Spacetime Metric Research Articles

Noether Symmetries and Conservation Laws in Static Cylindrically Symmetric Spacetimes via Rif Tree Approach

A new approach is adopted to completely classify the Lagrangian associated with the static cylindrically symmetric spacetime metric via Noether symmetries. The determining equations representing Noether symmetries are analyzed using a Maple algorithm that imposes different conditions on metric coefficients under which static cylindrically symmetric spacetimes admit Noether symmetries of different dimensions. These conditions are used to solve the determining equations, giving the explicit form of vector fields representing Noether symmetries. The obtained Noether symmetry generators are used in Noether’s theorem to find the expressions for corresponding conservation laws. The singularity of the obtained metrics is discussed by finding their Kretschmann scalar.

The no-hair theorem and black hole shadows

The successful observation of M87 supermassive black hole by the Black Hole Event Horizon Telescope(EHT) provides a very good opportunity to study the theory of gravity. In this work, we obtain the exact solution for the short hair black hole (BH) in the rotation situation, and calculate in detail how hairs affect the BH shadow. For the exact solution part, using the Newman-Janis algorithm, we generalize the spherically symmetric short-hair black hole metric to the rotation case (space-time lie element (2.25)). For the BH shadow part, we study two hairy BH models. In model 1, the properties of scalar hair are determined by the parameters α0 and L (We re-obtained the results in reference [48] for the convenience of discussion in this work). In model 2, the scalar hair of the BH is short hair. In this model, the shape of the BH shadow is determined by scalar charge Qm and k. The main results are as follows: (1) In the case of rotational short-hair BH, the value range of parameter k is k > 1 (2.25), the range of short-hair charge value Qm is greatly reduced due to the introduction of the BH spin a. When 0leqslant {Q}_mleqslant frac{2}{3}times {4}^{frac{1}{3}} , the rotational short-hair BH has two event horizons at this time. When {Q}_m>frac{2}{3}times {4}^{frac{1}{3}} , the rotational short-hair BH has three unequal event horizons, so the space-time structure of the BH is significantly different from that of Kerr BH. (2) For model 1, the effect of scalar hair on the BH shadows corresponds to that of ε > 0 in reference [38, 48], but the specific changes of the shadows in model 1 are different. This is because the BH hair in reference [38] is considered as a perturbation to the BH, while the space-time metric of model 1 is accurate and does not have perturbation property. For model 2, that is, the change of the BH shadow caused by short hairs, the main change trend is consistent with that of ε < 0 in reference [38]. Because of the special structure of the short-hair BH, the specific changes of BH shadows are different. (3) the variation of Rs and δs with L and α0 is not a monotone function in model 1, but in model 2, it is. These results show that scalar hairs (model 1) have different effects on Kerr BH shadows than short hairs (model 2), so it is possible to distinguish the types and properties of these hairs if they are detected by EHT observations. (4) as for the effects of the hairs on energy emissivity, the main results in model 1 [48], different energy emissivity curves have intersection phenomenon, while in model 2 (short-hair BH), there is no similar intersection phenomenon. In general, various BH hairs have different effects on the shadows, such as non-monotonic properties and intersection phenomena mentioned in this work. Using these characteristics, it is possible to test the no-hair theorem in future EHT observations, so as to have a deeper understanding of the quantum effect of BHs. In future work, we will use numerical simulations to study the effects of various hairs on BHs and their observed properties.

Barrow-type structure and thermodynamics of black holes

The thermodynamics of black holes is investigated in the case of fractal-like event horizon, initially introduced by Barrow. The nature of the horizon surface is controlled by a parameter 0≤Δ≤1 with 0 corresponding to continuous space and 1 to completely fractal one. It is found that the entropy, the Helmholtz free energy, the mass and the heat capacity of the black hole are altered in the presence of non-zero Δ. As the black hole produces Hawking radiation, the dependence of its mass on time becomes softer as Δ increases, resulting in longer evaporation times. This effect makes relatively small primordial black holes to survive long enough so as to be potential ingredients of cold dark matter. The concept of Barrow-type structure can be generalized to the entire space resulting in a modified space-time metric.

