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We consider physical parameters of Levin and Perez-Giz's `periodic table of orbits' around the Schwarzschild black hole, where each periodic orbit is classified according to three integers $(z,w,v)$. In particular, we chart its distribution in terms of its angular momenta $L$ and energy $E$. In the $(L,E)$-parameter space, the set of all periodic orbits can be partitioned into domains according to their whirl number $w$, where the limit of infinite $w$ approaches the branch of unstable circular orbits. Within each domain of a given whirl number $w$, the infinite zoom limit $\lim_{z\rightarrow\infty}(z,w,v)$ converges to the common boundary with the adjacent domain of whirl number $w-1$. The distribution of the periodic orbit branches can also be inferred from perturbing stable circular orbits, using the fact that every stable circular orbit is the zero-eccentricity limit of some periodic orbit, or arbitrarily close to one., Comment: 28 pages, 11 figures

The prime motivation behind this paper is to prove that any torus link $\mathcal{L}$ can be realized as the union of the one-dimensional connected components of the set of critical values of the argument map restricted to a complex algebraic plane curve. Moreover, given an isolated complex algebraic plane curve quasi-homogeneous singularity, we give an explicit topological and geometric description of the link $\mathcal{L}$ corresponding to this singularity. In other words, we realize this link as the union of the one-dimensional connected components of the set critical values of the argument map restricted to the intersection of the curve with a four-dimensional ball of a sufficiently small radius, centered at the given singularity. This established the first relationship between (co)amoebas and knot theory., Comment: 19 pages, 18 figures. arXiv admin note: text overlap with arXiv:0906.2729

We show that the free energy $\mathcal{F}$ and temperature $T$ of black holes, considered as a thermodynamic system, can be viewed as an $A$-discriminant of an appropriately-defined polynomial. As such, mathematical results about $A$-discriminants may lead to implications about black hole thermodynamics. In particular, for static spacetimes with spherical, planar, or hyperbolic symmetry, the number of distinct thermodynamic phases depend on the number of distinct terms in the metric component $g_{tt}$. We prove that if $g_{tt}$ consists of $N_f$ distinct terms, then the $\mathcal{F}$-$T$ curve consists of $N_f-2$ cusps, which in turn leads to $N_f-1$ distinct thermodynamic phases. This result is applied to explicit examples of the Schwarzschild-AdS, Reissner--Nordstr\"{o}m, power-law Maxwell, and Euler--Heisenberg black holes., Comment: 27 pages, 6 figures

Lee-Yang (LY) zeros, points on the complex plane of physical parameters where the partition function goes to zero, have found diverse applications across multiple disciplines like statistical physics, protein folding, percolation, complex networks etc. However, experimental extraction of the complete set of LY zeros for general asymmetrical classical systems remains a crucial challenge to put those applications into practice. Here, we propose a qubit-based method to simulate an asymmetrical classical Ising system, enabling the exploration of LY zeros at arbitrary values of physical parameters like temperature, internal couplings etc. Without assuming system symmetry, the full set of LY zeros forms an algebraic variety in a higher-dimensional complex plane. To determine this variety, we pro ject it into sets representing magnitudes (amoeba ) and phases (coamoeba ) of LY zeros. Our approach uses a probe qubit to initialize the system and to extract LY zeros without assuming any control over the system qubits. This is particularly important as controlling system qubits can get intractable with the increasing complexity of the system. Initializing the system at an amoeba point, coamoeba points are sampled by measuring probe qubit dynamics. Iterative sampling yields the entire algebraic variety. Experimental demonstration of the protocol is achieved through a three-qubit NMR register. This work expands the horizon of quantum simulation to domains where identifying LY zeros in general classical systems is pivotal. Moreover, by extracting abstract mathematical objects like amoeba and coamoeba for a given polynomial, our study integrates pure mathematical concepts into the realm of quantum simulations.

While a wide variety of astrophysical and cosmological phenomena suggest the presence of Dark Matter, all evidence remains via its gravitational effect on the known matter. As such, it is conceivable that this evidence could be explained by a modification to gravitation and/or concepts of inertia. Various formulations of modified gravity exist, each giving rise to several non-canonical outcomes. This motivates us to propose an experiment searching for departures from (quantum) Newtonian predictions in a bipartite setting with gravitational accelerations $\lesssim 10^{-10}$ m/s$^2$, i.e., where the effective force needs to be stronger than Newtonian to account for the Dark Matter effects. Since quantum particles naturally source weak gravitation, their non-relativistic dynamics offers opportunities to test this small acceleration regime. We show that two nearby mesoscopic quantum masses accumulate significantly larger entanglement in modified gravity models, such as the Modified Newtonian Dynamics. Our calculations include Casimir-Polder forces as well as tidal effects next to the surface of the earth, and confirm that entanglement is observable within the limits imposed by environmental decoherence. We demonstrate how the temperature can be fine-tuned such that modified gravity is certified simply by witnessing the entanglement generated from uncorrelated thermal states, eliminating the need for precise noise characterization. Overall, the required parameters could be realized in a tabletop experiment., Comment: Accepted for publication as a Letter in Physical Review D

