We determine the caloric curves of classical self-gravitating systems at statistical equilibrium in general relativity. In the classical limit, the caloric curves of a self-gravitating gas depend on a unique parameter $ν=GNm/Rc^2$, called the compactness parameter, where $N$ is the particle number and $R$ the system's size. Typically, the caloric curves have the form of a double spiral. The "cold spiral", corresponding to weakly relativistic configurations, is a generalization of the caloric curve of nonrelativistic classical self-gravitating systems. The "hot spiral'", corresponding to strongly relativistic configurations, is similar (but not identical) to the caloric curve of the ultrarelativistic self-gravitating black-body radiation. We introduce two types of normalization of energy and temperature in order to obtain asymptotic caloric curves describing respectively the cold and the hot spirals in the limit $ν\rightarrow 0$. As the number of particles increases, the cold and the hot spirals approach each other, merge at $ν'_S=0.128$, form a loop above $ν_S=0.1415$, reduce to a point at $ν_{\rm max}=0.1764$, and finally disappear. Therefore, the double spiral shrinks when the compactness parameter $ν$ increases, implying that general relativistic effects render the system more unstable. We discuss the nature of the gravitational collapse at low and high energies with respect to a dynamical (fast) or a thermodynamical (slow) instability.
We study the statistical mechanics of classical self-gravitating systems confined within a box of radius $R$ in general relativity. It has been found that the caloric curve $T_{\infty}(E)$ has the form of a double spiral whose shape depends on the compactness parameter $ν=GNm/Rc^2$. The double spiral shrinks as $ν$ increases and finally disappears when $ν_{\rm max}=0.1764$. Therefore, general relativistic effects render the system more unstable. On the other hand, the cold spiral and the hot spiral move away from each other as $ν$ decreases. Using a normalization $Λ=-ER/GN^2m^2$ and $η=GNm^2/R…
We study the nature of phase transitions between gaseous and condensed states in the self-gravitating Fermi gas at nonzero temperature in general relativity. The condensed states can represent compact objects such as white dwarfs, neutron stars, or dark matter fermion balls. The caloric curves depend on two parameters: the system size $R$ and the particle number $N$. When $N<N_{\rm OV}$, where $N_{\rm OV}$ is the Oppenheimer-Volkoff limit, there exists an equilibrium state for any value of the temperature $T$ and of the energy $E$ as in the nonrelativistic case [P.H. Chavanis, Int. J. Mod. Phy…
We develop a general formalism to determine the statistical equilibrium states of self-gravitating systems in general relativity and complete previous works on the subject. Our results are valid for an arbitrary form of entropy but, for illustration, we explicitly consider the Fermi-Dirac entropy for fermions. The maximization of entropy at fixed mass-energy and particle number determines the distribution function of the system and its equation of state. It also implies the Tolman-Oppenheimer-Volkoff equations of hydrostatic equilibrium and the Tolman-Klein relations. Our paper provides all th…
We review the history of the self-gravitating Fermi gas in Newtonian gravity and general relativity. We mention applications to white dwarfs, neutron stars and dark matter halos. We describe the nature of instabilities and phase transitions in the self-gravitating Fermi gas as energy (microcanonical ensemble) or temperature (canonical ensemble) is reduced. When $N<N_{\rm OV}$, where $N_{\rm OV}$ is the Oppenheimer-Volkoff critical particle number, the self-gravitating Fermi gas experiences a gravothermal catastrophe at $E_c$ stopped by quantum mechanics (Pauli's exclusion principle). The equil…
By a spherical gravitating condenser we mean two concentric charged shells made of perfect fluids restricted by the condition that the electric field is nonvanishing only between the shells. Flat space is assumed inside the inner shell. By using Israel's formalism we first analyze the general system of N shells and then concentrate on the two-shell condensers. Energy conditions are taken into account; physically interesting cases are summarized in two tables, but also more exotic situations in which, for example, the inner shell may occur below the inner horizon of the corresponding Reissner-N…
The Thomas-Fermi model is extended at finite temperature, to describe the gravitational phase transition occurring in massive fermionic systems in a general-relativistic framework. It is shown that, when a nondegenerate fermionic gas (for $N < N_{OV}$, where $N_{OV}$ is the Oppenheimer-Volkoff limit) is cooled down below a critical temperature, a condensed phase emerges and the gravitational collapse is prevented by quantum mechanics. If $N > N_{OV}$, by contrast, the system is destined to collapse towards a Black Hole because no equilibrium states exist.