A shadows from the static black hole mimickers

In this work, we study shadows from the naked singularity spacetime. The most analytical solutions of black hole shadows only investigated the case that the geodesic equations for photons can separate variables. We review the spherical null naked singularity metric and this spherically symmetric naked singularity spacetime metric is the solution of Einstein equations with an anisotropic fluid source which has no photon sphere. We also review a static, axially-symmetric singular solution of the vacuum Einstein’s equations without an event horizon which is can be used to describe the exterior gravitational field of a mass distribution with quadrupole moment. Moreover, the corresponding spacetime is characterized by the presence of naked singularities. It is theoretically known that not only a black hole can cast shadow, but other compact objects such as naked singularities, gravastar or boson stars can also cast shadows. We present the analytical calculation of shadows for both naked singularities spacetime and compare with the shadow of Schwarzschild static black hole, we show that this can serve as a black hole mimicker. Keywords: compact object, naked singularity, shadow.

Black hole shadow in f(R) gravity with nonlinear electrodynamics* *Supported by the National Key R&D Program of China (2020YFC2201400), the Major Program of the National Natural Science Foundation of China (11690021), and the National Natural Science Foundation of China (11505066 and 11805166). We also gratefully acknowledge financial support from Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP),

By analyzing the propagation of discontinuity in nonlinear electrodynamics, we numerically investigate the related black hole shadows of recently derived rotating black hole solutions in gravity. In this context, the geodesic motion of the relevant perturbations is governed by an effective geometry, which is closely related to the underlying spacetime metric. We derive the effective geometry, and the latter is used to determine the trajectory of the propagation vector of an arbitrary finite discontinuity in the electrodynamic perturbations, namely, the photon. Subsequently, the image of the black hole is evaluated using the ray-tracing technique. Moreover, we discuss the physical relevance of metric parameters, such as the nonlinear coupling, spin, and charge, by studying their impact on the resultant black hole shadows.

The 2+1-Dimensional Special Relativity

In the new mathematical description of special relativity in terms of the relativistic velocity space, many physical aspects acquire new geometric meanings. Performing conformal deformations upon the 2-dimensional relativistic velocity space for the (2+1)-dimensional special relativity, we find that these conformal deformations correspond to the generalized Lorentz transformations, which are akin to the ordinary Lorentz transformation, but are morphed by a global rescaling of the polar angle and correspondingly characterized by a topological integral index. The generalized Lorentz transformations keep the two fundamental principles of special relativity intact, suggesting that the indexed generalization may be related to the Bondi–Metzner–Sachs (BMS) group of the asymptotic symmetries of the spacetime metric. Furthermore, we investigate the Doppler effect of light, the Planck photon rocket, and the Thomas precession, affirming that they all remain in the same forms of the standard special relativity under the generalized Lorentz transformation. Additionally, we obtain the general formula of the Thomas precession, which gives a clear geometric meaning from the perspective of the gauge field theory in the relativistic velocity space.

Quantum field simulator for dynamics in curved spacetime.

In most cosmological models, rapid expansion of space marks the first moments of the Universe and leads to the amplification of quantum fluctuations1. The description of subsequent dynamics and related questions in cosmology requires an understanding of the quantum fields of the standard model and dark matter in curved spacetime. Even the reduced problem of a scalar quantum field in an explicitly time-dependent spacetime metric is a theoretical challenge2-5, and thus a quantum field simulator can lead to insights. Here we demonstrate such a quantum field simulator in a two-dimensional Bose-Einstein condensate with a configurable trap6,7 and adjustable interaction strength to implement this model system. We explicitly show the realization of spacetimes with positive and negative spatial curvature by wave-packet propagation and observe particle-pair production in controlled power-law expansion of space, using Sakharov oscillations to extract amplitude and phase information of the produced state. We find quantitative agreement with analytical predictions for different curvatures in time and space. This benchmarks and thereby establishes a quantum field simulatorof a new class. In the future, straightforward upgrades offer the possibility to enter unexplored regimes that give further insight into relativistic quantum field dynamics.