A general relativistic model of a parallel-plate electrostatic capacitor is presented. The spacetime is a solution to the Einstein--Maxwell equations and involves class of solution previously studied by Vesel\'{y} and \v{Z}ofka (V\v{Z}). In particular, the parts containing curvature singularities are cut out and the remaining regular section is glued to asymptotically-Minkowski spacetimes. In essence, this results in a curved electro-vacuum V\v{Z} spacetime sandwiched on both sides by exterior spacetimes with vanishing electromagnetic fields. Junction conditions require the presence of charged matter on the boundaries. We interpret this configuration as a parallel-plate capacitor with gravitational effects induced by the strong electric fields. The spherical capacitor is briefly considered., Comment: 17 pages, 3 figures. New section about spherical capacitor is added. A coordinate-independent notion of capacitance is introduced

The motion of charged particles in spacetimes containing a submanifold of constant positive or negative curvature is considered, with the electromagnetic tensor proportional to the volume two-form form of the submanifold. In the positive curvature case, this describes spherically symmetric spacetimes with a magnetic monopole, while in the negative curvature case, it is a hyperbolic spacetime with magnetic field uniform along hyperbolic surfaces. Constants of motion are found by considering Poisson brackets defined on a phase space with gauge-covariant momenta. In the spherically-symmetric case, we find a correspondence between the trajectories on the Poincar\'{e} cone with equatorial geodesics in a conical defect spacetime. In the hyperbolic case, the analogue of the Poincar\'{e} cone is defined as a surface in an auxiliary Minkowski spacetime. Explicit examples are solved for the Minkowski, $\mathrm{AdS}_4\times S^2$, and the hyperbolic AdS-Reissner--Nordstr\"{o}m spacetimes., Comment: 34 pages, 8 figures. Typos corrected, Section 4 expanded thanks to the suggestion of the anonymous referee

A configuration of a test magnetic field in Hayward spacetime is obtained by solving Maxwell's equation with the Hayward metric as the background. The magnetic field lines show a dipole loop-like configuration in the regular Hayward interior, and tends to an asymptotically uniform structure away from the cylindrical axis. The motion of charged particles is then studied in this spacetime. The parameters and stability of circular orbits on the equatorial plane are studied. Aspects of non-equatorial motion are also studied., Comment: 32 pages, 12 figures. Sec. 2 expanded, typos corrected

In this paper we study a Kaluza-Klein bubble-type spacetime with a magnetic field first obtained by Sarbach and Lehner in 2005. It is shown that this solution can be derived via a Harrison transformation of a vacuum Kaluza-Klein bubble, and its physical and geometrical quantities are determined. For certain ranges of the magnetic field parameter, this solution carries negative mass. Furthermore, for a given value of magnetic flux, there exist two branches of solutions, and the branch with lower mass is thermodynamically favoured. In light of these properties we interpret the spacetime as another analogue to the Anti-de Sitter-Melvin solution., Comment: 17 pages, 3 figures

In this paper we explore a solenoid configuration involving a magnetic universe solution embedded in an empty Anti-de Sitter (AdS) spacetime. This requires a non-trivial surface current at the interface between the two spacetimes which can be provided by a charged scalar field. When the interface is taken to the AdS boundary, we recover the full AdS--Melvin spacetime. The stability of the AdS--Melvin solution is also studied by computing the gravitational free energy from the Euclidean action., Comment: 16 pages, 1 figure

When geodesic equations are formulated in terms of an effective potential $U$, circular orbits are characterised by $U=\partial_a U=0$. In this paper we consider the case where $U$ is an algebraic function. Then the condition for circular orbits defines an $A$-discriminantal variety. A theorem by Rojas and Rusek, suitably interpreted in the context of effective potentials, gives a precise criteria for certain types of spacetimes to contain at most two branches of light rings (null circular orbits), where one is stable and the other one unstable. We identify a few classes of static, spherically-symmetric spacetimes for which these two branches occur and show that the spacetimes with non-degenerate horizons do not have stable light rings., Comment: 46 pages, 8 figures. Typos corrected and a new section added

