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.
We revisit and generalize, to the Einstein-Yang-Mills-Higgs system, previous results of D. Christodoulou and D. Chae concerning global solutions for the Einstein-scalar field and the Einstein-Maxwell-Higgs equations. The novelty of the present work is twofold. For one thing the assumption on the self-interaction potential is improved. For another thing explanation is furnished why the solutions obtained here and those proved by Chae for the Einstein-Maxwell-Higgs decay more slowly than those established by Christodoulou in the case of self-gravitating scalar fields. Actually this latter phenom…
The Johannsen-Psaltis spacetime explicitly violates the no hair theorem. It describes rotating black holes with scalar hair in the form of parametric deviations from the Kerr metric. In principle, black hole solutions in any modified theory of gravity could be written in terms of the Johannsen-Psaltis metric. We study the accretion of gas onto a static limit of this spacetime. We utilise a recently proposed pseudo-Newtonian formulation of the dynamics around arbitrary static, spherically symmetric spacetimes. We obtain a potential that generalises the Paczyński-Wiita potential to the static Jo…
Two components of spherically symmetric, inhomogeneous dust penetrating each other are introduced as a generalization of the well-known Tolman-Bondi dust solution. The field equations of this model are formulated and general properties are discussed. inhomogeneous Special solutions with additional symmetries - an extra Killing- or homothetic vector - and their matching to the corresponding Tolman-Bondi solution are investigated.
We use zero angular momentum null geodesics in the Kerr-de Sitter spacetime to transform the metric in a generalised Bondi coordinate system. We write the metric components explicitly. Next, we choose the radial coordinate to be the areal coordinate and write the asymptotic metric in the Bondi-Sachs gauge.
We analyze Lemaitre-Tolman-Bondi models in presence of the cosmological constant Λthrough the classical Weierstrass criterion. Precisely, we show that the Weierstrass approach allows us to classify the dynamics of these inhomogeneous spherically symmetric Universes taking into account their relationship with the sign of Λ.
The asymptotic properties of the solutions to the Einstein-Maxwell equations with boost-rotation symmetry and Petrov type D are studied. We find series solutions to the pertinent set of equations which are suitable for a late time descriptions in coordinates which are well adapted for the description of the radiative properties of spacetimes (Bondi coordinates). By calculating the total charge, Bondi and NUT mass and the Newman-Penrose constants of the spacetimes we provide a physical interpretation of the free parameters of the solutions. Additional relevant aspects on the asymptotics and rad…
We provide a formula for estimating the redshift and its secular change (redshift drift) in Lemaître-Tolman-Bondi (LTB) spherically symmetric universes. We compute the scaling of the redshift drift for LTB models that predict Hubble diagrams indistinguishable from those of the standard cosmological model, the flat $Λ$ Cold Dark Matter ($Λ$CDM) model. We show that the redshift drift for these degenerate LTB models is typically different from that predicted in the $Λ$CDM scenario. We also highlight and discuss some unconventional redshift-drift signals that arise in LTB universes and give them d…
We investigate the occurrence and nature of naked singularity for the inhomogeneous gravitational collapse of Tolman-Bondi dust clouds.It is shown that the naked singularities form at the center of the collapsing cloud in a wide class of collapse models which includes the earlier cases considered by Eardley and Smarr and Christodoulou. This class also contains self-similar as well as non-self-similar models. The structure and strength of this singularity is examined and the question is investigated as to when a non-zero measure set of non-spacelike trajectories could be emitted from the singul…
We give the necessary and sufficient (local) conditions for a metric tensor to be a non conformally flat spherically symmetric solution. These conditions exclusively involve explicit concomitants of the Riemann tensor. As a direct application we obtain the {\em ideal} labeling of the Schwarzschild, Reissner-Nordström and Lemaître-Tolman-Bondi solutions.
We review our recent work on a construction of spherically symmetric global solutions to the Einstein--scalar field system with large bounded variation norms and large Bondi masses. We show that similar ideas, together with Christodoulou's short pulse method, allow us to prove the following result: Given $M_i \geq M_f>0$ and $ε>0$, there exists a spherically symmetric (black hole) solution to the Einstein--scalar field system such that up to an error of size $ε$, the initial Bondi mass is $M_i$ and the final Bondi mass is $M_f$. Moreover, if one assumes a continuity property of the final Bondi…
Utilizing the ADM equations, we derive a metric and reduced field equations describing a general, spherically symmetric perfect fluid. The metric describes both the interior perfect fluid region and exterior vacuum Schwarzschild spacetime in a single coordinate patch. The exterior spacetime is in generalized Painleve-Gullstrand coordinates which is an infinite class of coordinate systems. In the static limit the system reduces to a Tolman-Oppenheimer-Volkoff equation on the interior with the exterior in Schwarzschild coordinates. We show the coordinate transformation for the non-static cases t…
We revisit the gravitational collapse of spherically symmetric Lemaître - Tolman - Bondi (LTB) dust models. A sufficient condition for global visibility of singularity is given. This condition also allows us to extend the condition of local visibility to mass functions which are not Taylor expandable near the centre.
