We present a numerical method for solving Weyl's embedding problem which consists of finding a global isometric embedding of a positively curved and positive-definite spherical 2-metric into the Euclidean three space. The method is based on a construction introduced by Weingarten and was used in Nirenberg's proof of Weyl's conjecture. The target embedding results as the endpoint of an embedding flow in R^3 beginning at the unit sphere's embedding. We employ spectral methods to handle functions on the surface and to solve various (non)-linear elliptic PDEs. Possible applications in 3+1 numerical relativity range from quasi-local mass and momentum measures to coarse-graining in inhomogeneous cosmological models.
We present a new approach to coarse-graining of variables describing dust flow in GR. It is based on assigning quasi-local shear, twist and expansion to 2-dimensional surfaces with the help of isometric embeddings into the 3-dimensional Euclidean space and deriving the time evolution equations for them. In contrast to the popular Buchert's scheme it allows to coarse-grain tensorial quantities in a coordinate-independent way. The framework can be used to estimate backreaction in inhomogeneous cosmological models.
The Killing equation for a general Finsler space is set up. It is showed that the Killing equation of $(α,β)$ space can be divided into two parts. One is the same with Killing equation of a Riemannian metric, another equation can be regarded as a constraint. The solutions of Killing equations present explicitly the isometric symmetry of Finsler space. We find that the isometric group of a special case of $(α,β)$ space is the same with the symmetry of Very Special Relativity (VSR). The Killing vectors of Finsler-Funk space are given. Unlike Riemannian constant curvature space, the 4 dimensional…
We consider gravitational field equations which are Einstein equations written in terms of embedding coordinates in some higher dimensional Minkowski space. Our main focus is to address some tricky issues relating to the Cauchy problem and possible non-equivalence with the intrinsic Einstein theory. The well known theory introduced by Regge and Teitelboim in 9+1 dimensions is cast in Cauchy-Kowalevskaya form and therefore local existence and uniqueness results follow for \emph{analytic} initial data. In seeking a weakening of the regularity conditions for initial data, we are led naturally to …
In the first part of this paper, we extend the result of Li-Wang on the linearized embedding problem to a compact manifold of arbitrary dimension. Using this, we then show that any metric perturbation of a embedded $n$-sphere is also isometrically embedded up to a solution of the homogenous Codazzi equation, irrespective of the ambient geometry. In the second part we specialize to dimension two, and study these results within an ambient Null Cone. Specifically, given a path of metrics on the 2-sphere and an initial isometric embedding, we develop a small parameter existence and uniqueness theo…
The mathematical theory of isometric embedding is applied to study the notion of quasilocal mass in general relativity. In particular, I shall report some recent progress of quasilocal mass with reference to a cosmological spacetime, such as the de Sitter or the Anti-de Sitter spacetime, or a blackhole spacetime, such as the Schwarzschild spacetime. This article is based on joint work with Po-Ning Chen, Ye-Kai Wang, and Shing-Tung Yau.
The definition of quasi-local mass for a bounded space-like region in space-time is essential in several major unsettled problems in general relativity. The quasi-local mass is expected to be a type of flux integral on the boundary two-surface and should be independent of whichever space-like region it bounds. An important idea which is related to the Hamiltonian formulation of general relativity is to consider a reference surface in a flat ambient space with the same first fundamental form and derive the quasi-local mass from the difference of the extrinsic geometries. This approach has been …
We construct a toy model for isothermal spheres in Horava gravity, which includes Einstein's gravity if a parameter is appropriately chosen. The equations for the isothermal spheres are derived from the partition function of the gravitating particle system. We confirm that the Newtonian limit of the system coincides with the model of the well-known isothermal sphere. The stability of the isothermal sphere is found to be sensitive to the energy density at the center of the sphere.
Interior perfect fluid solutions for the Reissner-Nordstrom metric are studied on the basis of a new classification scheme. General formulas are found in many cases. Explicit new global solutions are given as illustrations. Known solutions are briefly reviewed.
