這里翻譯一篇由栗夫席茲與哈拉特尼科夫?qū)τ谠诠矂?dòng)參考系內(nèi)偽奇點(diǎn)產(chǎn)生的幾何學(xué)機(jī)理解釋，作為二卷《場論》的 97的P 323的第2個(gè)注釋給出。

引力場方程的宇宙學(xué)解的奇異性

I.M. Khalatnikov, E. M. Lifshiftz, and V. V. Sudakov

蘇聯(lián)科學(xué)院物理問題研究所，莫斯科，美國。(1961年2月28日)

?

在廣義相對論的宇宙學(xué)應(yīng)用中，廣泛使用了眾所周知的愛因斯坦引力場方程的（弗里德曼模型）解，它是基于物質(zhì)空間分布的完全均勻性和各向同性的假設(shè)。這一假設(shè)在數(shù)學(xué)方面的準(zhǔn)確度有限，更不用說它在實(shí)際宇宙中的實(shí)現(xiàn)最多只能是近似的。因此，自然就有疑問：所得模型的重要屬性 —— 時(shí)空奇點(diǎn)的存在，在多大程度上取決于這個(gè)特定的假設(shè)？解決這個(gè)問題，對整個(gè)宇宙學(xué)來說是最重要的，需要對物質(zhì)和引力場在空間的相當(dāng)任意的分布所產(chǎn)生的情況進(jìn)行考察。這里對這種研究的結(jié)果做了一個(gè)簡短的總結(jié)。

?

在處理這個(gè)問題時(shí)，參考系的自然選擇是一個(gè)系統(tǒng)，受制于條件,?,?（我們將稱這樣的系統(tǒng)為同步參考系/共動(dòng)參考系，因?yàn)樗试S沿整個(gè)空間的時(shí)鐘同步）。很久以前，朗道就指出，由于引力場方程之一（方程），度規(guī)行列式一定在有限時(shí)間內(nèi)不可避免地變成零。然而，這個(gè)結(jié)果（最近也被其他作者獨(dú)立發(fā)現(xiàn)[1]）絕不是證明 ——（與其他文獻(xiàn)中表達(dá)的觀點(diǎn)相反）—— 度規(guī)中真正的（物理）奇點(diǎn)的存在是不可避免的，不能被參考系的任何變換所排除。這個(gè)奇點(diǎn)可能是虛構(gòu)的、非物理的，僅僅與所選參考系的具體性質(zhì)有關(guān)。

?

這個(gè)問題的答案來自于對同步參考系中時(shí)空特性的幾何分析。

?

很容易看出，在一個(gè)同步參考系中，時(shí)間線是四維時(shí)空的測地線。這一特性可用于在任何時(shí)空中對這種參考系進(jìn)行幾何構(gòu)造。我們選擇一個(gè)任意的空間超曲面，并構(gòu)建一組對該超曲面法線的測地線。如果我們現(xiàn)在把時(shí)間坐標(biāo)定義為一個(gè)給定的世界點(diǎn)和超曲面的交點(diǎn)之間的測地線的長度，那么，我們就可以很容易地看到，我們構(gòu)造了一個(gè)同步參考系。

?

但任意一簇測地線一般在一些包絡(luò)超曲面上相互交錯(cuò) —— 類比與幾何光學(xué)的四維超曲面。因此，出現(xiàn)一個(gè)奇點(diǎn)存在的幾何原因，是由于同步參考系的特定屬性，因此顯然具有非物理性質(zhì)。然而，需要強(qiáng)調(diào)的是，一般來說，四維時(shí)空的任意度量也允許存在不相交的時(shí)間狀的測地線集。但是，引力場方程的上述特性意味著它們所承認(rèn)的性質(zhì)排除了這種測地線集存在的可能性，因此，在任何同步參考系中，時(shí)間線必然相互交錯(cuò)。

?

這意味著，從分析的角度來看，在同步參考系中，愛因斯坦場方程有一個(gè)相對于時(shí)間的虛構(gòu)奇點(diǎn)的一般解。

?

因此，與這個(gè)一般解一起存在的另一個(gè)一般解的任何偽奇點(diǎn)都被消除了，這個(gè)一般解也將是一個(gè)一般解，但將有一個(gè)真正的奇點(diǎn)。解的一般性的標(biāo)準(zhǔn)是它所包含的（空間坐標(biāo)的）任意函數(shù)的數(shù)量。在這些函數(shù)中，一般也有這樣的函數(shù)，其任意性僅僅是由于方程所允許的參考系的自由選擇所造成的。重要的只是"物理上不同的"任意函數(shù)的數(shù)量，它不能因?yàn)閰⒖枷档娜魏尉唧w選擇而減少。對于一般的解決方案，這個(gè)數(shù)字必須是8個(gè)；這些函數(shù)必須規(guī)定有可能提出任意的初始條件，確定物質(zhì)的密度和三個(gè)速度分量的初始空間分布，以及確定自由引力場的四個(gè)數(shù)量。(后者的數(shù)量可以通過考慮弱引力波而得出；因?yàn)檫@些波是橫向的，它們的場由兩個(gè)服從二階微分方程的量來描述，因此，這個(gè)場的初始條件必須由四個(gè)空間函數(shù)給出）。

?

