Static Spacetimes Research Articles

On scalar charges and black hole thermodynamics

We revisit the first law of black hole thermodynamics in 4-dimensional theories containing scalar and Abelian vector fields coupled to gravity using Wald’s formalism and a new definition of scalar charge as an integral over a 2-surface which satisfies a Gauss law in the background of stationary black-hole spacetimes. We focus on ungauged supergravity-inspired theories with symmetric sigma models whose symmetries generate electric-magnetic dualities leaving invariant their equations of motion. Our manifestly duality-invariant form of the first law is compatible with the one obtained by of Gibbons, Kallosh and Kol. We also obtain the general expression for the scalar charges of a stationary black hole in terms of the other physical parameters of the solution and the position of the horizon, generalizing the expression obtained by Pacilio for dilaton black holes.

Mengontruksi Kontrol Optimal Berkendala Pada Sistem LTI Dengan Keadaan Berkendala Menggunakan Metode Fungsi Penalti

The optimal control problem is defined as a problem in determining the control u(t) that depends on time t, such that it produces the optimum value for the objective function. The optimal control system with constrained conditions can be changed to an optimal control system without constraints by constructing the value of u(t) so that u(t) is not constrained. In this research, we will examine how to determine an optimal control of a system of time-independent state space or Linear Time Invariant (LTI) with constrained states and minimize a given objective function. In addition, how to construct an optimal controller that is constrained to become an optimal controller without constraints using the penalty function method and its implementation using Matlab. The results of this study are that the optimal control is:.

Static perfect fluid spacetimes on GRW spacetimes

Static perfect fluid spacetimes on GRW spacetimes

Slowly rotating Kerr metric derived from the Einstein equations in affine-null coordinates

Using a quasispherical approximation of an affine-null metric adapted to an asymptotic Bondi inertial frame, we present high order approximations of the metric functions in terms of the specific angular momentum for a slowly rotating stationary and axisymmetric vacuum spacetime. The metric is obtained by following the procedure of integrating the hierarchy of Einstein equations in a characteristic formulation utilizing master functions for the perturbations. It further verified its equivalence with the Kerr metric in the slowly rotating approximation by carrying out an explicit transformation between the Boyer-Lindquist coordinates to the employed affine-null coordinates. A peculiar feature of the derivation is that in the solution of the perturbation equations for every order a new integration constant appears which cannot be set to zero using asymptotical flatness or regularity arguments. However, these additional integration constants can be absorbed into the overall Komar mass and Komar angular momentum of a slowly rotating black hole.

A quintessence type interior solution with Karmarkar condition

In this report, we present an interior solution to Einstein’s equations in a spherically symmetric and static spacetime filled by two sources with anisotropic pressures, one of these of ordinary matter for which the radial pressure is described by the MIT Bag state equation associated to the presence of quarks and the other by non-ordinary quintessence type matter. The solution is obtained from imposing the Karmarkar condition considering a metric function [Formula: see text] resulting in a physically acceptable, stable and adequate model to represent compact objects with compactness rate [Formula: see text]. Which allows to take different value of mass and radius in the range of the observational data of mass [Formula: see text] and radius [Formula: see text] of the star 4U1608-52 with compactness [Formula: see text], from these data we determine the range of [Formula: see text].

Gravitational lensing by using the 0th order of affine perturbation series of the deflection angle of a ray near a photon sphere

The 0th order of affine perturbation series of the deflection angle of a ray near a photon sphere is more accurate than a deflection angle in a strong deflection limit, which is used often, because the later has hidden error terms. We investigate gravitational lensing by using 0th order affine perturbation series of the deflection angle in a general asymptotically-flat, static, and spherical symmetric spacetime with the photon sphere. We apply our formula to Schwarzschild black hole, Reissner–Nordström black hole, and Ellis–Bronnikov wormhole spacetimes as examples. By comparing observables by using the deflection angles, we show that we can ignore the effect of the hidden error terms in the the deflection angle in the strong deflection limit on the observables in a usual lens configuration with the photon sphere since the hidden error terms are tiny. On the other hand, in a retro lensing configuration, the deflection angle in the strong-deflection-limit analysis have error of several percent and the 0th order of affine perturbation series of the deflection angle has almost half of the error. Thus, in the retro lensing configuration, we should use the 0th order of affine perturbation series of the deflection angle rather than the deflection angle in the strong-deflection-limit analysis. The 0th order of affine perturbation series of the deflection angle can give a brighter magnification by a dozen percent than the one by using the deflection angle in the strong-deflection-limit analysis.

