Unified first law of black-hole dynamics and relativistic thermodynamics

A unified first law of black-hole dynamics and relativistic thermodynamics is derived in spherically symmetric general relativity. This equation expresses the gradient of the active gravitational energy E according to the Einstein equation, divided into energy-supply and work terms. Projecting the equation along the flow of thermodynamic matter and along the trapping horizon of a blackhole yield, respectively, first laws of relativistic thermodynamics and black-hole dynamics. In the black-hole case, this first law has the same form as the first law of black-hole statics, with static perturbations replaced by the derivative along the horizon. There is the expected term involving the area and surface gravity, where the dynamic surface gravity is defined as in the static case but using the Kodama vector and trapping horizon. This surface gravity vanishes for degenerate trapping horizons and satisfies certain expected inequalities involving the area and energy. In the thermodynamic case, the quasi-local first law has the same form, apart from a relativistic factor, as the classical first law of thermodynamics, involving heat supply and hydrodynamic work, but with E replacing the internal energy. Expanding E in the Newtonian limit shows that it incorporates the Newtonian mass, kinetic energy, gravitational potential energy and thermal energy. There is also a weak type of unified zeroth law: a Gibbs-like definition of thermal equilibrium requires constancy of an effective temperature, generalising the Tolman condition and the particular case of Hawking radiation, while gravithermal equilibrium further requires constancy of surface gravity. Finally, it is suggested that the energy operator of spherically symmetric quantum gravity is determined by the Kodama vector, which encodes a dynamic time related to E.Comment: 18 pages, TeX, expanded somewhat, to appear in Class. Quantum Gra

Generalized inverse mean curvature flows in spacetime

Motivated by the conjectured Penrose inequality and by the work of Hawking, Geroch, Huisken and Ilmanen in the null and the Riemannian case, we examine necessary conditions on flows of two-surfaces in spacetime under which the Hawking quasilocal mass is monotone. We focus on a subclass of such flows which we call uniformly expanding, which can be considered for null as well as for spacelike directions. In the null case, local existence of the flow is guaranteed. In the spacelike case, the uniformly expanding condition leaves a 1-parameter freedom, but for the whole family, the embedding functions satisfy a forward-backward parabolic system for which local existence does not hold in general. Nevertheless, we have obtained a generalization of the weak (distributional) formulation of this class of flows, generalizing the corresponding step of Huisken and Ilmanen's proof of the Riemannian Penrose inequality.Comment: 21 pages, 1 figur

Energy conservation for dynamical black holes

An energy conservation law is described, expressing the increase in mass-energy of a general black hole in terms of the energy densities of the infalling matter and gravitational radiation. For a growing black hole, this first law of black-hole dynamics is equivalent to an equation of Ashtekar & Krishnan, but the new integral and differential forms are regular in the limit where the black hole ceases to grow. An effective gravitational-radiation energy tensor is obtained, providing measures of both ingoing and outgoing, transverse and longitudinal gravitational radiation on and near a black hole. Corresponding energy-tensor forms of the first law involve a preferred time vector which plays the role for dynamical black holes which the stationary Killing vector plays for stationary black holes. Identifying an energy flux, vanishing if and only if the horizon is null, allows a division into energy-supply and work terms, as in the first law of thermodynamics. The energy supply can be expressed in terms of area increase and a newly defined surface gravity, yielding a Gibbs-like equation, with a similar form to the so-called first law for stationary black holes.Comment: 4 revtex4 pages. Many (mostly presentational) changes; emphasizes the definition of gravitational radiation in the strong-field regim

Spacetimes foliated by Killing horizons

It seems to be expected, that a horizon of a quasi-local type, like a Killing or an isolated horizon, by analogy with a globally defined event horizon, should be unique in some open neighborhood in the spacetime, provided the vacuum Einstein or the Einstein-Maxwell equations are satisfied. The aim of our paper is to verify whether that intuition is correct. If one can extend a so called Kundt metric, in such a way that its null, shear-free surfaces have spherical spacetime sections, the resulting spacetime is foliated by so called non-expanding horizons. The obstacle is Kundt's constraint induced at the surfaces by the Einstein or the Einstein-Maxwell equations, and the requirement that a solution be globally defined on the sphere. We derived a transformation (reflection) that creates a solution to Kundt's constraint out of data defining an extremal isolated horizon. Using that transformation, we derived a class of exact solutions to the Einstein or Einstein-Maxwell equations of very special properties. Each spacetime we construct is foliated by a family of the Killing horizons. Moreover, it admits another, transversal Killing horizon. The intrinsic and extrinsic geometry of the transversal Killing horizon coincides with the one defined on the event horizon of the extremal Kerr-Newman solution. However, the Killing horizon in our example admits yet another Killing vector tangent to and null at it. The geometries of the leaves are given by the reflection.Comment: LaTeX 2e, 13 page

Monte Carlo flight performance reserve program

Computer program for evaluating flight performance reserve requirements for Centaur vehicle using Monte Carlo metho

Deformation of Codimension-2 Surface and Horizon Thermodynamics

The deformation equation of a spacelike submanifold with an arbitrary codimension is given by a general construction without using local frames. In the case of codimension-1, this equation reduces to the evolution equation of the extrinsic curvature of a spacelike hypersurface. In the more interesting case of codimension-2, after selecting a local null frame, this deformation equation reduces to the well known (cross) focusing equations. We show how the thermodynamics of trapping horizons is related to these deformation equations in two different formalisms: with and without introducing quasilocal energy. In the formalism with the quasilocal energy, the Hawking mass in four dimension is generalized to higher dimension, and it is found that the deformation of this energy inside a marginal surface can be also decomposed into the contributions from matter fields and gravitational radiation as in the four dimension. In the formalism without the quasilocal energy, we generalize the definition of slowly evolving future outer trapping horizons proposed by Booth to past trapping horizons. The dynamics of the trapping horizons in FLRW universe is given as an example. Especially, the slowly evolving past trapping horizon in the FLRW universe has close relation to the scenario of slow-roll inflation. Up to the second order of the slowly evolving parameter in this generalization, the temperature (surface gravity) associated with the slowly evolving trapping horizon in the FLRW universe is essentially the same as the one defined by using the quasilocal energy.Comment: Latex, 61 pages, no figures; v2, type errors corrected; v3, references and comments are added, English is improved, to appear in JHE

