Schwarzschild-de Sitter Spacetime Research Articles

Nonminimally coupled scalar field in Schwarzschild–de Sitter spacetime: Geodesic synchrotron radiation

Nonminimally coupled scalar field in Schwarzschild–de Sitter spacetime: Geodesic synchrotron radiation

Two regimes of asymptotic fall-off of a massive scalar field in the Schwarzschild–de Sitter spacetime

Two regimes of asymptotic fall-off of a massive scalar field in the Schwarzschild–de Sitter spacetime

Possible influence of a cosmic repulsion on large-scale jets: Geometric viewpoint

Cosmic repulsion represented by a small positive value of the cosmological constant changes significantly properties of central gravitational fields at large distances, leading to existence of a static (or turnaround) radius where gravitational attraction of a center is just balanced by cosmic repulsion. Analyzing behavior of radial timelike geodesics in the Schwarzschild–de Sitter spacetime near its static radius we show that the particles with specific energy close to unity have tendency to slow down and cluster just below the static radius, forming clumps which, subsequently, start to expand uniformly due to cosmic repulsion. For central masses of (106–1011)M⊙ and current value of the cosmological constant 1.1×10−52m−2, this phenomenon takes place at distances of tens to hundreds of kiloparsecs from the center, being comparable with distances in which huge radio-lobes from some active galaxies were observed.

Static sphere observers and geodesics in Schwarzschild-de Sitter spacetime

We analyze null- and spacelike radial geodesics in Schwarzschild-de Sitter spacetime connecting two conjugate static sphere observers, i.e. free-falling observers at a fixed radius in between the two horizons. We explicitly determine the changes in the causal structure with respect to these natural observers as a result of the inward bending of the black hole singularity, as well as the outward bending of asymptotic infinity. Notably, the inward and outward bending changes as a function of the black hole mass, first increasing towards a maximum and then decreasing to vanish in the extreme Nariai limit. For a generic mass of the black hole this implies the existence of finite size (temporal) windows for the presence of symmetric radial geodesics between the static sphere observers probing the interior region of the black hole, as well as the exterior de Sitter region. We determine the size of the interior (black hole) and exterior (de Sitter) temporal windows in 4, 5 and 6 spacetime dimensions, finding that they are equal in D = 5, and compute the proper lengths of the symmetric radial geodesics. We comment on the implications for information exchange and the potential role of the symmetric radial geodesics in a geodesic approximation of static sphere correlators in Schwarzschild-de Sitter spacetime.

Black hole solutions with constant Ricci scalar in a model of Finsler gravity

Ricci scalar being zero is equivalent to the vacuum field equation in Finsler space-time. The Schwarzschild metric can be concluded from the field equation's solution if the space-time conserves spherical symmetry. This research aims to investigate Finslerian Schwarzschild-de Sitter space-time. Recent studies based on Finslerian space-time geometric models are becoming more prevalent because the local anisotropic structure of space-time influences the gravitational field and gives rise to modified cosmological relations. We suggest a gravitational field equation with a non-zero cosmological constant in Finslerian geometry and apprehend that the presented Finslerian gravitational field equation corresponds to the non-zero Ricci scalar. In Finsler geometry, the peer of spherical symmetry is the Finslerian sphere. Assuming space-time to conserve the “Finslerian sphere” symmetry, the counterpart of the Riemannian sphere (Finslerian sphere) must have a constant flag curvature (λ). It is demonstrated that the Finslerian covariant derivative of the geometric part of the gravitational field equation is preserved under a condition using the Chern connection. According to the string theory, string clouds can be defined as a pool of strings made due to symmetry breaking in the universe's early stages. We find that for λ ≠ 1, this solution resembles a black hole surrounded by a cloud of strings. Furthermore, we investigate null and time-like geodesics for λ = 1. In this regard, the photon geodesics are obtained that are the closest paths to the photon sphere of the first photons visible at the black hole shadow limit. Also, circular orbit conditions are obtained for the effective potential.

Conformal geodesics and the evolution of spacetimes with positive Cosmological constant.