The spherically symmetric Einstein-Vlasov system in Schwarzschild coordinates (i.e. polar slicing and areal radial coordinate) is considered. An improved continuation criterion for global existence of classical solutions is given. Two other types of criteria which prevent finite time blow-up are also given.
There are two important statements regarding the Trautman-Bondi mass at null infinity: one is the positivity, and the other is the Bondi mass loss formula, which are both global in nature. In this note, we compute the limit of the Wang-Yau quasi-local mass on unit spheres at null infinity of an asymptotically flat spacetime in the Bondi-Sachs coordinates. The quasi-local mass leads to a local description of the radiation that is purely gravitational at null infinity. In particular, the quasi-local mass is evaluated in terms of the news function of the Bondi-Sachs coordinates.
The Bondi solution, which describes the radial inflow of a gas onto a non-rotating black hole, provides a powerful test for numerical relativistic codes. However, the Bondi solution is usually derived in Schwarzschild coordinates, which are not well suited for dynamical spacetime evolutions. Instead, many current numerical relativistic codes adopt moving-puncture coordinates, which render black holes in trumpet geometries. Here we transform the Bondi solution into trumpet coordinates, which result in regular expressions for the fluid flow extending into the black-hole interior. We also evolve …
By studying the Hawking radiation of the most general static spherically symmetric black hole arising from scalar and Dirac particles tunnelling, we find the Hawking temperature is invariant in the general coordinate representation (\ref{arbitrary1}), which satisfies two conditions: a) its radial coordinate transformation is regular at the event horizon; and b) there is a time-like Killing vector.
We describe general-relativistically a spherically symmetric stationary fluid accretion onto a black hole. Relativistic effects enhance mass accretion, in comparison to the Bondi model predictions, in the case when backreaction is neglected. That enhancement depends on the adiabatic index and the asymptotic gas temperature and it can magnify accretion by one order in the ultrarelativistic regime.
This paper investigates the global properties of a class of spherically symmetric spacetimes. The class contains the maximal development of asymptotically flat spherically symmetric initial data for a wide variety of coupled Einstein-matter systems. For this class, it is proven here that the existence of a single trapped surface or marginally trapped surface implies the completeness of future null infinity and the formation of an event horizon whose area radius is bounded by twice the final Bondi mass.
The Tolman-Bondi (TB) model is defined up to some transformation of a co-moving coordinate but the transformation is not fixed. The use of an arbitrary co-moving system of coordinates leads to the solution dependent on three functions $f, F, {\bf F}$ which are chosen independently in applications. The article studies the transformation rule which is given by the definition of an invariant mass. It is shown that the addition of the TB model by the definition of the transformation rule leads to the separation of the couples of functions ($f, F$) into nonintersecting classes. It is shown that eve…
Generally speaking, there is a negative kinetic energy term in the Lagrangian of the Einstein-Hilbert action of general relativity; On the other hand, the negative kinetic energy term can be vanished by designating a special coordinate system. For general spherically symmetric metric, the question that seeking special coordinate system that satisfies the positive kinetic energy coordinate condition is referred to solving a linear first-order partial differential equation. And then, we present a metric corresponding to the Reissner-Nordstrom solution that satisfies the positive kinetic energy c…
We study the spherically symmetric collapse of a perfect fluid using area-radial coordinates. We show that analytic mass functions describe a static regular centre in these coordinates. In this case, a central singularity can not be realized without an infinite discontinuity in the central density. We construct mass functions involving fluid dynamics at the centre and investigate the relationship between those and the nature of the singularities.
Here we consider our universe as inhomogeneous spherically symmetric Lemaitre-Tolman-Bondi Model and analyze the thermodynamics of this model of the universe. The trapping horizon is calculated and is found to coincide with the apparent horizon. The Einstein field equations are shown to be equivalent with the unified first law of thermodynamics. Finally assuming the first law of thermodynamics validity of the generalized second law of thermodynamics is examined at the apparent horizon for the perfect fluid and at the event horizon for holographic dark energy.