Ever since Karl Schwarzschild's 1916 discovery of the spacetime geometry describing the interior of a particular idealized general relativistic star -- a static spherically symmetric blob of fluid with position-independent density -- the general relativity community has continued to devote considerable time and energy to understanding the general-relativistic static perfect fluid sphere. Over the last 90 years a tangle of specific perfect fluid spheres has been discovered, with most of these specific examples seemingly independent from each other. To bring some order to this collection, in thi…
We discuss the physical features of two recent classes of analytical solutions of the Einstein equations sourced by an exotic perfect fluid with equation of state $ P=-ρ/5$. These geometries depend on up to four parameters and are static and spherically symmetric. They describe compact spaces with naked central singularities.
We study the problem of construction of explicit isometric embeddings of (pseudo)-Riemannian manifolds. We discuss the method which is based in the idea that the exterior symmetry of the embedded surface and the interior symmetry of the metric on it must be the same. In case of high enough symmetry of the metric such method allows to transform the metric inducedness condition, which is the one to be solved in order to construct an embedding, into a system of ODEs. It turns out that this method can be generalized to allow the surface to have lower symmetry as long as the above simplification oc…
In this paper we study the isotropic cases of static charged fluid spheres in general relativity. For this purpose we consider two different specialization and under these we solve the Einstein-Maxwell field equations in isotropic coordinates. The analytical solutions thus we obtained are matched to the exterior Reissner-Nordström solutions which concern with the values for the metric coefficients $e^ν$ and $e^μ$. We derive the pressure, density, pressure-to-density ratio at the centre of the charged fluid sphere and boundary $R$ of the star. Our conclusion is that static charged fluid spheres…
We investigate harmonic maps in the context of isometric embeddings when the target space is Ricci-flat and has codimension one. With the help of the Campbell-Magaard theorem we show that any $n$-dimensional ($n\geqslant 3$) Lorentzian manifold can be isometrically and harmonically embedded in a (n+1)-dimensional semi-Riemannian Ricci-flat space. We then extend our analysis to the case when the target space is an Einstein space. Finally, as an example, we work out the harmonic and isometric embedding of a Friedmann-Robertson-Walker spacetime in a five-dimensional Ricci-flat space and proceed t…
In the context of Lorentz-Finsler spacetime theories the relativity principle holds at a spacetime point if the indicatrix (observer space) is homogeneous. We point out that in four spacetime dimensions there are just three kinematical models which respect an exact form of the relativity principle and for which all observers agree on the spacetime volume. They have necessarily affine sphere indicatrices. For them every observer which looks at a flash of light emitted by a point would observe, respectively, an expanding (a) sphere, (b) tetrahedron, or (c) cone, with barycenter at the point. The…
We perturbatively construct asymptotically $\mathbb{R}^{1,3}\times\mathbb{T}^2$ black branes with multiple inhomogeneous directions and show that some of them are thermodynamically preferred over uniform branes in both the microcanonical and canonical ensembles. This demonstrates that, unlike five-dimensional black strings, the instability of some unstable black branes has a plausible endpoint that does not require a violation of cosmic censorship.
It is shown that unlike the perfect fluid case, anisotropic fluids (principal stresses unequal) may be geodesic, without this implying the vanishing of (spatial) pressure gradients. Then the condition of vanishing four acceleration is integrated in non-comoving coordinates. The resulting models are necessarily dynamic, and the mass function is expressed in terms of the fluid velocity as measured by a locally Minkowskian observer. An explicit example is worked out.
