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The Schwarzschild-Melvin spacetime is an exact solution of the Einstein electrovacuum equations describing a black hole immersed in a magnetic field which is asymptotically aligned with the z-axis. It plays an important role in our understanding of the interplay between geometry and matter, and is often used as a proxy for astrophysical environments. Here, we construct the scalar counterpart to the Schwarzschild-Melvin spacetime: a non-asymptotically flat black hole geometry with an everywhere regular scalar field whose gradient is asymptotically aligned with the z-axis., Comment: 2 pages

We study electromagnetic radiation reaction on a charged particle around a weakly magnetized Kerr black hole. We solve numerically the Teukolsky equation to find energy fluxes in electromagnetic radiation at the horizon and at spatial infinity. We also employ analytical methods in the low-frequency limit, finding excellent agreement with the numerical results. For a wide range of parameters, energy fluxes on the horizon are negative for all orbital radii: the modes are amplified through superradiance. More interestingly, the flux on the horizon is larger (in absolute value) than the flux at infinity for orbits with orbital radius $r_0 \gtrsim 9M$ -- floating orbits are a generic outcome of having magnetic fields around black holes. Particles on these orbits extract energy from the BH through their radiation field; the net effect is an increase in the particles' kinetic energy, so that they outspiral to infinity. Although in realistic accretion disks viscous forces are expected to dominate, floating orbits remain an interesting feature of the existence of ergoregions in black hole spacetimes., Comment: 9 pages, 5 figures. Matches published version in PRD

We derive the equations of motion for relativistic elastic membranes, that is, two-dimensional elastic bodies whose internal energy depends only on their stretching, starting from a variational principle. We show how to obtain conserved quantities for the membrane's motion in the presence of spacetime symmetries, determine the membrane's longitudinal and transverse speeds of sound in isotropic states, and compute the coefficients of linear elasticity with respect to the relaxed configuration. We then use this formalism to discuss two physically interesting systems: a rigidly rotating elastic disk, widely discussed in the context of Ehrenfest's paradox, and a Dyson sphere, that is, a spherical membrane in equilibrium in Schwarzschild's spacetime, with the isotropic tangential pressure balancing the gravitational attraction. Surprisingly, although spherically symmetric perturbations of this system are linearly stable, the axi-symmetric dipolar mode is already unstable. This may be taken as a cautionary tale against misconstruing radial stability as true stability., Comment: 18 pages, 1 figure; v2: typos corrected, more details and references added, now 19 pages, matches final published version. arXiv admin note: text overlap with arXiv:1712.05416

Linearised relativistic elasticity equations of motion are considered for a rod and a spinning ring encountering a gravitational wave. In the case of the rod, the equations reduce to a wave equation with appropriate boundary conditions. Using Fourier transforms, the resonant frequencies are found and an explicit distributional solution is given, both for a plus- and a cross-polarised gravitational wave. In the case of the spinning ring, the equations are coupled wave equations with periodic boundary conditions. Using a Fourier series expansion, the system of wave equations is recast as a family of ordinary differential equations for the Fourier coefficients, which are then solved via Fourier transforms. The resonant frequencies are found, including simple approximate expressions for slowly rotating rings, and an explicit distributional solution is obtained in the case of the non-spinning ring. Interestingly, it is possible to tune the resonant frequencies by adjusting the angular velocity of the spinning ring., Comment: 29 pages, 1 figure

We study the motion of an elastic rigid rod which is radially free-falling towards a Schwarzschild black hole. This is accomplished by reducing the corresponding free-boundary PDE problem to a sequence of ODEs, which we integrate numerically. Starting with a rod at rest, we show that it is possible to choose its initial compression profile so that its midpoint falls substantially faster, or slower, than a free-falling particle with the same initial conditions. This seems to be a purely kinematic effect, since on average there is no net transfer of elastic to mechanical energy., Comment: 10 pages, 5 figures; v2: more details added, now 12 pages and 7 figures, matches final published version

