arXiv subjects

Filipe C. Mena

Publications and source records attributed to Filipe C. Mena.

At least 19 recordsLinked to original sources

arXiv2026

A first-order formulation of f(R) gravity in spherical symmetry

We develop an augmented characteristic, first-order formulation of the field equations in f(R) gravity governing the global evolution of a (possibly) massive scalar field phi under spherical symmetry. This formulation is designed to isolate the genuine dynamical degrees of freedom while preserving the geometric structure of the theory. By treating the spacetime scalar curvature as an independent unknown, we obtain a closed first-order nonlocal system for the pair (phi,R). This augmentation eliminates the higher-derivative character of the original equations at the level of the principal part. Our formulation allows us to pose the characteristic initial value problem and to establish several structural properties of solutions. More precisely, we work in generalized Bondi-Sachs coordinates and prescribe initial data on an asymptotically flat, future light cone with vertex at the center of symmetry, and we identify the minimal regularity conditions required at the center. These regularity conditions are shown to be precisely those ensuring equivalence between the reduced system and the full f(R) equations. Extending Christodoulou's method for the Einstein-scalar-field system, we recast the f(R) field equations as an integro-differential system of two coupled, first-order, nonlocal, nonlinear hyperbolic equations, whose principal unknowns are the scalar field and the spacetime scalar curvature.

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arXiv2026

Dynamics of quadratic f(R) cosmology with a perfect fluid: Jordan and Einstein frames

We investigate the global dynamics of the field equations of (pure) quadratic theories of gravity which generalise Einstein's theory in spatially flat homogeneous and isotropic cosmological models with a perfect fluid. We introduce global and regular 3-dimensional dynamical systems' formulations, on both the Jordan frame and the conformally related Einstein frame. The analysis in the Jordan frame explores the monotonicity properties of the interior flow which, together with the characterisation of the orbit structure on the 2-dimensional invariant boundaries and the desingularisation of non-hyperbolic fixed points, provides a global description of the flow and its limit sets. In the Einstein frame, the analysis uses the skew-product structure of the Einstein state space and the characterisation of the orbit structure on the 2-dimensional invariant boundaries. Furthermore, by obtaining asymptotic expansions we identify the solutions that are global conformally mapped from the Jordan frame to the Einstein frame and those that are not.

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arXiv2025

Energy bounds and ergoregion instability in Einstein-Maxwell-Scalar field models

We investigate energy bounds and the stability of stationary asymptotically flat spacetimes with an ergoregion and no future horizon in the context of Einstein-Maxwell-Scalar field models which naturally arise in Kaluza-Klein and String theories. In order to do that we consider scalar field perturbations non-minimally coupled to a background electromagnetic field. We show that there is compactly supported initial data such that the initial energy flux across the initial Cauchy hypersurface is negative and that the energy flux decreases with time. Then, considering arguments due to J. Friedman and G. Moschidis, this indicates that the energy flux diverges for those solutions and such spacetimes with an ergoregion are unstable.

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arXiv2025

Local well-posedness of the initial value problem in Einstein-Cartan theory

We study the initial value problem in Einstein-Cartan theory which includes torsion and, therefore, a non-symmetric connection on the spacetime manifold. Generalizing the path of a classical theorem by Choquet-Bruhat and York for the Einstein equations, we use a $n+1$ splitting of the manifold and compute the evolution and constraint equations for the Einstein-Cartan system. In the process, we derive the Gauss-Codazzi-Ricci equations including torsion. We prove that the constraint equations are preserved during evolution. Imposing a generalized harmonic gauge, it is shown that the evolution equations can be cast as a quasilinear system in a Cauchy regular form with a characteristic determinant having a non-trivial multiplicity of characteristics. Using the Leray-Ohya theory for non-strictly hyperbolic systems we then establish the local geometric well-posedness of the Cauchy problem, for sufficiently regular initial data. For vanishing torsion we recover the classical results for the Einstein equations.