This article provides a discussion on the construction of conformal Gaussian gauge systems to study the evolution of solutions to the Einstein field equations with positive Cosmological constant. This is done by means of a gauge based on the properties of conformal geodesics. The use of this gauge, combined with the extended conformal Einstein field equations, yields evolution equations in the form of a symmetric hyperbolic system for which standard Cauchy stability results can be employed. This strategy is used to study the global properties of de Sitter-like spacetimes with constant negative scalar curvature. It is then adapted to study the evolution of the Schwarzschild-de Sitter spacetime in the static region near the conformal boundary. This review is based on Minucci et al. 2021 Class. Quantum Grav. 38, 145026. (doi:10.1088/1361-6382/ac0356) and Minucci et al. 2023 Class. Quantum Grav. 40, 145005. (doi:10.1088/1361-6382/acdb3f). This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

Dark energy as a critical period in binary motion: Bounds from multi-scale binaries

We study the two-body problem in the context of both dark energy and post-Newtonian modifications. In this unified framework, we demonstrate that dark energy plays the role of a critical period with TΛ = 2π/c√Λ ≈ 60 Gyr. We also show that the ratio between the orbital and critical periods naturally emerges from the Kretschmann scalar, which is a quadratic curvature invariant characterizing all binary systems effectively represented by de Sitter-Schwarzschild space-time. The suitability of a binary system in constraining dark energy is determined by the ratio between its Keplerian orbital period, TK, and the critical period, TΛ. Systems with TK ≈ TΛ are optimal for constraining the cosmological constant, Λ, such as the Local Group and the Virgo Cluster. Systems with TK ≪ TΛ are dominated by attractive gravity (which are best suited for studying modified gravity corrections). Systems with TK ≫ TΛ are dominated by repulsive dark energy and can thus be used to constrain Λ from below. We used our unified framework of post-Newtonian and dark-energy modifications to calculate the precession of bounded and unbounded astrophysical systems and infer constraints on Λ from them. We analyzed pulsars, the solar system, S stars around Sgr A*, the Local Group, and the Virgo Cluster, having orbital periods of days to gigayears. Our results reveal that the upper bound on the cosmological constant decreases when the orbital period of the system increases, emphasizing that Λ is a critical period in binary motion.

Gaussian quantum steering in multi-event horizon spacetime

We study Gaussian quantum steering in the Schwarzschild–de Sitter (SdS) spacetime that is endowed with both a black hole event horizon (BEH) and a cosmological event horizon (CEH), giving rise to two different Hawking temperatures. It is shown that the Hawking effect of the black hole always reduces the quantum steering, but the Hawking effect of the expanding universe does not always play the same role. For the first time, we find that the Hawking effect can improve quantum steering. We also find that the observer who locates in the BEH has stronger steerability than the observer who locates in CEH. Further, we study the steering asymmetry, and the conditions for two-way, one-way and no-way steering in the SdS spacetime. Finally, we study the Gaussian quantum steering in the scenario of effective equilibrium temperature. We show that quantum steering reduces monotonically with the effective temperature but now increases monotonically with the Hawking temperature of the black hole, which banishes the belief that the Hawking effect can only destroy quantum steering.

Entanglement area law violation from field-curvature coupling

We investigate the ground state entanglement entropy of a massive scalar field nonminimally coupled to spacetime curvature, assuming a static, spherically symmetric background. We first discretize the field Hamiltonian by introducing a lattice of spherical shells and imposing a cutoff in the radial direction. We then study the ground state of the field and quantify deviations from area law due to nonminimal coupling, focusing in particular on Schwarzschild-de Sitter and Hayward spacetimes, also discussing de Sitter spacetime as a limiting case. We show that large positive coupling constants can significantly alter the entropy scaling with respect to the boundary area, in case of coordinate-dependent spacetime curvature. Our outcomes are interpreted in view of black hole entropy production and early universe scenarios.

