The standard "static" spherically symmetric ansatz with perfect fluid source revisited
Ibrahim Semiz
备注：
24 standard LaTeX pages, no figures. Some change in emphasis; changes in parametrizations of some of the solutions (ND2, TD2, TD3, NS1); one new solution (TS4); removal of an incorrect statement (about ND4)
Considering the standard "static" spherically symmetric ansatz ds2 = -B(r) dt2 + A(r) dr2 + r2 dOmega2 for Einstein's Equations with perfect fluid source, we ask how we can interpret solutions where A(r) and B(r) are not positive, as they must be for the static matter source interpretation to be valid.
Noting that the requirement of Lorentzian signature implies A(r) B(r) >0, we find two possible interpretations:
(i) The nonzero component of the source four-velocity does not have to be u0. This provides a connection from the above ansatz to the Kantowski-Sachs (KS) spacetimes.
(ii) Regions with negative A(r) and B(r) of "static" solutions in the literature must be interpreted as corresponding to tachyonic source.
The combinations of source type and four-velocity direction result in four possible cases. One is the standard case, one is identical to the KS case, and two are tachyonic. The dynamic tachyonic case was anticipated in the literature, but the static tachyonic case seems to be new. We derive Oppenheimer-Volkoff-like equations for each case, and find some simple solutions. We conclude that new "simple" black hole solutions of the above form, supported by a perfect fluid, do not exist.
Conformally flat spherically symmetric cosmological models representing a charged perfect fluid as well as a bulk viscous fluid distribution have been obtained. The cosmological constant Λis found positive and is a decreasing function of time which is consistent with the recent supernovae observations. The physical and geometrical properties of the models are discussed.
In this paper we present a detailed calculation of an Ansatz that allows to obtain spherically symmetric Einstein-Dirac configurations in $d$-dimensions. We show that this is possible by combining $2^{\lfloor \frac{d-2}{2} \rfloor}$ Dirac fields, making use of the properties of the angular dependence of the spinors in a spherical background. By applying this Ansatz, we investigate some simple analytical solutions. One of them is a regular wormhole supported by the Dirac fields. Other solutions include a pathological black hole and a naked singularity. We analyze the domain of existence and pro…
We consider a space-time of a spherically symmetric black hole with one simple horizon. As a standard coordinate frame fails in its vicinity, this requires continuation across the horizon and constructing frames which are regular there. Up to now, several standard frames of such a kind are known. It was shown in literature before, how some of them can be united in one picture as different limits of a general scheme. However, some types of frames (the Kruskal-Szekeres and Lema\^ıtre ones) and transformations to them from the original one remained completely disjoint. We show that the Kruskal-Sz…
Recently, Rahaman et al [ Nuovo.Cim 119B, 1115(2004)] have shown that the static spherically symmetric solutions in presence of C-field give rise to wormhole geometry. We highlight some of the characteristics of this wormhole, which have not been considered in the previous study.
Spherically symmetric solutions of generic gravitational models are optimally, and legitimately, obtained by expressing the action in terms of the two surviving metric components. This shortcut is not to be overdone, however: a one-function ansatz invalidates it, as illustrated by the incorrect solutions of [1].
We will show that the Nariai metric, i.e. the static spherically symmetric vacuum spacetime with a cosmological constant, admits a conformally Kerr-Schild spacetime representation. We find the vacuum solutions of the Einstein-Maxwell equations for the Nariai metric using the Horsky-Mitskievich generating conjecture.