Using a lattice equation of state combined with the D-dimensional Tolman-Oppenheimer-Volkoff equation and the Friedmann equations, we investigate the possibility of the formation of compact objects as well as the time evolution of the scale factor and the density profile of a self-gravitating material cluster. The numerical results show that in a $2+1$ dimensional spacetime, the mass is independent of the central pressure. Hence, the formation of only compact objects with a finite constant mass similar to the white dwarf is possible. However, in a $3+1$ dimensional spacetime, self-gravity lead…
The motion with intersections of relativistic gravitating shells in the Schwarzschild gravitational field of a central body is considered. Formulas are derived for calculating parameters of the shells after intersection via their parameters before intersection. Such special cases as the Newtonian approximation, intersections of light shells, and intersections of a test shell with a gravitating shell are also considered. The ejection of one of the shells to infinity in the relativistic region is described. The equations of motion for the shells are analyzed numerically.
We investigate stationary, self-gravitating, magnetised disks (or tori) around black holes. The models are obtained by numerically solving the coupled system of the Einstein equations and the equations of ideal general-relativistic magnetohydrodynamics. The mathematical formulation and numerical aspects of our approach are similar to those reported in previous works modeling stationary self-gravitating perfect-fluid tori, but the inclusion of magnetic fields represents a new ingredient. Following previous studies of purely hydrodynamical configurations, we construct our models assuming Kepleri…
We discuss the occurrence of gravitational phase transitions and instabilities in a gas of self-gravitating fermions within the framework of general relativity. In the classical (nondegenerate) limit, the system undergoes a gravitational collapse at low energies $E<E_c$ and low temperatures $T<T_c$. This is called "gravothermal catastrophe" in the microcanonical ensemble and "isothermal collapse" in the canonical ensemble. When quantum mechanics is taken into account and when the particle number is below the Oppenheimer-Volkoff limit ($N<N_{\rm OV}$), complete gravitational collapse is prevent…
The problem of construction of a continuous (macroscopic) matter model for a given point-like (microscopic) matter distribution in general relativity is formulated. The existing approaches are briefly reviewed and a physical analogy with the similar problem in classical macroscopic electrodynamics is pointed out. The procedure by Szekeres in the linearized general relativity on Minkowski background to construct a tensor of gravitational quadruple polarization by applying Kaufman's method of molecular moments for derivation of the polarization tensor in macroscopic electrodynamics and to derive…
Vortons can be viewed as (flat space-) field theory analogs of black rings in general relativity. They are made from loops of vortices, being sustained against collapse by the centrifugal force. In this work we discuss such configurations in the global version of Witten's U(1)xU(1) theory. We first consider solutions in a flat spacetime background and show their non-uniqueness. The inclusion of gravity leads to new features. In particular, an ergoregion can occur. Also, similar to boson stars, we show that the vortons exist only in a limited frequency range. The coupling to gravity gives rise …
We consider the dynamics of a self-gravitating spinor field and a self-gravitating rotating perfect fluid. It is shown that both these matter distributions can induce a vortex field described by the curl 4-vector of a tetrad: $ω^i = \frac12\eps^{iklm}e_{(a)k}e^{(a)}_{l;m}$, where $e^{(a)}_k$ are components of the tetrad. The energy-momentum tensor $T_{ik}(ω)$ of this field has been found and shown to violate the strong and weak energy conditions which leads to possible formation of geometries with nontrivial topology like wormholes. The corresponding exact solutions to the equations of general…
Solar system observations have traditionally allowed for very stringent tests of Einstein's theory of general relativity. We here revisit the possibility of using these observations to constrain gravitational parity violation as encapsulated in dynamical Chern-Simons gravity. Working in the small-coupling and post-Newtonian approximations, we calculate analytically the scalar field and the gravitomagnetic sector of the gravitational field in the interior and the exterior of an isolated, weakly-gravitating object with uniform rotation and a quadrupolar mass deformation. We find that the asympto…
We revisit the problem of the equations of motion of a system of $N$ self-interacting massive particles (without spins) in the first post-Minkowskian (1PM) approximation of general relativity. We write the equations of motion, gravitational field and associated conserved integrals of the motion in a form suitable for comparison with recently published post-Newtonian (PN) results at the 4PN order. We show that the Lagrangian associated with the equations of motion in harmonic coordinates is a generalized one, and compute all the terms linear in $G$ up to 5PN order. We discuss the Hamiltonian in…
In a recent paper, Herrera \cite{2} (L. Herrera: Phys. Rev. D97, 044010(2018)) have proposed a new definition of complexity for static self-gravitating fluid in General Relativity. In the present article, we implement this definition of complexity for static self-gravitating fluid to case of $f(R)$ gravity. Here, we found that in the frame of $f(R)$ gravity the definition of complexity proposed by Herrera, entirely based on the quantity known as complexity factor which appears in the orthogonal splitting of the curvature tensor. It has been observed that fluid spheres possessing homogenous ene…
We find a self-gravitating monopole and its black hole solution in Brans-Dicke theory. We mainly discuss the properties of these solutions in the Einstein frame and compare the solutions with those in general relativity.