Lie symmetry approach to the time-dependent Karmarkar condition

We obtain solutions of the time-dependent Einstein Field Equations which satisfy the Karmarkar condition via the method of Lie symmetries. Spherically symmetric spacetime metrics are used with metric functions set to impose conformal flatness, Weyl-free collapse and shear-free collapse. In particular, a solution was found which satisfies the heat-flux boundary condition of Santos, and a radiating stellar model was then obtained and investigated. Solutions obtained which do not allow for the application of the junction conditions at a boundary surface may lend themselves to cosmological models. This is a first attempt in generating solutions satisfying the Karmarkar condition via the method of Lie symmetries and our example of a radiating model highlights the viability of this method.

Spacetime Metrics and Ringdown Waveforms for Galactic Black Holes Surrounded by a Dark Matter Spike

Theoretical models suggest the existence of a dark matter spike surrounding the supermassive black holes at the core of galaxies. The spike density is thought to obey a power law that starts at a few times the black hole horizon radius and extends to a distance, R sp, of the order of a kiloparsec. We use the Tolman–Oppenheimer–Volkoff equations to construct the spacetime metric representing a black hole surrounded by such a dark matter spike. We consider the dark matter to be a perfect fluid, but make no other assumption about its nature. The assumed power-law density provides in principle three parameters with which to work: the power-law exponent γ sp, the external radius R sp, and the spike density at R sp. These in turn determine the total mass of the spike. We focus on Sagittarius A* and M87 for which some theoretical and observational bounds exist on the spike parameters. Using these bounds in conjunction with the metric obtained from the Tolman–Oppenheimer–Volkoff equations, we investigate the possibility of detecting the dark matter spikes surrounding these black holes via the gravitational waves emitted at the ringdown phase of black hole perturbations. Our results suggest that if the spike to black hole mass ratio is roughly constant, greater mass black holes require relatively smaller spike densities to yield potentially observable signals. We find that is unlikely for the spike in M87 to be detected via the ringdown waveform with currently available techniques unless its mass is roughly an order of magnitude larger than existing observational estimates. However, given that the signal increases with black hole mass, dark matter spikes might be observable for more massive galactic black holes in the not too distant future.

Noether Symmetries and Conservation Laws in Non-Static Plane Symmetric Spacetime

In this paper, we find all nonstatic plane symmetric spacetime metrics whose corresponding Lagrangians possess Noether symmetries. The set of determining equations is analyzed through a Maple algorithm that restricts the metric coefficients to satisfy certain conditions. These restrictions on metric coefficients, while using them to solve the determining equations, give rise to a number of plane symmetric metrics admitting 4-, 5-, 6-, 7-, 8-, 9-, 11-, and 17-dimensional Noether algebras. The Noether theorem is used to find a conserved quantity corresponding to each Noether symmetry. Some physical implications are discussed by finding bounds for different energy conditions for the obtained metrics.

Cosmic censorship and charged radiation in second order Lovelock gravity

The conditions for naked singularity formation are considered for a radiating metric of Boulware–Deser type within an electromagnetic field in second order Lovelock (or Einstein–Gauss–Bonnet) gravity. The spacetime metric remains real only up to certain maximum charge contribution. This differs from general relativity. Beyond a certain maximal charge, there exists no real and physical spacetime since the metric becomes complex. We establish that, under certain parameters and for specific values of the mass function and charge contribution, this branch singularity is indeed a naked singularity. This is in contrast to the neutral case where the spacetime metric is always real for a positive mass function, and further, a weak, initially naked singularity always occurs before it becomes covered by an event horizon for all future time. We highlight that both neutral and charged collapse under gravity in Einstein–Gauss–Bonnet gravity differ significantly to their general relativistic counterparts.