Characterizations of a Spacetime of Quasi‐Constant Sectional Curvature and F(R)$\mathcal {F}(\mathcal {R})$‐Gravity

AbstractThe main aim of this article is to investigate a spacetime of quasi‐constant sectional curvature. At first, the existence of such a spacetime is established by several examples. We have shown that a spacetime of quasi‐constant sectional curvature agrees with the present state of the universe and it represents a Robertson Walker spacetime. Moreover, if the spacetime is Ricci semi‐symmetric or Ricci symmetric, then either the spacetime represents a spacetime of constant sectional curvature, or the spacetime represents phantom era. Also, we prove that a Ricci symmetric spacetime of quasi‐constant sectional curvature represents a static spacetime and the spacetime under consideration is of Petrov type I, D or O. Finally, we concentrate on a quasi‐constant sectional curvature spacetime solution in ‐gravity. As a result, various energy conditions are studied and analyzed our obtained outcomes in terms of a ‐gravity model.

Tolman-Ehrenfest’s criterion of thermal equilibrium extended to conformally static spacetimes

With insight from examples and physical arguments, the Tolman-Ehrenfest criterion of thermal equilibrium for test fluids in static spacetimes is extended to local thermal equilibrium in conformally static geometries. The temperature of the conformally rescaled fluid scales with the inverse of the conformal factor, reproducing the evolution of the cosmic microwave background in Friedmann universes, the Hawking temperature of the Sultana-Dyer cosmological black hole, and a heuristic argument by Dicke.

EFFECTS OF GRAVITY’S RAINBOW ON A RELATIVISTIC SPIN-1 OSCILLATOR

We consider a relativistic spin-1 particle with non-minimal coupling in the context of gravity’s rainbow in the three dimensional background spacetime spanned by static cosmic string. In this context, we acquire an exact solution of the associated spin-1 equation in the modified three dimensional static cosmic string-spanned background spacetime. This relativistic wave equation includes a reducible spinor and this allows us to acquire a non-perturbative expression including the modification functions in the energy domain. In the low energy limit, our results agree well with current literature and provide a basis to discuss the fundamental features of the relativistic spin-1 oscillator. Afterwards, we try to discuss the effects of gravity rainbow functions on the considered spin-1 oscillator in three different scenarios for the modification functions.

Discrete potential mean field games: duality and numerical resolution

We propose and investigate a general class of discrete time and finite state space mean field game (MFG) problems with potential structure. Our model incorporates interactions through a congestion term and a price variable. It also allows hard constraints on the distribution of the agents. We analyze the connection between the MFG problem and two optimal control problems in duality. We present two families of numerical methods and detail their implementation: (i) primal-dual proximal methods (and their extension with nonlinear proximity operators), (ii) the alternating direction method of multipliers (ADMM) and a variant called ADM-G. We give some convergence results. Numerical results are provided for two examples with hard constraints.

Joint scheduling of barges and tugboats for river–sea intermodal transport

In this paper, we study a barge and tugboat joint scheduling problem arising from a river–sea intermodal transport system, where non-self-propelled barges travel with the help of tugboats between an inland river bulk port and sea transshipment platforms to transport cargo. Due to limited channels and the sequential scheduling among ships, transit inefficiency and resource under-utilization often occur, thus it is essential to design barge timetables and tugboat paths. This paper jointly schedules barges and tugboats in a river–sea transshipment port with a tidally influenced channel, and focuses on barge round trips with multiple legs. We construct a state-space–time network from the perspective of tugboats, in which the state implies whether a barge is served by a tugboat. The problem is formulated as an integer programming model based on the network to minimize time taken for bulk cargo transshipment. A tailored variable neighborhood search algorithm is designed and its efficiency, effectiveness, and stability are demonstrated by numerical experiments, which are based on real data from the Boffa port. It is demonstrated that, by joint scheduling barges and tugboats, transfer efficiency can be significantly improved under limited tugboat resources and shorter tidal windows. The results also provide managerial insights on schedule and valuable conclusions about the resource allocation in the port.