We show that, when we study the coexistence of general relativity with thermodynamics, some physical properties that are usually thought of as holographic and lying in the domain of quantum gravity can actually be accessed even at the classical level. In particular, we demonstrate that the thermodynamics of gravitating systems in equilibrium is fully specified by variables defined on the system's boundary, namely, the boundary's geometry and extrinsic curvature. Hence, information is non-trivially incorporated in boundary variables because of the structure (the symmetries) of the classical gra…
We study the stability of three-dimensional numerical evolutions of the Einstein equations, comparing the standard ADM formulation to variations on a family of formulations that separate out the conformal and traceless parts of the system. We develop an implementation of the conformal-traceless (CT) approach that has improved stability properties in evolving weak and strong gravitational fields, and for both vacuum and spacetimes with active coupling to matter sources. Cases studied include weak and strong gravitational wave packets, black holes, boson stars and neutron stars. We show under wh…
This paper deals with the spherically symmetric self-gravitating star which is considered to be expansion free dissipative perfect fluids distribution. Some recent research reveals that expansion free dynamical star must be accelerating and dissipating. We adopted some conjectures to obtain the analytical solution for the dynamical model of such stars. Firstly, it has shown that density of dynamical star is homogeneous and \Lamada-dominated under quasi-static diffusion approximation. Secondly, the self-similar solution is also discussed to describe the dynamical model.
We prove the local existence of solutions to the Einstein-Elastic equations that represent self-gravitating, relativistic elastic bodies with compact support.
Using the the concept of cracking we explore the influence of density fluctuations and local anisotropy have on the stability of local and non-local anisotropic matter configurations in general relativity. This concept, conceived to describe the behaviour of a fluid distribution just after its departure from equilibrium, provides an alternative approach to consider the stability of selfgravitating compact objects. We show that potentially unstable regions within a configuration can be identify as a function of the difference of propagations of sound along tangential and radial directions. In f…
Mathematical modeling of gravitating configurations of physical fields is one of the priority directions of the modern theory of gravity. Most of the exact solutions constructed within the framework of the general relativity are static or stationary configurations. This is due to the objective complexity of solving the Einstein equations under the assumption of nonstationarity. We present an approach to constructing nonstationary configurations of a spherically symmetric nonlinear real scalar field and the electromagnetic field, which areassumed both to be minimally coupled to gravity. It is b…
We show that for any perfect fluid in a static spacetime, if the Einstein constraint equation is satisfied and the temperature of the fluid obeys Tolman's law, then the other components of Einstein's equation are implied by the assumption that the total entropy of the fluid achieves an extremum for fixed total particle number and for all variations of metric with certain boundary conditions. Conversely, one can show that the extrema of the total entropy of the fluid are implied by Einstein's equation. Compared to previous works on this issue, we do not require spherical symmetry for the spacet…
We propose the almost-geodesic motion of self-gravitating test bodies as a possible selection rule among metric theories of gravity. Starting from a heuristic statement, the "gravitational weak equivalence principle", we build a formal, operative test able to probe the validity of the principle for any metric theory of gravity, in an arbitrary number of spacetime dimensions. We show that, if the theory admits a well-posed variational formulation, this test singles out only the purely metric theories of gravity. This conclusion reproduces known results in the cases of general relativity (also w…
Recent work has shown that scalar fields around black holes can form long-lived, quasistationary configurations surviving for cosmological timescales. With this requirement, scalar fields cannot be discarded as viable candidates for dark matter halo models in galaxies around central supermassive black holes (SMBH). One hypothesis for the formation of most SMBHs at high redshift is the gravitational collapse of supermassive stars (SMS) with masses of $\sim10^5 \rm {M_{\odot}}$. Therefore, a constraint for the existence of quasi-bound states of scalar fields is their survival to such dynamic eve…
