Construction of Initial Data Sets for Einstein-Scalar and Einstein-Maxwell Equations by Conformally Covariant Split System

We revisit Winicour's affine-null metric initial value formulation of General Relativity, where the characteristic initial value formulation is set up with a null metric having two affine parameters. In comparison to past work, where the application of the formulation was aimed for the timelike-null initial value problem, we consider here a boundary surface that is a null hypersurface. All of the initial data are either metric functions or first derivatives of the metric. Given such a set of initial data, Einstein equations can be integrated in a hierarchical manner, where first a set of equations is solved hierarchically on the null hypersurface serving as a boundary. Second, with the obtained boundary values, a set of differential equations, similar to the equations of the Bondi-Sachs formalism, comprising of hypersurface and evolution equations is solved hierarchically to find the entire space-time metric. An example is shown how the double null Israel black hole solution arises after specification to spherical symmetry and vacuum. This black hole solution is then discussed to with respect to Penrose conformal compactification of spacetime.

Barrow and Sonoda (1986) have investigated the stability, at large time, of certain Bianchi universes. Their method involves studying a set of first-order differential equations which governs the evolution of the three principal expansion rates, and the conservation equations. This set of equations is not in general equivalent to Einstein's equations; in some cases it may be a subset, or an asymptotic approximation. The results given by Barrow and Sonoda refer to the stability of exact solutions of the Einstein equations with respect to perturbations which are governed by this set of first-order equations, rather than by Einstein's equations. The authors show that the results obtained in this way do not reliably determine whether a spacetime is stable to perturbations which evolve according to the full and exact Einstein equations.

The main task of numerical relativity is to solve Einstein equations with the aid of supercomputer. There are two main schemes to write Einstein equations explicitly as differential equations. One is based on 3 + 1 decomposition and reduces the Einstein equations to a Cauchy problem. The another takes the advantage of the characteristic property of Einstein equations and reduces them to a set of ordinary differential equations. The latter scheme is called characteristic formalism which is free of constraint equations in contrast to the corresponding Cauchy problem. Till now there is only one well developed code (PITT code) for characteristic formalism. In PITT code, special finite difference algorithm is adopted for the numerical calculation. And it is this special difference algorithm that restricts the numerical accuracy order to second-order. In addition, this special difference algorithm makes the popular Runge–Kutta method used in Cauchy problem not available. In this paper, we modify the equations for characteristic formalism. Based on our new set of equations, we can use usual finite difference method as done in usual Cauchy evolution. And Runge–Kutta method can also be adopted naturally. We develop a set of code in the framework of AMSS-NCKU code based on our new method and some numerical tests are done.

It is our purpose here to show an unusual relationship between the Einstein equations and the Yang-Mills equations. We give a correspondence between solutions of the self-dual Einstein vacuum equations and the self-dual Yang-Mills equations with a special choice of gauge group. The extension of the argument to the full Yang-Mills equations yields Einstein's unifield equations. We try to incorporate the full Einstein vacuum equations, but the approach is incomplete. We first consider Yang-Mills theory for an arbitrary Lie-algebra with the condition that the connection 1-form and curvature are constant on Minkowski space. This leads to a set of algebraic equations on the connection components. We then specialize the Lie-algebra to be the (infinite dimensional) Lie-algebra of a group of diffeomorphisms of some manifold. The algebraic equations then become differential equations for four vector fields on the manifold on which the diffeomorphisms act. In the self-dual case, if we choose the connection components from the Lie-algebra of the volume preserving 4-dimensional diffeomorphism group, the resulting equations are the same as those obtained by Ashtekar, Jacobsen and Smolin, in their remarkable simplification of the self-dual Einstein vacuum equations. (An alternative derivation of the same equations begins with the self-dual Yang-Mills connection now depending only on the time, then choosing the Lie algebra as that of the volume preserving 3-dimensional diffeomorphisms.) When the reduced full Yang-Mills equations are used in the same context, we get Einstein's equations for his unified theory based on absolute parallelism. To incorporate the full Einsteinvacuum equations we use as the Lie group the semi-direct product of the diffeomorphism group of a 4-dimensional manifold with the group of frame rotations of anSO(1, 3) bundle over the 4-manifold. This last approach, however, yields equations more general than the vacuum equations.

