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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 Born coordinates ： ウィキペディア英語版 Born coordinates In relativistic physics, the Born coordinate chart is a coordinate chart for (part of) Minkowski spacetime, the flat spacetime of special relativity. It is often used to analyze the physical experience of observers who ride on a ring or disk rigidly rotating at relativistic speeds. This chart is often attributed to Max Born, due to his 1909 work on the relativistic physics of a rotating body – see Born rigidity. ==Langevin observers in the cylindrical chart== To motivate the Born chart, we first consider the family of Langevin observers represented in an ordinary cylindrical coordinate chart for Minkowski spacetime. The world lines of these observers form a timelike congruence which is ''rigid'' in the sense of having a vanishing expansion tensor. They represent observers who rotate rigidly around an axis of cylindrical symmetry. From the line element : we can immediately read off a frame field representing the local Lorentz frames of stationary (inertial) observers :$\backslash vec\_0\; =\; \backslash partial\_T,\; \backslash ;\; \backslash ;\; \backslash vec\_1\; =\; \backslash partial\_Z,\; \backslash ;\; \backslash ;\; \backslash vec\_2\; =\; \backslash partial\_R,\; \backslash ;\; \backslash ;\; \backslash vec\_3\; =\; \backslash frac\; \backslash ,\; \backslash partial\_\backslash Phi$ Here, $\backslash vec\_0$ is a timelike unit vector field while the others are spacelike unit vector fields; at each event, all four are mutually orthogonal and determine the infinitesimal Lorentz frame of the static observer whose world line passes through that event. Simultaneously boosting these frame fields in the $\backslash vec\_3$ direction, we obtain the desired frame field describing the physical experience of the Langevin observers, namely :$\backslash vec\_0\; =\; \backslash frac\; \backslash partial\_\backslash Phi$ :$\backslash vec\_1\; =\; \backslash partial\_Z,\; \backslash ;\; \backslash ;\; \backslash vec\_2\; =\; \backslash partial\_R$ :$\backslash vec\_3\; =\; \backslash frac\; \backslash ,\; \backslash partial\_\backslash Phi\; +\; \backslash frac\_0$ appears in the cylindrical chart as a helix with constant radius (such as the red curve in the figure at right). Suppose we choose one Langevin observer and consider the other observers who ride on a ''ring'' of radius R which is rigidly rotating with angular velocity ω. Then if we take an integral curve (blue helical curve in the figure at right) of the spacelike basis vector $\backslash vec\_3$, we obtain a curve which we might hope can be interpreted as a "line of simultaneity" for the ring-riding observers. But as we see from the figure, ideal clocks carried by these ring-riding observers cannot be synchronized. This is our first hint that it is not as easy as one might expect to define a satisfactory notion of ''spatial geometry'' even for a ''rotating ring'', much less a ''rotating disk!'' Computing the kinematic decomposition of the Langevin congruence, we find that the acceleration vector is :$\backslash nabla\_\; \backslash vec\_0\; =\; \backslash frac\; \backslash ;\; \backslash vec\_2$ This points radially inward and it depends only on the (constant) radius of each helical world line. The ''expansion tensor'' vanishes identically, which means that nearby Langevin observers maintain constant distance from each other. The ''vorticity vector'' is :$\backslash vec\; =\; \backslash frac\; \backslash ;\; \backslash vec\_1$ which is parallel to the axis of symmetry. This means that the world lines of the nearest neighbors of each Langevin observer are ''twisting about its own world line'', as suggested by the figure at right. This is a kind of ''local notion'' of "swirling" or vorticity. In contrast, note that projecting the helices onto any one of the spatial hyperslices $T=T\_0$ orthogonal to the world lines of the static observers gives a circle, which is of course a closed curve. Even better, the coordinate basis vector $\backslash partial\_\backslash Phi$ is a ''spacelike'' Killing vector field whose integral curves are closed spacelike curves (circles, in fact), which moreover degenerate to zero length closed curves on the axis R = 0. This expresses the fact that our spacetime exhibits ''cylindrical symmetry'', and also exhibits a kind of ''global notion'' of the rotation of our Langevin observers. In the figure, the magenta curve shows how the spatial vectors $\backslash vec\_2,\; \backslash ;\; \backslash vec\_3$ are spinning about $\backslash vec\_1$ (which is suppressed in the figure since the Z coordinate is inessential). That is, the vectors $\backslash vec\_2,\; \backslash ;\; \backslash vec\_3$ are not Fermi–Walker transported along the world line, so the Langevin frame is ''spinning'' as well as non-inertial. In other words, in our straightforward derivation of the Langevin frame, we kept the frame aligned with the radial coordinate basis vector $\backslash partial\_R$. By introducing a constant rate rotation of the frame carried by each Langevin observer about $\backslash vec\_1$, we could, if we wished "despin" our frame to obtain a gyrostabilized version. 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「Born coordinates」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.