In self-gravitating $N$-body systems, small perturbations introduced at the start, or infinitesimal errors that are produced by the numerical integrator or are due to limited precision in the computer, grow exponentially with time. For Newton's gravity, we confirm earlier results that for relatively homogeneous systems, this rate of growth per crossing time increases with $N$ up to $N \sim 30$, but that for larger systems, the growth rate has a weaker scaling with $N$. For concentrated systems, however, the rate of exponential growth continues to scale with $N$. In relativistic self-gravitatin…
This is a short review of a series of papers which, in collaboration with Yue Ma, establish several novel existence results for systems of coupled wave-Klein-Gordon equation. Our method, the Hyperbolic Hyperboloidal Method, has allowed us to address the global evolution problem for the Einstein equations of general relativity and investigate the global geometry of matter spacetimes that are initially close to Minkowski spacetime.
Recent perturbative studies have shown the existence of long-lived, quasi-stationary configurations of scalar fields around black holes. In particular, such configurations have been found to survive for cosmological timescales, which is a requirement for viable dark matter halo models in galaxies based on such type of structures. In this paper we perform a series of numerical relativity simulations of dynamical non-rotating black holes surrounded by self-gravitating scalar fields. We solve numerically the coupled system of equations formed by the Einstein and the Klein-Gordon equations under t…
In this paper, we discuss general relativistic, self-gravitating and uniformly rotating perfect fluid bodies with a toroidal topology (without central object). For the equations of state describing the fluid matter we consider polytropic as well as completely degenerate, perfect Fermi gas models. We find that the corresponding configurations possess similar properties to the homogeneous relativistic Dyson rings. On the one hand, there exists no limit to the mass for a given maximal mass-density inside the body. On the other hand, each model permits a quasistationary transition to the extreme K…
We develop the large $D$ limit of general relativity for spherically symmetric scalar fields in both asymptotically flat and asymptotically anti-de Sitter spaces. The leading order equations in the $1/D$ expansion can be solved analytically, providing a large $D$ description of oscillating soliton stars. When the amplitude reaches a critical threshold, certain divergences occur which we interpret as signal of horizon formation. We estimate the size of the resulting black hole and obtain, with respect to that definition, a Choptuik scaling exponent for our family of solutions.
The theory presented in this monograph establishes the first mathematically rigorous result on the global nonlinear stability of self-gravitating matter under small perturbations of an asymptotically flat, spacelike hypersurface of Minkowski spacetime. It allows one to exclude the existence of dynamically unstable, self-gravitating massive fields and, therefore, solves a long-standing open problem in General Relativity. By a significant extension of the Hyperboloidal Foliation Method they introduced in 2014, the authors establish global-in-time existence for the Einstein equations expressed as…
We summarize the general formalism describing surface flows in three-dimensional space in a form which is suitable for various astrophysical applications. We then apply the formalism to the analysis of non-radial perturbations of self-gravitating spherical fluid shells. Spherically symmetric gravitating shells (or bubbles) have been used in numerous model problems especially in general relativity and cosmology. A radially oscillating shell was recently suggested as a model for a variable cosmic object. Within Newtonian gravity we show that self-gravitating static fluid shells are unstable with…
The cataclysmic observations of event GW170817, first as gravitational waves along the inspiral motion of two neutron stars, then as a short $γ$-ray burst, and later as a kilonova, launched the era of multimessenger astronomy, and played a pivotal role in furthering our understanding on a number of longstanding questions. Numerical modelling of such multimessenger sources is an important tool to understand the physics of compact objects and, more generally, the physics of matter under extreme conditions. In this review we present a unified view of various techniques used to obtain equilibrium …
This paper summarises a number of new, potentially significant, results, obtained recently by the author and his collaborators, which impact on various issues related to the gravitational N-body problem, both Newtonianly and in the context of general relativity. Topics addressed include: (1) direct N-body simulations and their interpretation, with reference to the observed exponential instability towards small changes in initial conditions and the phenomenon of ``nonviolent relaxation;'' (2) the Hamiltonian structure of the collisionless Boltzmann equation of general relativity, i.e., the Vlas…