In this short note we comment about some criticisms - appeared in a recent paper by Iguchi et al - to our previous works on gravitational collapse of perfect fluids. We show that those criticisms are incorrect on their own.
We obtain an energy inequality on null surfaces $u=const$ in the Bondi-Sachs formalism. We show that for a sufficiently regular event horizon $H$ there is an affine radial coordinate which is constant on $H$. Then the energy inequality can be prolongated to the horizon giving an estimation which is closely related to the Penrose inequality. We test it for the Kerr solution written in the Fletcher-Lun coordinates.
We transform the metric of an isolated matter source in the multipolar post-Minkowskian approximation from harmonic (de Donder) coordinates to radiative Newman-Unti (NU) coordinates. To linearized order, we obtain the NU metric as a functional of the mass and current multipole moments of the source, valid all-over the exterior region of the source. Imposing appropriate boundary conditions we recover the generalized Bondi-van der Burg-Metzner-Sachs residual symmetry group. To quadratic order, in the case of the mass-quadrupole interaction, we determine the contributions of gravitational-wave ta…
In this paper I study stationary, spherically symmetric accretion of fluids onto a charged black hole in the presence of the cosmological constant. For some isothermal equations of state it is possible to obtain analytic solutions. In the case of a radiation fluid I point out a connection between locations of the horizons and the sonic (critical) points. In specific cases the solutions form closed, binoculars-like trajectories in a phase diagram of the velocity vs. radius.
In the Bondi formulation of the axisymmetric vacuum Einstein equations, we argue that the ``surface area'' coordinate condition determining the ``radial'' coordinate can be considered as part of the initial data and should be chosen in a way that gives information about the physical problem whose solution is sought. For the two-body problem, we choose this coordinate by imposing a condition that allows it to be interpreted, near infinity, as the (inverse of the) Newtonian potential. In this way, two quantities that specify the problem -- the separation of the two particles and their mass ratio…
The characteristic initial (boundary) value problem has numerous applications in general relativity (GR) involving numerical studies, and is often formulated using Bondi-like coordinates. Recently it was shown that several prototype formulations of this type are only weakly hyperbolic. Presently we examine the root cause of this result. In a linear analysis we identify the gauge, constraint and physical blocks in the principal part of the Einstein field equations in such a gauge, and show that the subsystem related to the gauge variables is only weakly hyperbolic. Weak hyperbolicity of the ful…
We present in this paper a new application of the geodesic light-cone (GLC) gauge for weak lensing calculations. Using interesting properties of this gauge, we derive an exact expression of the amplification matrix - involving convergence, magnification and shear - and of the deformation matrix - involving the optical scalars. These expressions are simple and non-perturbative as long as no caustics are created on the past light-cone and are, by construction, free from the thin lens approximation. We apply these general expressions on the example of an Lemaître-Tolman-Bondi (LTB) model with an …
The definition of well-behaved coordinate charts for black hole spacetimes can be tricky, as they can lead for example to either unphysical coordinate singularities in the metric (e.g. $r=2M$ in the Schwarzschild black hole) or to an implicit dependence of the chosen coordinate to physical relevant coordinates (e.g. the dependence of the null coordinates in the Kruskal metric). Here we discuss two approaches for coordinate choices in spherical symmetry allowing us to discuss explicitly "solitary" and spherically symmetric black holes from a regular horizon to null infinity. The first approach …
A metric representing the Kerr geometry has been obtained by Pretorius and Israel. We make a coordinate transformation on this metric, thereby bringing it into Bondi-Sachs form. In order to validate the metric, we evaluate it numerically on a regular grid of the new coordinates. The Ricci tensor is then computed, for different discretizations, and found to be convergent to zero. We also investigate the behaviour of the metric near the axis of symmetry and confirm regularity. Finally we investigate a Bondi-Sachs representation of the Kerr geometry reported by Fletcher and Lun; we confirm numeri…
It is well known that static spherically symmetric spacetimes can admit foliations by flat spacelike hypersurfaces, which are best described in terms of the Painlevè--Gullstrand coordinates. The uniqueness and existence of such foliations were addressed earlier. In this paper, we prove, purely geometrically, that any possible foliation of a static spherically symmetric spacetime by an arbitrary codimension-one spherical spacelike geometry, up to time translation and rotation, is unique, and we find the algebraic condition under which it exists. This leads us to what can be considered as the mo…