The hedgehog ansatz for spherically symmetric spacetimes in self-gravitating nonlinear sigma models and Skyrme models is revisited and its generalization for non-spherically symmetric spacetimes is proposed. The key idea behind our construction is that, even if the matter fields depend on the Killing coordinates in a nontrivial way, the corresponding energy-momentum tensor can still be compatible with spacetime symmetries. Our generalized hedgehog ansatz reduces the Skyrme equations to coupled differential equations for two scalar fields together with several constraint equations between them.…
The hierarchies of both Lovelock gravity and power-Yang-Mills field are combined through gravity in a single theory. In static, spherically symmetric ansatz exact particular integrals are obtained in all higher dimensions. The advantage of such hierarchies is the possibility of choosing coefficients, which are arbitrary otherwise, to cast solutions into tractable forms. To our knowledge the solutions constitute the most general spherically symmetric metrics that incorporate complexities both of Lovelock and Yang-Mills hierarchies within the common context. A large portion of our general class …
Multidimensional model describing the "cosmological" and/or spherically symmetric configuration with n+1 Einstein spaces in the theory with several scalar fields and forms is considered. When electro-magnetic composite p-brane ansatz is adopted, n "internal" spaces are Ricci-flat, one space M_0 has a non-zero curvature, and all p-branes do not "live" in M_0, a class of exact solutions is obtained if certain block-orthogonality relations on p-brane vectors are imposed. A subclass of spherically-symmetric solutions containing non-extremal p-brane black holes is considered. Post-Newtonian paramet…
We discuss the nonlinear origin of the power-law tail in the long-time evolution of a spherically symmetric self-gravitating massless scalar field in even-dimensional spacetimes. Using third-order perturbation method, we derive explicit expressions for the tail (the decay rate and the amplitude) for solutions starting from small initial data and we verify this prediction via numerical integration of the Einstein-scalar field equations in four and six dimensions. Our results show that the coincidence of decay rates of linear and nonlinear tails in four dimensions (which has misguided some tail …
We study the effect of the cosmological constant $Λ$ on the bending of light by a concentrated spherically symmetric mass. Contrarily to previous claims, we show that when the Schwarzschild-de Sitter geometry is taken into account, $Λ$ does indeed contribute to the bending.
We prove uniform decay estimates in the entire exterior of the Schwarzschild black hole for gauge invariant norms on the Yang-Mills fields valued in the Lie algebra associated to the Lie group $SU(2)$. We assume that the initial data are spherically symmetric satisfying a certain Ansatz, and have small energy, which eliminates the stationary solutions which do not decay. In particular, there don't exist any Coulomb type solutions satisfying this Ansatz. We first prove a Morawetz type estimate for the Yang-Mills fields within this setting, using the Yang-Mills equations directly. We then adapt …
We derive spherically symmetric solutions of the classical λ-R model, a minimal, anisotropic modification of general relativity with a preferred foliation and two local degrees of freedom. Starting from a 3 + 1 decomposition of the four-metric in a general spherically symmetric ansatz, we perform a phase space analysis of the reduced model. We show that its constraint algebra is consistent with that of the full λ-R model, and also yields a constant mean curvature or maximal slicing condition as a tertiary constraint. Although the solutions contain the standard Schwarzschild geometry for the ge…
We investigate the spherical accretion process for general static spherically symmetric fluids. We analyze this process by using the general metric ansatz for spherically symmetric black holes. We specialize to the case of normal and phantom isothermal fluids and investigate their accretion process onto normal and phantom Einstein-Maxwell-dilaton black holes. Backreaction effects are discussed.
We investigate the existence of the skyrmion field in the background of an asymptotically flat stationary reflecting star. For this purpose, we consider the Einstein Skyrme system for which there is a skyrmion hair in the black hole case. We discuss spherically symmetric skyrmions and employ the hedgehog ansatz for the skyrmion field. We show that, in contrast to the black hole there is no skyrmion hair for a reflecting star.
