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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Nordström's theory of gravitation ： ウィキペディア英語版 Nordström's theory of gravitation In theoretical physics, Nordström's theory of gravitation was a predecessor of general relativity. Strictly speaking, there were actually ''two'' distinct theories proposed by the Finnish theoretical physicist Gunnar Nordström, in 1912 and 1913 respectively. The first was quickly dismissed, but the second became the first known example of a metric theory of gravitation, in which the effects of gravitation are treated entirely in terms of the geometry of a curved spacetime. Neither of Nordström's theories are in agreement with observation and experiment. Nonetheless, the first remains of interest insofar as it led to the second. The second remains of interest both as an important milestone on the road to the current theory of gravitation, general relativity, and as a simple example of a self-consistent relativistic theory of gravitation. As an example, this theory is particularly useful in the context of pedagogical discussions of how to derive and test the predictions of a metric theory of gravitation. ==Development of the theories== Nordström's theories arose at a time when several leading physicists, including Nordström in Helsinki, Max Abraham in Milan, Gustav Mie in Greifswald, Germany, and Albert Einstein in Prague, were all trying to create competing relativistic theories of gravitation. All of these researchers began by trying to suitably modify the existing theory, the field theory version of Newton's theory of gravitation. In this theory, the field equation is the Poisson equation $\backslash Delta\; \backslash phi\; =\; 4\; \backslash pi\; \backslash rho$, where $\backslash phi$ is the gravitational potential and $\backslash rho$ is the density of matter, augmented by an equation of motion for a test particle in an ambient gravitational field, which we can derive from Newton's force law and which states that the acceleration of the test particle is given by the gradient of the potential :$\backslash frac\; =\; -\backslash nabla\; \backslash phi$ This theory is not relativistic because the equation of motion refers to coordinate time rather than proper time, and because, should the matter in some isolated object suddenly be redistributed by an explosion, the field equation requires that the potential everywhere in "space" must be "updated" ''instantaneously'', which violates the principle that any "news" which has a physical effect (in this case, an effect on test particle motion far from the source of the field) cannot be transmitted faster than the speed of light. Einstein's former calculus professor, Hermann Minkowski had sketched a vector theory of gravitation as early as 1908, but in 1912, Abraham pointed out that no such theory would admit stable planetary orbits. This was one reason why Nordström turned to scalar theories of gravitation (while Einstein explored tensor theories). Nordström's first attempt to propose a suitable relativistic scalar field equation of gravitation was the simplest and most natural choice imaginable: simply replace the Laplacian in the Newtonian field equation with the D'Alembertian or wave operator, which gives $\backslash Box\; \backslash phi\; =\; 4\; \backslash pi\; \backslash ,\; \backslash rho$. This has the result of changing the vacuum field equation from the Laplace equation to the wave equation, which means that any "news" concerning redistribution of matter in one location is transmitted at the speed of light to other locations. Correspondingly, the simplest guess for a suitable equation of motion for test particles might seem to be $\backslash dot\_a\; =\; -\backslash phi\_$ where the dot signifies differentiation with respect to proper time, subscripts following the comma denote partial differentiation with respect to the indexed coordinate, and where $u^a$ is the velocity four-vector of the test particle. This force law had earlier been proposed by Abraham, and Nordström knew that it wouldn't work. Instead he proposed $\backslash dot\_a\; =\; -\backslash phi\_\; -\; \backslash dot\; \backslash ,\; u\_a$. However, this theory is unacceptable for a variety of reasons. Two objections are theoretical. First, this theory is not derivable from a Lagrangian, unlike the Newtonian field theory (or most metric theories of gravitation). Second, the proposed field equation is linear. But by analogy with electromagnetism, we should expect the gravitational field to carry energy, and on the basis of Einstein's work on relativity theory, we should expect this energy to be equivalent to mass and therefore, to gravitate. This implies that the field equation should be ''nonlinear''. Another objection is more practical: this theory disagrees drastically with observation. Einstein and von Laue proposed that the problem might lie with the field equation, which, they suggested, should have the linear form $F\; T\_\; =\; \backslash rho$, where F is some yet unknown function of $\backslash phi$, and where Tmatter is the trace of the stress–energy tensor describing the density, momentum, and stress of any matter present. In response to these criticisms, Nordström proposed his second theory in 1913. From