We consider the lagrangian of a self-interacting complex scalar field admitting generically Q-balls solutions. This model is extended by minimal coupling to electromagnetism and to gravity. A stationnary, axially-symmetric ansatz for the different fields is used in order to reduce the classical equations. The system of non-linear partial differential equations obtained becomes a boundary value problem by supplementing a suitable set of boundary conditions. We obtain numerical evidences that the angular excitations of uncharged Q-balls, which exist in flat space-time, get continuously deformed by the Maxwell and the Einstein terms. The electromagnetic and gravitating properties of several solutions, including the spinning Q-balls, are emphasized.
The 3.5 post-Newtonian (PN) order is tackled by extending the canonical formalism of Arnowitt, Deser, and Misner to spinning objects. This extension is constructed order by order in the PN setting by utilizing the global Poincare invariance as the important consistency condition. The formalism is valid to linear order in the single spin variables. Agreement with a recent action approach is found. A general formula for the interaction Hamiltonian between matter and transverse-traceless part of the metric at 3.5PN is derived. The wave equation resulting from this Hamiltonian is considered in the…
We investigate self-gravitating rotating solutions in the Einstein-Skyrme theory. These solutions are globally regular and asymptotically flat. We present a new kind of solutions with zero baryon number, which possess neither a flat limit nor a static limit.
In general relativity, systems of spinning classical particles are implemented into the canonical formalism of Arnowitt, Deser, and Misner [1]. The implementation is made with the aid of a symmetric stress-energy tensor and not a 4-dimensional covariant action functional. The formalism is valid to terms linear in the single spin variables and up to and including the next-to-leading order approximation in the gravitational spin-interaction part. The field-source terms for the spinning particles occurring in the Hamiltonian are obtained from their expressions in Minkowski space with canonical va…
We obtain from the first principles a general-relativistic Keplerian rotation law for self-gravitating disks around spinning black holes. This is an extension of a former rotation law that was designed mainly for toroids around spin-less black holes. We integrate numerically axial stationary Einstein equations with self-gravitating disks around spinless or spinning black holes; that includes the first ever integration of the Keplerian selfgravitating tori. This construction can be used for the description of tight black hole-torus systems produced during coalescences of two neutron stars or mo…
In the AdS/CFT correspondence, single trace operators of large-$N$ gauge theories at large spin $J$ can be described by classical spinning strings, giving a geometric and classical description of their spectrum at strong coupling. We observe that in AdS$_3$ these strings have significant gravitational back-reaction at sufficiently large spin, since the gravitational force does not decay at long distances. We construct solutions for folded spinning strings coupled to gravity in AdS$_3$ and compute their spectrum, corresponding to the leading Regge trajectory of Virasroro primary operators. Thes…
We derive the post-Newtonian next-to-leading order conservative spin-orbit and spin(a)-spin(b) gravitational interaction Hamiltonians for arbitrary many compact objects. The spin-orbit Hamiltonian completes the knowledge of Hamiltonians up to and including 2.5PN for the general relativistic three-body problem. The new Hamiltonians include highly nontrivial three-body interactions, in contrast to the leading order consisting of two-body interactions only. This may be important for the study of effects like Kozai resonances in mergers of black holes with binary black holes.