When using the black hole exclusion (horizon boundary condition) technique, $K$ is usually nonzero and spatially variable, so none of the special cases of York's conformal-decomposition algorithm apply, and the full 4-vector nonlinear York equations must be solved numerically. We discuss the construction of dynamic black hole initial data slices using this technique: We perturb a known black hole slice via some Ansatz, apply the York decomposition (using another Ansatz for the inner boundary conditions) to project the perturbed field variables back into the constraint hypersurface, and finally…
Spherical reduction of generic four-dimensional theories is revisited. Three different notions of "spherical symmetry" are defined. The following sectors are investigated: Einstein-Cartan theory, spinors, (non-)abelian gauge fields and scalar fields. In each sector a different formalism seems to be most convenient: the Cartan formulation of gravity works best in the purely gravitational sector, the Einstein formulation is convenient for the Yang-Mills sector and for reducing scalar fields, and the Newman-Penrose formalism seems to be the most transparent one in the fermionic sector. Combining …
We investigate the accretion process for different spherically symmetric space-time geometries for a static fluid. We analyse this procedure using the most general black hole metric ansatz. After that, we examine the accretion process for specific spherically symmetric metrics obtaining the velocity of the sound during the process and the critical speed of the flow of the fluid around the black hole. In addition, we study the behaviour of the rate of change of the mass for each chosen metric for a barotropic fluid.
The Einstein-Yang-Mills equations are the source of many interesting solutions within general relativity, including families of particle-like and black hole solutions, and critical phenomena of more than one type. These solutions, discovered in the last thirty years, all assume a restricted form for the Yang-Mills gauge potential known as the "magnetic" ansatz. In this thesis we relax that assumption and investigate the most general solutions of the Einstein-Yang-Mills system assuming spherically symmetry, a Yang-Mills gauge group of SU(2), and zero cosmological constant. We proceed primarily …
In this work we study a more restricted class of quasi-topological gravity theories where the higher curvature terms have no contribution to the equation of motion on general static spherically symmetric metric where $g_{tt} g_{rr} \ne \mathrm{constant}$. We construct such theories up to quintic order in Riemann tensor and observe an important property of these theories: the higher order term in the Lagrangian vanishes identically when evaluated on the most general non-stationary spherically symmetric metric ansatz. This not only signals the higher terms could only have non-trivial effects whe…
In this work spherically symmetric solutions to 5D Kaluza-Klein theory, with "electric" and/or "magnetic" fields are examined. Different relative strengths of the "electric" and "magnetic" charges of the solutions are studied by varying certain parameters in our metric ansatz. As the strengths of these two charges are varied the resultant spacetime exhibits an interesting "evolution".
We study critical behavior in gravitational collapse of a general spherically symmetric Yang-Mills field coupled to the Einstein equations. Unlike the magnetic ansatz used in previous numerical work, the general Yang-Mills connection has two degrees of freedom in spherical symmetry. This fact changes the phenomenology of critical collapse dramatically. The magnetic sector features both type I and type II critical collapse, with universal critical solutions. In contrast, in the general system type I disappears and the critical behavior at the threshold between dispersal and black hole formation…
Brans-Dicke cosmology with an (inverse) power-law potential is revisited in the light of modern quintessence and inflation models. A simple ansatz relating scale factor and scalar field recovers most of the known solutions and generates new ones. A phase space interpretation of the ansatz is provided and these universes are mapped into solutions of f(R) cosmology.
We discuss a procedure to build new sequences of magnetised, equilibrium tori around Kerr black holes which combines two approaches previously considered in the literature. For simplicity we assume that the test-fluid approximation holds, and hence we neglect the self-gravity of the fluid. The models are built assuming a particular form of the angular momentum distribution from which the location and morphology of equipotential surfaces can be computed. This ansatz includes, in particular, the constant angular momentum case originally employed in the construction of thick tori - or Polish doug…
In the companion paper [Phys. Rev. D 103 (2021) 10, [2101.02951]] we have derived the short-ranged potentials for the Teukolsky equations for massless spins $(0,1/2,1,2)$ in general spherically-symmetric and static metrics. Here we apply these results to numerically compute the Hawking radiation spectra of such particles emitted by black holes (BHs) in three different ansatz: charged BHs, higher-dimensional BHs, and polymerized BHs arising from models of quantum gravity. In order to ensure the robustness of our numerical procedure, we show that it agrees with newly derived analytic formulas fo…
We investigate the general relativistic collapse of spherically symmetric, massless spin-1/2 fields at the threshold of black hole formation. A spherically symmetric system is constructed from two spin-1/2 fields by forming a spin singlet with no net spin-angular momentum. We study the system numerically and find strong evidence for a Type II critical solution at the threshold between dispersal and black hole formation, with an associated mass scaling exponent $γ~ 0.26$. Although the critical solution is characterized by a continuously self-similar (CSS) geometry, the matter fields exhibit dis…
We present a new shear free model for the gravitational collapse of a spherically symmetric charged body. We propose a dissipative contraction with radiation emitted outwards. The Einstein field equations, using the junction conditions and an ansatz, are integrated numerically. A check of the energy conditions is also performed. We obtain that the charge delays the black hole formation and it can even halt the collapse.
