Timelike Vector Field Research Articles

General mass formulas for charged Kerr-AdS black holes

It is well-known that the mass of a non-asymptotically flat spacetime cannot be uniquely defined. Some mass formulas for the Kerr-AdS black hole have been found and used in studying black hole thermodynamics. However, the derivations usually need a background subtraction to eliminate the divergence at infinity. It is also unknown whether the mass depends on the choice of coordinates. In this paper, we provide a more straightforward derivation for the mass formula, only demanding that the first law of black hole thermodynamics and Smarr formula are satisfied. We first make use of the Iyer-Wald formalism to derive a first law which avoids the divergence at infinity. Then we apply this formula to charged Kerr-AdS black hole expressed in the coordinates rotating at infinity. However, the first law associated with the timelike Killing vector field ∂∂t is not integrable. Then, by making use of the gauge freedom of t, we find a favorite parameter t′ which just makes the mass integrable. Applying the scaling argument, we show that the mass satisfies the Smarr formula and takes the form M/Ξ3/2. Moreover, applying the conformal method with ∂∂t′ , we obtain the same mass. By applying the first law to the coordinates which is not rotating at infinity, we find a preferred time T that makes the first law integrable and the mass is just the familiar mass M/Ξ2 in the literature. This mass is also confirmed by the conformal method. We find that the two mass formulas correspond to different families of observers and the preferred Killing times. So our work clarifies the ambiguities of mass in Kerr-AdS spacetimes.

Even- and odd-parity stabilities of black holes in Einstein-Aether gravity

In Einstein-Aether theories with a timelike unit vector field, we study the linear stability of static and spherically symmetric black holes against both even- and odd-parity perturbations.For this purpose, we formulate a gauge-invariant black hole perturbation theory in the background Aether-orthogonal frame where the spacelike property of hypersurfaces orthogonal to the timelike Aether field is always maintained even inside the metric horizon. Using a short-wavelength approximation with large radial and angular momenta, we show that, in general, there are three dynamical degreesof freedom arising from the even-parity sector besides two propagating degrees of freedom present in the odd-parity sector. The propagation speeds of even-parity perturbations and their no-ghost conditions coincide with those of tensor, vector, and scalar perturbations on the Minkowski background, while the odd sector contains tensor and vector modes with the same propagation speeds as those in the even-parity sector (and hence as those on the Minkowski background).Thus, the consistent study of black hole perturbations in the Aether-orthogonal frame on static and spherically symmetric backgrounds does not add new small-scale stability conditions to those known for the Minkowski background in the literature.

Kodama-like vector fields in axisymmetric spacetimes

We extend the concept of the Kodama symmetry, a quasi-local time translation symmetry for dynamical spherically symmetric spacetimes, to a specific class of dynamical axisymmetric spacetimes, namely the families of Kerr–Vaidya and Kerr–Vaidya–de Sitter spacetimes. We study some geometrical properties of the asymptotically flat Kerr–Vaidya metric, such as the Brown–York mass and the Einstein tensor. Furthermore, we propose a generalization of the Kerr–Vaidya metric to an asymptotic de Sitter background. We show that for these classes of dynamical axisymmetric black hole spacetimes, there exists a timelike vector field that exhibits similar properties to the Kodama vector field in spherical symmetry. This includes the construction of a covariantly conserved current and a corresponding locally conserved charge, which in the Kerr–Vaidya case converges to the Brown–York mass in the asymptotically flat region.

A generalization of photon sphere based on escape/capture cone

In general asymptotically flat spacetimes, bearing the null geodesics reaching the future null infinity in mind, we propose new concepts, the “dark horizons” (outer dark horizon and inner dark horizon) as generalizations of the photon sphere. They are defined in terms of the structure of escape/capture cones of photons with respect to a unit timelike vector field to capture the motion of light sources. More specifically, considering a two-sphere that represents a set of emission directions of photons, the dark horizons are located at positions where a hemisphere is marginally included in the capture and escape cones, respectively. In addition, our definition succeeds in incorporating relativistic beaming effect. We show that the dark horizon is absent in the Minkowski spacetime, while they exist in spacetimes with black hole(s) under a certain condition. We derive the general properties of the dark horizons in spherically symmetric spacetimes and explicitly calculate the locations of the dark horizons in the Vaidya spacetime and the Kerr spacetime. In particular, in the Kerr spacetime, the outer dark horizon coincides with the shadow observed from infinity on the rotation axis.