The Magnetization of Cu_2(C_5H_{12}N_2)_2Cl_4 : A Heisenberg Spin Ladder System

We study the magnetization of a Heisenberg spin ladder using exact diagonalization techniques, finding three distinct magnetic phases. We consider the results in relation to the experimental behaviour of the new copper compound Cu_2(C_5H_{12}N_2)_2Cl_4 and deduce that the compound is well described by such a model with a ratio of `chain' to `rung' bond strengths (J/J^\prime) of the order of 0.2, consistent with results from the magnetic susceptibility. The effects of temperature, spin impurities and additional diagonal bonds are presented and we give evidence that these diagonal bonds are indeed of a ferromagnetic nature.Comment: Latex file (4 pages), related figures (encapsulated postscript) appende

Hamiltonians for Reduced Gravity

A generalised canonical formulation of gravity is devised for foliations of spacetime with codimension $n\ge1$. The new formalism retains n-dimensional covariance and is especially suited to 2+2 decompositions of spacetime. It is also possible to use the generalised formalism to obtain boundary contributions to the 3+1 Hamiltonian.Comment: 18 pages, revtex, 3 postscript figures include

Antibiotic prescribing for lower UTI in elderly patients in primary care and risk of bloodstream infection: A cohort study using electronic health records in England

BACKGROUND: Research has questioned the safety of delaying or withholding antibiotics for suspected urinary tract infection (UTI) in older patients. We evaluated the association between antibiotic treatment for lower UTI and risk of bloodstream infection (BSI) in adults aged ≥65 years in primary care. METHODS AND FINDINGS: We analyzed primary care records from patients aged ≥65 years in England with community-onset UTI using the Clinical Practice Research Datalink (2007-2015) linked to Hospital Episode Statistics and census data. The primary outcome was BSI within 60 days, comparing patients treated immediately with antibiotics and those not treated immediately. Crude and adjusted associations between exposure and outcome were estimated using generalized estimating equations. A total of 147,334 patients were included representing 280,462 episodes of lower UTI. BSI occurred in 0.4% (1,025/244,963) of UTI episodes with immediate antibiotics versus 0.6% (228/35,499) of episodes without immediate antibiotics. After adjusting for patient demographics, year of consultation, comorbidities, smoking status, recent hospitalizations, recent accident and emergency (A&E) attendances, recent antibiotic prescribing, and home visits, the odds of BSI were equivalent in patients who were not treated with antibiotics immediately and those who were treated on the date of their UTI consultation (adjusted odds ratio [aOR] 1.13, 95% CI 0.97-1.32, p-value = 0.105). Delaying or withholding antibiotics was associated with increased odds of death in the subsequent 60 days (aOR 1.17, 95% CI 1.09-1.26, p-value < 0.001), but there was limited evidence that increased deaths were attributable to urinary-source BSI. Limitations include overlap between the categories of immediate and delayed antibiotic prescribing, residual confounding underlying differences between patients who were/were not treated with antibiotics, and lack of microbiological diagnosis for BSI. CONCLUSIONS: In this study, we observed that delaying or withholding antibiotics in older adults with suspected UTI did not increase patients' risk of BSI, in contrast with a previous study that analyzed the same dataset, but mortality was increased. Our findings highlight uncertainty around the risks of delaying or withholding antibiotic treatment, which is exacerbated by systematic differences between patients who were and were not treated immediately with antibiotics. Overall, our findings emphasize the need for improved diagnostic/risk prediction strategies to guide antibiotic prescribing for suspected UTI in older adults

Production and decay of evolving horizons

We consider a simple physical model for an evolving horizon that is strongly interacting with its environment, exchanging arbitrarily large quantities of matter with its environment in the form of both infalling material and outgoing Hawking radiation. We permit fluxes of both lightlike and timelike particles to cross the horizon, and ask how the horizon grows and shrinks in response to such flows. We place a premium on providing a clear and straightforward exposition with simple formulae. To be able to handle such a highly dynamical situation in a simple manner we make one significant physical restriction, that of spherical symmetry, and two technical mathematical restrictions: (1) We choose to slice the spacetime in such a way that the space-time foliations (and hence the horizons) are always spherically symmetric. (2) Furthermore we adopt Painleve-Gullstrand coordinates (which are well suited to the problem because they are nonsingular at the horizon) in order to simplify the relevant calculations. We find particularly simple forms for surface gravity, and for the first and second law of black hole thermodynamics, in this general evolving horizon situation. Furthermore we relate our results to Hawking's apparent horizon, Ashtekar et al's isolated and dynamical horizons, and Hayward's trapping horizons. The evolving black hole model discussed here will be of interest, both from an astrophysical viewpoint in terms of discussing growing black holes, and from a purely theoretical viewpoint in discussing black hole evaporation via Hawking radiation.Comment: 25 pages, uses iopart.cls V2: 5 references added; minor typos; V3: some additional clarifications, additional references, additional appendix on the Viadya spacetime. This version published in Classical and Quiantum Gravit