外文翻譯---允許的角速度模式和預(yù)地平線制度

外文文獻(xiàn)及翻譯 外文資料原文： 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 外文資料 翻譯 ： 允許的角速度模式和預(yù)地平線制度 費爾南多德費利切 摘要 本文假設(shè) 允許存在一個觀察員及其周邊參數(shù)回歸源 認(rèn)識到事件的時候可能產(chǎn)生 力學(xué)效應(yīng)。當(dāng)這些影響體現(xiàn) 在 時空 中的時候就被認(rèn)為是在預(yù)地平線制度。在這里角速度 這個概念會更加精確度量。 簡介 物理學(xué)中有 興趣的 一個 問題是在黑洞周圍的軌道觀測器是否 可以實現(xiàn)從框架內(nèi)的構(gòu)想，無論它是是多么接近 事件視界。 但當(dāng) 觀察員 在圓形軌道上移動時，至少有三個力學(xué)效應(yīng)該信號可能接近事件視界。它們是： （ 1）需要保持圓形軌道推力，隨模向外 延伸 的軌道頻率。 （ 2）地方慣性羅盤（陀螺 儀） 關(guān)于轉(zhuǎn)發(fā)軌道頻率的 延伸 頻率。 （ 3）與軌道 向前陀螺延伸度 模量增加 的頻率 。 如果我們調(diào)用 f 軌道的空間加速度（具體推力），假設(shè) 角 速度的軌道測量頻率 將在無窮遠(yuǎn)， 那么 R 的角頻率慣性和 Q2相關(guān) ， 這時 條件（ 1）至（ 3）可分別表示為： 當(dāng)所有這些屬性都處于時空的時候，就說是在預(yù)地平線制度。顯然條件（ 1）才有意義，假設(shè)觀察者還能夠到測量工作部分，那么假設(shè) z2 是最困難的數(shù)量，就能夠確定一些遙遠(yuǎn)的觀察者的相互信息交流 ;掌握每個人都應(yīng)該知道它的模量及其與尊重當(dāng)?shù)氐捻槙r針方向的標(biāo)志。很可能在它 的軌道幀的觀察員能夠測量 的只有 一個函數(shù)，在這種情況下，將取決于 對所調(diào)查的具體情況 是否能夠驗證觀測條件 （ 1）。 在史瓦西時空，例如，觀察員 在 圓形軌道上移動 ，那么 通過適當(dāng)?shù)? 局部應(yīng)力測量來估算 軌道方向 的延伸和適當(dāng)?shù)念l率，其模量 即由紅移乘以 z2 因子。由于紅移因素是積極的，預(yù)條件（ 1）視野中 Schwanschild可以重新在適當(dāng)?shù)念l率上，允許觀察員的軌道，以確定他沒有模棱兩可的空間位置。 相反， 對于克爾度量來說， 目前尚不清楚應(yīng)如何確定角頻率的軌道革命 Q或它的一個方便的功能，因此我應(yīng)當(dāng)承擔(dān)下列討論的目的，所有的參數(shù)（ 1）是已知先驗。 條件（ 1）告訴軌道觀察員，如果一個事件視界的存在，它將會是接近，而不是如何結(jié)束 ;此外，他們可能在沒有真實存在的一個事件視界。這種情況發(fā)生在多收的前庭，諾世全時空，在一個靜態(tài)的，不可壓縮，球明星，由內(nèi)部史瓦西解（德費利切描述 1991年），并在克爾裸奇點的解決方案（德菲菲等 1991）。 當(dāng) prehorizon 度量的鑒定在德菲菲等人（ 1991）不包括在條件（ 1）所有地區(qū)都滿意。我將在這里演繹正確的預(yù)層結(jié)構(gòu)（第 3 條），并建議允許一個作業(yè)程序在軌觀測承認(rèn) ，它 的軌道是合作與度量源旋轉(zhuǎn)，即本地向外方向（方向遠(yuǎn)離源）與諧和 感物理推力 ，它 的位置就光子環(huán)形軌道的。我會進(jìn)一步的證據(jù)，使一個陀螺儀的軌道行為模式，這可能是解釋為一個由靜態(tài)情況下，它背離了背景幾何內(nèi)存。顯然，假設(shè)觀察者先驗地知道一個背景幾何由科爾給了我 在他來說 認(rèn)為這是赤道的軌道是圓形空間 的假設(shè) 。單位是使得 C= 1=克 ;希臘從 0到 3 指數(shù) 2?？藸柖攘浚簣A形軌道，并允許角速度模式在博耶和林德基斯特坐標(biāo)（米斯納等人 1973年），由肯度量描述線元素： 其中