We consider a system representing self-gravitating balls of dust in an expanding Universe. It is demonstrated that one can prescribe data for such a system at infinity and evolve it backward in time without the development of shocks or singularities. The resulting solution to the Einstein-lambda-dust equations exists for an infinite amount of time in the asymptotic region of the spacetime. Furthermore, we find that if the density is small compared to the Cosmological constant, then it is possible to construct Cosmological solutions to the Einstein constraint equations on a standard Cauchy hype…
Based on the Arnowitt-Deser-Misner (ADM) canonical formulation of general relativity, a canonical formulation of gravitationally interacting classical spinning-object systems is given to linear order in spin. The constructed position, linear momentum and spin variables fulfill standard Poisson bracket relations. A spatially symmetric time gauge for the tetrad field is introduced. The achieved formulation is of fully reduced form without unresolved constraints, supplementary, gauge, or coordinate conditions. The canonical field momentum is not related to the extrinsic curvature of spacelike hyp…
We perform general relativistic simulations of self-gravitating black hole-disks in which the spin of the black hole is significantly tilted ($45^\circ$ and $90^\circ$) with respect to the angular momentum of the disk and the disk-to-black hole mass ratio is $16\%-28\%$. The black holes are rapidly spinning with dimensionless spins up to $\sim 0.97$. These are the first self-consistent hydrodynamic simulations of such systems, which can be prime sources for multimessenger astronomy. In particular tilted black hole-disk systems lead to: i) black hole precession; ii) disk precession and warping …
We introduce a formulation for spinning gravitating objects in the effective field theory in the post-Newtonian scheme in the context of the binary inspiral problem. We aim at an effective action, where all field modes below the orbital scale are integrated out. We spell out the relevant degrees of freedom, in particular the rotational ones, and the associated symmetries. Building on these symmetries, we introduce the minimal coupling part of the point particle action in terms of gauge rotational variables, and construct the spin-induced nonminimal couplings, where we obtain the leading order …
Two new classes of exact interior static solutions of the Einstein equations in homogeneous coordinates for a gravitating ball filled by a Pascal perfect fluid are obtained. Schwarzschild's interior solution of is a special case of these solutions.
We investigate how gravity affects Q-balls by exemplifying the case of the Affleck-Dine potential $V(ϕ):=m^4 \ln (1+\frac{ϕ^2}{m^2})$. Surprisingly, stable Q-balls with arbitrarily small charge exist, no matter how weak gravity is, contrary to the case of flat spacetime. We also show analytically that this feature holds true for general models as long as the leading order term of the potential is a positive mass term in its Maclaurin series.
We consider the $\mathrm{O}(3)$ non-linear sigma-model, composed of three real scalar fields with a standard kinetic term and with a symmetry breaking potential in four spacetime dimensions. We show that this simple, geometrically motivated model, admits both self-gravitating, asymptotically flat, non-topological solitons and hairy black holes, when minimally coupled to Einstein's gravity, $without$ the need to introduce higher order kinetic terms in the scalar fields action. Both spherically symmetric and spinning, axially symmetric solutions are studied. The solutions are obtained under a an…
We study compact gravitating $Q$-ball, $Q$-shell solutions in a sigma model with the target space $\mathbb{C}P^N$. Models with odd integer $N$ and suitable potential can be parameterized by $N$-th complex scalar fields and they support compact solutions. A coupling with gravity allows for harboring of the Schwarzschild black holes for the $Q$-shell solutions. The energy of the solutions behaves as $E\sim |Q|^{5/6}$, where $Q$ stands for the $U(1)$ Noether charge, for both the gravitating and the black hole solutions.Notable difference from the solutions of the flat space is that upper bound of…
We present the next-to-next-to-leading order post-Newtonian (PN) spin-orbit Hamiltonian for two self-gravitating spinning compact objects. If at least one of the objects is rapidly rotating, then the corresponding interaction is comparable in strength to a 3.5PN effect. The result in the present paper in fact completes the knowledge of the post-Newtonian Hamiltonian for binary spinning black holes up to and including 3.5PN. The Hamiltonian is checked via known results for the test-spin case and via the global Poincare algebra with the center-of-mass vector uniquely determined by an ansatz.
Einstein's gravity minimally coupled to free, massive, classical fundamental fields admits particle-like solutions. These are asymptotically flat, everywhere non-singular configurations that realise Wheeler's concept of a geon: a localised lump of self-gravitating energy whose existence is anchored on the non-linearities of general relativity, trivialising in the flat spacetime limit. In arXiv:1708.05674 the key properties for the existence of these solutions (also referred to as stars or self-gravitating solitons) were discussed - which include a harmonic time dependence in the matter field -…
We investigate how gravity affects "Q-balls" with the Affleck-Dine potential $V_{AD}(ϕ):=\frac{m^2}{2}ϕ^2[ 1+K\ln (\fracϕ{M})^2]$. Contrary to the flat case, in which equilibrium solutions exist only if $K<0$, we find three types of gravitating solutions as follows. In the case that $K<0$, ordinary Q-ball solutions exist; there is an upper bound of the charge due to gravity. In the case that K=0, equilibrium solutions called (mini-)boson stars appear due to gravity; there is an upper bound of the charge, too. In the case that $K>0$, equilibrium solutions appear, too. In this case, these soluti…
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…
The existence of static self-gravitating Newtonian elastic balls is proved under general assumptions on the constitutive equations of the elastic material. The proof uses methods from the theory of finite-dimensional dynamical systems and the Euler formulation of elasticity theory for spherically symmetric bodies introduced recently by the authors. Examples of elastic materials covered by the results of this paper are Saint Venant-Kirchhoff, John and Hadamard materials.
