Emergent time in Hamiltonian general relativity

A limiting diagram for the Segre classification in 5-dimensional space–times is obtained, extending a recent work on limits of the energy–momentum tensor in general relativity. Some of Geroch’s results on limits of space–times in general relativity are also extended to the context of five-dimensional Kaluza–Klein space–times.

For an asymptotically flat space–time in general relativity there exist certain integrals, called linkages, over cross sections of null infinity, which represent the energy, momentum, or angular momentum of the system. A new formulation of the linkages is introduced and applied. It is shown that there exists a flux, representing the contribution of gravitational and matter radiation to the linkage. A uniqueness conjecture for the linkages is formulated. The ambiguities due to the possible presence of supertranslations in asymptotic rotations are studied using the behavior of the linkages under first-order perturbations in the metric. While in certain situations these ambiguities disappear in the first-order treatment, an example is given which suggests that they are an essential feature of general relativity and its asymptotic structure.

Recent attempts at explaining free-fall motion (i.e. motion only under the influence of gravity) in the context of the General Theory of Relativity have invoked a concept of time passing/progressing. Although students who have studied the Special Theory of Relativity will likely know that this concept does not feature in Special Relativity, they may incorrectly accept the concept of time passing/progressing as an aspect of General Relativity if they have encountered these accounts of free-fall motion. It is pointed out that time passing/progressing is not part of General Relativity theory and is briefly shown that free-fall motion in General Relativity is quantitatively described without such a concept.

It is shown - in Ashtekar's canonical framework of General Relativity - that the constraints of spherically symmetric (Schwarzschild) gravity in 4 dimensional space-time can be solved completely yielding two canonically conjugate observables for asymptotically flat spacetimes, namely mass and - surprisingly - time. The emergence of the time observable is a consequence of the Hamiltonian formulation and its subtleties concerning the slicing of space and time and is not in contradiction to Birkhoff's theorem. Our results can be expressed within the ADM formalism, too, and their relation to the equivalent ones Kuchař obtained recently are briefly discussed. Quantization of the system and the associated Schrödinger equation depend on the allowed spectrum of the masses.

Science sets itself apart from other paths to truth by recognizing that even its greate practitioners sometimes err.

We investigate the potential energy surfaces of supercooled water, both intrabasin structures and distributions of the potential energy basins in configuration space by the steepest descent quenching and the normal mode excitation. The paths from liquid configurations to the corresponding local energy minima and the root mean square distance exhibit different characters below and above the temperature 213 K where the thermodynamics and structural properties change abruptly. The root mean square distance in the temperature range (298 K to 173 K) is larger than the corresponding harmonic oscillator system and it drastically increases above 223 K. The complexity also increases along the steepest descent path by increasing the number of inflection points. In order to investigate the other potential energy basins distributed in the vicinity of the (central) basin that the molecular dynamics trajectory visits, the distributions of these basins are examined by the normal mode excitation. The number of distinct basins increases with increasing temperature in the same volume of configuration space. The minimum potential energy of the adjacent basin at low temperature is almost always higher than the minimum potential energy of the central basin while that of the adjacent basin at high temperature is comparable to the central basin. The locations of the other basin centers are mostly orthogonal to the normal mode excitation. The potential energy surfaces are also examined by the basin hopping technique to seek for lower energy configurations started with a random and high energy molecular arrangement. It is found that energetically more stable molecules aggregate rather heterogeneously in the intermediate energy levels, which are hardly observed in the simulation, while the stable molecules distribute homogeneously in the lowest energy levels.

The successes of f(R) gravitational theory as a logical extension of Einstein’s theory of general relativity (GR) encourage us to delve deep into this theory and continue our study to attempt to derive an extension of the Schwarzschild black hole (BH) solution. In this study, in order to solve the output nonlinear differential equation, we closed the form of the system by assuming the derivative of f(R) with respect to the scalar curvature R to have the form F(r)=df(R(r))dR(r)=1-αr4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(r)=\\frac{\ extrm{d}f(R(r))}{\ extrm{d}R(r)}=1- \\frac{\\alpha }{r^4}$$\\end{document}, where α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} is a dimensional constant. Our study shows that when α→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\rightarrow 0$$\\end{document}, we obtain the Schwarzschild BH solution of GR assuming some constraints on the constant of integration, and if these constraints are bounded, we obtain the anti-de Sitter (AdS)/de Sitter (dS) spacetime. For the general case, i.e., when α≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \ e 0$$\\end{document}, we obtain a BH solution that tends asymptotically to AdS/dS spacetime. Moreover, we derive the timelike and null particle geodesics of the BH solution studied in this article. The equation of motion and effective potential of test particles are calculated to study the stability of radial orbits (trajectories). The energy and angular momentum are calculated to study the circular motion and stability of circular orbits. We also derive the stability condition using the geodesic deviation. Moreover, we discuss the physics of the output BH solutions through calculation of the thermodynamic quantities including entropy, the Hawking temperature, and Gibbs free energy. Finally, we check the validity of the first law of thermodynamics applied to the BH of this study. Although we can derive a Schwarzschild black hole solution in the lower order of f(R), specifically when f(R)=R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(R)=R$$\\end{document}, where the gravitational mass is generated from the source of gravity, we demonstrate that in the higher orders of f(R), when f(R)≠R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(R)\ e R$$\\end{document}, the source of gravity is attributed primarily to higher-order corrections, and the source of gravity that was originally derived from the Schwarzschild black hole has ceased to be dominant.