The present article considers time symmetric initial data sets for the vacuum Einstein field equations which in a neighbourhood of infinity have the same massless part as that of some static initial data set. It is shown that the solutions to the regular finite initial value problem at spatial infinity for this class of initial data sets extend smoothly through the critical sets where null infinity touches spatial infinity if and only if the initial data sets coincide with static data in a neighbourhood of infinity. This result highlights the special role played by static data among the class of initial data sets for the Einstein field equations whose development gives rise to a spacetime with a smooth conformal compactification at null infinity.

Anisotropic solutions of the Einstein-Boltzmann equations: I. General formalism

It is argued that the full set of Einstein's field equations has more information than the spatial components of Einstein's equation plus the energy conservation law. Applying the former approach to the D-dimensional decrumpling FRW cosmology recently proposed, it is shown that the spacetime singularity cannot be avoided and that turning points are absent.

This paper extends the work of the preceding paper by including the gravitational (general relativistic) aspects of a magnetic field such as results from an initial uniform current which has been switched off. The to the entire set of Einstein-Maxwell equations are expressed simply in terms of the Maxwell solutions obtained by neglecting gravitation. The latter solutions, in both graphical and analytical form, were found and discussed in the preceding paper; they are not simple functions of space and time. Appropriate initial conditions for the space outside the wire ($\ensuremath{\rho}g1$) are established by setting time derivatives and electric field equal to zero in the Einstein-Maxwell equations, and solving. We then show that the time-dependent are given by $B=\frac{{B}_{M}{h}_{3}}{{h}_{0}}$, and $E=\frac{{E}_{M}{h}_{3}}{{h}_{0}}$, where ${B}_{M}(\ensuremath{\rho}, \ensuremath{\tau})$ and ${E}_{M}(\ensuremath{\rho}, \ensuremath{\tau})$ are the time-dependent Maxwell fields given in the preceding paper, ${h}_{0}$, ${h}_{1}$, ${h}_{2}$, and ${h}_{3}$ are the gravitational-field-determining scale factors ${h}_{\ensuremath{\alpha}}\ensuremath{\equiv}{|{g}_{\ensuremath{\alpha}\ensuremath{\alpha}}|}^{\frac{1}{2}}$ associated with $\ensuremath{\tau}\ensuremath{\equiv}\frac{\mathrm{ct}}{a}$, $\ensuremath{\rho}\ensuremath{\equiv}\frac{r}{a}$, $\ensuremath{\varphi}$, and $\ensuremath{\zeta}\ensuremath{\equiv}\frac{z}{a}$, respectively ($r$, $\ensuremath{\varphi}$, and $z$ correspond to standard cylindrical coordinates); $a$ is the wire radius. The ${h}_{\ensuremath{\alpha}}$ are expressed in terms of the pure Maxwell field quantities ${A}_{M}$ and $V$ and a small universal constant $K\ensuremath{\equiv}{(\frac{4\ensuremath{\pi}G}{{\ensuremath{\mu}}_{0}{c}^{4}})}^{\frac{1}{2}}\ensuremath{\simeq}2.91\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}19}$ m/Wb by the relations ${h}_{3}=\frac{1}{cosh(K{A}_{M})}, {h}_{2}=\frac{{a}_{\ensuremath{\rho}}}{{h}_{3}}, {h}_{1}={h}_{0}=\mathrm{exp}({K}^{2}V){h}_{3},$ ${A}_{M}(\ensuremath{\rho}, \ensuremath{\tau})=\ensuremath{-}{(\mathrm{vector}\mathrm{potential})}_{M}={A}_{M}(\ensuremath{\rho}, 0)+\frac{a}{c}\ensuremath{\int}{0}^{\ensuremath{\tau}}{E}_{M}(\ensuremath{\rho}, s)ds,$ ${A}_{M}(\ensuremath{\rho}, 0)=\frac{1}{2}b({\ensuremath{\rho}}^{2}\ensuremath{-}1) (\ensuremath{\rho}\ensuremath{\le}1), {A}_{M}(\ensuremath{\rho}, 0)=b\mathrm{ln}\ensuremath{\rho} (\ensuremath{\rho}\ensuremath{\ge}1), b\ensuremath{\equiv}a{B}_{M}(1, 0)=\frac{{\ensuremath{\mu}}_{0}I}{2\ensuremath{\pi}},$ $V(\ensuremath{\rho}, t)=(\frac{{\ensuremath{\mu}}_{0}}{\ensuremath{\pi}}) (\mathrm{field}\mathrm{energy}\mathrm{out}\mathrm{to} \ensuremath{\rho} \mathrm{per}\mathrm{unit} z)\ensuremath{-}\frac{1}{4}{b}^{2}={a}^{2}\ensuremath{\int}{0}^{\ensuremath{\rho}}\mathrm{xdx}\left[\frac{{{E}_{M}}^{2}(x, \ensuremath{\tau})}{{c}^{2}}+{{B}_{M}}^{2}(x, \ensuremath{\tau})\right]\ensuremath{-}\frac{1}{4}{b}^{2}.$The methods and order of investigation presented in these two papers can be a practical modus operandi for finding realistic to the Einstein-Maxwell time-dependent equations.