Ricci Solitons on Spacelike Hypersurfaces of Generalized Robertson–Walker Spacetimes

Ricci Solitons on Spacelike Hypersurfaces of Generalized Robertson–Walker Spacetimes

Biconservative Riemannian Submanifolds in Minkowski 5-Space

In this study, we focus biconservative Riemannian submanifolds with parallel normalized mean curvature vector field (PNMCV) in $\mathbb{E}^5_1$. We obtain explicit classifications for these submanifolds with exactly two distinct principal curvatures of the shape operator along the mean curvature vector field. In particular, we investigate these submanifolds which have time-like and space-like mean curvature vector field in $\mathbb{E}^5_1$.

Existence of gradient CKV and gradient conformally stationary LRS spacetimes

In this work, we study the existence of gradient (proper) CKVs in locally rotationally symmetric spacetimes (LRS), those CKVs in the space spanned by the tangent to observers’ congruence and the preferred spatial direction, allowing us to provide a (partial) characterization of gradient conformally static (GCSt) LRS solutions. Irrrotational solutions with non-zero spatial twist admit an irrotational timelike gradient conformal Killing vector field and hence are GCSt. In the case that both the vorticity and twist vanish, that is, restricting to the LRS II subclass, we obtain the necessary and sufficient condition for the spacetime to admit a gradient CKV. This is given by a single wave-like PDE, whose solutions are in bijection to the gradient CKVs on the spacetime. We then introduce a characterization of these spacetimes as GCSt using the character of the divergence of the CKV, provided that the metric functions of the spacetimes obey certain inequalities.

Interaction of the Cosmic Dark Fluid with Dynamic Aether: Parametric Mechanism of Axion Generation in the Early Universe

We consider an isotropic homogeneous cosmological model with five interacting elements: first, the dynamic aether presented by a unit timelike vector field; second, the pseudoscalar field describing an axionic component of the dark matter; third, the cosmic dark energy, described by a rheologic fluid; fourth, the non-axionic dark matter coupled to the dark energy; and fifth, the gravity field. We show that the early evolution of the Universe described by this model can include two specific epochs: the first one can be characterized as a super-inflation epoch; the second epoch is associated with an oscillatory regime. The dynamic aether carries out a regulatory mission; the rheologic dark fluid provides the specific features of the spacetime evolution. The oscillations of the scale factor and of the Hubble function are shown to switch on the parametric (Floquet-type) mechanism of the axion number growth.

On the rigidity of cosmological space-times

In this paper, we analyse a family of geometrically well-behaved cosmological space-times (V^{n+1},g), which are foliated by intrinsically isotropic space-like hypersurfaces {M_t}_{tin mathbb {R}}, which are orthogonal to a family of co-moving observers defined by a global time-like vector field U. In particular, this implies such space-times satisfy several of the well-known criteria for isotropic cosmological space-times, although, in the family in question, the simultaneity spaces (M_t,g_t) associated with U can have as sectional curvature a sign-changing function k(t). Being this clearly impossible in the FLRW family of standard cosmological space-times, it motivates us to revisit the geometric rigidity consequences of different definitions of isotropy available in the literature. In this analysis, we divide such definition according to whether the isometries involved are taken to be (local) space-time (STI space-times) or (local) space isometries (SI space-times) of (M_t,g_t) for each t. This subtlety will be shown to be critical, proving that only when space-time isometries are considered one obtains the well-known rigidity properties associated with isotropic cosmological space-times. In particular, SI space-times will be shown to be a strictly larger class than the STI ones, allowing a family of basic cosmological curvature change models which are not even locally isometric to any FLRW space-time.

On the flow of perfect energy tensors

The necessary and sufficient conditions are obtained for a unit time-like vector field u to be the unit velocity of a divergence-free perfect fluid energy tensor. This plainly kinematic description of a conservative perfect fluid requires considering eighteen classes defined by differential concomitants of u. For each of these classes, we get the additional constraints that label the flow of a conservative energy tensor, and we obtain the pairs of functions , energy density and pressure, which complete a solution to the conservation equations.

Boundary and bulk notions of transport in the AdS3/CFT2 correspondence

We construct operators in holographic two-dimensional conformal field theory, which act locally in the code subspace as arbitrary bulk spacelike vector fields. Key to the construction is an interplay between parallel transport in the bulk spacetime and in kinematic space. We outline challenges, which arise when the same construction is extended to timelike vector fields. We also sketch several applications, including boundary formulations of the bulk Riemann tensor, dreibein, and spin connection, as well as an application to holographic complexity.