For a static, perfect fluid sphere with a general equation of state, we obtain the relativistic equation of hydrostatic equilibrium, namely the Tolman-Oppenheimer-Volkov equation, as the thermodynamical equilibrium in the microcanonical, as well as the canonical, ensemble. We find that the stability condition determined by the second variation of entropy coincides with the dynamical stability condition derived by variations to first order in the dynamical Einstein's equations. Thus, we show the equivalence of microcanonical thermodynamical stability with linear dynamical stability for a static…
In [1], the thermal equilibrium of static, spherically symmetric perfect fluids in General Relativity was studied. I would like to elaborate three points relevant to the results of [1]. The first point is only a clarification, summarized in theorem 1 below, of results that appear in [1]. The following two points correct the error in [1], stating that the condition for thermodynamic stability, found in [1], is referring to the microcanonical ensemble, while it was referring to the canonical one. In theorems 2 and 3, specific cases for which equivalence of dynamical and thermodynamic stability h…
Interior perfect fluid solutions for the Reissner-Nordstrom metric are studied on the basis of a new classification scheme. It specifies which two of the fluid's characteristics are given functions and picks up accordingly one of the three main field equations, the other two being universal. General formulae are found for charged de Sitter solutions, the case of constant energy component of the energy-momentum tensor, the case of known pressure (including charged dust) and the case of linear equation of state. Explicit new global solutions, mainly in elementary functions, are given as illustra…
The condition for the vanishing of the Weyl tensor is integrated in the spherically symmetric case. Then, the resulting expression is used to find new, conformally flat, interior solutions to Einstein equations for locally anisotropic fluids. The slow evolution of these models is contrasted with the evolution of models with similar energy density or radial pressure distribution but non-vanishing Weyl tensor, thereby bringing out the different role played by the Weyl tensor, the local anisotropy of pressure and the inhomogeneity of the energy density in the collapse of relativistic spheres.
In the present paper we consider convection and cracking instabilities as well as their interplay. We develop a simple criterion to identify equations of state unstable to convection, and explore the influence of buoyancy on cracking (or overturning) for isotropic and anisotropic relativistic spheres. We show that a density profile $ρ(r)$, monotonous, decreasing and concave , i.e. $ρ' < 0$ and $ρ'' < 0$, will be stable against convection, if the radial sound velocity monotonically decreases outward. We also studied the cracking instability scenarios and found that isotropic models can be unsta…
The limiting mass of cold white dwarfs was first calculated by E. Stoner in an approximate model of a uniform star and was soon reduced by ~20% in papers by S. Chandrasekhar and L. D. Landau based on an exact solution of the equations for the stellar equilibrium. Here we examine uniform models of white dwarfs taking general relativity effects and the influence of finite temperature into account. Solutions are obtained in the form of finite analytic formulas and, for masses differing by no more than ~20% from the exact solutions, found by numerical integration of the differential equations for …
In this article, Einstein-Maxwell space-time has been considered in connection to some of the astrophysical solutions as previously obtained by Tolman (1939) and Bayin (1978). The effect of inclusion of charge into these solutions has been investigated thoroughly and also the nature of fluid pressure and mass density throughout the sphere have been discussed. Mass-radius and mass-charge relations have been derived for various cases of the charged matter distribution. Two cases are obtained where perfect fluid with positive pressures give rise to electromagnetic mass models such that gravitatio…
The Florides solution, proposed as an alternative to the interior Schwarzschild solution, represents a static and spherically symmetric geometry with vanishing radial stresses. It is regular at the center, and is matched to an exterior Schwarzschild solution. The specific case of a constant energy density has been interpreted as the field inside an Einstein cluster. In this work, we are interested in analyzing the geometry throughout the permitted range of the radial coordinate without matching it to the Schwarzschild exterior spacetime at some constant radius hypersurface. We find an interest…
In a static spherically symmetric Einstein-Maxwell spacetime the class of astrophysical solution found out by Ray and Das (2002) and Pant and Sah (1979) are revisited here in connection to the phenomenological relationship between the gravitational and electromagnetic fields. It is qualitatively shown that the charged relativistic stars of Tolman (1939) and Bayin (1978) type are of purely electromagnetic origin. The existence of this type of astrophysical solutions is a probable extension of Lorentz's conjecture that electron-like extended charged particle possesses only `electromagnetic mass'…
We investigate spacetimes whose light cones could be anisotropic. We prove the equivalence of the structures: (a) Lorentz-Finsler manifold for which the mean Cartan torsion vanishes, (b) Lorentz-Finsler manifold for which the indicatrix (observer space) at each point is a convex hyperbolic affine sphere centered on the zero section, and (c) pair given by a spacetime volume and a sharp convex cone distribution. The equivalence suggests to describe {\em (affine sphere) spacetimes} with this structure, so that no algebraic-metrical concept enters the definition. As a result, this work shows how t…
Alexandrov proved that any simplicial complex homeomorphic to a sphere with strictly non-negative Gaussian curvature at each vertex can be isometrically embedded uniquely in $\mathbb{R}^3$ as a convex polyhedron. Due to the nonconstructive nature of his proof, there have yet to be any algorithms, that we know of, that realizes the Alexandrov embedding in polynomial time. Following his proof, we developed the adiabatic isometric mapping (AIM) algorithm. AIM uses a guided adiabatic pull-back procedure to produce "smooth" embeddings. Tests of AIM applied to two different polyhedral metrics sugges…
In the present paper we prove a uniqueness theorem for the static and asymptotically flat solutions to the Einstein-scalar field equations which possess a photon sphere. We show that such solutions are uniquely specified by their mass $M$ and scalar charge $q$ and that they are isometric to the Janis-Newman-Winicour solution with the same mass and scalar charge subject to the inequality $\frac{q^2}{M^2}<3$.