Existence and structure of past asymptotically simple solutions of Einstein's field equations with positive cosmological constant

The energy tensor for a mixture of matter and outflowing radiation is derived, and a set of equations following from Einstein's field equations are written down whose solutions would represent nonstatic radiating spherical distributions. A few explicit analytical solutions are obtained, which describe a distribution of matter and outflowing radiation for $r\ensuremath{\le}a(t)$, an ever-expanding zone of pure radiation for $a(t)\ensuremath{\le}r\ensuremath{\le}b(t)$ and empty space beyond $r=b(t)$. Since $\frac{\mathrm{db}(t)}{\mathrm{dt}}$ is almost equal to 1 and $\frac{\mathrm{da}(t)}{\mathrm{dt}}$ is negative, the solutions obtained represent contracting distributions, but the contraction is not gravitational because $\frac{m}{r}$ is a constant on the boundary $r=a(t)$, $m$ being the mass. The contraction is a purely relativistic effect, the corresponding newtonian distributions being equilibrium distributions. It is hoped that the scheme developed here will be useful in working out solutions which would help in a clear understanding of the initial or the final stages of stellar evolution.

Many evolution problems in physics are described by partial differential equations on an infinite domain; therefore, one is interested in the solutions to such problems for a given initial dataset. A prominent example is the binary black-hole problem within Einstein’s theory of gravitation, in which one computes the gravitational radiation emitted from the inspiral of the two black holes, merger and ringdown. Powerful mathematical tools can be used to establish qualitative statements about the solutions, such as their existence, uniqueness, continuous dependence on the initial data, or their asymptotic behavior over large time scales. However, one is often interested in computing the solution itself, and unless the partial differential equation is very simple, or the initial data possesses a high degree of symmetry, this computation requires approximation by numerical discretization. When solving such discrete problems on a machine, one is faced with a finite limit to computational resources, which leads to the replacement of the infinite continuum domain with a finite computer grid. This, in turn, leads to a discrete initial-boundary value problem. The hope is to recover, with high accuracy, the exact solution in the limit where the grid spacing converges to zero with the boundary being pushed to infinity.The goal of this article is to review some of the theory necessary to understand the continuum and discrete initial boundary-value problems arising from hyperbolic partial differential equations and to discuss its applications to numerical relativity; in particular, we present well-posed initial and initial-boundary value formulations of Einstein’s equations, and we discuss multi-domain high-order finite difference and spectral methods to solve them.