We consider the Einstein-Yang-Mills Lagrangian in a (4+n)-dimensional space-time. Assuming the matter and metric fields to be independent of the n extra coordinates, a spherical symmetric Ansatz for the fields leads to a set of coupled ordinary differential equations. We find that for n > 1 only solutions with either one non-zero Higgs field or with all Higgs fields constant exist. We construct the analytic solutions which fulfill this conditions for arbitrary n, namely the Einstein-Maxwell-dilaton solutions. We also present generic solutions of the effective 4-dimensional Einstein-Yang-Mills-…
Exact solutions of the Einstein's field equations describing a spherically symmetric cosmological model without a big bang or any other kind of singularity recently obtained by Dadhich and Patel (2000) are revisited. The matter content of the model is a shear-free perfect fluid with isotropic pressure and a radial heat flux. Three different exact solutions are obtained for both perfect fluid and fluid with bulk viscosity. It turns out that the cosmological rerm $Λ(t)$ is a decreasing function of time, which is consistent with recent observations of type Ia supernovae.
We look for "static" spherically symmetric solutions of Einstein's Equations for perfect fluid source with equation of state $p=wρ$. In order to include the possibilities of recently popularized dark energy and phantom energy possibly pervading the spacetime, we put no constraints on the constant $w$. We consider all four cases compatible with the standard ansatz for the line element, discussed in previous work. For each case we derive the equation obeyed by the mass function or its analogs. For these equations, we find {\em all} finite-polynomial solutions, including possible negative powers.…
Effective gravitational field equations on a four dimensional brane embedded in a five dimensional bulk have been considered. Using the Einstein-Hilbert action along with the Gauss-Bonnet correction term, we have derived static spherically symmetric vacuum solution to the effective field equations, first order in the Gauss-Bonnet coupling parameter. The solution so obtained, has one part corresponding to general relativity with an additional correction term, proportional to the Gauss-Bonnet coupling parameter. The correction term modifies the spacetime structure, in particular, the location of…
A class of solutions describing the interior of a static spherically symmetric compact anisotropic star is reported. The analytic solution has been obtained by utilizing the Finch and Skea ({\it Class. Quant. Grav.} {\bf 6} (1989) 467) ansatz for the metric potential $g_{rr}$ which has a clear geometric interpretation for the associated background space-time. Based on physical grounds appropriate bounds on the model parameters have been obtained and it has been shown that the model admits an equation of state (EOS) which is quadratic in nature.