In the present paper we are willing to model anisotropic star by choosing a new grr metric potential. All the physical parameters like the matter density, radial and transverse pressure and are regular inside the anisotropic star, with the speed of sound less than the speed of light. So the new solution obtained by us gives satisfactory description of realistic astrophysical compact stars. The model of the present paper is compatible with observational data of compact objects like RX J1856-37, Her X-1, Vela X-12 and Cen X-3. A particular model of Her X-1 (Mass 0.98 times solar mass and radius=…
We show that it is possible to obtain credible static anisotropic spherically symmetric matter configurations starting from known density profiles and satisfying a nonlocal equation of state. These particular types of equation of state describe, at a given point, the components of the corresponding energy-momentum tensor not only as a function at that point, but as a functional throughout the enclosed configuration. To establish the physical plausibility of the proposed family of solutions satisfying nonlocal equation of state, we study the constraints imposed by the junction and energy condit…
Nontrivial isometric embeddings for flat metrics (i.e., those which are not just planes in the ambient space) can serve as useful tools in the description of gravity in the embedding gravity approach. Such embeddings can additionally be required to have the same symmetry as the metric. On the other hand, it is possible to require the embedding to be unfolded so that the surface in the ambient space would occupy the subspace of the maximum possible dimension. In the weak gravitational field limit, such a requirement together with a large enough dimension of the ambient space makes embedding gra…
We briefly revisit the Schwarzschild-de Sitter solution in the context of five-dimensional general relativity. We obtain a class of five-dimensional solutions of Einstein vacuum field equations into which the four-dimensional Schwarzschild-de Sitter space can be locally and isometrically embedded. We show that this class of solutions is well-behaved in the limit of lambda approaching zero. Applying the same procedure to the de Sitter cosmological model in five dimensions we obtain a class of embedding spaces which are similarly well-behaved in this limit. These examples demonstrate that the pr…
In this thesis four separate problems in general relativity are considered, divided into two separate themes: coordinate conditions and perfect fluid spheres. Regarding coordinate conditions we present a pedagogical discussion of how the appropriate use of coordinate conditions can lead to simplifications in the form of the spacetime curvature -- such tricks are often helpful when seeking specific exact solutions of the Einstein equations. Regarding perfect fluid spheres we present several methods of transforming any given perfect fluid sphere into a possibly new perfect fluid sphere. This is …
We argue that an arbitrary general relativistic static anisotropic fluid sphere, (static and spherically symmetric but with transverse pressure not equal to radial pressure), can nevertheless be successfully mimicked by suitable linear combinations of theoretically attractive and quite simple classical matter: a classical (charged) isotropic perfect fluid, a classical electromagnetic field, and a classical (minimally coupled) scalar field. While the most general decomposition is not unique, a preferred minimal decomposition can be constructed that is unique. We show how the classical energy co…
We characterize the worldvolume theories on symmetric D-branes in a six-dimensional Cahen-Wallach pp-wave supported by a constant Neveu-Schwarz three-form flux. We find a class of flat noncommutative euclidean D3-branes analogous to branes in a constant magnetic field, as well as curved noncommutative lorentzian D3-branes analogous to branes in an electric background. In the former case the noncommutative field theory on the branes is constructed from first principles, related to dynamics of fuzzy spheres in the worldvolumes, and used to analyse the flat space limits of the string theory. The …
Non-vacuum static spherically-symmetric solutions in Palatini f(R) gravity are examined. It is shown that for generic choices of f(R), there are commonly-used equations of state for which no satisfactory physical solution of the field equations can be found within this framework, apart from in the special case of General Relativity, casting doubt on whether Palatini f(R) gravity can be considered as giving viable alternatives to General Relativity.