Buchdahl-inspired spacetimes and wormholes: Unearthing Hans Buchdahl’s other “hidden” treasure trove

Circa 1962 Hans A. Buchdahl pioneered a program — and made significant progress — seeking vacuo configurations in pure [Formula: see text] gravity (H. A. Buchdahl, Nuovo Cimento 23 (1962) 141). Unfortunately, he deemed the final step in his calculations impassable and prematurely suspended his pursuit. Since then, his achievements on this front have faded into dormancy. Unbeknownst to Buchdahl himself, the [Formula: see text] vacua that he sought were within his striking distance. In our recent three-paper body of work published in Nguyen (Phys. Rev. D 106 (2022) 104004; Phys. Rev. D 107 (2023) 104008; Phys. Rev. D 107 (2023) 104009), we broke this six-decades-old impasse and accomplished his goal: A novel class of metrics, describing non-Schwarzschild spacetimes in quadratic gravity and carrying a hallmark of higher-derivative characteristic. Intriguing properties of Buchdahl-inspired spacetimes and their associated Morris–Thorne–Buchdahl wormholes — summarized herein — embody a new branch of phenomenology that transcends the Einstein–Hilbert paradigm. May this unsung legacy of Hans A. Buchdahl long “forsaken” and “forgotten” finally live on Our essay is a passageway bridging the past and the future. The past is rooted in a research program originated but left unfinished by Buchdahl going back threescore years ago. The future is the new horizons his program brings forth. In between is our labor — summarized in a three-installment “Beyond Schwarzschild–de Sitter spacetimes” series[1–3] — to advance the Buchdahl program and achieve his ultimate goal.

On the Global Temperature of the Schwarzschild–de Sitter Spacetime

It is shown that the Schwarzschild–de Sitter spacetime has the universal temperature. This temperature describes the thermal processes of decay of the composite particles and the other processes, which are energetically forbidden in the Minkowski spacetime, but are allowed in the de Sitter and in Schwarzschild–de Sitter backgrounds. In particular, this temperature describes the probability of ionization of the atom in the Schwarzschild–de Sitter, which is observed by the stationary observer at the point where the shift function (velocity) in the Arnowitt–Deser–Misner formalism changes sign. This activation temperature does not depend on the black hole mass and is fully determined by the Hubble parameter, {{T}_{{{text{SdS}}}}} = sqrt 3 H{text{/}}pi . This temperature is twice the Bousso–Hawking temperature {{T}_{{{text{BH}}}}}, which characterizes the limit of degenerate Lorentzian Schwarzschild–de Sitter universe, when the cosmological and black hole horizons are close to each other, {{T}_{{{text{SdS}}}}} = 2{{T}_{{{text{BH}}}}}. The similar doubling of the temperature of Hawking radiation is known in the pure de Sitter spacetime, where the corresponding local temperature describing the ionization of atoms is twice the Gibbons–Hawking temperature, {{T}_{{{text{dS}}}}} = 2{{T}_{{{text{GH}}}}} = H{text{/}}pi . We suggest that the activation temperature {{T}_{{{text{dS}}}}} can be considered as the thermodynamic temperature of the de Sitter state, which determines the local entropy in this state, s = 3H{text{/}}4G.

On the non-linear stability of the Cosmological region of the Schwarzschild-de Sitter spacetime

The non-linear stability of the sub-extremal Schwarzschild-de Sitter spacetime in the stationary region near the conformal boundary is analysed using a technique based on the extended conformal Einstein field equations and a conformal Gaussian gauge. This strategy relies on the observation that the Cosmological stationary region of this exact solution can be covered by a non-intersecting congruence of conformal geodesics. Thus, the future domain of dependence of suitable spacelike hypersurfaces in the Cosmological region of the spacetime can be expressed in terms of a conformal Gaussian gauge. A perturbative argument then allows to prove existence and stability results close to the conformal boundary and away from the asymptotic points where the Cosmological horizon intersects the conformal boundary. In particular, we show that small enough perturbations of initial data for the sub-extremal Schwarzschild-de Sitter spacetime give rise to a solution to the Einstein field equations which is regular at the conformal boundary. The analysis in this article can be regarded as a first step towards a stability argument for perturbation data on the Cosmological horizons.

