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However it is not always the case. Imagine a bug travelling across a 2-D paper folded into a cone. For him space is not infinite. Mathematically, curvature of a space is given by Riemann Curvature Tensor, whose contraction is Ricii Tensor, and taking its trace yields a scalar called Ricci Scalar or Curvature Scalar. Imagine driving a car on a hilly terrain keeping the steering absolutely straight. The trajectory followed by the car, gives us the notion of geodesics. Geodesics are like straight lines in higher dimensional maybe curved space. Mathematically, geodesics are calculated by solving set of differential equation for each space time component using the equation:.
We are considering an IBVP of the following form,. As before, we assume for simplicity that all coefficients belong to the class of bounded, smooth functions with bounded derivatives.
Continuum and Discrete Initial-Boundary Value Problems and Einstein’s Field Equations
The data consists of the initial data and the boundary data. Compared to the initial-value problem discussed in Section 3 the following new issues and difficulties appear when boundaries are present:. Assuming that u is continuous, for instance, Eqs. If u is continuously differentiable, then taking a time derivative of Eq. Assuming higher regularity of u , one obtains additional compatibility conditions by taking further time derivatives of Eq.
In particular, for an infinitely-differentiable solution u , one has an infinite family of such compatibility conditions at S , and one must make sure that the data f, g satisfies each of them if the solution u is to be reproduced by the IBVP. If an exact solution u 0 of the partial-differential equation 5.
However, depending on the problem at hand, this might be too restrictive. The next issue is the question of what class of boundary conditions 5. In particular, one would like to know, which are the restrictions on the matrix b t, x, u implying existence of a unique solution, provided the compatibility conditions hold.
Additional difficulties appear when the system has constraints, like in the case of electromagnetism and general relativity. There are two common techniques for analyzing an IBVP. The first, discussed in Section 5. This approach, called the Laplace method, is very useful for finding necessary conditions for the well-posedness of linear, constant coefficient IBVPs. Likely, these conditions are also necessary for the quasilinear IBVP, since small-amplitude high-frequency perturbations are essentially governed by the corresponding linearized, frozen coefficient problem.
Based on the Kreiss symmetrizer construction [ ] and the theory of pseudo-differential operators, the Laplace method also gives sufficient conditions for the linear, variable coefficient problem to be well posed; however, the general theory is rather technical. For a discussion and interpretation of this approach in terms of wave propagation we refer to [ ]. The second method, which is discussed in Section 5. It provides a class of boundary conditions, called maximal dissipative, which leads to a well-posed IBVP.
Essentially, these boundary conditions specify data to the incoming normal characteristic fields, or to an appropriate linear combination of the in- and outgoing normal characteristic fields. Although technically less involved than the Laplace one, this method requires the evolution equations 5. In Section 5. Upon linearization and localization, the IBVP 5. We are then back into the case of Section 3 , and we conclude that a necessary condition for the IBVP 5.
In particular, the equation 5. This is the subject of this subsection. Here, with and denoting the Laplace-Fourier and Fourier transform, respectively, of F 0 and f, and is the Laplace-Fourier transform of the boundary data g. In the following, we assume for simplicity that the boundary matrix A is invertible, and that the equation 5. An interesting example with a singular boundary matrix is mentioned in Example 26 below. If A can be inverted, then we rewrite Eq.
Properties of Einstein's equation
We solve this equation subject to the boundary conditions 5. For this, it is useful to have information about the eigenvalues of M s, k.
Lemma 3 [ , , ]. Suppose the equation 5. The eigenvalues are counted according to their algebraic multiplicity. Since the equation 5. According to this lemma, the Jordan normal form of the matrix M s, k has the following form:. Furthermore, N s, k commutes with D s, k. Having cast the IBVP into the ordinary differential system 5. Accordingly, we split. When the most general solution of Eq. In view of the boundary conditions 5. Let us make the following observations:. The condition 5. The violation of condition 5. Therefore, an equivalent necessary condition for well-posedness is that no such simple wave solutions exist.
This is known as the Lopatinsky condition. If such a simple wave solution exists for some s 0 , k 0 , then the homogeneity of the problem implies the existence of a whole family,. In particular, it follows that. Therefore, one has solutions growing exponentially in time at an arbitrarily large rate. Section 8. Introduced into the boundary condition 5. These fields are the ones defined in Eq. In this example, we apply the Lopatinsky condition in order to find necessary conditions for the resulting IBVP to be well posed. With these assumptions and definitions, Laplace-Fourier transformation of the system 3.
The last three equations are purely algebraic and can be used to eliminate the zero speed fields and Open in a separate window from the remaining equations.
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The result is the ordinary differential system. The first block gives. The second block is. On the other hand, Laplace-Fourier transformation of the boundary conditions 5. Introducing into this solutions 5. In the second case, the determinant of the system is. We conclude that the strongly hyperbolic evolution system 3. This case is covered by the results in Section 5.
Next, let us discuss sufficient conditions for the linear, constant coefficient IBVP 5.
Then, we obtain the IBVP. By applying the Laplace-Fourier transformation to it, one obtains the boundary-value problem 5.
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However, in view of the generalization to variable coefficients, one would like to have a method that does not rely on the explicit representation of the solution in Fourier space. Definition 6. The inequality 5.
In view of Eq. Definition 7. Clearly, strong well-posedness in the generalized sense implies boundary stability. The main result is that, modulo technical assumptions, the converse is also true: boundary stability implies strong well-posedness in the generalized sense. Theorem 5. Assume that equation 5. Then, the problem is strongly well posed in the generalized sense if and only if it is boundary stable. Maybe the importance of Theorem 5 is not so much its statement, which concerns only the linear, constant coefficient case for which the solutions can also be constructed explicitly, but rather the method for its proof, which is based on the construction of a smooth symmetrizer symbol, and which is amendable to generalizations to the variable coefficient case using pseudo-differential operators.
Then, we have,. Theorem 6. Furthermore, H can be chosen to be a smooth function of the matrix coefficients of A j and b. First, using Eq. Let us go back to Example 25 of the 2D Dirac equation on the halfspace with boundary condition 5. The solution of Eqs.
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This, together with the results obtained in Example 25, yields the following conclusions: the IBVP 5. Before discussing second-order systems, let us make a few remarks concerning Theorem The boundary stability condition 5.