The Initial Boundary Value Problem for Einstein’s Vacuum Field Equation
نویسندگان
چکیده
We study the initial boundary value problem for Einstein’s vacuum field equation. We prescribe initial data on an orientable, compact, 3-dimensional manifold S with boundary 6 6= ∅ and boundary conditions on the manifold T = R0 × 6. We assume the boundaries 6 and {0} × 6 of S and T to be identified in the natural way. Furthermore, we prescribe certain gauge source functions which determine the evolution of the fields. Provided that all data are smooth and certain consistency conditions are met on 6, we show that there exists a smooth solution to Einstein’s equation Ric[g] = 0 on a manifold which has (after an identification) a neighbourhood of S in T ∪ S as a boundary. The solution is such that S is space-like, the initial data are induced by the solution on S, and, in the region of T where the solution is defined, T is time-like and the boundary conditions are satisfied.
منابع مشابه
An analytic solution for a non-local initial-boundary value problem including a partial differential equation with variable coefficients
This paper considers a non-local initial-boundary value problem containing a first order partial differential equation with variable coefficients. At first, the non-self-adjoint spectral problem is derived. Then its adjoint problem is calculated. After that, for the adjoint problem the associated eigenvalues and the subsequent eigenfunctions are determined. Finally the convergence ...
متن کاملSolving Some Initial-Boundary Value Problems Including Non-classical Cases of Heat Equation By Spectral and Countour Integral Methods
In this paper, we consider some initial-boundary value problems which contain one-dimensional heat equation in non-classical case. For this problem, we can not use the classical methods such as Fourier, Laplace transformation and Fourier-Birkhoff methods. Because the eigenvalues of their spectral problems are not strictly and they are repeated or we have no eigenvalue. The presentation of the s...
متن کاملSolution of Vacuum Field Equation Based on Physics Metrics in Finsler Geometry and Kretschmann Scalar
The Lemaître-Tolman-Bondi (LTB) model represents an inhomogeneous spherically symmetric universefilledwithfreelyfallingdustlikematterwithoutpressure. First,wehaveconsideredaFinslerian anstaz of (LTB) and have found a Finslerian exact solution of vacuum field equation. We have obtained the R(t,r) and S(t,r) with considering establish a new solution of Rµν = 0. Moreover, we attempttouseFinslergeo...
متن کاملA minimization problem for the lapse and the initial-boundary value problem for Einstein’s field equations
We discuss the initial-boundary value problem of General Relativity. Previous considerations for a toy model problem in electrodynamics motivate the introduction of a variational principle for the lapse with several attractive properties. In particular, it is argued that the resulting elliptic gauge condition for the lapse together with a suitable condition for the shift and constraint-preservi...
متن کاملA model problem for the initial-boundary value formulation of Einstein’s field equations
In many numerical implementations of the Cauchy formulation of Einstein’s field equations one encounters artificial boundaries which raises the issue of specifying boundary conditions. Such conditions have to be chosen carefully. In particular, they should be compatible with the constraints, yield a well posed initial-boundary value formulation and incorporate some physically desirable properti...
متن کامل