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arXiv2025

Asymptotics of spin-0 fields and conserved charges on n-dimensional Minkowski spaces

We use conformal geometry methods and the construction of Friedrich's cylinder at spatial infinity to study the propagation of spin-$0$ fields (solutions to the wave equation) on $n$-dimensional Minkowski spacetimes in a neighbourhood of spatial and null infinity. We obtain formal solutions written in terms of series expansions close to spatial and null infinity and use them to compute non-trivial asymptotic spin-$0$ charges. It is shown that if one considers the most general initial data within the class considered in this paper, the expansion is poly-homogeneous and hence of restricted regularity at null infinity. Furthermore, we derive the conditions on the initial data needed to obtain regular solutions and well-defined limits for the asymptotic charges at the critical sets where null infinity and spatial infinity meet. In even dimensions, we find that there are infinitely many well-defined asymptotic charges at the critical sets, while for odd higher dimensions there exists no non-trivial asymptotic charges that remain regular at the critical sets.

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arXiv2023

Asymptotic structure and stability of spatially homogeneous space-times with a positive cosmological constant

We investigate the future asymptotics of spatially homogeneous space-times with a positive cosmological constant by using and further developing geometric conformal methods in General Relativity. For a large class of source fields, including fluids with anisotropic stress, we prove that the space-times are future asymptotically simple and geometrically conformally regular. We use that result in order to show the global conformal regularity of the Einstein-Maxwell system as well as the Einstein-radiation, Einstein-dust, massless Einstein-Vlasov and particular Einstein-scalar field systems for Bianchi space-times. Taking into account previous results, this implies the future non-linear stability of some of those space-times in the sense that, for small perturbations, the space-times approach locally the de Sitter solution asymptotically in time. This extends some cosmic no-hair theorems to almost spatially homogeneous space-times. However, we find that the conformal Einstein field equations preserve the Bianchi type even at conformal infinity, so the resulting asymptotic space-times have conformal hair.

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arXiv2022

Global evolution in spherical symmetry for self-gravitating massive fields

We are interested in the global dynamics of a massive scalar field evolving under its own gravitational field and, in this paper, we study spherically symmetric solutions to Einstein's field equations coupled with a Klein-Gordon equation with quadratic potential. For the initial value problem we establish a global existence theory when initial data are prescribed on a future light cone with vertex at the center of symmetry. A suitably generalized solution in Bondi coordinates is sought which has low regularity and possibly large but finite Bondi mass. A similar result was established first by Christodoulou for massless fields. In order to deal with massive fields, we must overcome several challenges and significantly modify Christodoulou's original method. First of all, we formulate the Einstein-Klein-Gordon system in spherical symmetry as a non-local and nonlinear hyperbolic equation and, by carefully investigating the global dynamical behavior of the massive field, we establish various estimates concerning the Einstein operator, the Hawking mass, and the Bondi mass, including positivity and monotonicity properties. Importantly, in addition to a regularization at the center of symmetry we find it necessary to also introduce a regularization at null infinity. We also establish new energy and decay estimates for, both, regularized and generalized solutions.

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arXiv2022

Improved decay estimates and $C^2$-asymptotic stability of solutions to the Einstein-scalar field system in spherical symmetry

We investigate the asymptotic stability of solutions to the characteristic initial value problem for the Einstein (massless) scalar field system with a positive cosmological constant. We prescribe spherically symmetric initial data on a future null cone with a wider range of decaying profiles than previously considered. New estimates are then derived in order to prove that, for small data, the system has a unique global classical solution. We also show that the solution decays exponentially in (Bondi) time and that the radial decay is essentially polynomial, although containing logarithmic factors in some special cases. This improved asymptotic analysis allows us to show that, under appropriate and natural decaying conditions on the initial data, the future asymptotic solution is differentiable, up to and including spatial null-infinity, and approaches the de Sitter solution, uniformly, in a neighborhood of infinity. Moreover, we analyze the decay of derivatives of the solution up to second order showing the (uniform) $C^2$-asymptotic stability of the de Sitter attractor in this setting. This corresponds to a surprisingly strong realization of the cosmic no-hair conjecture.