the proportionality of inertial and gravitational mass, he deduced that the field equation should be $\backslash phi\; \backslash ,\; \backslash Box\; \backslash phi\; =\; -4\; \backslash pi\; \backslash ,\; T\_$, which is nonlinear. Nordström now took the equation of motion to be :$\backslash frac\; =\; -\backslash phi\_$ or $\backslash phi\; \backslash ,\; \backslash dot\_a\; =\; -\backslash phi\_\; -\; \backslash dot\; \backslash ,\; u\_a$. Einstein took the first opportunity to proclaim his approval of the new theory. In a keynote address to the annual meeting of the Society of German Scientists and Physicians, given in Vienna on September 23, 1913, Einstein surveyed the state of the art, declaring that only his own work with Marcel Grossmann and the second theory of Nordström were worthy of consideration. (Mie, who was in the audience, rose to protest, but Einstein explained his criteria and Mie was forced to admit that his own theory did not meet them.) Einstein considered the special case when the only matter present is a cloud of ''dust'' (that is, a perfect fluid in which the pressure is assumed to be negligible). He argued that the contribution of this matter to the stress–energy tensor should be: :$\backslash left(\; T\_\; \backslash right)\_\; =\; \backslash phi\; \backslash ,\; \backslash rho\; \backslash ,\; u\_a\; \backslash ,\; u\_b$ He then derived an expression for the stress–energy tensor of the gravitational field in Nordström's second theory, :$4\; \backslash pi\; \backslash ,\; \backslash left(\; T\_\; \backslash right)\_\; =\; \backslash phi\_\; \backslash ,\; \backslash phi\_\; -\; 1/2\; \backslash ,\; \backslash eta\_\; \backslash ,\; \backslash phi\_\; \backslash ,\; \backslash phi^$ which he proposed should hold in general, and showed that the sum of the contributions to the stress–energy tensor from the gravitational field energy and from matter would be ''conserved'', as should be the case. Furthermore, he showed, the field equation of Nordström's second theory follows from the Lagrangian :$L\; =\; \backslash frac\; \backslash ,\; \backslash eta^\; \backslash ,\; \backslash phi\_\; \backslash ,\; \backslash phi\_\; -\; \backslash rho\; \backslash ,\; \backslash phi$ Since Nordström's equation of motion for test particles in an ambient gravitational field also follows from a Lagrangian, this shows that Nordström's second theory can be derived from an action principle and also shows that it obeys other properties we must demand from a self-consistent field theory. Meanwhile, a gifted Dutch student, Adriaan Fokker had written a Ph.D. thesis under Hendrik Lorentz in which he derived what is now called the Fokker-Planck equation. Lorentz, delighted by his former student's success, arranged for Fokker to pursue post-doctoral study with Einstein in Prague. The result was a historic paper which appeared in 1914, in which Einstein and Fokker observed that the Lagrangian for Nordström's equation of motion for test particles, $L\; =\; \backslash phi^2\; \backslash ,\; \backslash eta\_\; \backslash ,\; \backslash dot^a\; \backslash ,\; \backslash dot^b$, is the geodesic Lagrangian for a curved Lorentzian manifold with metric tensor $g\_\; =\; \backslash phi^2\; \backslash ,\; \backslash eta\_$. If we adopt Cartesian coordinates with line element $d\backslash sigma^2\; =\; \backslash eta\_\; \backslash ,\; dx^a\; \backslash ,\; dx^b$ with corresponding wave operator $\backslash Box$ on the flat background, or Minkowski spacetime, so that the line element of the curved spacetime is $ds^2\; =\; \backslash phi^2\; \backslash ,\; \backslash eta\_\; \backslash ,\; dx^a\; \backslash ,\; dx^b$, then the Ricci scalar of this curved spacetime is just :$R\; =\; -\backslash frac$ Therefore Nordström's field equation becomes simply :$R\; =\; 24\; \backslash pi\; \backslash ,\; T$ where on the right hand side, we have taken the trace of the stress–energy tensor (with contributions from matter plus any non-gravitational fields) using the metric tensor $g\_$. This is a historic result, because here for the first time we have a field equation in which on the left hand side stands a purely geometrical quantity (the Ricci scalar is the trace of the Ricci tensor, which is itself a kind of trace of the fourth rank Riemann curvature tensor), and on the right hand stands a purely physical quantity, the trace of the stress–energy tensor. Einstein gleefully pointed out that this equation now takes the form which he had earlier proposed with von Laue, and gives a concrete example of a class of theories which he had studied with Grossmann. Some time later, Hermann Weyl introduced the Weyl curvature tensor $C\_$, which measures the deviation of a Lorentzian manifold from being ''conformally flat'', i.e. with metric tensor having the form of the product of some scalar function with the metric tensor of flat spacetime. This is exactly the special form of the metric proposed in Nordström's second theory, so the entire content of this theory can be summarized in the following two equations: :$R\; =\; 24\; \backslash pi\; \backslash ,\; T,\; \backslash ;\; \backslash ;\; \backslash ;\; C\_\; =\; 0$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Nordström's theory of gravitation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.