Constraining the Lorentz-violating bumblebee vector field with big bang nucleosynthesis and gravitational baryogenesis

By assuming the cosmological principle i.e., an isotropic and homogeneous universe, we consider the cosmology of a vector-tensor theory of gravitation known as the bumblebee model. In this model a single Lorentz-violating timelike vector field with a nonzero vacuum expectation value (VEV) couples to the Ricci tensor and scalar, as well. Taking the ansatz B(t)sim t^beta for the time evolution of the vector field, where beta is a free parameter, we derive the relevant dynamic equations of the Universe. In particular, by employing observational data coming from the Big Bang Nucleosynthesis (BBN) and the matter–antimatter asymmetry in the baryogenesis era, we impose some constraints on the VEV of the bumblebee timelike vector field i.e., xi b^2, and the exponent parameter beta . The former and the latter limit the size of Lorentz violation, and the rate of the time evolution of the background Lorentz-violating bumblebee field, respectively.

Ricci solitons on general relativistic spacetimes

The main aim of this manuscript is to characterize the general relativistic spacetimes with Ricci and gradient Ricci solitons. It is proven that if the metric of a general relativistic spacetime (M 4, ξ) admitting a special unit timelike vector field ξ is an almost Ricci soliton (g, ξ, λ), then (M 4, ξ) is a perfect fluid spacetime, and almost Ricci soliton (g, ξ, λ) on (M 4, ξ) becomes shrinking Ricci soliton. We prove that a general relativistic perfect fluid spacetime equipped with a special unit timelike vector field together with a Ricci soliton is an Einstein spacetime. In this sequel, we also prove that the Ricci soliton is shrinking, soliton vector field is Killing and the scalar curvature of the perfect fluid spacetime is constant. It is proven that a general relativistic perfect fluid spacetime together with a Ricci soliton is a generalized Robertson-Walker (GRW) spacetime. The existence of gradient Ricci solitons on general relativistic spacetimes are established. We also construct a non-trivial example of general relativistic spacetime equipped with a special unit timelike vector field, and verify some of our theorems.

Solitonic Aspect of Relativistic Magneto-Fluid Spacetime with Some Specific Vector Fields

The target of the current research article is to investigate the solitonic attributes of relativistic magneto-fluid spacetime (MFST) if its metrics are Ricci–Yamabe soliton (RY-soliton) and gradient Ricci–Yamabe soliton (GRY-soliton). We exhibit that a magneto-fluid spacetime filled with a magneto-fluid density ρ, magnetic field strength H, and magnetic permeability μ obeys the Einstein field equation without the cosmic constant being a generalized quasi-Einstein spacetime manifold (GQE). In such a spacetime, we obtain an EoS with a constant scalar curvature R in terms of the magnetic field strength H and magnetic permeability μ. Next, we achieve some cauterization of the magneto-fluid spacetime in terms of Ricci–Yamabe solitons with a time-like torse-forming vector field ξ and a φ(Ric) vector field. We establish the existence of a black hole in the relativistic magneto-fluid spacetime by demonstrating that it admits a shrinking Ricci–Yamabe soliton and satisfies the time-like energy convergence criteria. In addition, we examine the magneto-fluid spacetime with a gradient Ricci–Yamabe soliton and deduce some conditions for an equation of state (EoS) ω=−15 with a Killing vector field. Furthermore, we demonstrate that the EoS ω=−15 of the magneto-fluid spacetime under some constraints represents a star model and a static, spherically symmetric perfect fluid spacetime. Finally, we prove that a gradient Ricci–Yamabe soliton with the conditions μ=0 or H=2; μ≠0, H>2 and obeying the equation of state ω=−15 is conceded in a magneto-fluid spacetime, and a naked singularity with a Cauchy horizon subsequently emerges, respectively.

Covariant Wick rotation: action, entropy, and holonomies

Given an arbitrary Lorentzian metric g_{ab} and a nowhere vanishing, timelike vector field u^a, one can construct a class of metrics {widehat{g}}_{ab} which have Euclidean signature in a specific domain, with a transition to Lorentzian regime occurring on some hypersurface Sigma orthogonal to u^a. Geometry associated with {{{widehat{g}}}}_{ab} has been shown to yield some remarkable insights for classical and quantum gravity. In this work, we focus on studying the implications of this geometry for thermal effects in curved spacetimes and compare and contrast the results with those obtained through conventional Euclidean methods. We show that the expression for entropy computed using {widehat{g}}_{ab} for simple field theories and Lanczos–Lovelock actions differ from Wald entropy by additional terms depending on extrinsic curvature. We also compute the holonomy associated with loops lying partially or wholly in the Euclidean regime in terms of extrinsic curvature and acceleration and compare it with the well-known expression for temperature.