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arXiv2022

Dynamics of interacting monomial scalar field potentials and perfect fluids

Motivated by cosmological models of the early universe we analyse the dynamics of the Einstein equations with a minimally coupled scalar field with monomial potentials $V(ϕ)=\frac{(λϕ)^{2n}}{2n}$, $λ>0$, $n\in\mathbb{N}$, interacting with a perfect fluid with linear equation of state $p_\mathrm{pf}=(γ_\mathrm{pf}-1)ρ_\mathrm{pf}$, $γ_\mathrm{pf}\in(0,2)$, in flat Robertson-Walker spacetimes. The interaction is a friction-like term of the form $Γ(ϕ)=μϕ^{2p}$, $μ>0$, $p\in\mathbb{N}\cup\{0\}$. The analysis relies on the introduction of a new regular 3-dimensional dynamical systems' formulation of the Einstein equations on a compact state space, and the use of dynamical systems' tools such as quasi-homogeneous blow-ups and averaging methods involving a time-dependent perturbation parameter. We find a bifurcation at $p=n/2$ due to the influence of the interaction term. In general, this term has more impact on the future (past) asymptotics for $p n/2$). For $p n/2$ the future asymptotics is similar to the case with no interaction. Finally, we show that irrespective of the parameters, an inflationary quasi-de-Sitter solution always exists towards the past, and therefore the cases with $p\leq(n-2)/2$ may provide new cosmological models of quintessential inflation.

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arXiv2022

Primordial magnetic fields in theories of gravity with non-minimal coupling between curvature and matter

The existence of magnetic fields in the universe is unmistakable. They are observed at all scales from stars to galaxy clusters. However, the origin of these fields remains enigmatic. It is believed that magnetic field seeds may have emerged from inflation, under certain conditions. This possibility is analised in the context of an alternative theory of gravity with non-minimal coupling between curvature and matter. We find, through the solution of the generalised Maxwell equations in the context of non-minimal models, that for general slow-roll inflationary scenarios with low reheating temperatures, $T_{RH}\simeq10^{10}$GeV, the generated magnetic fields can be made compatible with observations at large scales, $λ\sim 1 Mpc$.

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arXiv2022

Quasinormal modes and stability of boosted Reissner-Nordström AdS black holes

We investigate the numerical stability of accelerated AdS black holes against linear scalar perturbations. In particular, we study the evolution of a probe non-minimally coupled scalar field on Schwarzschild and Reissner-Nordström AdS black holes with small accelerations by computing the quasinormal modes of the perturbation spectrum. We decompose the scalar field Klein-Gordon equation and study the eigenvalue problem for its angular and radial-temporal parts using different numerical methods. The angular part is written in terms of the Heun solution and expanded through the Frobenius method which turns out to give eigenvalues qualitatively similar to the ones obtained through the spherical harmonics representation. The radial-temporal evolution renders a stable field profile which is decomposed in terms of damped and purely imaginary oscillations of the quasinormal modes. We calculate the respective frequencies for different spacetime parameters showing the existence of a fine-structure in the modes, for both real and imaginary parts, which is not present in the non-accelerated AdS black holes. Our results indicate that the Schwarzschild and Reissner-Nordström AdS black holes with small accelerations are stable against linear scalar perturbations.

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arXiv2020

Global solutions to the spherically symmetric Einstein-scalar field system with a positive cosmological constant in Bondi coordinates

We consider a characteristic initial value problem, with initial data given on a future null cone, for the Einstein (massless) scalar field system with a positive cosmological constant, in Bondi coordinates. We prove that, for small data, this system has a unique global classical solution which is causally geodesically complete to the future and decays polynomially in radius and exponentially in Bondi time, approaching the de Sitter solution.

