A comparison of Noether charge and Euclidean methods for Computing the Entropy of Stationary Black Holes

The entropy of stationary black holes has recently been calculated by a number of different approaches. Here we compare the Noether charge approach (defined for any diffeomorphism invariant Lagrangian theory) with various Euclidean methods, specifically, (i) the microcanonical ensemble approach of Brown and York, (ii) the closely related approach of Ba\~nados, Teitelboim, and Zanelli which ultimately expresses black hole entropy in terms of the Hilbert action surface term, (iii) another formula of Ba\~nados, Teitelboim and Zanelli (also used by Susskind and Uglum) which views black hole entropy as conjugate to a conical deficit angle, and (iv) the pair creation approach of Garfinkle, Giddings, and Strominger. All of these approaches have a more restrictive domain of applicability than the Noether charge approach. Specifically, approaches (i) and (ii) appear to be restricted to a class of theories satisfying certain properties listed in section 2; approach (iii) appears to require the Lagrangian density to be linear in the curvature; and approach (iv) requires the existence of suitable instanton solutions. However, we show that within their domains of applicability, all of these approaches yield results in agreement with the Noether charge approach. In the course of our analysis, we generalize the definition of Brown and York's quasilocal energy to a much more general class of diffeomorphism invariant, Lagrangian theories of gravity. In an appendix, we show that in an arbitrary diffeomorphism invariant theory of gravity, the ``volume term" in the ``off-shell" Hamiltonian associated with a time evolution vector field $t^a$ always can be expressed as the spatial integral of $t^a {\cal C}_a$, where ${\cal C}_a = 0$ are the constraints associated with the diffeomorphism invariance.Comment: 29 pages (double-spaced) late

Duality between Electric and Magnetic Black Holes

A number of attempts have recently been made to extend the conjectured $S$ duality of Yang Mills theory to gravity. Central to these speculations has been the belief that electrically and magnetically charged black holes, the solitons of quantum gravity, have identical quantum properties. This is not obvious, because although duality is a symmetry of the classical equations of motion, it changes the sign of the Maxwell action. Nevertheless, we show that the chemical potential and charge projection that one has to introduce for electric but not magnetic black holes exactly compensate for the difference in action in the semi-classical approximation. In particular, we show that the pair production of electric black holes is not a runaway process, as one might think if one just went by the action of the relevant instanton. We also comment on the definition of the entropy in cosmological situations, and show that we need to be more careful when defining the entropy than we are in an asymptotically-flat case.Comment: 23 pages, revtex, no figures. Major revision: two sections on the electric Ernst solution adde

Towards a Qatar Cybersecurity Capability Maturity Model with a Legislative Framework

في هذا العصر، يجب على الدول وضع التشريعات التي تقيس قدرات أمنها السيبراني وتطوير برامجها، بالأخص عندما تُستخدم ثغرات الأمن السيبراني كذريعة لفرض الحصار، كما هو الحال في دولة قطر، وذلك بعد أن تم اختراق وكالة الأنباء القطرية. يقترح هذا البحث نموذجًا لتعزيز قدرات الأمن السيبراني (Q-C2M2) في دولة قطر ضمن إطار تشريعي. ويتناول البحث نموذجًا أصيلًا لتعزيز قدرات الأمن السيبراني مع تسليط الضوء على غرضه وخصائصه واعتماده. كما يعرض البحث نماذجًا لتعزيز قدرات الأمن السيبراني الحالية والمعترف بها عالميًا، ودراسة عن الأمن السيبراني في دولة قطر باستخدام الوثائق المتاحة، وذلك بناء على منهجية التحليل الموضوعي للوثائق. كما يقدم هذا البحث تحليلًا مقارنًا لنماذج تعزيز قدرات الأمن السيبراني في ضوء الأمن السيبراني القطري. وفي هذا الإطار، ساعد التحليل المقارن للوثائق في تحديد الثغرات الموجودة في سياسة تأمين المعلومات الوطنية القطرية بشكل عام، ودليل تأمين المعلومات الوطنية القطرية بشكل خاص. يهدف نموذج  (Q-C2M2) المقترح إلى تعزيز إطار عمل الأمن السيبراني في قطر من خلال توفير نموذج عملي مع عنصر تشريعي يمكن استخدامه لقياس أداء الأمن السيبراني وتطويره. كما يقترح هذا النموذج مجالات للمستخدمين “USERS” التي تتكون من الفهم (Understand)، والأمن(Secure) ، والكشف(Expose) ، والاستعادة(Recover) ، والاستدامة(Sustain) ، حيث يتضمن كل مجال مجالات فرعية، والتي بموجبها يمكن للمؤسسة إنشاء أنشطة للأمن السيبراني عند التقييم الأولي. يستخدم نموذج (Q-C2M2) المستويات الخمسة التالية لقياس تعزيز قدرات الأمن السيبراني للمنظمات: البدء والتطبيق والتطوير والتكيف والمرونة.In an age when cybersecurity vulnerabilities can be used as a pretext for a blockade, as in the case of Qatar prompted by a hack of the Qatar News Agency, it becomes incumbent upon states to consider legislating the capability maturity measurement and the development of their cybersecurity programs across the community. This paper proposes a Qatar Cybersecurity Capability Maturity Model (Q-C2M2) with a legislative framework. The paper discusses the origin, purpose and characteristics of a capability maturity model and its adoption in the cybersecurity domain. Driven by a thematic analysis under the document analysis methodology, the paper examines existing globally recognized cybersecurity capability maturity models and Qatar’s cybersecurity framework using publicly available documents. This paper also conducts a comparative analysis of existing cybersecurity capability maturity models in light of the Qatari cybersecurity framework, including a comparative analysis of cybersecurity capability maturity model literature. The comparative document analysis helped identify gaps in the existing Qatar National Information Assurance Policy and specifically the Qatar National Information Assurance Manual. The proposed Q-C2M2 aims to enhance Qatar’s cybersecurity framework by providing a workable Q-C2M2 with a legislative component that can be used to benchmark, measure and develop Qatar’s cybersecurity framework. The Q-C2M2 proposes the USERS domains consisting of Understand, Secure, Expose, Recover and Sustain. Each domain consists of subdomains, under which an organization can create cybersecurity activities at initial benchmarking. The Q-C2M2 uses the following five levels to measure the cybersecurity capability maturity of an organization: Initiating, Implementing, Developing, Adaptive and Agile