We consider a self-gravitating system consisting of perfect fluid with spherical symmetry. Using the general expression of entropy density, we extremize the total entropy $S$ under the constraint that the total number of particles is fixed. We show that extrema of $S$ coincides precisely with the relativistic Tolman-Oppenheimer-Volkoff (TOV) equation of hydrostatic equilibrium. Furthermore, we apply the maximum entropy principle to a charged perfect fluid and derive the generalized TOV equation. Our work provides a strong evidence for the fundamental relationship between general relativity and…
In this paper we will study for the first time how the wave-packet of a self-gravitating meso-scopic system spreads in theories beyond Einstein's general relativity. In particular, we will consider a ghost-free infinite derivative gravity, which resolves the $1/r$ singularity in the potential - such that the gradient of the potential vanishes within the scale of non-locality. We will show that a quantum wave-packet spreads faster for a ghost-free and singularity-free gravity as compared to the Newtonian case, therefore providing us a unique scenario for testing classical and quantum properties…
The full set of equations governing the structure and the evolution of self--gravitating spherically symmetric dissipative fluids with anisotropic stresses, is written down in terms of five scalar quantities obtained from the orthogonal splitting of the Riemann tensor, in the context of general relativity. It is shown that these scalars are directly related to fundamental properties of the fluid distribution, such as: energy density, energy density inhomogeneity, local anisotropy of pressure, dissipative flux and the active gravitational mass. It is also shown that in the static case, all poss…
We consider a self-gravitating string generated by a global vortex solution in general relativity. We investigate the Einstein and field equations of a global vortex in the region of its central line and at a distance from the centre of the order of the inverse of its Higgs boson mass. By combining the two we establish by a limiting process of large Higgs mass the dynamics of a self-gravitating global string. Under our assumptions the presence of gravitation restricts the world sheet of the global string to be totally geodesic.
We review some recent results concerning the properties of a spherically symmetric global monopole in $(D=d+2)$-dimensional general relativity. Some common features of monopole solutions are found independently of the choice of the symmetry-breaking potential. Thus, the solutions show six types of qualitative behavior and can contain at most one simple horizon. For the standard Mexican hat potential, we analytically find the $D$-dependent range of $γ$ (the gravitational field strength parameter) in which there exist globally regular solutions with a monotonically growing Higgs field, containin…
Shape Dynamics is a 3D conformally invariant theory of gravity which possesses a large set of solutions in common with General Relativity. When looked closely, these solutions are found to behave in surprising ways, so in order to probe the fitness of Shape Dynamics as a viable alternative to General Relativity one must find and understand increasingly more complex, less symmetrical exact solutions, on which to base perturbative studies and numerical analyses in order to compare them with data. Spherically symmetric exact solutions have been studied, but only in a static vacuum setup. In this …
Static spherically-symmetric matter distributions whose energy-momentum tensor is characterized by a non-negative trace are studied analytically within the framework of general relativity. We prove that such field configurations are necessarily highly relativistic objects. In particular, for matter fields with $T\geqα\cdotρ\geq0$ (here $T$ and $ρ$ are respectively the trace of the energy-momentum tensor and the energy density of the fields, and $α$ is a non-negative constant), we obtain the lower bound $\text{max}_r\{2m(r)/r\}>(2+2α)/(3+2α)$ on the compactness (mass-to-radius ratio) of regular…
We present a new covariant, gauge-invariant formalism describing linear metric perturbation fields on any spherically symmetric background in general relativity. The advantage of this formalism relies in the fact that it does not require a decomposition of the perturbations into spherical tensor harmonics. Furthermore, it does not assume the background to be vacuum, nor does it require its staticity. In the particular case of vacuum perturbations, we derive two master equations describing the propagation of arbitrary linear gravitational waves on a Schwarzschild black hole. When decomposed int…
The evolution of self-gravitating collision-less matter and scalar waves within the general relativity context is described by Einstein and Vlasov equations. The sources of Einstein equations are generated by a distribution function and a scalar field, respectively subject to the Vlasov and wave equations. We prove in contracting T^2 symmetry case, a local in time existence of solutions.