Learning Approximate Isometric Embeddings for Efficient Template Placement
Nikhef, National Institute for Nuclear Physics and High-Energy Physics（荷兰国家核物理与高能物理研究所）；CAISR Health, Halmstad University（哈尔姆斯塔德大学）；Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University（乌得勒支大学引力与亚原子物理研究所）；University of Massachusetts Dartmouth（马萨诸塞大学达特茅斯分校）
We consider a way to avoid black hole singularities by gluing a black hole exterior to an interior with a tube-like geometry consisting of a direct product of two-dimensional AdS, dS, or Rindler spacetime with a two-sphere of constant radius. As a result we obtain a spacetime with either "cosmological" or "acceleration" (event) horizons but without an apparent horizon. The inner region is everywhere regular and supported by matter with the vacuum-like equation of state $p_{r}+ρ=0$ where $p_{r}=T_{r}^{r}$ is the longitudinal pressure, $ρ=-T_{0}^{0}$ is the energy density, $T_μ^ν$ is the stress-…
Recent numerical relativity simulations of mergers of binary black holes suggest that the maximum final value of $a/M$ is $\sim 0.95$ for the coalescence of two equal mass black holes with aligned spins of the same magnitude $a/M=0.994$ which is close to the upper limit $a/M=0.998$ of accretion spin-up shown by Thorne. Using the WKB method, we suggest that if quasinormal modes with $a/M \sim 0.95$ are detected by the second generation gravitational wave detectors, we could confirm the strong gravity space-time based on Einstein's general relativity up to $1.33M$ which is only $\sim 1.014$ time…
Analytical and numerical solutions for the integral curves of the velocity field (streamlines) of a steady-state flow of an ideal fluid with $p = ρ$ equation of state are presented. The streamlines associated with an accelerate black hole and a rigid sphere are studied in some detail, as well as, the velocity fields of a black hole and a rigid sphere in an external dipolar field (constant acceleration field). In the latter case the dipole field is produced by an axially symmetric halo or shell of matter. For each case the fluid density is studied using contour lines. We found that the presence…
In relativity, the energy of a moving particle depends on the observer, and the rest mass is the minimal energy seen among all observers. The Wang-Yau quasi-local mass for a surface in spacetime introduced in [7] and [8] is defined by minimizing quasi-local energy associated with admissible isometric embeddings of the surface into the Minkowski space. A critical point of the quasi-local energy is an isometric embedding satisfying the Euler-Lagrange equation. In this article, we prove results regarding both local and global minimizing properties of critical points of the Wang-Yau quasi-local en…
We look for physically realistic initial data in numerical relativity which are in agreement with post-Newtonian approximations. We propose a particular solution of the time-symmetric constraint equation, appropriate to two momentarily static black holes, in the form of a conformal decomposition of the spatial metric. This solution is isometric to the post-Newtonian metric up to the 2PN order. It represents a non-linear deformation of the solution of Brill and Lindquist, i.e. an asymptotically flat region is connected to two asymptotically flat (in a certain weak sense) sheets, that are the im…
If general relativity is spontaneously induced, that is if the reciprocal Newton constant serves as a VEV, the electrically charged black hole limit is governed by a Davidson-Gurwich phase transition which occurs precisely at the would have been outer horizon. The transition profile which connects the exterior Reissner-Nordstrom solution with the novel interior is analytically derived. The inner core is characterized by a vanishing spatial volume and constant surface gravity, and in some respects, resembles a maximally stretched horizon. The Komar mass residing inside any concentric interior s…
