外文翻译---允许的角速度模式和预地平线制度

外文文献及翻译 外文资料原文： The Permitted Angular Velocity Pattern And The Pre-horizon Regime* Fernando de Felice Abstract There exist mechanical effects which allow an observer on circular its around ametric source to recognize the likely closeness of an event horizon. When these effects manifest lhemselves the spacetime is said to be in a pre-horizon regime. Here this concept will be made more precise in the case of the Kerr metric. Introduction A question of physical interest is whether an observer in orbit around a black hole can realize from within his rest frame, how close he is to the event horizon. If the observer is moving on spatially circular trajectories then there are at least three mechanical effects which signal the likely proximity to an event horizon. These are: (i) The thrust needed to keep the orbit circular, increases outward with the modulus of the orbital frequency of revolution. (ii) The local inertial compass (a gyroscope) precesses forwards with respect to the orbital frequency of revolution. (iii) The forward gyroscopic precession increases with the modulus of the orbital frequency of revolution. If we call f the spatial acceleration of the orbit (the specific thrust), R the angular frequency of the orbital revolution as would be measured at infinity, Q2 the angular frequency of precession of the local compass of inertia and S2* QG/S2l, conditions (i) to (iii) can be expressed respectively as: 1 When all these properties are satisfied the spacetime is then said to be in a pre-horizon regime. Evidently conditions (1) are meaningful only if the observer is able to measure the Work partially supparted by the Italian Space Agency under contract AS191-RS-49.three parameters f, z2 and z2 and also able to distinguish between inwards and outwards with respect to the metric source. z2 is the most difficult quantity to determine without some interaction with a distant observer; to know C in fact one should know its modulus and its sign with respect to a local clockwise direction. It is likely that within his orbiting frame the observer is able to measure only some function of a, in which case it will depend on the specific case under investigation whether the observer is able to verify conditions (1) or not. In Schwarzschild spacetime, for example, observers moving on a circular orbit can determine, with a suitable measurement of the local stresses, the direction of the orbital revolution and the modulus of its proper frequency, namely z2 multiplied by the redshift factor (de Felice et a1 1993). Since the redshift factor is positive, conditions (1) for a pre horizon in Schwanschild spacetime can be reformulated in terms of the proper frequency, allowing the orbiting observer to identify without ambiguities his spatial position. In the Kerr metric, on the contrary, it is not yet clear how one should determine the angular frequency of the orbital revolution Q or a convenient function of it, therefore I shall assume, for the purpose of the following discussion, that all the parameters in (I) are known a priori. Conditions (1) tell the orbiting observer that if an event horizon exists it is going to be close, but not how close; moreover, they may hold without an event horizon really existing. This situation occurs in the over-charged Reissner-Nordstrom spacetime and inside a static, incompressible, spherical star, described by the internal Schwarzschild solution (de Felice 1991) and 2 also in the Kerr naked singularity solution (de Felice et al 1991). In the Ken metric the identification of the prehorizon made in de Felice et al (1991) included a region where not all of conditions (1) were satisfied. I will deduce here the correct pre-horizon structure (section 3) and suggest an operational procedure which allows the orbiting observer to recognize that his orbit is co-rotating with the metric source, that the local outward direction (direction away from the source) is concordant with the sense of the physical thrust and what his position is with respect to the photon circular orbits. I will further bring into evidence a mode of behaviour of the orbiting gyroscope, which could be interpreted as a memory by the background geometry of the static case it deviates from. Evidently the observer is assumed to know a priori that the background geometry is given by the Kerr m e acn d that he is on a spatially circular equatorial orbit. The units are such that c = 1 = G; Greek indices run from 0 to 3. Kerr metric: circular orbits and the permitted angular velocity pattern In Boyer and Lindquist coordinates (Misner et al 1973), the Ken metric is described by the line element: where a and m are the specific