Homothetic Symmetries of Static Cylindrically Symmetric Spacetimes—A Rif Tree Approach

In this paper, we find all static, cylindrically symmetric spacetime metrics admitting homothetic symmetries. For this purpose, first we analyze the homothetic symmetry equations by an algorithm developed in Maple which gives all possible static, cylindrically symmetric metrics that may possess proper homothetic symmetry. After that, we have solved the homothetic symmetry equations for all these metrics to get the final form of homothetic symmetry vector fields. Comparing the obtained results with those of direct integration technique, it is observed that the Rif tree approach not only recovers the metrics already found by direct integration technique, but it also produces some new metrics.

Compact star coupled with dark energy in the background of Tolman–Kuchowicz spacetime

This paper provides a simplified description for simulating the coupling of dark energy with baryonic matter by considering a super-dense pulsar PSRJ1614-2230 as the model star. The starting equation of state for modeling the dark energy is motivated by the MIT Bag model for spherically confined hadrons and the observational evidences of its repulsive nature. Einstein field equations are solved in the stellar interior using the generalized framework of Tolman–Kuchowicz spacetime metric. The solutions are then analyzed for various physical parameters such as metric potential, pressure, density, energy conditions, etc. The physical analysis of multiple parameters indicates stable star formation. The proposed stellar model is free from all singularities and also meets the requisite stability criteria. The numerical results obtained for relativistic adiabatic index indicate that the model star is stiff and stable against radially induced adiabatic perturbations.

Dynamical symmetries of homogeneous minisuperspace models

We investigate the phase space symmetries and conserved charges of homogeneous gravitational minisuperspaces. These ($0+1$)-dimensional reductions of general relativity are defined by spacetime metrics in which the dynamical variables depend only on a time coordinate and are formulated as mechanical systems with a nontrivial field space metric (or supermetric) and effective potential. We show how to extract conserved charges for those minisuperspaces from the homothetic Killing vectors of the field space metric. In the case of two-dimensional field spaces, we exhibit a universal eight-dimensional symmetry algebra $\mathcal{A}=(\mathfrak{sl}(2,\mathbb{R})\ensuremath{\bigoplus}\mathbb{R})⨭{\mathfrak{h}}_{2}$, based on the two-dimensional Heisenberg algebra ${\mathfrak{h}}_{2}\ensuremath{\simeq}{\mathbb{R}}^{4}$. We apply this to the systematic study of the Bianchi models for homogeneous cosmology. This extends the previous results on the $\mathfrak{sl}(2,\mathbb{R})$ algebra for Friedmann-Lema\^{\i}tre-Robertson-Walker cosmology and the Poincar\'e symmetry for Kantowski-Sachs metrics describing the black hole interior. The presence of this rich symmetry structure already in minisuperspace models opens new doors toward quantization and the study of solution generating mechanisms.

Noether Symmetries of Lemaitre-Tolman-Bondi Metric via Rif Tree Approach

In this paper, we have explored Noether symmetries for the Lagrangian corresponding to the Lemaitre-Tolman-Bondi (LTB) spacetime metric via a Rif tree approach. Instead of the frequently used method of directly integrating the Noether symmetry equations, a MAPLE algorithm is used to convert these equations to the reduced involutive form (Rif). The interesting feature of this algorithm is that it provides all possible metrics admitting different dimensional Noether symmetries. These metrics are given in the form of branches of a tree, known as a Rif tree. These metrics are used to solve the determining equations and the explicit form of symmetry vector fields are found, giving 4, 5, 6, 7, 8, 9, 11, and 17-dimensional Noether algebras. To add some physical implications, Einstein’s field equations are used to find the stress-energy tensor for all the explicitly known metrics, and the parameters appearing in the metrics are used to find bounds for different energy conditions.