We study radiation from charged particles in circular motion around a Schwarzschild black hole immersed in an asymptotically uniform magnetic field. In curved space, the radiation reaction force is described by the DeWitt-Brehme equation, which includes a complicated, non-local tail term. We show that, contrary to some claims in the literature, this term cannot, in general, be neglected. We account for self-force effects directly by calculating the electromagnetic energy flux at infinity and on the horizon. The radiative field is obtained using black hole perturbation theory. We solve the relevant equations analytically, in the low-frequency and slow-motion approximation, as well as numerically in the general case. Our results show that great care must be taken when neglecting the tail term, which is often fundamental to capture the dynamics of the particle: in fact, it only seems to be negligible when the magnetic force greatly dominates the gravitational force, so that the motion is well described by the Abraham--Lorentz--Dirac equation. We also report a curious "horizon dominance effect" that occurs for a radiating particle in a circular orbit around a black hole (emitting either scalar, electromagnetic or gravitational waves): for fixed orbital radius, the fraction of energy that is absorbed by the black hole can be made arbitrarily large by decreasing the particle velocity., Comment: 18 pages, 6 figures. Matches published version in PRD

It has been claimed in I. Arraut, 2023 EPL 144 29003 [arXiv:2311.05652] that General Relativity can generate effects equivalent to dark matter, based on a "new modified symmetry that emerges at galactic scale". We show that (i) the spacetime considered is unphysical, violating energy conditions everywhere; (ii) actually no new symmetry emerges, which results from incorrect definitions of angular momentum, velocity, and arclength in the proposed geometry; (iii) no Tully-Fisher law, or flat rotation curves are obtained either, they simply result from a series of errors in the equations., Comment: 2 pages, no figures. Two minor typos in the text corrected, matches the final published version

It has been suggested in recent literature that nonlinear and/or gravitomagnetic general relativistic effects can play a leading role in galactic dynamics, partially or totally replacing dark matter. Using the 1+3 "quasi-Maxwell" formalism, we show, on general grounds, such hypothesis to be impossible., Comment: 14 pages, 8 figures. Improved version with new Figs 4 and 8 (lensing images obtained with the GYOTO ray-tracing code), updated Fig. 3, a sharper version of the main proof, improved structure, improved and extended explanations, references added. Matches the final published version

The purpose of this review it to present a renewed perspective of the problem of self-gravitating elastic bodies under spherical symmetry. It is also a companion to the papers [Phys. Rev. D105, 044025 (2022)], [Phys. Rev. D106, L041502 (2022)], and [arXiv:2306.16584 [gr-qc]], where we introduced a new definition of spherically symmetric elastic bodies in general relativity, and applied it to investigate the existence and physical viability, including radial stability, of static self-gravitating elastic balls. We focus on elastic materials that generalize fluids with polytropic, linear, and affine equations of state, and discuss the symmetries of the energy density function, including homogeneity and the resulting scale invariance of the TOV equations. By introducing invariant characterizations of physical admissible initial data, we numerically construct mass-radius-compactness diagrams, and conjecture about the maximum compactness of stable physically admissible elastic balls., Comment: v2: Published version. 132 pages, 18 figures; Invited review for the Special Issue of Class. Quantum Gravity on 'Focus on the Mathematics of Gravitation in the Non-Vacuum Regime'

We present a model of relativistic elastic stars featuring scale invariance. This implies a linear mass-radius relation and the absence of a maximum mass. The most compact spherically symmetric configuration that is radially stable and satisfies all energy and causality conditions has a slightly smaller radius than the Schwarzschild light ring radius. To the best of our knowledge, this is the first material compact object with such remarkable properties in General Relativity, which makes it a unique candidate for a black-hole mimicker., Comment: v2: Published version. 8 pages, 3 figures. Minor changes to the discussion. Inclusion of Fig.3 and appendix

Quantum gravity phenomenology suggests the interesting possibility that black holes are not eternal. Collapse could be halted by some unknown mechanism, or Hawking radiation might leave behind a regular spacetime. Here we investigate a simple bouncing geometry, with (outer and inner) apparent horizons but no event horizon. We show that the inner horizon blueshifts radiation, which can lead to a gigantic amplification of energy observable from far away regions. Thus, if such phenomena exists in our universe, they can power high-energy bursts at late stages in their lives, when the horizons disappear and spacetime bounces back to a flat geometry., Comment: 6 pages, 3 figures. v2: references added