In General Relativity in Hamiltonian form, change has seemed to be missing, defined only asymptotically, or otherwise obscured at best, because the Hamiltonian is a sum of first-class constraints and a boundary term and thus supposedly generates gauge transformations. By construing change as essential time dependence (lack of a time-like Killing vector field), one can find change locally in vacuum GR in the Hamiltonian formulation just where it should be. But what if spinors are present? This paper is motivated by the tendency in space-time philosophy tends to slight fermionic/spinorial matter, the tendency in Hamiltonian GR to misplace changes of time coordinate, and the tendency in treatments of the Einstein-Dirac equation to include a gratuitous local Lorentz gauge symmetry along with the physically significant coordinate freedom. Spatial dependence is dropped in most of the paper, both restricting the physical situation and largely fixing the spatial coordinates. In the interest of including all and only the coordinate freedom, the Einstein-Dirac equation is investigated using the Schwinger time gauge and Kibble-Deser symmetric triad condition are employed as a 3+1 version of the DeWitt-Ogievetsky-Polubarinov nonlinear group realization formalism that dispenses with a tetrad and local Lorentz gauge freedom. Change is the lack of a time-like stronger-than-Killing field for which the Lie derivative of the metric-spinor complex vanishes. An appropriate 3+1-friendly form of the Rosenfeld-Anderson-Bergmann-Castellani gauge generator G, a tuned sum of first class-constraints, is shown to change the canonical Lagrangian by a total derivative, implying the preservation of Hamilton’s equations. Given the essential presence of second-class constraints with spinors and their lack of resemblance to a gauge theory (unlike, say, massive photons), it is useful to have an explicit physically interesting example. This gauge generator implements changes of time coordinate for solutions of the equations of motion, showing that the gauge generator makes sense even with spinors.

We study the scalar radiation emitted by a source orbiting a Schwarzschild--anti--de Sitter black hole in circular geodesics. We obtain the emitted power using quantum field theory in curved spacetimes at tree level with reflective boundary conditions at infinity. We compare our results with the ones obtained in Schwarzschild and Schwarzschild--de Sitter ($\mathrm{\ensuremath{\Lambda}}>0$) spacetimes. In the Schwarzschild--de Sitter spacetime we observe an enhancement of the emitted power associated to the lower multipole modes, when compared with the Schwarzschild case. In contrast, for the Schwarzschild--anti--de Sitter spacetime, the enhancement is rather of the higher multipole modes.

We apply the well known Kovacic algorithm to generate Liouvillian (i.e., closed-form) solutions to the ordinary differential equation that governs the radial part of the separable, coupled to scalar curvature, massive scalar wave equation on a D-dimensional de Sitter background spacetime. The radial equation is simply related to the famous hypergeometric equation, and our work is carried out in this general setting. We find several families of infinite sets of special parameter values for which Liouvillian solutions exist, and we demonstrate how to generate these solutions recursively. The solutions are related to hypergeometric polynomials and in some cases can be constructed explicitly. For some of the families, the Liouvillian solutions exactly satisfy the boundary conditions for quasinormal modes, a fundamental set of damped, outgoing wave-functions that depend only on the background spacetime and scalar field parameters. In these families we obtain agreement with published results on the scalar quasinormal modes of the pure de Sitter spacetime. For other families, we establish conjectures regarding the factorization of the parameter constraints required for existence of Liouvillian solutions. Our results are consistent with published results on the hypergeometric equation and the well known Pöschl–Teller equation. Our new contributions are to demonstrate how to generate the hypergeometric Liouvillian solutions recursively, to completely solve the quasinormal mode problem with Liouvillian solutions, and to analyze the Liouvillian parameter constraints for the hypergeometric equation, including the case of the physical de Sitter parameters. This work contributes new knowledge to the fields of special functions and general relativity.

Since the black hole and cosmological horizons in the de Sitter (dS) spacetime with the Reissner–Nordström (RN) black hole are not independent with each other, which is caused by the gravitational effect, the interplay between two horizons should be considered. Based on this, by introducing the interactive entropy the RN–dS spacetime is analogous to a thermodynamic system with various thermodynamic quantities, in which the laws of thermodynamics still hold on. In our work, the thermodynamic properties of the RN–dS spacetime are mapped out in the induced phase space, which are similar to that in AdS black holes. The phase transition of the RN–dS spacetime between the high-potential and the low-potential black hole phases is observed. Compared with an ordinary thermodynamic system, the similar behaviors about the Joule–Thomson expansion and the critical exponents are also checked out. Finally, the scalar curvatures of two existent phases are presented to reveal the underlying microstructure and nature of phase transition in the RN–dS spacetime, which opens a new window to investigate the dS spacetime with black holes from an observational perspective.