Action-angle coordinates are a tool commonly used in celestial mechanics to systematically parametrize and store general solutions of the equations of motion of astrophysical bodies. I perturbatively construct action-angle coordinates for bound test particle motion in static, spherically symmetric space-times using a post-circular expansion. Then I specialise the expressions to the motion in the gravitational fields of Schwarzschild black holes and give explicit formulas for the Hamiltonian to the 10th power in the radial action (20th power in eccentricity), and the transformation to angle coo…
We perform a numerical study of black hole formation from the spherically symmetric collapse of a massless scalar field. The calculations are done in Painlevé-Gullstrand (PG) coordinates that extend across apparent horizons and allow the numerical evolution to proceed until the onset of singularity formation. We generate spacetime maps of the collapse and illustrate the evolution of apparent horizons and trapping surfaces for various initial data. We also study the critical behaviour and find the expected Choptuik scaling with universal values for the critical exponent and echoing period consi…
Formulating a perfect fluid filled spherically symmetric metric utilizing the 3+1 formalism for general relativity, we show that the metric coefficients are completely determined by the mass-energy distribution, and its time rate of change on an initial spacelike hypersurface. Rather than specifying Schwarzschild coordinates for the exterior of the collapsing region, we let the interior dictate the form of the solution in the exterior, and thus both regions are found to be written in one coordinate patch. This not only alleviates the need for complicated matching schemes at the interface, but …
Lemaitre-Tolman-Bondi models as specific spherically symmetric solutions of general relativity simplify in their reduced form some of the mathematical ingredients of black hole or cosmological applications. The conditions imposed in addition to spherical symmetry turn out to take a simple form at the kinematical level of loop quantum gravity, which allows a discussion of their implications at the quantum level. Moreover, the spherically symmetric setting of inhomogeneity illustrates several non-trivial properties of lattice refinements of discrete quantum gravity. Nevertheless, the situation a…
Using the framework for solving the spherically symmetric field equations in observational coordinates given in Araujo and Stoeger (1999), their formulation and solution in the perturbed FLRW sperically symmetric case with observational data representing galaxy redshifts, number counts and observer area distances, both as functions of redshift on our past light cone, are presented. The importance of the central conditions, those which must hold on our world line C, is emphasized. In detailing the solution for these perturbations, we discuss the gauge problem and its resolution in this context,…
In all 2d theories of gravity a conservation law connects the (space-time dependent) mass aspect function at all times and all radii with an integral of the matter fields. It depends on an arbitrary constant which may be interpreted as determining the initial value together with the initial values for the matter field. We discuss this for spherically reduced Einstein-gravity in a diagonal metric and in a Bondi-Sachs metric using the first order formulation of spherically reduced gravity, which allows easy and direct fixations of any type of gauge. The relation of our conserved quantity to the …
We propose a new scheme for extracting gravitational radiation from a characteristic numerical simulation of a spacetime. This method is similar in conception to our earlier work but analytical and numerical implementation is different. The scheme is based on direct transformation to the Bondi coordinates and the gravitational waves are extracted by calculating the Bondi news function in Bondi coordinates. The entire calculation is done in a way which will make the implementation easy when we use uniform Bondi angular grid at $\mathcal I^+$. Using uniform Bondi grid for news calculation has ad…
We review recent results concerning the spherically symmetric Einstein-scalar field system with positive cosmological constant. We do so by comparing with the classical results of Christodoulou concerning the asymptotically flat case (vanishing cosmological constant) and by discussing some of the issues which have emerged since the publication of our main results. Concerning the positive cosmological constant case, we also sketch an alternative proof of global in (Bondi) time existence, based on energy estimates, which is presumably more flexible and, consequently, amenable to generalizations;…
Einstein, Landau-Lifshitz, Papapetrou, Weinberg, and Møller energy-momentum prescriptions in general spherically symmetric space-times are investigated. It is shown that for two special but not unusual classes of general spherically symmetric space-times several energy-momentum prescriptions in Schwarzschild Cartesian coordinates lead to some coincidences in energy distribution. It is also obtained that for a special class of spherically symmetric metrics Møller and Einstein energy-momentum prescriptions give the same result for energy distribution if and only if it has a specific dependence o…
We consider static spherically symmetric solutions of Einstein's equations coupled to an SU(2) Yang Mills field that are smooth at the center of spherical symmetry. We prove that with small cosmological constant there exist solutions that possess a coordinate singularity at some r that is not maximum. The singularity can be removed with a Kruskal-like change of coordinates.