The physical interpretation of the exact solutions of the Einstein field equations is, in general, a challenging task, part of the difficulties lying in the significance of the coordinate system. We discuss the extension of the International Astronomical Union (IAU) reference system to the exact theory. It is seen that such an extension, retaining some of its crucial properties, can be achieved in a special class of spacetimes, admitting nonshearing congruences of observers which, at infinity, have zero vorticity and acceleration. As applications, we consider the Friedmann-Lema\^itre-Robertson-Walker (FLRW), Kerr and Newman-Unti-Tamburino (NUT) spacetimes, the van Stockum rotating dust cylinder, spinning cosmic strings and, finally, we debunk the so-called Balasin-Grumiller (BG) model, and the claims that the galaxies' rotation curves can be explained through gravitomagnetic effects without the need for dark matter. The BG spacetime is shown to be completely inappropriate as a galactic model: its dust is actually static with respect to the asymptotic inertial frame, its gravitomagnetic effects arise from unphysical singularities along the axis (a pair of NUT rods, combined with a spinning cosmic string), and the rotation curves obtained are merely down to an invalid choice of reference frame -- the congruence of zero angular momentum observers, which are being dragged by the singularities., Comment: 30 pages, 10 figures. Slightly improved Section III.D, typos corrected, references added. Supplemental material is provided in the ancillary Mathematica files "NUTmetrics.nb" and "BGmetric.nb". Matches the final version published in PRD as an Editor's Suggestion

We study electromagnetic radiation reaction in curved space and the dynamics of radiating charged particles. The equation of motion for such particles is the DeWitt-Brehme equation, and it contains a particularly complicated, non-local, tail term. It has been claimed that the tail term can be neglected in certain magnetized black hole spacetimes, and that radiation reaction may then lead to energy extraction ("orbital widening") in the absence of an ergoregion. We show that such claims are incorrect, at least in the Newtonian limit: the tail term can never be neglected consistently in the relevant scenarios, and when it is included the reported energy extraction no longer occurs. Thus, previous results are called into question by our work., Comment: 7 pages, 1 figure. v2: minor changes, references added, matches final published version in PRD

Stationary compact astrophysical objects such as black holes and neutron stars behave as classical systems from the gravitational point of view. Their (observable) curvature is everywhere "small". Here we investigate whether mergers of such objects, or other strongly dynamical spacetimes such as collapsing configurations, may probe the strong-curvature regime of General Relativity. Our results indicate that dynamical black hole spacetimes always result in a modest increase $\sim 3$ in the Kretschmann scalar, relative to the stationary state. Our results show that the Kretschmann scalar can dynamically increase by orders of magnitude, during the gravitational collapse of scalar fields, and that the (normalized) peak curvature does not correspond to that of the critical solution. Nevertheless, without fine tuning of initial data, this increase lies far below that needed to render quantum-gravity corrections important., Comment: 8 pages, 5 figures

A foundational theorem due to Buchdahl states that, within General Relativity (GR), the maximum compactness $\mathcal{C}\equiv GM/(Rc^2)$ of a static, spherically symmetric, perfect fluid object of mass $M$ and radius $R$ is $\mathcal{C}=4/9$. As a corollary, there exists a compactness gap between perfect fluid stars and black holes (where $\mathcal{C}=1/2$). Here we generalize Buchdahl's result by introducing the most general equation of state for elastic matter with constant longitudinal wave speeds and apply it to compute the maximum compactness of regular, self-gravitating objects in GR. We show that: (i) the maximum compactness grows monotonically with the longitudinal wave speed; (ii) elastic matter can exceed Buchdahl's bound and reach the black hole compactness $\mathcal{C}=1/2$ continuously; (iii) however, imposing subluminal wave propagation lowers the maximum compactness bound to $\mathcal{C}\approx0.462$, which we conjecture to be the maximum compactness of \emph{any} static elastic object satisfying causality; (iv) imposing also radial stability further decreases the maximum compactness to $\mathcal{C}\approx 0.389$. Therefore, although anisotropies are often invoked as a mechanism for supporting horizonless ultracompact objects, we argue that the black hole compactness cannot be reached with physically reasonable matter within GR and that true black hole mimickers require either exotic matter or beyond-GR effects., Comment: v2: 4 pages, 4 figures; Version submitted to PRD: Major revision extending the class of elastic materials to describe rigid materials beyond spherical-symmetry. Bounds on the compactness modified accordingly but discussion qualitatively similar to the previous version