Considering a strong field limit, we investigate the retrolensing phenomenon in the vicinity of a Janis-Newman-Winicour (JNW) naked singularity embedded in a scalar field. We assume that the light rays from a nearby source are reflected by the photon sphere of the naked singularity, acting as a lens, to create a pair of images. The analytic expressions of the lensing coefficients $\bar{a}$ and $\bar{b}$ are obtained, which are scalar dependent and discordant with \cite{Bozza:2002b}. Moreover, considering the powerful supermassive black hole candidates Sgr A* and M87*, we examined the influence of the scalar field on the apparent brightness and angular positions of the parity images. Our results are highlighted in correspondence with a non-scalar field gravity, specifically the Schwarzschild gravity. We have found that the brightness increases in the presence of the scalar field, while, the lensing coefficients $\bar{a}$, $\bar{b}$, the angular positions and the angular separation of the relativistic images experience a reverse effect., Comment: 16 pages, 7 figures

We study the motion of charged particles in a family of five-dimensional solutions describing either a black hole or topological star with a fifth compact dimension stabilised by a magnetic flux. The particle's trajectory is shown to move along the surface of a Poincar\'{e} cone. The radial motion shows a rich structure where the existence of various bound, plunging, or escaping trajectories depend on the constants of motion. Curves of energy and angular momentum corresponding to spherical orbits show a swallow-tail structure highly reminiscent to phase transitions of thermodynamics. When the momentum along the compact direction is varied, the is a critical point beyond which the swallow-tail kink disappears and becomes a smooth curve., Comment: References added. Typos corrected. 24 pages, 10 figures

In this paper we study the motion of photons or massless particles in the C-metric with cosmological constant. The Hamilton--Jacobi equations are known to be completely separable, giving a Carter-like quantity $Q$ which is a constant of motion. All possible trajectories are classified according to a two-dimensional parameter space representing the particle's angular momentum and energy scaled in units of $Q$. Exact solutions are given in the C-metric coordinates in terms of Jacobi elliptic functions. Using the exact solutions, we find examples of periodic orbits on the photon surface., Comment: 27 pages, 9 figures. Typos corrected and references added

The trajectory of a charged test particle in the Melvin magnetic universe is shown to take the form of hypocycloids in two different regimes, first of which is the class of perturbed circular orbits, and secondly in the weak-field approximation. In the latter case we find a simple relation between the charge of the particle and the number of cusps. These two regimes are within a continuously connected family of deformed hypocycloid-like orbits parametrised by the magnetic flux strength of the Melvin spacetime., Comment: 18 pages, 4 figures. Typos corrected. Introduction and conclusion improved thanks to the suggestions of the anonymous referee

In this paper, we study the motion of test particles around two exact charged black-hole solutions in Einstein-{\AE}ther theory. Specifically, we first consider the quasi-periodic oscillations (QPOs) and their resonances generated by the particle moving in the Einstein-{\AE}ther black hole and then turn to study the periodic orbits of the massive particles. For QPOs, we drop the usually adopted assumptions $\nu_U=\nu_\theta$, $\nu_L=\nu_r$, and $\nu_U/\nu_L=3/2$ with $\nu_U$ ($\nu_L$) and $\nu_r$ ($\nu_\theta$) being the upper (lower) frequency of QPOs and radial (vertical) epicyclic frequency of the orbiting particles, respectively. Instead, we put-forward a new working ansatz for which the Keplerian radius is much closer to that of the innermost stable circular orbit and explore in detail the effects of the {\ae}ther field on the frequencies of QPOs. We then realize good curves for the frequencies of QPOs, which fit to data of three microquasars very well by ignoring any effects of rotation and magnetic fields. The innermost stable circular orbits (isco) of timelike particles are also analyzed and we find the isco radius increases with increasing $c_{13}$ for the first type black hole while decreases with increasing $c_{14}$ for the second one. We also obtain several periodic orbits and find that they share similar taxonomy schemes as the periodic equatorial orbits in the Schwarzschild/Kerr metrics, in addition to exact solutions for certain choices of the Einstein-\AE ther parameters. The equations for null geodesics are also briefly considered, where we study circular photon orbits and bending angles for gravitational lensing., Comment: 21 pages, 17 captioned figures, accepted for publication in Phys.Rev.D

A solution to an Einstein--Maxwell--dilaton-type theory with $M$ Liouville potentials and $N$ gauge fields is presented, where $M$ and $N$ are arbitrary integers. This exact solution interpolates between the Lifshitz black hole and the topological dilaton black hole. The thermodynamic behaviour of the solution is found to be similar to that of the Lifshitz black hole, where a phase transition may occur for sufficiently small charge in the canonical ensemble, or sufficiently small potential in the grand canonical ensemble., Comment: 30 pages, 5 figures. Typos corrected and more details have been added

The spacetime satisfying the flat curve condition for galaxies is shown to be the zero mass limit of the dilaton blackhole with an exponential potential. We derive the geodesic equations and by studying rotational curves and light deflection, and find that the presence of the dilaton increases the gravitational effects overall but diminishes the contribution from the central mass when the distances are close to the black hole. Also, the dilaton reduces the size of innermost stable circular time-like orbits, while the radius of the photon sphere may be larger or smaller than its Schwrazschild counterpart, depending on the strength of the dilaton field. We also show that a generalisation of the flat-curve-condition spacetime considered elsewhere in the literature does not solve the Einstein equation., Comment: 16 pages, 6 figures