Trajectories of Affine Control Systems and Geodesics of a Spacetime with a Causal Killing Vector Field

We study the geodesic connectedness of a globally hyperbolic spacetime (M,g) admitting a complete smooth Cauchy hypersurface S and endowed with a complete causal Killing vector field K. The main assumptions are that the kernel distribution D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal D$\\end{document} of the one-form induced by K on S is non-integrable and that the gradient of g(K,K) is orthogonal to D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {D}$\\end{document}. We approximate the metric g by metrics gε smoothly depending on a real parameter ε and admitting K as a timelike Killing vector field. A known existence result for geodesics of such type of metrics provides a sequence of approximating solutions, joining two given points, of the geodesic equations of (M,g) and whose Lorentzian energy turns out to be bounded thanks to an argument involving trajectories of some affine control systems related with D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {D}$\\end{document}.

A variational setting for an indefinite Lagrangian with an affine Noether charge

We introduce a variational setting for the action functional of an autonomous and indefinite Lagrangian on a finite dimensional manifold M. Our basic assumption is the existence of an infinitesimal symmetry whose Noether charge is the sum of a one-form and a function on M. Our setting includes different types of Lorentz–Finsler Lagrangians admitting a timelike Killing vector field.

A Gutzwiller Trace Formula for Dirac Operators on a Stationary Spacetime

A Duistermaat–Guillemin–Gutzwiller trace formula for Dirac-type operators on a globally hyperbolic spatially compact stationary spacetime is achieved by generalising the recent construction by Strohmaier and Zelditch (Adv Math 376:107434, 2021) to a vector bundle setting. We have analysed the spectrum of the Lie derivative with respect to a global timelike Killing vector field on the solution space of the Dirac equation and found that it consists of discrete real eigenvalues. The distributional trace of the time evolution operator has singularities at the periods of induced Killing flow on the space of lightlike geodesics. This gives rise to the Weyl law asymptotic at the vanishing period.

Spin Hall effects and the localization of massless spinning particles

The spin Hall effects of light represent a diverse class of polarization-dependent physical phenomena involving the dynamics of electromagnetic wave packets. In a medium with an inhomogeneous refractive index, wave packets can be effectively described by massless spinning particles following polarization-dependent trajectories. Similarly, in curved spacetime the gravitational spin Hall effect of light is represented by polarization-dependent deviations from null geodesics. In this paper, we analyze the equations of motion describing the gravitational spin Hall effect of light. We show that these equations are a special case of the Mathisson-Papapetrou equations for spinning objects in general relativity. This allows us to use several known results for the Mathisson-Papapetrou equations, and apply them to the study of electromagnetic wave packets. We derive conservation laws, we discuss the limits of validity of the spin Hall equations, and we study how the energy centroids of wave packets, effectively described as massless spinning particles, depend on the external choice of a timelike vector field, representing a family of observers. In flat spacetime, the relativistic Hall effect and the Wigner(-Souriau) translations are recovered, while our equations also provide a generalization of these effects in arbitrary spacetimes. We construct a large class of wave packets that can be described by the spin Hall equations, but also find its limits by giving examples of wave packets which are more general and are not described by the spin Hall equations. Lastly, we examine the assumption that electromagnetic wave packets are massless. While this is approximately true in many contexts, it is not exact. We show that failing to carefully account for the limitations of the massless approximation results in the appearance of unphysical ``centroids'' which are nowhere near the wave packet itself.

On the Scattering of Waves inside Charged Spherically Symmetric Black Holes

In this paper, we show that there is a breakdown of scattering between the event horizon (or the Cauchy horizon) and an intermediate Cauchy hypersurface in the dynamic interior of a Reissner–Nordström-like black hole. More precisely, we show that the trace operators and their analytic counterparts, the inverse wave operators, do not have bounded inverses, even though these operators themselves are bounded. This result holds for the natural energy given by the energy–momentum tensor of the wave equation using the timelike vector field of the Regge–Wheeler variable, which asymptotically becomes normal to the horizons. The behaviour of solutions at low spatial frequencies and their behaviour at high angular momenta are the only obstructions causing this breakdown of scattering. The breakdown follows from an analysis of a $$1+1$$ -dimensional wave equation with exponentially decaying potential which we treat for general potentials, and we show that the breakdown is generic.