We investigate the properties of interacting Q-balls and boson stars that sit on top of each other in great detail. The model that describes these solutions is essentially a (gravitating) two-scalar field model where both scalar fields are complex. We construct interacting Q-balls or boson stars with arbitrarily small charges but finite mass. We observe that in the interacting case - where the interaction can be either due to the potential or due to gravity - two types of solutions exist for equal frequencies: one for which the two scalar fields are equal, but also one for which the two scalar…
A numerical solution of Einstein field equations for a spherical symmetric and stationary system of identical and auto-gravitating particles in phase transition is presented. The fluid possess a perfect fluid energy momentum tensor, and the internal interactions of the system are represented by a van der Walls like equation of state able to describe a first order phase transition of the type gas-liquid. We find that the space-time curvature, the radial component of the metric, and the pressure and density show discontinuities in their radial derivatives in the phase coexistence region. This re…
We present the next-to-next-to-leading order post-Newtonian (PN) spin(1)-spin(2) Hamiltonian for two self-gravitating spinning compact objects. If both objects are rapidly rotating, then the corresponding interaction is comparable in strength to a 4PN effect. The Hamiltonian is checked via the global Poincare algebra with the center-of-mass vector uniquely determined by an ansatz.
We consider development of the models of the source of the Kerr-Newman (KN) solution and new aspects related with the obtained recently field model based on a domain wall bubble with superconducting interior, arXiv:1003.2928[hep-th]. The internal Higgs field regularizes the KN solution, expelling electromagnetic field from interior to the boundary of bubble. The KN source forms a gravitating soliton, interior of which is similar to oscillating solitons (Q-balls, oscillons), while exterior is consistent with the KN solution. We obtain that a closed Wilson loop appears on the edge of the bubble,…
Propagation of light in the gravitational field of self-gravitating spinning bodies moving with arbitrary velocities is discussed. The gravitational field is assumed to be "weak" everywhere. Equations of motion of a light ray are solved in the first post-Minkowskian approximation that is linear with respect to the universal gravitational constant $G$. We do not restrict ourselves with the approximation of gravitational lens so that the solution of light geodesics is applicable for arbitrary locations of source of light and observer. This formalism is applied for studying corrections to the Sha…
We present spinning Q-balls and boson stars in four dimensional anti-de Sitter spacetime. These are smooth, horizonless solutions for gravity coupled to a massive complex scalar field with a harmonic dependence on time and the azimuthal angle. Similar to the flat spacetime configurations, the angular momentum is quantized. We find that a class of solutions with a self-interaction potential has a limit corresponding to static solitons with axial symmetry only. An exact solution describing spherically symmetric Q-balls in a fixed AdS background is also discussed.
We integrate numerically axially symmetric stationary Einstein equations describing self-gravitating disks around spinless black holes. The numerical scheme is based on a method developed by Shibata, but contains important new ingredients. We derive a new general-relativistic Keplerian rotation law for self-gravitating disks around spinning black holes. Former results concerning rotation around spin-less black holes emerge in the limit of a vanishing spin parameter. These rotation curves might be used for the description of rotating stars, after appropriate modification around the symmetry axi…
The charged, spinning and gravitating soliton is realized as a regular solution of the Kerr-Newman field coupled with a chiral Higgs model. A regular core of the solution is formed by a domain wall bubble interpolating between the external Kerr-Newman solution and a flat superconducting interior. An internal electromagnetic (em) field is expelled to the boundary of the bubble by the Higgs field. The solution reveals two new peculiarities: (i) the Higgs field is oscillating, similar to the known oscillon models, (ii) the em field forms on the edge of the bubble a Wilson loop, resulting in quant…
The next-to-next-to-leading order spin-squared interaction potential for generic compact binaries is derived for the first time via the effective field theory for gravitating spinning objects in the post-Newtonian scheme. The spin-squared sector is an intricate one, as it requires the consideration of the point particle action beyond minimal coupling, and mainly involves the spin-squared worldline couplings, which are quite complex, compared to the worldline couplings from the minimal coupling part of the action. This sector also involves the linear in spin couplings, as we go up in the nonlin…
Extending the Proca Lagrangian of a massive complex-valued vector field by self-interaction potential, we construct a large class of spherically symmetric solutions in flat Minkowski background as well as in the self-gravitating case. Our solutions encompass Proca Q-balls and Proca stars known in the literature, but present new features and go beyond the limited region of parameter space charted so far. A special emphasis is set to the domain of existence of the solutions in relation with the coupling constants of the potential and to the critical phenomena limiting this domain.