Global Charges in Chern-Simons theory and the 2+1 black hole

We use the Regge-Teitelboim method to treat surface integrals in gauge theories to find global charges in Chern-Simons theory. We derive the affine and Virasoro generators as global charges associated with symmetries of the boundary. The role of boundary conditions is clarified. We prove that for diffeomorphisms that do not preserve the boundary there is a classical contribution to the central charge in the Virasoro algebra. The example of anti-de Sitter 2+1 gravity is considered in detail.Comment: Revtex, no figures, 26 pages. Important changes introduced. One section added

Duality of Quasilocal Gravitational Energy and Charges with Non-orthogonal Boundaries

We study the duality of quasilocal energy and charges with non-orthogonal boundaries in the (2+1)-dimensional low-energy string theory. Quasilocal quantities shown in the previous work and some new variables arisen from considering the non-orthogonal boundaries as well are presented, and the boost relations between those quantities are discussed. Moreover, we show that the dual properties of quasilocal variables such as quasilocal energy density, momentum densities, surface stress densities, dilaton pressure densities, and Neuve-Schwarz(NS) charge density, are still valid in the moving observer's frame.Comment: 19pages, 1figure, RevTe

Symmetries, Horizons, and Black Hole Entropy

Black holes behave as thermodynamic systems, and a central task of any quantum theory of gravity is to explain these thermal properties. A statistical mechanical description of black hole entropy once seemed remote, but today we suffer an embarrassment of riches: despite counting very different states, many inequivalent approaches to quantum gravity obtain identical results. Such ``universality'' may reflect an underlying two-dimensional conformal symmetry near the horizon, which can be powerful enough to control the thermal characteristics independent of other details of the theory. This picture suggests an elegant description of the relevant degrees of freedom as Goldstone-boson-like excitations arising from symmetry breaking by the conformal anomaly.Comment: 6 pages; first prize essay, 2007 Gravity Research Foundation essay contes

Lie symmetries of Einstein's vacuum equations in N dimensions

We investigate Lie symmetries of Einstein's vacuum equations in N dimensions, with a cosmological term. For this purpose, we first write down the second prolongation of the symmetry generating vector fields, and compute its action on Einstein's equations. Instead of setting to zero the coefficients of all independent partial derivatives (which involves a very complicated substitution of Einstein's equations), we set to zero the coefficients of derivatives that do not appear in Einstein's equations. This considerably constrains the coefficients of symmetry generating vector fields. Using the Lie algebra property of generators of symmetries and the fact that general coordinate transformations are symmetries of Einstein's equations, we are then able to obtain all the Lie symmetries. The method we have used can likely be applied to other types of equations

Conservation Laws and 2D Black Holes in Dilaton Gravity

A very general class of Lagrangians which couple scalar fields to gravitation and matter in two spacetime dimensions is investigated. It is shown that a vector field exists along whose flow lines the stress-energy tensor is conserved, regardless of whether or not the equations of motion are satisfied or if any Killing vectors exist. Conditions necessary for the existence of Killing vectors are derived. A new set of 2D black hole solutions is obtained for one particular member within this class of Lagrangians. One such solution bears an interesting resemblance to the 2D string-theoretic black hole, yet contains markedly different thermodynamic properties.Comment: 11 pgs. WATPHYS-TH92/0

Black hole entropy from classical Liouville theory

In this article we compute the black hole entropy by finding a classical central charge of the Virasoro algebra of a Liouville theory using the Cardy formula. This is done by performing a dimensional reduction of the Einstein Hilbert action with the ansatz of spherical symmetry and writing the metric in conformally flat form. We obtain two coupled field equations. Using the near horizon approximation the field equation for the conformal factor decouples. The one concerning the conformal factor is a Liouville equation, it posses the symmetry induced by a Virasoro algebra. We argue that it describes the microstates of the black hole, namely the generators of this symmetry do not change the thermodynamical properties of the black hole.Comment: LaTeX, 11 pages, to appear on JHE