We introduce a methodology for quantitatively measuring at all times in its evolution how close our physical spatially flat [Formula: see text] CDM universe with cosmological constant [Formula: see text] is to the de Sitter spacetime [Formula: see text] with de Sitter radius [Formula: see text]. The main idea in this study is to align the respective scale factors [Formula: see text] and [Formula: see text] of these two spacetimes, where de Sitter spacetime is taken with respect to a spatially flat foliation. This goal is accomplished by fine-tuning an adjustable parameter [Formula: see text] that arises naturally in the de Sitter scale factor by requiring that these scale factors be future-asymptotically convergent. Once this parameter is adjusted and the scale factors are aligned, we define a relative error function [Formula: see text] that computes as a function of time [Formula: see text] how close the scale factors of these two spacetimes are to one another. Our results quantify how close our physical [Formula: see text]CDM universe is to its corresponding de Sitter spacetime as both spacetimes converge as they expand. As an example of our results, we show that at the present time [Formula: see text][Formula: see text]Gy, to an accuracy of [Formula: see text], and at [Formula: see text][Formula: see text]Gy, to an accuracy of [Formula: see text], we can use de Sitter spacetime to model our own [Formula: see text]CDM universe. Our results also show by statistical analysis that with a confidence level of 68.3%, for [Formula: see text][Formula: see text]Gy, the scale factor [Formula: see text] of our [Formula: see text] universe and the scale factor [Formula: see text] of the corresponding de Sitter spacetime are indistinguishable to within the accuracy of current cosmological measurements.

We study and look for similarities between the response rates ${R}^{\mathrm{dS}}{(a}_{0},\ensuremath{\Lambda})$ and ${R}^{\mathrm{SdS}}{(a}_{0},\ensuremath{\Lambda},M)$ of a static scalar source with constant proper acceleration ${a}_{0}$ interacting with a massless, conformally coupled Klein-Gordon field (i) in de Sitter spacetime, in the Euclidean vacuum, which describes a thermal flux of radiation emanating from the de Sitter cosmological horizon and (ii) in Schwarzschild--de Sitter spacetime, in the Gibbons-Hawking vacuum, which describes thermal fluxes of radiation emanating from both the hole and the cosmological horizons, respectively, where $\ensuremath{\Lambda}$ is the cosmological constant and M is the black hole mass. After performing the field quantization in each of the above spacetimes, we obtain the response rates at the tree level in terms of an infinite sum of zero-energy field modes possessing all possible angular momentum quantum numbers. In the case of de Sitter spacetime, this formula is worked out and a closed, analytical form is obtained. In the case of Schwarzschild--de Sitter spacetime such a closed formula could not be obtained, and a numerical analysis is performed. We conclude, in particular, that ${R}^{\mathrm{dS}}{(a}_{0},\ensuremath{\Lambda})$ and ${R}^{\mathrm{SdS}}{(a}_{0},\ensuremath{\Lambda},M)$ do not coincide in general, but tend to each other when $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Lambda}}0$ or ${a}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}.$ Our results are also contrasted and shown to agree (in the proper limits) with related ones in the literature.

The geodesic deviation equation (GDE) describes the tendency of objects to accelerate towards or away from each other due to spacetime curvature. The GDE assumes that nearby geodesics have a small rate of separation, which is formally treated as the same order in smallness as the separation itself. This assumption is discussed in various papers but is not articulated in any standard textbooks on general relativity. Relaxing this assumption leads to the generalized geodesic deviation equation (GGDE). We elucidate the distinction between the GDE and the GGDE by explicitly computing the relative acceleration between timelike geodesics in two-dimensional de Sitter spacetime. We do this by considering a fiducial geodesic and a secondary geodesic (both timelike) that cross with nonzero speed. These geodesics are spanned by a spacelike geodesic, whose tangent evaluated at the fiducial geodesic defines the separation. The second derivative of the separation describes the relative acceleration between the fiducial and secondary geodesics. Near the crossing point, where the separation between the timelike geodesics is small but their rates of separation can be large, we show that the GGDE holds but the GDE fails to apply.

The entropy of a free scalar field is calculated in the Reissner–Nordström–(anti-)de Sitter spacetimes. Due to the presence of the cosmological horizon in the Reissner–Nordström–de Sitter spacetime, we introduce a cutoff at the cosmological horizon, besides the cutoff at the horizon of black holes in the brick wall model. The entropy is found to be the sum of two terms, which are proportional to the area of the cosmological horizon and of black hole horizon, respectively. In the Reissner–Nordström–anti-de Sitter spacetime the contribution of the anti-de Sitter background to the entropy of scalar fields vanishes when an infinite volume is taken. The entropy of scalar fields is also evaluated in some special backgrounds described by solutions of Einstein–Maxwell equations with a cosmological constant, such as the cold black holes, lukewarm black holes, ultracold solutions, a naked singularity in de Sitter space, and the de Sitter space. The physical meaning of some results is briefly discussed.