angular momentum and total mass of the metric source respectively, and: The family of timelike spatially circular orbits in spacetime (1) is given by: where k and m are the time and the axial Killing vectors respectively; R = u+/u is, as stated, the angular frequency of revolution, and e* is the redshift factor which is given by: 3 I shall limit my considerations to the equatorial plane (6 = x/2). he requirement that U=be timelike constrains R to the range: where: are solutions of the equation e-u = 0. It is convenient to introduce a new variable: The plots of the functions y = yc+(r,a), limited to the region outside the outer event horizon, are shown in figure 2. The extrema of yc+ occur at the photon circular orbits whose radial coordinates will be denoted as r+ and whose location is fixed by the equation: where： 4 Figure 1 which is the locus of the event horimn To complete the analysis of f(y; r ) we need to find its critical points. From (10) we have: Figure 2. The plots limited to the region in the equatorial plane which is outside 5 the angular frequency of gravitational drag. From (7) and (15) this occurs when: As stated in the introduction, in a pre-horizon R* should increase with y ,hence we search for where this is true. From (17) and the definition of R*, we have: The solutions of (20) are given by: In order to recognize the analytical behaviour of (23) we need to compare (21) and (22)with the other functions a2(r) which were essential in order to draw the PAv-pattern of figure 2. It is straightforward to find that: 6 Conclusion The main question which motivated the present investigation is bow an observer orbiting around a black hole can identify his spatial position with respect to the metric source, by means of measurements performed only within his rest frame (a space ship, say). More precisely, one may ask what is the minimum amount of a priori information necessary in order to set up, with local experiments, a non-ambiguous orientation in spacetime without interacting with a set of different observers.Evidently it is essential to know how mechanical systems, such as a physical thrust and a gyroscope, respond to a change of the orbital frequency of revolution for the localization of the orbit in the PAV-pattern. For this reason I have discussed the structure of the PAV-pattern in the equatorial plane of the Kerr metric. The pre-horizon is the most interesting part of this structure, since it hosts, loosely speaking, a gravitational field, the strength of which foreshadows the likely closeness to an event horizon. While this criterion is unambiguous in Schwarzschild and Reissner-Nordstrom spacetimes (de Felice 1991; de Felice et al 1993), in the Kerr metric the rotation of the source contributes to an apparent weakening of the gravitational strength for co-rotating orbits, causing a shrinking of the pre-horizon regime. However, monitoring the behaviour of the physical thrust with the angular frequency of revolution S2, one can still recover the necessary information about the observers position with respect to an event horizon. For this to be possible one has to 7 determine, from within the orbiting frame, the direction of the orbital revolution with respect to a local clockwise direction and the modulus of its proper frequency. This and the extention to electromagnetic types of measurements is now a matter of investigation. 8 外文资料 翻译 ： 允许的角速度模式和预地平线制度 费尔南多德费利切 摘要 本文假设 允许存在一个观察员及其周边参数回归源 认识到事件的时候可能产生 力学效应。当这些影响体现 在 时空 中的时候就被认为是在预地平线制度。在这里角速度 这个概念会更加精确度量。 简介 物理学中有 兴趣的 一个 问题是在黑洞周围的轨道观测器是否 可以实现从框架内的构想，无论它是是多么接近 事件视界。 但当 观察员 在圆形轨道上移动时，至少有三个力学效应该信号可能接近事件视界。它们是： （ 1）需要保持圆形轨道推力，随模向外 延伸 的轨道频率。 （ 2）地方惯性罗盘（陀螺 仪） 关于转发轨道频率的 延伸 频率。 （ 3）与轨道 向前陀螺延伸度 模量增加 的频率 。 如果我们调用 f 轨道的空间加速度（具体推力），假设 角 速度的轨道测量频率 将在无穷远， 那么 R 的角频率惯性和 Q2相关 ， 这时 条件（ 1）至（ 3）可分别表示为： 当所有这些属性都处于时空的时候，就说是在预地平线制度。显然条件（ 1）才有意义，假设观察者还能够到测量工作部分，那么假设 z2 是最困难的数量，就能够确定一些遥远的观察者的相互信息交流 ;掌握每个人都应该知道它的模量及其与尊重当地的顺时针方向的标志。很可能在它 的轨道帧的观察员能够测量 的只有 一个函数，在这种情况下，将取决于 对所调查的具体情况 是否能够验证观测条件 （ 1）。 在史瓦西时空，例如，观察员 在 圆形轨道上移动 ，那么 通过适当的9 局部应力测量来估算 轨道方向 的延伸和适当的频率，其模量 即由红移乘以 z2 因子。由于红移因素是积极的，预条件（ 1）视野中 Schwanschild可以重新在适当的频率上，允许观察员的轨道，以确定他没有模棱两可的空间位置。 相反， 对于克尔度量来说， 目前尚不清楚应如何确定角频率的轨道革命 Q或它的一个方便的功能，因此我应当承担下列讨论的目的，所有的参数（ 1）是已知先验。 条件（ 1）告诉轨道观察员，如果一个事件视界的存在，它将会是接近，而不是如何结束 ;此外，他们可能在没有真实存在的一个事件视界。这种情况发生在多收的前庭，诺世全时空，在一个静态的，不可压缩，球明星，由内部史瓦西解（德费利切描述 1991年），并在克尔裸奇点的解决方案（德菲菲等 1991）。 当 prehorizon 度量的鉴定在德菲菲等人（ 1991）不包括在条件（ 1）所有地区都满意。我将在这里演绎正确的预层结构（第 3 条），并建议允许一个作业程序在轨观测承认 ，它 的轨道是合作与度量源旋转，即本地向外方向（方向远离源）与谐和 感物理推力 ，它 的位置就光子环形轨道的。我会进一步的证据，使一个陀螺仪的轨道行为模式，这可能是解释为一个由静态情况下，它背离了背景几何内存。显然，假设观察者先验地知道一个背景几何由科尔给了我 在他来说 认为这是赤道的轨道是圆形空间 的假设 。单位是使得 C= 1=克 ;希腊从 0到 3 指数 2。克尔度量：圆形轨道，并允许角速度模式在博耶和林德基斯特坐标（米斯纳等人 1973年），由肯度量描述线元素： 其中