We implement the effective field theory for gravitating spinning objects in the post-Newtonian scheme at the next-to-next-to-leading order level to derive the gravitational spin-orbit interaction potential at the third and a half post-Newtonian order for rapidly rotating compact objects. From the next-to-next-to-leading order interaction potential, which we obtain here in a Lagrangian form for the first time, we derive straightforwardly the corresponding Hamiltonian. The spin-orbit sector constitutes the most elaborate spin dependent sector at each order, and accordingly we encounter a prolife…
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…
Cosmic strings are considered in two types of gauged sigma models, which generalize the gravitating Abelian Higgs model. The two models differ by whether the U(1) kinetic term is of the Maxwell or Chern-Simons form. We obtain the self-duality conditions for a general two-dimensional target space defined in terms of field dependent "dielectric functions". In particular, we analyze analytically and numerically the equations for the case of O(3) models (two-sphere as target space), and find cosmic string solutions of several kinds as well as gravitating vortices. We classify the solutions by thei…
We analyze stationary self-gravitating disks around spinning black holes that satisfy the recently found general-relativistic Keplerian rotation law. There is a numerical evidence that the angular velocity, circumferential radius and angular momenta yield a bound onto the asymptotic mass of the system. This bound is proven analytically in the special case of massless disks of dust in the Kerr spacetime.
The equation of state of SU(3) Yang-Mills theory can be modelled by an effective $Z_3-$symmetric potential $V(\vertϕ\vert,ϕ^3+ϕ^{3*}, T)$ depending on the temperature $T$ and on a scalar field $ϕ$ -- the averaged Polyakov loop. Allowing $ϕ$ to be dynamical opens the way to the study of spatially localized classical configurations of the Polyakov loop. We first show that spherically symmetric static Q-balls exist in the range $(1-1.21)\times T_c$, $T_c$ being the deconfinement temperature. Then we argue that Q-holes solutions, if any are unphysical within our framework. Finally we couple the Po…
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…
It is shown that at the center of gravitating fluid ball there is the phase transition of second kind of gravitational field as a change of field's algebraic type.
The Ellis-Bronnikov solution provides a simple toy model for the study of various aspects of wormhole physics. In this work we solve the Klein-Gordon equation in this background and find an exact solution in terms of Heun's function. This may describe 'scalar clouds' ($i.e.$ localized, particle-like configuration) or scalar waves. However, in the former case, the radial derivative of the scalar field is discontinuous at the wormhole's throat (except for the spherical case). This pathology is absent for a suitable scalar field self-interaction, and we provide evidence for the existence of spher…
Self-gravitating non-topological solitons whose potential admits multiple vacua are promising candidates for exotic compact objects. Such objects can arise in several extensions of the Standard Model and could be produced in the early Universe. In this work, we focus on objects made from complex scalars (gravitating Q-balls/soliton boson stars), deriving analytic solutions in spherical symmetry and comparing them with fully numerical ones. In the high-compactness limit we find that these objects present an effectively linear equation of state, thus saturating the Buchdahl limit with the causal…
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…
We consider a self-gravitating collisionless gas as described by the Vlasov-Poisson or Einstein-Vlasov system or a self-gravitating fluid ball as described by the Euler-Poisson or Einstein-Euler system. We give a simple proof for the finite extension of spherically symmetric equilibria, which covers all these models simultaneously. In the Vlasov case the equilibria are characterized by a local growth condition on the microscopic equation of state, i.e., on the dependence of the particle distribution on the particle energy, at the cut-off energy E_0, and in the Euler case by the corresponding g…
A new exact static interior solution of the Einstein equations is obtained for a gravitating ball filled with a Pascal perfect fluid . The solution is an extension of the well-known interior solution with a parabolic distribution of mass density and describes more compact astrophysical objects such as neutron and hyperon stars. It is shown that a behaviour of the new mass density distribution near the stellar centre can be described as a cusp catastrophe.