Gradient flow of Einstein-Maxwell theory and Reissner-Nordström black holes

Ricci flow is a natural gradient flow of the Einstein-Hilbert action. Here we consider the analog for the Einstein-Maxwell action, which gives Ricci flow with a stress tensor contribution coupled to a Yang-Mills flow for the Maxwell field. We argue that this flow is well-posed for static spacetimes with pure electric or magnetic potentials and show it preserves both non-extremal and extremal black hole horizons. In the latter case we find the flow of the near horizon geometry decouples from that of the exterior. The Schwarzschild black hole is an unstable fixed point of Ricci flow for static spacetimes. Here we consider flows of the Reissner-Nordström (RN) fixed point. The magnetic RN solution becomes a stable fixed point of the flow for sufficient charge. However we find that the electric RN black hole is always unstable. Numerically solving the flow starting with a spherically symmetric perturbation of a non-extremal RN solution, we find similar behaviour in the electric case to the Ricci flows of perturbed Schwarzschild, namely the horizon shrinks to a singularity in finite time or expands forever. In the magnetic case, a perturbed unstable RN solution has a similar expanding behaviour, but a perturbation that decreases the horizon size flows to a stable black hole solution rather than a singularity. For extremal RN we solve the near horizon flow for spherical symmetry exactly, and see in the electric case two unstable directions which flow to singularities in finite flow time. However, even turning these off, and fixing the near horizon geometry to be that of RN, we numerically show that the flows appear to become singular in the vicinity of its horizon.

Einstein-Vlasov system with equal-angular momenta in AdS5

We investigate solutions of the five-dimensional rotating Einstein-Vlasov system with an $R\ifmmode\times\else\texttimes\fi{}SU(2)\ifmmode\times\else\texttimes\fi{}U(1)$ isometry group. In a five-dimensional spacetime, there are two independent planes of rotation; thus, considering $U(1)$ symmetry on each rotation plane, we may impose an $R\ifmmode\times\else\texttimes\fi{}U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ isometry to a stationary spacetime. Furthermore, when the values of the two angular momenta are equal to each other, the spatial symmetry gets enhanced to $R\ifmmode\times\else\texttimes\fi{}SU(2)\ifmmode\times\else\texttimes\fi{}U(1)$ symmetry, and the spacetime has a cohomogeneity-1 structure. Imposing the same symmetry to the distribution function of the particles of which the Vlasov system consists, the distribution function can be dependent on three mutually independent and commutative conserved charges for particle motion [energy, total angular momentum on $SU(2)$ and $U(1)$ angular momentum]. We consider the distribution function which exponentially depends on the $U(1)$ angular momentum and reduces to the thermal equilibrium state in spherical symmetry. Then, in this paper, we numerically construct solutions of the asymptotically anti--de Sitter Einstein-Vlasov system.

Dyonic black holes in the theory of two electromagnetic potentials. II

The results obtained in our previous paper are now extended to the case of stationary axially symmetric dyonic black boles within the theory of two electromagnetic potentials. We slightly enlarge the classical Ernst formalism by introducing, with the aid of the $t$- and $\varphi$-components of the dual potential $B_\mu$, the magnetic potential $\Phi_m$ which, similar to the known electric potential $\Phi_e$, also takes constant value on the black hole horizon. We analyze in detail the case of the dyonic Kerr-Newman black hole and show how the Komar mass must be evaluated correctly in this stationary dyonic model. In particular, we rigorously prove the validity of the standard Tomimatsu mass formula and point out that attempts to "improve" it made in recent years are explained by misunderstanding of the auxiliary role that singular potentials play in the description of magnetic charges. Our approach is symmetrical with respect to electric and magnetic charges and, like in the static case considered earlier, Dirac strings of all kind are excluded from the physical picture of the stationary black hole dyonic spacetimes.

Hidden freedom in the mode expansion on static spacetimes

We review the construction of ground states focusing on a real scalar field whose dynamics is ruled by the Klein–Gordon equation on a large class of static spacetimes. As in the analysis of the classical equations of motion, when enough isometries are present, via a mode expansion the construction of two-point correlation functions boils down to solving a second order, ordinary differential equation on an interval of the real line. Using the language of Sturm–Liouville theory, most compelling is the scenario when one endpoint of such interval is classified as a limit circle, as it often happens when one is working on globally hyperbolic spacetimes with a timelike boundary. In this case, beyond initial data, one needs to specify a boundary condition both to have a well-defined classical dynamics and to select a corresponding ground state. Here, we take into account boundary conditions of Robin type by using well-known results from Sturm–Liouville theory, but we go beyond the existing literature by exploring an unnoticed freedom that emerges from the intrinsic arbitrariness of secondary solutions at a limit circle endpoint. Accordingly, we show that infinitely many one-parameter families of sensible dynamics are admissible. In other words, we emphasize that physical constraints guaranteeing the construction of ground states do not, in general, fix one such state unambiguously. In addition, we provide, in full detail, an example on (1 + 1)-half Minkowski spacetime to spell out the rationale in a specific scenario where analytic formulae can be obtained.