Let us now follow Einstein's suggestion that gravity is a manifestation of spacetime curvature induced by the presence of matter. We must therefore obtain a set of equations that describe quantitatively how the curvature of spacetime at any event is related to the matter distribution at that event. These will be the gravitational field equations , or Einstein equations , in the same way that the Maxwell equations are the field equations of electromagnetism. Maxwell's equations relate the electromagnetic field F at any event to its source, the 4-current density j at that event. Similarly, Einstein's equations relate spacetime curvature to its source, the energy–momentum of matter. As we shall see, the analogy goes further. In any given coordinate system, Maxwell's equations are second-order partial differential equations for the components F µν of the electromagnetic field tensor (or equivalently for the components A µ of the electromagnetic potential). We shall find that Einstein's equations are also a set of second-order partial differential equations, but instead for the metric coefficients g µν of spacetime. The energy–momentum tensor To construct the gravitational field equations, we must first find a properly relativistic (or covariant ) way of expressing the source term . In other words, we must identify a tensor that describes the matter distribution at each event in spacetime.

Exact solutions to Einstein's field equations, which give rise to a Stackel-separable Hamilton-Jacobi equation of the form $$,y,z)\left[ {X(x)\left( {\frac{{\partial S}}{{\partial x}}} \right)^2 - 2\left( {\frac{{\partial S}}{{\partial x}}} \right)\left( {\frac{{\partial S}}{{\partial t}}} \right) - 2\left( {\frac{{\partial S}}{{\partial y}}} \right)\left( {\frac{{\partial S}}{{\partial t}}} \right) + Z(z)\left( {\frac{{\partial S}}{{\partial z}}} \right)^2 - 2\left( {\frac{{\partial S}}{{\partial z}}} \right)\left( {\frac{{\partial S}}{{\partial t}}} \right) - F(x,y,z)\left( {\frac{{\partial S}}{{\partial t}}} \right)^2 } \right] = \lambda $$ are considered. It is shown that there are no solutions for whichD is a function ofx orz, orx andz. The exact solutions are of Petrov typeN and are plane polarized waves without rotation. Some of the solutions are given explicitly, up to two arbitary functions. For these solutions the Hamilton-Jacobi equation is reduced to an uncoupled set of first-order ordinary differential equations.

In this paper we present a new unified theory of electromagnetic and gravitational interactions. By considering a four-dimensional spacetime as a hypersurface embedded in a five-dimensional bulk spacetime, we derive the complete set of field equations in the four-dimensional spacetime from the five-dimensional Einstein field equation. Besides the Einstein field equation in the four-dimensional spacetime, an electromagnetic field equation is derived: $\nabla_a F^{ab}-\xi R^b_{\;\,a}A^a=-4\pi J^b$ with $\xi=-2$, where $F^{ab}$ is the antisymmetric electromagnetic field tensor defined by the potential vector $A^a$, $R_{ab}$ is the Ricci curvature tensor of the hypersurface, and $J^a$ is the electric current density vector. The electromagnetic field equation differs from the Einstein-Maxwell equation by a curvature-coupled term $\xi R^b_{\;\,a}A^a$, whose presence addresses the problem of incompatibility of the Einstein-Maxwell equation with a universe containing a uniformly distributed net charge as discussed in a previous paper by the author [L.-X. Li, Gen. Relativ. Gravit. {\bf 48}, 28 (2016)]. Hence, the new unified theory is physically different from the Kaluza-Klein theory and its variants where the Einstein-Maxwell equation is derived. In the four-dimensional Einstein field equation derived in the new theory, the source term includes the stress-energy tensor of electromagnetic fields as well as the stress-energy tensor of other unidentified matter. Under some conditions the unidentified matter can be interpreted as a cosmological constant in the four-dimensional spacetime. We argue that, the electromagnetic field equation and hence the unified theory presented in this paper can be tested in an environment with a high mass density, e.g., inside a neutron star or a white dwarf, and in the early epoch of the universe.

The existence of outer trapped surfaces in conformally flat, axisymmetric, momentarily static initial data sets for Einstein's equations is investigated. It is shown that none of the level surfaces of the conformal factor can be outer trapped, whenever the minimum value of the circumferences (or of the square roots of the areas) of all the surfaces surrounding the source region is greater than a constant times the Arnowitt-Deser-Misner mass. This result is along the lines of the hoop conjecture. It also provides evidence in favor of the conclusion of Shapiro and Teukolsky, drawn from recent numerical relativity calculations, that the gravitational field on a spacelike hypersurface can become arbitrarily singular without the appearance of an apparent horizon.