We propose a new numerical technique for following the evolution of a self-gravitating collisionless system in general relativity. Matter is modeled as a scalar field obeying the coupled Klein-Gordon and Einstein equations. A phase space distribution function, constructed using covariant coherent states, obeys the relativistic Vlasov equation provided the de Broglie wavelength for the field is very much smaller than the scales of interest. We illustrate the method by solving for the evolution of a system of particles in a static, plane-symmetric, background spacetime.
We consider self-gravitating fluids in cosmological spacetimes with Gowdy symmetry on the torus $T^3$ and, in this class, we solve the singular initial value problem for the Einstein-Euler system of general relativity, when an initial data set is prescribed on the hypersurface of singularity. We specify initial conditions for the geometric and matter variables and identify the asymptotic behavior of these variables near the cosmological singularity. Our analysis of this class of nonlinear and singular partial differential equations exhibits a condition on the sound speed, which leads us to the…
The holostar is an exact spherically symmetric solution to the field equations of general relativity with anisotropic interior pressure. Its properties are similar to a black hole. It has an internal temperature inverse proportional to the square root of the radial coordinate value, from which the Hawking temperature law follows. The number of particles within any concentric region of the holostar's interior is proportional to the proper area of its boundary. The holostar-metric is static throughout the whole space-time. There are no trapped surfaces, no singularity and no event horizon. Infor…
We consider static, cylindrically symmetric configurations in general relativity coupled to nonlinear electrodynamics (NED) with an arbitrary gauge-invariant Lagrangian of the form $L_{em}= Φ(F)$, $F =F_{mn}F^{mn}$. We study electric and magnetic fields with three possible orientations: radial (R), longitudinal (L) and azimuthal (A), and try to find solitonic stringlike solutions, having a regular axis and a flat metric at large $r$, with a possible angular defect. Assuming the function $Φ(F)$ to be regular at small $F$, it is shown that a regular axis is impossible in R-fields if there is a n…
We emphasize that the pressure related work appearing in a general relativistic first law of thermodynamics should involve {\em proper volume element} rather than coordinate volume element. This point is highlighted by considering both local energy momentum conservation equation as well as particle number conservation equation. It is also emphasized that we are considering here a {\em non-singular} fluid governed by purely classical general relativity. Therefore, we are not considering here any semi-classical or quantum gravity which apparently suggests thermodynamical properties even for a (s…
We calculated the dimensionless gyromagnetic ratio ("$g$-factor") of self-gravitating, uniformly rotating disks of dust with a constant specific charge $ε$. These disk solutions to the Einstein-Maxwell equations depend on $ε$ and a "relativity parameter" $γ$ ($0<γ\le 1$) up to a scaling parameter. Accordingly, the $g$-factor is a function $g=g(γ,ε)$. The Newtonian limit is characterized by $γ\ll 1$, whereas $γ\to 1$ leads to a black-hole limit. The $g$-factor, for all $ε$, approaches the values $g=1$ as $γ\to 0$ and $g=2$ as $γ\to 1$.
We consider the collision of self-gravitating n-branes in a (n+2)-dimensional spacetime. We show that there is a geometrical constraint which can be expressed as a simple sum rule for angles characterizing Lorentz boosts between branes and the intervening spacetime regions. This constraint can then be re-interpreted as either energy or momentum conservation at the collision.
We put forward a new definition of complexity, for static and spherically symmetric self--gravitating systems, based on a quantity, hereafter referred to as complexity factor, that appears in the orthogonal splitting of the Riemann tensor, in the context of general relativity. We start by assuming that the homogeneous (in the energy density) fluid, with isotropic pressure is endowed with minimal complexity. For this kind of fluid distribution, the value of complexity factor is zero. So, the rationale behind our proposal for the definition of complexity factor stems from the fact that it measur…