This paper presents a classifier ensemble for Facial Expression Recognition (FER) based on models derived from transfer learning. The main experimentation work is conducted for facial action unit detection using feature extraction and fine-tuning convolutional neural networks (CNNs). Several classifiers for extracted CNN codes such as Linear Discriminant Analysis (LDA), Support Vector Machines (SVMs) and Long Short-Term Memory (LSTM) are compared and evaluated. Multi-model ensembles are also used to further improve the performance. We have found that VGG-Face and ResNet are the relatively optimal pre-trained models for action unit recognition using feature extraction and the ensemble of VGG-Net variants and ResNet achieves the best result., Comment: 6 pages, Humanoids 2017 IEEE RAS International Conference workshop Cooperative Autonomous Robot Experience (Presentation)

A spacetime consisting of parallel electric/magnetic fields held together by its own gravity in the presence of a cosmological constant $\Lambda$ is derived as a limit of the de Sitter/anti-de Sitter C-metric. The limiting procedure is similar to the $\Lambda=0$ case where the Melvin universe is derived from the C-metric. Under an appropriate coordinate transformation, we show that this solution is equivalent to the solution obtained by Astorino. Some physical and geometrical properties of the solution are studied, as well as its geodesics. For $\Lambda<0$, the solution is asymptotically locally anti-de Sitter, and can be derived as a double-Wick rotation of a charged anti-de Sitter black hole with a planar horizon., Comment: 21 pages, 2 figures. References added and typos corrected

We derive the full set of field equations based on Hossenfelder's recent covariant formulation of the emergent gravity model, along with perturbative and exact solutions. The exact solution describes a static, spherically-symmetric spacetime with a non-trivial vector field which plays the role of dark matter under the emergent gravity paradigm. Equations of motion of relativistic test masses are derived and are shown to reduce to Modified Newtonian Dynamics with additional relativistic corrections. It is also shown that the presence of the vector field gives an additional positive contribution to the bending angle in the deflection of light., Comment: New section added. Accepted for publication in Phys. Rev. D

The motion of time-like test particles in the Fisher/Janis-Newman-Winicour (F/JNW) spacetime is studied with the Hamiltonian formulation of the geodesic equations. The spacetime is characterised by its mass parameter $r_g$ and scalar field parameter $\nu$. The innermost bound and stable circular orbits are calculated and the effective potential is analysed. Consistent with numerical results in earlier literature, for $\nu<1/2$, particles with non-zero angular momentum encounter an infinite potential barrier, preventing them from reaching the naked singularity at $r=r_g$. Periodic orbits in the spacetime are also obtained. Compared to the periodic orbits around the Schwarzschild black hole, it is found that typically lower energies are required for the same orbits in the F/JNW spacetime., Comment: 14 pages, 7 figures

We present an exact solution in Einstein-Maxwell-dilaton gravity describing a spacetime with an anisotropic Kasner-type singularity and Lifshitz asymptotics. This configuration can also be supported by a phantom scalar while still satisfying the Null Energy Condition. For certain parameters of this solution, null geodesics can have an infinitely deep effective potential, thus trapping photons in a finite region along the radial direction. Some examples of periodic null geodesics are obtained. A particularly interesting special case of this solution is a regular, soliton-type metric that retains its Lifshitz scaling in the time coordinate., Comment: 26 pages, 6 figures. Typos corrected, new paragraph added to Sec. 2

The bounce solutions of self-interacting scalar fields coupled to gravity are studied using a semi-classical approach. We found that bounce solutions have a maximum required barrier curvature, in addition to the known minimum required barrier curvature. In particular, as the maximum barrier curvature is approached, the scale factor of the well-known Colemen-De Luccia (CDL) bounce solutions become divergent. Unlike the CDL or its more general oscillating bounce counterparts, this cannot be considered as a subset of the Hawking-Turok solution., Comment: 15 pages, 5 figures. Added references and typos corrected. Accepted for publication in Classical and Quantum Gravity

The field equations for Einstein-Maxwell-dilaton gravity in $D$ dimensions are reduced to an effective one-dimensional system under the influence of exponential potentials. Various cases where exact solutions can be found are explored. With this procedure, we present interesting solutions such as a one-parameter generalisation of the dilaton-Melvin spacetime and a three-parameter solution that interpolates between the Reissner-Nordstr\"{o}m and Bertotti-Robinson solution. This procedure also allows simple, alternative derivations of known solutions such as the Lifshitz spacetime and the planar Anti-de Sitter naked singularity. In the latter case, the metric is cast in a simpler form which reveals the presence of an additional curvature singularity., Comment: Typos corrected, 20 pages, no figures