Morawetz Estimates Without Relative Degeneration and Exponential Decay on Schwarzschild–de Sitter Spacetimes

We use a novel physical space method to prove relatively non-degenerate integrated energy estimates for the wave equation on subextremal Schwarzschild–de Sitter spacetimes with parameters (M,Lambda ). These are integrated decay statements whose bulk energy density, though degenerate at highest order, is everywhere comparable to the energy density of the boundary fluxes. As a corollary, we prove that solutions of the wave equation decay exponentially on the exterior region. The main ingredients are a previous Morawetz estimate of Dafermos–Rodnianski and an additional argument based on commutation with a vector field which can be expressed in the form r1-2Mr-Λ3r2∂∂r,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} r\\sqrt{1-\\frac{2M}{r}-\\frac{\\Lambda }{3}r^2}\\frac{\\partial }{\\partial r}, \\end{aligned}$$\\end{document}where partial _r here denotes the coordinate vector field corresponding to a well-chosen system of hyperboloidal coordinates. Our argument gives exponential decay also for small first-order perturbations of the wave operator. In the limit Lambda =0, our commutation corresponds to the one introduced by Holzegel–Kauffman (A note on the wave equation on black hole spacetimes with small non-decaying first-order terms, 2020. arXiv:2005.13644).

Infrared effects and the Unruh state

Detailed behaviors of the modes of quantized scalar fields in the Unruh state for various eternal black holes in two dimensions are investigated. It is shown that the late-time behaviors of some of the modes of the quantum fields and of the symmetric two-point function are determined by infrared effects. The nature of these effects depends upon whether there is an effective potential in the mode equation and what form this potential takes. Here, three cases are considered, one with no potential and two with potentials that are nonnegative everywhere and are zero on the event horizon of the black hole and zero at either infinity or the cosmological horizon. Specifically, the potentials are a delta function potential and the potential that occurs for a massive scalar field in Schwarzschild–de Sitter spacetime. In both cases, scattering effects remove infrared divergences in the mode functions that would otherwise arise from the normalization process. When such infrared divergences are removed, it is found that the modes that are positive frequency with respect to the Kruskal time on the past black hole horizon approach zero in the limit that the radial coordinate is fixed and the time coordinate goes to infinity. In contrast, when there is no potential and thus infrared divergences occur, the same modes approach nonzero constant values in the late-time limit when the radial coordinate is held fixed. The behavior of the symmetric two-point function when the field is in the Unruh state is investigated for the case of a delta function potential in certain asymptotically flat black hole spacetimes in two dimensions. The removal of the infrared divergences in the mode functions results in the elimination of terms that grow linearly in time.

Upon the horizon’s verge: Thermal particle creation between and approaching horizons

Quantum particle creation from spacetime horizons, or accelerating boundaries in the dynamical Casimir effect, can have an equilibrium, or thermal, distribution. Using an accelerating boundary in flat spacetime (moving mirror), we investigate the production of thermal energy flux despite non-equilibrium accelerations, the evolution between equilibrium states, and the ‘interference’ between horizons. In particular, this allows us to give a complete solution to the particle spectrum of the accelerated boundary correspondence with Schwarzschild–de Sitter spacetime.

Beyond Schwarzschild–de Sitter spacetimes: A new exhaustive class of metrics inspired by Buchdahl for pure R2 gravity in a compact form

Some sixty years ago Buchdahl pioneered a program in search of static spherically symmetric vacua for pure $R^{2}$ gravity (Nuovo Cimento, Vol 23, No 1, pp 141-157 (1962); [https://link.springer.com/article/10.1007/BF02733549]). Surpassing several obstacles, his work culminated in a non-linear second-order ordinary differential equation (ODE) which required being solved. However Buchdahl deemed the ODE intractable and prematurely abandoned his pursuit for an analytical solution. We have finally managed to break this six-decades-old impasse and accomplish his goal. Reformulating the Buchdahl ODE, we obtain a novel exhaustive class of metrics (which we shall call the Buchdahl-inspired metrics hereafter) in a compact form. We are able to verify ex post, via direct inspection, that the metric obtained herein satisfies the $R^{2}$ vacuo field equation, hence establishing its validity. We show why the Buchdahl-inspired metrics, which exhibit non-constant scalar curvature, defeat a Lichnerowicz-type no-go theorem, previously proved for quadratic gravity, by evading the overly strong restriction on the asymptotic falloff of the metric at spatial infinity. Our novel solution thereby completes Buchdahl's six-decades-old program. We also explore the mathematical properties of the Buchdahl-inspired metrics in the limit of small k and in the region around the coordinate origin.