In this work we give a complete picture of how to in a direct simple way define the mass at null infinity in harmonic coordinates in three different ways that we show satisfy the Bondi mass loss law. The first and second way involve only the limit of metric (Trautman mass) respectively the null second fundamental forms along asymptotically characteristic surfaces (asymptotic Hawking mass) that only depend on the ADM mass. The last in an original way involves construction of special characteristic coordinates at null infinity (Bondi mass). The results here rely on asymptotics of the metric deri…
We consider time dependent problem in perturbative approach for a nonrelativistic inviscid spherically symmetric accretion model where the effect of the gravity of the medium is considered in Newtonian gravity framework. We consider spherically symmetric nonrelativistic accretion, i.e. Bondi accretion with self-gravity. Our approach is perturbative in the linear order of perturbation regime. We introduce linear perturbation over the existing steady state solution of the system. The analysis has two features, one is that the linear perturbation in mass accretion rate in such irrotational invisc…
We investigate static and spherically symmetric solutions in the Minimal Theory of Bigravity (MTBG). First, we show that a pair of Schwarzschild-de Sitter spacetimes with different cosmological constants and black hole masses written in the spatially-flat Gullstrand-Painlevé (GP) coordinates is a solution in the self-accelerating branch of MTBG, while it cannot be a solution in the normal branch. We then illustrate how Schwarzschild-de Sitter solutions can become compatible with the normal branch when using different coordinates. We also confirm that the self-accelerating branch of MTBG admits…
We discuss the implications of a wave function for quantum gravity, which involves nothing but 3-dimensional geometries as arguments and is invariant under general coordinate transformations. We derive an analytic wave function from the Wheeler-DeWitt equation for spherically symmetric space-time with the coordinate system arbitrary. The de Broglie-Bohm interpretation of quantum mechanics is applied to the wave function. In this interpretation, deterministic dynamics can be yielded from a wave function in fully quantum regions as well as in semiclassical ones. By introducing a coordinate syste…
The Schwarzschild-deSitter metric is the known solution of Einstein field equations with cosmological constant term for vacuum spherically symmetric space around a point mass M. Recently it has been reported that in a $Lamda$-dominant world the Schwarzschild type coordinate systems are disqualified by redshift-magnitude test as a proper frame of reference(gr-qc/9812092). We derive the solution in a FRW type coordinate system which is qualified according to the mentioned test. Asymptotically it approachs to the non-static form of deSitter metric. The obtained metric is transformable to Schwarzs…
We survey many of the important properties of spherically symmetric spacetimes as follows. We present several different ways of describing a spherically symmetric spacetime and the resulting metrics. We then focus our discussion on an especially useful form of the metric of a spherically symmetric spacetime in polar-areal coordinates and its properties. In particular, we show how the metric component functions chosen are extremely compatible with notions in Newtonian mechanics. We also show the monotonicity of the Hawking mass in these coordinates. As an example, we discuss how these coordinat…
Geometries with horizons offer insights into relationships between general relativity and quantum physics. For static spherically symmetric space-times, the event horizon is coincident with a coordinate anomaly that introduces complications in descriptions of near horizon physics. Naive introduction of dynamics using coordinates with anomalous behavior coincident with the horizon also introduces invariant singular physical content at that horizon. However, the introduction of a temporal coordinate that is non-orthogonal to spatial coordinates near the horizon, but asymptotically orthogonal, pr…
A contribution linear in r to the gravitational potential can be created by a suitable conformal duality transformation: the conformal factor is 1/(1+r)^2 and r will be replaced by r/(1+r), where r is the Schwarzschild radial coordinate. Thus, every spherically symmetric solution of conformal Weyl gravity is conformally related to an Einstein space. This result finally resolves a long controversy about this topic. As a byproduct, we present an example of a spherically symmetric Einstein space which is a limit of a sequence of Schwarzschild-de Sitter space-times but which fails to be expressabl…
We compare relativistic approximation methods, which describe gravitational instability in the expanding universe, in a spherically symmetric model. Linear perturbation theory, second-order perturbation theory, relativistic Zel'dovich approximation, and relativistic post-Zel'dovich approximation are considered and compared with the Lema\^ıtre-Tolman-Bondi solution in order to examine the accuracy of these approximations. We consider some cases of inhomogeneous matter distribution while the homogeneous top-hat model has been usually taken in the previous Newtonian works. It is found that the Ze…
The motion of a spherical dust cloud is described by the Lemaitre-Tolman-Bondi solution and is completely specified by initial values of distributions of the rest mass density and specific energy of the dust fluid. From generic initial conditions of this spherically symmetric collapse, there appears a naked singularity at the symmetric center in the course of the gravitational collapse of the dust cloud. So this might be a counter example to the cosmic censorship hypothesis. To investigate the genericity of this example, we examine the stability of the `nakedness' of this singularity against o…