We derive the equations governing static, spherically symmetric vacuum solutions to the Einstein equations, as modified by the frame-dependent effective action arising from trace dynamics. We give analytic and numerical results for the solutions of these equations, first in polar coordinates, and then in isotropic coordinates. General features of the static case are that (i) there is no horizon, since $g_{00}$ is non-vanishing for finite values of the polar radius, and only vanishes (in isotropic coordinates) at the internal singularity, (ii) the Ricci scalar $R$ vanishes identically, and (iii…
In this article, we present a class of relativistic solutions describing spherically symmetric and static anisotropic stars in hydrostatic equilibrium. For this purpose, we consider a particularized matric potential, namely, Buchdahl ansatz [Phys. Rev. D \textbf{116}, 1027 (1959).] which encompasses almost all the known analytic solutions to the spherically symmetric, static Einstein field equations(EFEs) with a perfect fluid source, including in particular the Vaidya-Tikekar and Finch-Skea. We here developed the model by considering an anisotropic spherically symmetric static general relativi…
We study static, spherically symmetric vacuum solutions to Quadratic Gravity, extending considerably our previous Rapid Communication [Phys. Rev. D 98, 021502(R) (2018)] on this topic. Using a conformal-to-Kundt metric ansatz, we arrive at a much simpler form of the field equations in comparison with their expression in the standard spherically symmetric coordinates. We present details of the derivation of this compact form of two ordinary differential field equations for two metric functions. Next, we apply analytical methods and express their solutions as infinite power series expansions. We…
We study classical solutions in the SU(2) Einstein-Yang-Mills-Higgs theory. The spherically symmetric ansätze for all fields are given and the equations of motion are derived as a system of ordinary differential equations. The asymptotics and the boundary conditions at space origin for regular solutions and at event horizon for black hole solutions are studied. Using the shooting method, we found numerical solutions to the theory. For regular solutions, we find two new sets of asymptotically flat solutions. Each of these sets contains continua of solutions in the parameter space spanned by the…
We present a detailed study of the spherically symmetric solutions in Lorentz breaking massive gravity. There is an undetermined function $\mathcal{F}(X, w_1, w_2, w_3)$ in the action of Stückelberg fields $S_ϕ=Λ^4\int{d^4x\sqrt{-g}\mathcal{F}}$, which should be resolved through physical means. In the general relativity, the spherically symmetric solution to the Einstein equation is a benchmark and its massive deformation also play a crucial role in Lorentz breaking massive gravity. $\mathcal{F}$ will satisfy the constraint equation $T_0^1=0$ from the spherically symmetric Einstein tensor $G_0…
We present a detailed study of the static spherically symmetric solutions in de Rham-Gabadadze-Tolley (dRGT) theory. Since the diffeomorphism invariance can be restored by introducing the Stückelberg fields $ϕ^a$, there is new invariant $I^{ab}=g^{μν}\partial_μϕ^a\partial_νϕ^b$ in the massive gravity, which adds to the ones usually encountered in general relativity (GR). In the unitary gauge $ϕ^a=x^μδ_μ^a$, any inverse metric $g^{μν}$ that has divergence including the coordinate singularity in GR would exhibit a singularity in the invariant $I^{ab}$. Therefore, there is no conventional Schwarz…
We study a self-interacting scalar field theory coupled to gravity and are interested in spherically symmetric solutions with a regular origin surrounded by a horizon. For a scalar potential containing a barrier, and using the most general spherically symmetric ansatz, we show that in addition to the known static, oscillating solutions discussed earlier in the literature there exist new classes of solutions which appear in the strong field case. For these solutions the spatial sphere shrinks either beyond the horizon, implying a collapsing universe outside of the cosmological horizon, or it sh…
Noether and Mei symmetries in static spherically symmetric quadratic gravity: variational consistency, constraints, and conserved curvature flux
The British University in Egypt（埃及英国大学）；North-West University（西北大学）；Imam Mohammad Ibn Saud Islamic University (IMSIU)（伊马姆穆罕默德·本·沙特伊斯兰大学）；Fukushima University（福岛大学）
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'…
Spherically symmetric relativistic stars with the polytropic equation of state, which possess the local pressure anisotropy, are considered in the context of general relativity. The modified Lane-Emden equations are derived for the special ansatz for the anisotropy parameter $Δ$ in the form of the differential relation between $Δ$ and the metric function $ν$. The analytical solutions of the obtained equations are found for incompressible fluid stars. The dynamical stability of incompressible anisotropic fluid stars against radial oscillations is studied.
We study spherically symmetric, asymptotically flat black hole solutions in the low-energy effective heterotic string theory, which is the Einstein gravity with Gauss-Bonnet term and the dilaton, in various dimensions. We derive the field equations for suitable ansatz for general D dimensions and construct black hole solutions of various masses numerically in D=4,5,6 and 10 dimensional spacetime with (D-2)-dimensional hypersurface with positive constant curvature. A detailed comparison with the non-dilatonic solutions is made. We also examine the thermodynamic properties of the solutions. It i…
