Schwarzschild-anti de Sitter within an Isothermal Cavity: Thermodynamics, Phase Transitions and the Dirichlet Problem

The thermodynamics of Schwarzschild black holes within an isothermal cavity and the associated Euclidean Dirichlet boundary-value problem are studied for four and higher dimensions in anti-de Sitter (AdS) space. For such boundary conditions classically there always exists a unique hot AdS solution and two or no Schwarzschild-AdS black-hole solutions depending on whether or not the temperature of the cavity-wall is above a minimum value, the latter being a function of the radius of the cavity. Assuming the standard area-law of black-hole entropy, it was known that larger and smaller holes have positive and negative specific heats and hence are locally thermodynamically stable and unstable respectively. In this paper we present the first derivation of this by showing that the standard area law of black-hole entropy holds in the semi-classical approximation of the Euclidean path integral for such boundary conditions. We further show that for wall-temperatures above a critical value a phase transition takes hot AdS to the larger Schwarzschild-AdS within the cavity. The larger hole thus can be globally thermodynamically stable above this temperature. The phase transition can occur for a cavity of arbitrary radius above a (corresponding) critical temperature. In the infinite cavity limit this picture reduces to that considered by Hawking and Page. The case of five dimensions is found to be rather special since exact analytic expressions can be obtained for the masses of the two holes as functions of cavity radius and temperature thus solving exactly the Euclidean Dirichlet problem. This makes it possible to compute the on-shell Euclidean action as functions of them from which other quantities of interest can be evaluated exactly.Comment: 23 pages, Late

Thermodynamic Interpretation of Field Equations at Horizon of BTZ Black Hole

A spacetime horizon comprising with a black hole singularity acts like a boundary of a thermal system associated with the notions of temperature and entropy. In case of static metric of BTZ black hole, the field equations near horizon boundary can be expressed as a thermal identity $dE = TdS + P_{r}dA$, where $E = M$ is the mass of BTZ black hole, $dA$ is the change in the area of the black hole horizon when the horizon is displaced infinitesimally small, $P_{r}$ is the radial pressure provided by the source of Einstein equations, $S= 4\pi a$ is the entropy and $T = \kappa / 2\pi$ is the Hawking temperature associated with the horizon. This approach is studied further to generalize it for non-static BTZ black hole and show that it is also possible to interpret the field equation near horizon as a thermodynamic identity $dE = TdS + P_{r}dA + \Omega_{+} dJ$, where $\Omega_{+}$ is the angular velocity and $J$ is the angular momentum of BTZ black hole. These results indicate that the field equations for BTZ black hole possess intrinsic thermodynamic properties near horizon.Comment: 8 page

Comparative Study of Mathematics Learning Students Outcomes Taught by Cooperative Learning Model Teams Games Tournament Type (TGT) and Talking Stick Type (TS)

This research method was Experimental Research. The Research aimed at determining whether there were differences on the student learning outcomes taught by cooperative learning model Teams Games Tournament type (TGT) and the students who were taught by cooperative learning model Talking Stick type (TS). This research was conducted at class VIII SMP Negeri 2 Kolaka in the academic year 2016/2017 consisting of 8 classes with total students was 199 students. VIII5 Class was the first experimental class taught by cooperative learning model TGT type and VIII3 Class was the second experimental class taught by cooperative learning TS type.  Technique of Data Analysis used descriptive statistics and inferential statistics. The research findings were: (1) Mathematics learning outcomes of students taught by cooperative learning model TGT  type consisting of 28 students obtained mean = 81,29, median = 84, mode = 85, standard deviation = 8,814, and variance = 77,693 . In addition, 23 of 28 students (82.15%) had learning outcomes above KKM = 75, and 19 of 28 students (57.14%) had scores above mean = 81.29. (2) The learning outcomes of mathematics student taught by cooperative learning model of TS type consisting of 28 students obtained mean = 81,64, median = 84,50, mode = 87, standard deviation = 9,306, and variance = 86,605. In addition, 22 of 28 students (78.57%) had learning outcomes above KKM = 75, and 20 of 28 students (71.42%) had scores above mean = 81.64. Based on the findings above, it can be concluded that there was no difference in the mean of mathematics learning outcomes of students taught by cooperative learning model both TGT type and TS type. The models provided good learning outcomes and improved the students’ engagement in the teaching and learning process

Ricci-flat Metrics with U(1) Action and the Dirichlet Boundary-value Problem in Riemannian Quantum Gravity and Isoperimetric Inequalities

The Dirichlet boundary-value problem and isoperimetric inequalities for positive definite regular solutions of the vacuum Einstein equations are studied in arbitrary dimensions for the class of metrics with boundaries admitting a U(1) action. We show that in the case of non-trivial bundles Taub-Bolt infillings are double-valued whereas Taub-Nut and Eguchi-Hanson infillings are unique. In the case of trivial bundles, there are two Schwarzschild infillings in arbitrary dimensions. The condition of whether a particular type of filling in is possible can be expressed as a limitation on squashing through a functional dependence on dimension in each case. The case of the Eguchi-Hanson metric is solved in arbitrary dimension. The Taub-Nut and the Taub-Bolt are solved in four dimensions and methods for arbitrary dimension are delineated. For the case of Schwarzschild, analytic formulae for the two infilling black hole masses in arbitrary dimension have been obtained. This should facilitate the study of black hole dynamics/thermodynamics in higher dimensions. We found that all infilling solutions are convex. Thus convexity of the boundary does not guarantee uniqueness of the infilling. Isoperimetric inequalities involving the volume of the boundary and the volume of the infilling solutions are then investigated. In particular, the analogues of Minkowski's celebrated inequality in flat space are found and discussed providing insight into the geometric nature of these Ricci-flat spaces.Comment: 40 pages, 3 figure