In the main article [CQG 38 (2021) 055003], a new "canonical" form for the Lewis metrics of the Weyl class has been obtained, depending only on three parameters -- Komar mass and angular momentum per unit length, plus the angle deficit -- corresponding to a coordinate system fixed to the "distant stars" and an everywhere timelike Killing vector field. Such form evinces the local but non-global static character of the spacetime, and striking parallelisms with the electromagnetic analogue. We discuss here its generality, main physical features and important limits (the Levi-Civita static cylinder, and spinning cosmic strings). We contrast it on geometric and physical grounds with the Kerr spacetime -- as an example of a metric which is locally non-static., Comment: 15 pages, 2 figures. To appear in the Proceedings of the sixteenth Marcel Grossmann Meeting (MG16), July 5-10 2021; based on the invited talk "Gravitomagnetism in the Lewis cylindrical metrics", given in the parallel session PT5 -- "Dragging is never draggy: MAss and CHarge flows in GR"

We obtain explicit formulas for the solution of the wave equation in certain Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetimes. Our method, pioneered by Klainerman and Sarnak, consists in finding differential operators that map solutions of the wave equation in these FLRW spacetimes to solutions of the conformally invariant wave equation in simpler, ultra-static spacetimes, for which spherical mean formulas are available. In addition to recovering the formulas for the dust-filled flat and hyperbolic FLRW spacetimes originally derived by Klainerman and Sarnak, and generalizing them to the spherical case, we obtain new formulas for the radiation-filled FLRW spacetimes and also for the de Sitter, anti-de Sitter and Milne universes. We use these formulas to study the solutions with respect to the Huygens principle and the decay rates, and to formulate conjectures about the general decay rates in flat and hyperbolic FLRW spacetimes. The positive resolution of the conjecture in the flat case is seen to follow from known results in the literature., Comment: 34 pages, 2 figures; v2: conjectures refined, improved introduction, proof of Theorem 6.2 corrected

It has been claimed in [arXiv:2110.09522] that the expression for the Sagnac coordinate time delay given in [arXiv:2109.14641] "differs from the standard interpretation described in the book by Landau-Lifshitz (LL)". We note that: 1) the Sagnac effect is not even discussed in LL; 2) the expression in [arXiv:2109.14641] is standard, given in countless papers and even textbooks; 3) the expression by LL quoted by the author consists of the (infinitesimal) two-way trip travel time for a light signal, which the author confuses with the Sagnac time delay (when they are actually very different things); 4) such confusion would negate the existence, both in special and general relativity, of the well-known and experimentally tested Sagnac effect; 5) the claims that it sheds doubt in any of the assertions made in [arXiv:2109.14641] are completely unfounded., Comment: 3+2 pages, 1 figure

Originally introduced in connection with general relativistic Coriolis forces, the term $\textit{frame-dragging}$ is associated today with a plethora of effects related to the off-diagonal element of the metric tensor. It is also frequently the subject of misconceptions leading to incorrect predictions, even of nonexistent effects. We show that there are three different levels of frame-dragging corresponding to three distinct gravitomagnetic objects: gravitomagnetic potential 1-form, field, and tidal tensor, whose effects are independent, and sometimes opposing. It is seen that, from the two analogies commonly employed, the analogy with magnetism holds strong where it applies, whereas the fluid-dragging analogy (albeit of some use, qualitatively, in the first level) is, in general, misleading. Common misconceptions (such as viscous-type "body-dragging") are debunked. Applications considered include rotating cylinders (Lewis-Weyl metrics), Kerr, Kerr-Newman and Kerr-dS spacetimes, black holes surrounded by disks/rings, and binary systems., Comment: 20+8 pages, 4 figures. Invited feature paper for Universe special issue on "Frame-Dragging and Gravitomagnetism"; matches the final published version

We introduce a rigorous and general framework to study systematically self-gravitating elastic materials within general relativity, and apply it to investigate the existence and viability, including radial stability, of spherically symmetric elastic stars. We present the mass-radius ($M-R$) diagram for various families of models, showing that elasticity contributes to increase the maximum mass and the compactness up to $\approx 22\%$, thus supporting compact stars with mass well above two solar masses. Some of these elastic stars can reach compactness as high as $GM/(c^2R)\approx 0.35$ while remaining stable under radial perturbations and satisfying all energy conditions and subluminal wave propagation, thus being physically realizable models of stars with a light ring. We provide numerical evidence that radial instability occurs for central densities larger than that corresponding to the maximum mass, as in the perfect-fluid case. Elasticity may be a key ingredient to build consistent models of exotic ultracompact objects and black-hole mimickers, and can also be relevant for a more accurate modelling of the interior of neutron stars., Comment: 5 pages, 3 figures; v3: Inclusion of discussion regarding missing sound speed. Updated plot of Fig. 2 with previously missing velocity. Conclusions and results unchanged