A frame based approach to computing symmetries with non-trivial isotropy groups

A frame approach to determining the most general solution admitting a desired symmetry group has previously been examined in Riemannian and teleparallel geometries with some success. In teleparallel geometries, one must determine the general form of the frame and spin connection to generate a general solution admitting the desired symmetry group. Current approaches often rely on the use of the proper frame, where the spin connection is zero. However, this leads to particular theoretical and practical problems. In this paper, we introduce an entirely general approach to determining the most general Riemann–Cartan geometries that admit a given symmetry group and apply these results to teleparallel geometries. To illustrate the approach, we determine the most general geometries, with the minimal number of arbitrary functions, for particular choices of symmetry groups with dimension one, three, six, and seven. In addition, we rigorously show how the teleparallel analog of the Robertson–Walker, de Sitter, and Einstein static spacetimes can be determined.

Counterexamples to a divergence lower bound for the covariant derivative of skew-symmetric 2-tensor fields

In Hwang and Yun (Ann Glob Anal Geom 62(3):507–532, 2022), an estimate for skew-symmetric 2-tensors was claimed. Soon after, this estimate has been exploited to claim powerful classification results: Most notably, it has been employed to propose a proof of a Black Hole Uniqueness Theorem for vacuum static spacetimes with positive scalar curvature (Xu and Ye in Invent Math 33(2):64, 2022) and in connection with the Besse conjecture (Yun and Hwang in Critical point equation on three-dimensional manifolds and the Besse conjecture). In the present note, we point out an issue in the argument proposed in Hwang and Yun (Ann Glob Anal Geom 62(3):507–532, 2022) and we provide a counterexample to the estimate.

Nonrelativistic limit of scalar and Dirac fields in curved spacetime

We give from first principles the non-relativistic limit of scalar and Dirac fields in curved spacetime. We aim to find general relativistic corrections to the quantum theory of particles affected by Newtonian gravity, a regime nowadays experimentally accessible. We believe that the ever-improving measurement accuracy and the theoretical interest in finding general relativistic effects in quantum systems require the introduction of corrections to the Schr\"{o}dinger-Newtonian theory. We rigorously determine these corrections by the non-relativistic limit of fully relativistic quantum theories in curved spacetime. For curved static spacetimes, we show how a non-inertial observer (equivalently, an observer in the presence of a gravitational field) can distinguish a scalar field from a Dirac field by particle-gravity interaction. We study the Rindler spacetime and discuss the difference between the resulting non-relativistic Hamiltonians. We find that for sufficiently large acceleration, the gravity-spin coupling dominates over the corrections for scalar fields, promoting Dirac particles as the best candidates for observing non-Newtonian gravity in quantum particle phenomenology.

Ricci-flat Finsler metrics by warped product

In this work, we consider a class of Finsler metrics using the warped product notion introduced by Chen et al. [Internat. J. Math. 29 (2018), 1850081], with another “warping”, one that is consistent with static spacetimes. We will give the PDE characterization for the proposed metrics to be Ricci-flat and explicitly construct two non-Riemannian examples.

Bi-level optimization for customized bus routing serving passengers with multiple-trips based on state–space–time network

The emerging customized bus (CB) can serve personalized trip requests in a more flexible and convenient way, especially for passengers who have multiple-trip requests in a short time period. Considering the CB service for passengers with multiple trips (CSPMT), a loading-state-oriented state–space–time network-based bi-level programming model is proposed to optimize the routing of CBs, with the objectives of maximizing the operational profit and minimizing the travel cost, while considering the characteristics of multiple trips, time windows, capacity and mixed loads. Besides, a nested algorithm combining the genetic algorithm (GA) and the augmented Lagrangian relaxation-based dynamic programming algorithm is proposed. Then, the proposed model and algorithm are verified and analyzed through a Sioux Falls network and a Beijing sketch network. It can be found from the results that the method can optimize a CB routing plan for passengers with multiple-trip requests and for different network scales. The proposed bi-level model and corresponding algorithm can better adapt to passengers’ personalized trip requests, and promote a higher level of public transport service which will attract more residents from private cars to public transportation, ultimately reducing energy consumption and exhaust emissions, and promoting the sustainable development of modern metropolises.