Stress and Coping Stress on Motorcycle Driver

The imbalance increasing number of motorcycle every year compared to the increasing street development causing traffic jam. The traffic jam also cause stress on motorcycle driver. The aim of this study is to know the stress and the coping stress on motorcycle driver, with one research subject which is a man who usually drive motorcycle every day. The factors that can cause stress are physic stressor; such as weather and traffic jam, social stressor; such as busy hours, crowded streets and reckless drivers. The psychological symptomps are exhausted physic and easy to get angry or mad on the street. The subject uses emotion-focused coping and problem-focused coping to handle those conditions

Implementasi Perlindungan Hak-hak Masyarakat Miskin sebagai Hak Konstitusional dalam Memdapatkan Pelayanan Kesehatan di Kabupaten Rokan Hilir

Constitutional development has shifted from centralized to decentralized marked with the enactment of the Local Government Act, resulting in the shift of some tasks of the central government becomes the burden of responsibility in the area of managing. The Act contains a provision stating that the health sector is fully submitted to the Local Government with authority to manage and administer all aspects of health. Accordingly, Act No. 32 of 1992 on Health necessary adjustments with the spirit of regional autonomy. In this case I will focus on research in the area Rokan Hilir Government in the implementation of health services to the poor are classified continue to face problems in implementation.This type of research can be classified in socio-juridical research, because in this study the authors directly conduct research on the location or place under study in order to give a complete and clear picture of the problem under study. This research was conducted in jurisdictions Government Rokan Hilir, while the population and the sample is a whole party relating to the issues examined in this study, the data sources used, the primary data, secondary data, and the data tertiary data collection techniques in this study with observation, interview and literature study.From the research, there are three main issues that can be inferred: first, the implementation of health services for the poor in Rokan Hilir the form of Community Health Insurance (Jamkesmas) and the Regional Health Insurance (Jamkesda); second, inhibiting factor in the implementation of health services include internal factors such as lack of medical personnel, limited medical facilities, and Abuse of authority medical personnel, and external factors such as infrastructure constraints of non-medical and Geographic Conditions; Third, efforts made by the Government yanng Rokan Hilir in meeting the implementation of health care for the poor in Rokan Hilir include monitoring, coaching and training actions. Suggestions author, first, should have been the implementation of the provision of health care and social programs, also supported by the human resources as a medical team in providing good service and is responsible not only to patients, but also to God, because it is a social work community , Second, Rokan Hilir government should also consider the factors supporting each program, particularly in health services in poor communities, to meminimallisir constraints in the implementation of the provision of health services, third, efforts taken by the government should continue to pay attention to the various needs based kendala- constraints faced by the government in implementing health care programs, not just focus on the things that are technical health services it is of also matters is the formation of character and social

\bbbc P^2 and \bbbc P^{1} Sigma Models in Supergravity: Bianchi type IX Instantons and Cosmologies

We find instanton/cosmological solutions with biaxial Bianchi-IX symmetry, involving non-trivial spatial dependence of the \bbbc P^{1}- and \bbbc P^{2}-sigma-models coupled to gravity. Such manifolds arise in N=1, $d=4$ supergravity with supermatter actions and hence the solutions can be embedded in supergravity. There is a natural way in which the standard coordinates of these manifolds can be mapped into the four-dimensional physical space. Due to its special symmetry, we start with \bbbc P^{2} with its corresponding scalar Ansatz; this further requires the spacetime to be $SU(2) \times U(1)$-invariant. The problem then reduces to a set of ordinary differential equations whose analytical properties and solutions are discussed. Among the solutions there is a surprising, special-family of exact solutions which owe their existence to the non-trivial topology of \bbbc P^{2} and are in 1-1 correspondence with matter-free Bianchi-IX metrics. These solutions can also be found by coupling \bbbc P^{1} to gravity. The regularity of these Euclidean solutions is discussed -- the only possibility is bolt-type regularity. The Lorentzian solutions with similar scalar Ansatz are all obtainable from the Euclidean solutions by Wick rotation

Entropy Corrections for Schwarzschild and Reissner-Nordstr\"om Black Holes

Schwarzschild black hole being thermodynamically unstable, corrections to its entropy due to small thermal fluctuations cannot be computed. However, a thermodynamically stable Schwarzschild solution can be obtained within a cavity of any finite radius by immersing it in an isothermal bath. For these boundary conditions, classically there are either two black hole solutions or no solution. In the former case, the larger mass solution has a positive specific heat and hence is locally thermodynamically stable. We find that the entropy of this black hole, including first order fluctuation corrections is given by: {\cal S} = S_{BH} - \ln[\f{3}{R} (S_{BH}/4\p)^{1/2} -2]^{-1} + (1/2) \ln(4\p), where $S_{BH}=A/4$ is its Bekenstein-Hawking entropy and $R$ is the radius of the cavity. We extend our results to four dimensional Reissner-Nordstr\"om black holes, for which the corresponding expression is: {\cal S} = S_{BH} - \f{1}{2} \ln [ {(S_{BH}/\p R^2) ({3S_{BH}}/{\p R^2} - 2\sqrt{{S_{BH}}/{\p R^2 -\a^2}}) \le(\sqrt{{S_{BH}}/{\p R^2}} - \a^2 \ri)}/ {\le({S_{BH}}/{\p R^2} -\a^2 \ri)^2} ]^{-1} +(1/2)\ln(4\p). Finally, we generalise the stability analysis to Reissner-Nordstr\"om black holes in arbitrary spacetime dimensions, and compute their leading order entropy corrections. In contrast to previously studied examples, we find that the entropy corrections in these cases have a different character.Comment: 6 pages, Revtex. References added, minor changes. Version to appear in Class. Quant. Gra