We define a notion of extrinsic black hole in pure Lovelock gravity of degree $k$ which captures the essential features of the so-called Lovelock-Schwarzschild solutions, viewed as rotationally invariant hypersurfaces with null $2k$-mean curvature in Euclidean space $\mathbb R^{n+1}$, $2\leq 2k\leq n-1$. We then combine a regularity argument with a rigidity result by Ara\'ujo-Leite to prove, under a natural ellipticity condition, a global uniqueness theorem for this class of black holes. As a consequence we obtain, in the context of the corresponding Penrose inequality for graphs established by Ge-Wang-Wu, a local rigidity result for the Lovelock-Schwarzschild solutions., Comment: 15 pages; no figures; this posting supersedes arXiv:1205.1132; published version

We clarify a number of issues that arise when extending the analysis of Strong Cosmic Censorship (SCC) to perturbations of highly charged Reissner-Nordstr\"{o}m de Sitter (RNdS) spacetimes. The linear stability of the Cauchy horizon can be determined from the spectral gap of quasinormal modes, thus giving a clear idea of the ranges of parameters that are likely to lead to SCC violations for infinitesimally small perturbations. However, the situation becomes much more subtle once nonlinear backreaction is taken into account. These subtleties have created a considerable amount of confusion in the literature regarding the conclusions one is able to derive about SCC from the available numerical simulations. Here we present new numerical results concerning charged self-gravitating scalar fields in spherical symmetry, correct some previous claims concerning the neutral case, and argue that the existing numerical codes are insufficient to draw conclusions about the potential failure of SCC for near extremal RNdS black hole spacetimes., Comment: 4 pages, 2 figures. Updates title to match published version

In electromagnetism, the concept of Poynting vector as measured by an observer is well known. A mathematical analogue in general relativity is the super-Poynting vector of the Weyl tensor. Observers for which the (super-)Poynting vector vanishes are called principal. When, at a given point, the electromagnetic field is non-null, or the gravitational field is of Weyl-Petrov type I or D, principal observers instantaneously passing through that point always exist. We survey characterizations of such observers and study their relation to arbitrary observers. In the non-null electromagnetic case it is known that, given any observer, there is a principal observer which moves relative to the first in the direction of his Poynting vector. Replacing Poynting by super-Poynting yields a possible gravitational analogue; we show that this analogy indeed holds for any observer when the Petrov type is D, but only for a one-dimensional variety of observers when the Petrov type is I. We provide algorithms to obtain the principal observers directly from the electric and magnetic fields (in the electromagnetic case) or electric and magnetic parts of the Weyl tensor (in the gravitational case) relative to an arbitrary observer. It is found that in Petrov type D doubly aligned non-null Einstein-Maxwell fields (which include all classical charged black hole solutions) the Poynting and super-Poynting vectors are aligned, at each point and for each observer, and the principal observers coincide. Our results are illustrated in simple examples., Comment: 42 pages, 6 figures. Minor improvements, typos corrected, references added. Matches the final published version

We establish a one-to-one correspondence between static spacetimes and Riemannian manifolds that maps causal geodesics to geodesics, as suggested by L. C. Epstein. We then explore constant curvature spacetimes - such as the de Sitter and the anti-de Sitter universes - and find that they map to constant curvature Riemannian manifolds, namely the Euclidean space, the sphere and the hyperbolic space. By imposing the conditions required to map to the sphere, we obtain the spherically symmetric metrics for which there is radial oscillatory motion with a period independent of the amplitude. We then consider the case of a perfect fluid and an Einstein cluster and determine the pressure and density profiles required to find this type of motion. Finally, we give examples of surfaces corresponding to certain types of motion for metrics that do not exhibit constant curvature, such as the Schwarzschild, Schwarzschild de Sitter and Schwarzschild anti-de Sitter solutions, and even for a simplified model of a wormhole., Comment: 18 pages, 9 figures; v2: minor changes, references added, matches final published version

We show explicitly, for different types of extremal black holes, that test fields satisfying the null energy condition at the event horizon cannot violate the weak cosmic censorship conjecture. This is done by checking, in each case, that the hypotheses for the general theorem in [7] are satisfied., Comment: 16 pages, no figures; v2: minor changes, references added, matches final published version