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arXiv2020

Stability of the Cauchy horizon in accelerating black-hole spacetimes

The extendibility of spacetime and the existence of weak solutions to the Einstein field equations beyond Cauchy horizons, is a crucial ingredient to examine the limits of General Relativity. Strong Cosmic Censorship serves as a firewall for gravitation by demanding inextendibility of spacetime beyond the Cauchy horizon. For asymptotically flat spacetimes, the predominance of the blueshift instability and the subsequent formation of a mass-inflation singularity at the Cauchy horizon have, so far, substantiated the conjecture. Accelerating black holes, described by the $C-$metric, are exact solutions of the field equations without a cosmological constant, which possess an acceleration horizon with similar causal properties to the cosmological horizon of de Sitter spacetime. Here, by considering linear scalar field perturbations, we provide numerical evidence for the stability of the Cauchy horizon of charged accelerating black holes. In particular, we show that the stability of Cauchy horizons in accelerating charged black holes is connected to quasinormal modes, we discuss the regularity requirement for which weak solutions to the field equations exist at the Cauchy horizon and show that Strong Cosmic Censorship may be violated near extremality.

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arXiv2020

Matching stationary spacetimes

Using the quasi-Maxwell formalism, we derive the necessary and sufficient conditions for the matching of two stationary spacetimes along a stationary timelike hypersurface, expressed in terms of the gravitational and gravitomagnetic fields and the 2-dimensional matching surface on the space manifold. We prove existence and uniqueness results to the matching problem for stationary perfect fluid spacetimes with spherical, planar, hyperbolic and cylindrical symmetry. Finally, we find an explicit interior for the cylindrical analogue of the NUT spacetime.

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arXiv2020

Accelerating black holes: quasinormal modes and late-time tails

Black holes found in binaries move at very high velocities relative to our own reference frame and can accelerate due to the emission of gravitational radiation. Here, we investigate the numerical stability and late-time behavior of linear scalar perturbations in accelerating black holes described by the $C-$metric. We identify a family of quasinormal modes associated with the photon surface and a brand new family of purely imaginary modes associated with the boost parameter of the accelerating black hole spacetime. When the accelerating black hole is charged, we find a third family of modes which dominates the ringdown waveform near extremality. Our frequency and time domain analysis indicate that such spacetimes are stable under scalar fluctuations, while the late-time behavior follows an exponential decay law, dominated by quasinormal modes. This result is in contrast with the common belief that such perturbations, for black holes without a cosmological constant, always have a power-law cutoff. In this sense, our results suggest that the asymptotic structure of black hole backgrounds does not always dictate how radiative fields behave at late times.

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arXiv2020

Global dynamics of Yang-Mills field and perfect-fluid Robertson-Walker cosmologies

We apply a new global dynamical systems formulation to flat Robertson-Walker cosmologies with a massless and massive Yang-Mills field and a perfect-fluid with linear equation of state as the matter sources. This allows us to give proofs concerning the global dynamics of the models including asymptotic source-dominance towards the past and future time directions. For the pure massless Yang-Mills field, we also contextualize well-known explicit solutions in a global (compact) state space picture.

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arXiv2019

Strong Cosmic Censorship in Horndeski Theory

The strong cosmic censorship hypothesis has recently regained a lot of attention in charged and rotating black holes immersed in de Sitter space. Although the picture seems to be clearly leaning towards the validity of the hypothesis in Kerr-de Sitter geometries, Reissner-Nordström-de Sitter black holes appear to be serious counter-examples. Here, we perform another test to the hypothesis by using a scalar field perturbation non-minimally coupled to the Einstein tensor propagating on Reissner-Nordström-de Sitter spacetimes. Such non-minimal derivative coupling is characteristic of Horndeski scalar-tensor theories. Although the introduction of higher-order derivative couplings in the energy-momentum tensor increases the regularity requirements for the existence of weak solutions beyond the Cauchy horizon, we are still able to find a small finite region in the black hole's parameter space where strong cosmic censorship is violated.

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arXiv2019

Singularity theorems and the inclusion of torsion in affine theories of gravity

We extend the scope of the Raychaudhuri-Komar singularity theorem of General Relativity to affine theories of gravity with and without torsion. We first generalize the existing focusing theorems using time-like and null congruences of curves which are hypersurface orthogonal, showing how the presence of torsion affects the formation of focal points in Lorentzian manifolds. Considering the energy conservation on a given affine gravity theory, we prove new singularity theorems for accelerated curves in the cases of Lorentzian manifolds containing perfect fluids or scalar field matter sources.

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