We use the model approach to the description of spherical gravitating static fluid ball with an electric charge in general relativity. The metric is written in Bondi's coordinates. The total energy-momentum tensor (EMT) is chosen as a sum of the EMT of a Pascal perfect fluid and that of the electromagnetic field. An exact solution of the Einstein-Maxwell equations is found, extending a similar solution with parabolic mass density distribution.
Gravitational and electromagnetic (EM) field of the Dirac electron is described by the Kerr-Newman (KN) solution. We elaborate a regular source of the KN solution which satisfies the requirement of flat space-time inside the source and realization of the exact KN solution outside the source. This requirement removes conflict between gravity and quantum theory and determines many details of the source structure. In particular, we obtain that the KN source should forms a gravitating bag model, similar to the known MIT and SLAC bag models. As opposite to the known bag models, the self-interacting…
In this paper we consider the possibility of measuring the corrections induced by the square of the parameter a_g of the Kerr metric to the general relativistic deflection of electromagnetic waves and time delay in an Earth based experiment. It turns out that, while at astronomical scale the well known gravitoelectric effects are far larger than the gravitomagnetic ones, at laboratory scale the situation is reversed: the gravitomagnetic effects exceed definitely the gravitoelectric ones which are totally negligible. By using a small rapidly rotating sphere as gravitating source on the Earth th…
Within the framework of General Relativity a model approach to a description of spherical gravitating static fluid balls with an electric charge is considered. The metric interval is written in Bondi's radiation coordinates. The total energy-momentum tensor as a direct sum of the perfect Pascal fluid energy-momentum tensor and the energy-momentum tensor of an electromagnetic field is chosen. The exact solution of the Einstein-Maxwell equations as an extension of similar solution with parabolic distribution of mass density is found.
A new kind of Q-balls is found: Q-balls in a non-linear sigma model. Their main properties are presented together with those of their self-gravitating generalization, sigma model Q-stars. A simple special limit of solutions which are bound by gravity alone (``sigma stars'') is also discussed briefly. The analysis is based on calculating the mass, global U(1) charge and binding energy for families of solutions parameterized by the central value of the scalar field. Two kinds (differing by the potential term) of the new sigma model Q-balls and Q-stars are analyzed. They are found to share some c…
We construct supersymmetric Q-balls and boson stars in (d+1) dimensions. These non-topological solitons are solutions of a scalar field model with global U(1) symmetry and a scalar field potential that appears in gauge-mediated supersymmetry (SUSY) breaking in the minimal supersymmetric extension of the Standard Model (MSSM). We are interested in both the asymptotically flat as well as in the asymptotically Anti-de Sitter (AdS) solutions. In particular, we show that for our choice of the potential gravitating, asymptotically flat boson stars exist in (2+1) dimensions. We observe that the behav…
In the system of a gravitating Q-ball, there is a maximum charge $Q_{\rm max}$ inevitably, while in flat spacetime there is no upper bound on $Q$ in typical models such as the Affleck-Dine model. Theoretically the charge $Q$ is a free parameter, and phenomenologically it could increase by charge accumulation. We address a question of what happens to Q-balls if $Q$ is close to $Q_{\rm max}$. First, without specifying a model, we show analytically that inflation cannot take place in the core of a Q-ball, contrary to the claim of previous work. Next, for the Affleck-Dine model, we analyze perturb…
We study solutions of the Einstein-Skyrme model. Firstly we consider test field Skyrmions on the Kerr background. These configurations -- hereafter dubbed Skerrmions -- can be in equilibrium with a Kerr black hole (BH) by virtue of a synchronisation condition. We consider two sectors for Skerrmions. In the sector with non-zero baryon charge, Skerrmions are akin to the known Skyrme solutions on the Schwarzschild background. These `topological' configurations reduce to flat spacetime Skyrmions in a vanishing BH mass limit; moreoever, they never become "small" perturbations on the Kerr background…