Light propagation in a plasma on an axially symmetric and stationary spacetime: Separability of the Hamilton–Jacobi equation and shadow

The properties of light rays around compact objects surrounded by a plasma are affected by both strong gravitational fields described by a general-relativistic spacetime and by a dispersive and refractive medium, characterized by the density distribution of the plasma. We study these effects employing the relativistic Hamiltonian formalism under the assumption of stationarity and axisymmetry. The necessary and sufficient conditions on the metric and on the plasma frequency are formulated such that the rays can be analytically determined from a fully separated Hamilton–Jacobi equation. We demonstrate how these results allow us to analytically calculate the photon region and the shadow if they exist. Several specific examples are discussed in detail: the “hairy” Kerr black holes, the Hartle–Thorne spacetime metrics, the Melvin universe, and the Teo rotating traversable wormhole. In all of these cases, a plasma medium is present as well.

Slowly rotating spacetimes with scalar hair

We consider asymptotically flat, rotating configurations of a selfgravitating real scalar field minimally coupled to Einstein gravity. For the sake of completeness and to introduce convenient notation, we formulate the EinsteinKlein-Gordon equations in the orthonormal tetrad uniquely associated with the metric of a nonstationary axisymmetric spacetime. First, in the stationary case, we study Bianchi identities to find appropriate variants for reducing the EinsteinKlein-Gordon system to a complete subsystem of independent equations. It turns out that this procedure can be performed in three ways. Second, assuming that the rotation is slow, the full system of these equations is linearized about a spherically symmetric scalar field configuration, which may be thought of as some exact solution. In doing so, we do not assume that the scalar field is small, and do not specify the type of the basic spherically symmetric configuration, which can be a black hole, a naked singularity, or a wormhole.

Generalized Kantowski–Sachs type spacetime metrics and their harmonicity

In this paper, we deal with harmonic metrics with respect to generalized Kantowski–Sachs type spacetime metrics. We also consider the Sasaki, horizontal and complete lifts of generalized Kantowski–Sachs type spacetime metrics to tangent bundle and study their harmonicity.

On generally covariant mathematical formulation of Feynman integral in Lorentz signature

It is widely accepted that the Feynman integral is one of the most promising methodologies for defining a generally covariant formulation of nonperturbative interacting quantum field theories (QFTs) without a fixed prearranged causal background. Recent literature suggests that if the spacetime metric is not fixed, e.g. because it is to be quantized along with the other fields, one may not be able to avoid considering the Feynman integral in the original Lorentz signature, without Wick rotation. Several mathematical phenomena are known, however, which are at some point showstoppers to a mathematically sound definition of Feynman integral in Lorentz signature. The Feynman integral formulation, however, is known to have a differential reformulation, called to be the master Dyson–Schwinger (MDS) equation for the field correlators. In this paper it is shown that a particular presentation of the MDS equation can be cast into a mathematically rigorously defined form: the involved function spaces and operators can be strictly defined and their properties can be established. Therefore, MDS equation can serve as a substitute for the Feynman integral, in a mathematically sound formulation of constructive QFT, in arbitrary signature, without a fixed background causal structure. It is also shown that even in such a generally covariant setting, there is a canonical way to define the Wilsonian regularization of the MDS equation. The main result of the paper is a necessary and sufficient condition for the regularized MDS solution space to be nonempty, for conformally invariant Lagrangians. This theorem also provides an iterative approximation algorithm for obtaining regularized MDS solutions, and is guaranteed to be convergent whenever the solution space is nonempty. The algorithm could eventually serve as a method for putting Lorentz signature QFTs onto lattice, in the original metric signature.

The spacetime geometry of fixed-area states in gravitational systems

The concept of fixed-area states has proven useful for recent studies of quantum gravity, especially in connection with gravitational holography. We explore the Lorentz-signature spacetime geometry intrinsic to such fixed-area states in this paper. This contrasts with previous treatments which focused instead on Euclidean-signature saddles for path integrals that prepare such states. We analyze general features of fixed-area state geometries and construct explicit examples. The spacetime metrics are real at real times and have no conical singularities. With enough symmetry the classical metrics are in fact smooth, though more generally their curvatures feature power-law divergences along null congruences launched orthogonally from the fixed-area surface. While we argue that such divergences are not problematic at the classical level, quantum fields in fixed-area states feature stronger divergences. At the quantum level we thus expect fixed-area states to be well-defined only when the fixed-area surface is appropriately smeared.