdS spacet…
We employ the method of comparison equations to study the propagation of a massless minimally coupled scalar field on the Schwarzschild background. In particular, we show that this method allows us to obtain explicit approximate expressions for the radial modes with energy below the peak of the effective potential which are fairly accurate over the whole region outside the horizon. This case can be of particular interest, for example, for the problem of black hole evaporation.
We present the first numerical construction of the scalar Schwarzschild Green function in the time-domain, which reveals several universal features of wave propagation in black hole spacetimes. We demonstrate the trapping of energy near the photon sphere and confirm its exponential decay. The trapped wavefront propagates through caustics resulting in echoes that propagate to infinity. The arrival times and the decay rate of these caustic echoes are consistent with propagation along null geodesics and the large l-limit of quasinormal modes. We show that the four-fold singularity structure of th…
Dynamical quasinormal mode excitation II: propagation and convergence in Schwarzschild
Perimeter Institute for Theoretical Physics（Perimeter理论物理研究所）；CENTRA, Departamento de Fisica, Instituto Superior Técnico – IST, Universidade de Lisboa – UL（里斯本大学高级技术学院物理系CENTRA中心）；School of Physics and Astronomy and Institute for Gravitational Wave Astronomy, University of Birmingham（伯明翰大学物理与天文学学院及引力波天文研究院）；School of Mathematical Sciences, University of Nottingham（诺丁汉大学数学科学学院）
In this paper we show that the equation of motion of a massless scalar particle near the black hole (BH) horizon in the Schwarzschild-deSitter (SdS) spacetime enjoys a hidden conformal symmetry when the two horizons satisfy a quadratic relation, not related with the Nariai limit. This hidden symmetry is $SL(2,\mathbb{R})\times SO(2)$ instead of $SO(2,1)\times SO(2)$, the expected symmetry of $dS_2\times S^2$. We present a structural analysis of SdS spacetime and compute the leading quantum corrections to the entropy of SdS spacetime in a semi-classical way.
We present heuristic arguments suggesting that if EM waves with wavelengths somewhat larger than the Schwarzschild radius of a black hole were fully absorbed by it, the second law of thermodynamics would be violated, under the Bekenstein interpretation of the area of a black hole as a measure of its entropy. Thus, entropy considerations make the well known fact that large wavelengths are only marginally absorbed by black holes, a natural consequence of thermodynamics. We also study numerically the ingoing radial propagation of a scalar field wave in a Schwarzschild metric, relaxing the standar…
Recent studies of asymptotic symmetries suggest, that a Hamiltonian phase space analysis in gravitational theories might be able to account for black hole microstates. In this context we explain, why the use of conventional Bondi fall-off conditions for the gravitational field is too restrictive in the presence of an event horizon. This implies an enhancement of physical degrees of freedom ($\mathcal{A}$-modes). They provide new gravitational hair and are responsible for black hole microstates. Using covariant phase space methods, for the example of a Schwarzschild black hole, we give a propos…
We derive the statistical entropy of the Schwarzschild black hole by considering the asymptotic symmetry algebra near the $\cal{I^{-}}$ boundary of the spacetime at past null infinity. Using a two-dimensional description and the Weyl invariance of black hole thermodynamics this symmetry algebra can be mapped into the Virasoro algebra generating asymptotic symmetries of anti-de Sitter spacetime. Using lagrangian methods we identify the stress-energy tensor of the boundary conformal field theory and we calculate the central charge of the Virasoro algebra. The Bekenstein-Hawking result for the bl…
We explicitly prove that the Weyl conformal symmetry solves the black hole singularity problem, otherwise unavoidable in a generally covariant local or non-local gravitational theory. Moreover, we yield explicit examples of local and non-local theories enjoying Weyl and diffeomorphism symmetry (in short co-covariant theories). Following the seminal paper by Narlikar and Kembhavi, we provide an explicit construction of singularity-free spherically symmetric and axi-symmetric exact solutions for black hole spacetimes conformally equivalent to the Schwarzschild or the Kerr spacetime. We first che…
In this paper, we address a neural field equation that characterizes spatio-temporal propagation of a neural population pulse. Due that the human brain is a complex system whose constituents interaction give rise to fundamental states of consciousness and behavior, it is crucial to gain insight into its functioning even at relativistic scales. To this end, we study the action of the relativistic conformal group on the accounted neural field propagation equation. In particular, we obtain an exact solution for the field propagation equation when the space-time is 3 or 4 dimensional. Furthermore,…
We study a general D-dimensional Schwarzschild-type black brane solution of the Einstein-dilaton theory and derive, by using the holographic renormalization, its thermodynamics consistent with the geometric results. Using the membrane paradigm, we calculate the several hydrodynamic transport coefficients and compare them with the results obtained by the Kubo formula, which shows the self-consistency of the gauge/gravity duality in the relativistic non-conformal theory. In order to understand more about the relativistic non-conformal theory, we further investigate the binding energy, drag force…