Entanglement harvesting from conformal vacuums between two Unruh-DeWitt detectors moving along null paths

It is well-known that the (1 + 1) dimensional Schwarzschild and spatially flat FLRW spacetimes are conformally flat. This work examines entanglement harvesting from the conformal field vacuums in these spacetimes between two Unruh-DeWitt detectors, moving along outgoing null trajectories. In (1 + 1) dimensional Schwarzschild spacetime, we considered the Boulware and Unruh vacuums for our investigations. In this analysis, one observes that while entanglement harvesting is possible in (1+1) dimensional Schwarzschild and (1 + 3) dimensional de Sitter spacetimes, it is not possible in the (1 + 1) dimensional de Sitter background for the same set of parameters when the detectors move along the same outgoing null trajectory. The qualitative results from the Boulware and the Unruh vacuums are alike. Furthermore, we observed that the concurrence depends on the distance d between the two null paths of the detectors periodically, and depending on the parameter values, there could be entanglement harvesting shadow points or regions. We also observe that the mutual information does not depend on d in (1 + 1) dimensional Schwarzschild and de Sitter spacetimes but periodically depends on it in (1 + 3) dimensional de Sitter background. We also provide elucidation on the origin of the harvested entanglement.

Static and spherically symmetric general relativity solutions in minimal theory of bigravity

We investigate static and spherically symmetric solutions in the Minimal Theory of Bigravity (MTBG). First, we show that a pair of Schwarzschild-de Sitter spacetimes with different cosmological constants and black hole masses written in the spatially-flat Gullstrand-Painlev\'e (GP) coordinates is a solution in the self-accelerating branch of MTBG, while it cannot be a solution in the normal branch. We then illustrate how Schwarzschild-de Sitter solutions can become compatible with the normal branch when using different coordinates. We also confirm that the self-accelerating branch of MTBG admits static and spherically symmetric general relativity solutions with matter written in the spatially-flat coordinates, including neutron stars with arbitrary matter equations of state. Finally, we show that in the self-accelerating branch nontrivial solutions are given by the Schwarzschild-de Sitter metrics written in nonstandard coordinates.

Energy for accelerating observers in black hole spacetimes

Quasi-local energies for constantly accelerating observers in Ba\~nados, Teitelboim, and Zanelli (BTZ), Schwarzschild and Schwarzschild-de Sitter spacetimes are derived. The energies are expressed in terms of acceleration, cosmological constant, and area, quantities measurable by the observers. Based on results from quantum fields in curved spacetime for the redshifted Hawking temperature, entropy and thermodynamic-like laws are briefly explored in the three spacetimes.

Red and blue shift in spherical and axisymmetric spacetimes and astrophysical constraints

We compute the red and blue shifts for astrophysical and cosmological sources. In particular, we consider low, intermediate and high gravitational energy domains. Thereby, we handle the binary system Earth–Mars as low energy landscape whereas white dwarfs and neutron stars as higher energy sources. To this end, we take into account a spherical Schwarzschild–de Sitter spacetime and an axially symmetric Zipoy–Voorhees metric to model all the aforementioned systems. Feasible outcomes come from modeling neutron stars and white dwarfs with the Zipoy–Voorhees metric, where quadrupole effects are relevant, and framing solar system objects using a Schwarzschild–de Sitter spacetime. In the first case, large \(\delta \) parameters seem to be favorite, leading to acceptable bounds mainly for neutron stars. In the second case, we demonstrate incompatible red and blue shifts with respect to lunar and satellite laser ranging expectations, once the cosmological constant is taken to Planck satellite’s best fit. To heal this issue, we suggest coarse-grained experimental setups and propose Phobos for working out satellite laser ranging in order to get more suitable red and blue shift intervals, possibly more compatible than current experimental bounds. Implications to cosmological tensions are also debated.