當(dāng)然，上述幾何學(xué)方面的考慮并不排除存在具有真正奇點(diǎn)的嚴(yán)格宇宙學(xué)解的可能性。事實(shí)上，對這種解的廣泛搜索（由我們兩人進(jìn)行[2]）表明，其中最寬松的解只包含七個(gè)物理上不同的任意函數(shù)，即比一般解所要求的少一個(gè)；因此，即使這個(gè)解盡管寬松，也只是一個(gè)特例。換句話說，這個(gè)解是不穩(wěn)定的；存在著導(dǎo)致其耗散的小擾動(dòng)。由于在同步參考系中，奇點(diǎn)不可能完全消失，這就意味著，作為擾動(dòng)的結(jié)果，它必須轉(zhuǎn)變成一個(gè)虛構(gòu)的奇點(diǎn)。

?

因此，我們得出的基本結(jié)論是，物理時(shí)間奇點(diǎn)的存在不是廣義相對論的宇宙學(xué)模型的強(qiáng)制性屬性。在物質(zhì)和引力場任意分布的一般情況下，會導(dǎo)致沒有這樣的奇點(diǎn)。

?

這個(gè)結(jié)果在形式上對朝向兩個(gè)時(shí)間方向的奇點(diǎn)同樣有效。然而，在物理上，這些方向當(dāng)然是不等價(jià)的，在問題本身的陳述中，這兩種情況就有本質(zhì)的區(qū)別。未來的奇點(diǎn)只有在它被先前任何時(shí)刻所給的相當(dāng)任意的條件所承認(rèn)時(shí)才有物理意義。另一方面，完全清楚的是，在宇宙演化過程中達(dá)到的物質(zhì)和場的分布，根本沒有理由符合實(shí)現(xiàn)具有物理奇點(diǎn)的特殊解決方案所必需的具體條件。即使人們承認(rèn)在某個(gè)時(shí)間點(diǎn)上實(shí)現(xiàn)了這種特定的分布，但由于不可避免的波動(dòng)，在接下來的時(shí)間里，它將不可避免地被違反。因此，上述結(jié)果排除了未來存在奇點(diǎn)的可能性；這意味著宇宙的收縮（如果它真的實(shí)現(xiàn)的話）之后必須再次轉(zhuǎn)變?yōu)閿U(kuò)張。至于過去的奇點(diǎn)，僅基于引力場方程的研究，只能對初始條件的可接受性施加某些限制，在現(xiàn)有理論的框架內(nèi)不可能完全闡明。

?

這項(xiàng)工作的詳細(xì)說明將發(fā)表在《實(shí)驗(yàn)和理論物理學(xué)雜志》（美國）。

?

我們衷心感謝朗道教授對我們工作的持續(xù)關(guān)注和多次討論。

[1] A. Komar, Phys. Rev. 104, 544 (1956).?

[2] E. M. Lifshitz and I. M. Khalatnikov, J. Exptl. ?

Theoret. Phys. {U.S.S.R.) 39, 149 and 800 (1960) [translations: Soviet Phys. —JETP 12, 108 (1961) 待發(fā)表].?

以下是英文原文

SINGULARITIES OF THE COSMOLOGICAL SOLUTIONS OF GRAVITATIONAL EQUATIONS

I. M. Khalatnikov, E. M. Lifshiftz, and V. V. Sudakov

The Institute of Physical Problems, Academy of Sciences, Moscow, U. S.S.R.

(Received February 28, 1961)

?

In cosmological applications of the general relativity theory extensive use is made of the well-known (Friedmann's) solution of the Einstein gravitational equation which is based on the assumption of complete homogeneity and isotropy of the space distribution of matter. This assumption is far reaching in its mathematical aspects, not to mention that its fulfillment in the actual universe could at best be only of approximate nature. Hence the question arises: To what extent does the important property of the resulting solution —the existence of the time singularity, depend on this specific assumption? The solution of this problem, which is of primary importance for the entire cosmology, requires an investigation of the situation arising for a quite arbitrary distribution of matter and gravitational field in space. A short summary is given here of the results of such an investigation.

?

The natural choice of the reference system in dealing with this problem turns out to be a system, subject to the conditions , , (we shall call such a system synchronous, since it allows of the synchronization of clocks along the entire space). It was long ago pointed out by Landau that due to one of the gravitational equations (the equation) the metric determinant must inevitably become zero in a finite time. However, this result (which was recently found independently also by other authors[1]) does by no means prove-contrary to the opinion expressed in the literature — the inevitability of the existence of a real (physical) singularity in the metric, which cannot be excluded by any transformation of the reference system. The singularity can turn out to be fictitious, nonphysical, being connected merely with the specific nature of the chosen reference system.

An answer to this question emerges from the geometrical analysis of the space-time properties in the synchronous system of reference.

?

It is easily seen that in a synchronous reference system the lines of time are geodesics in the 4-space. This property can be used for a geometrical construction of such a system in any space-time. We choose an arbitrary spacelike hypersurface and construct a set of geodesics normal to this hypersurface. If one defines now the time coordinate as the length of a geodesic between a given world point and the intersection with the hypersurface, one arrives, as it is easy to see, at a synchronous reference system.

But geodesic lines of an arbitrary set in general intersect each other on some envelope hypersurfaces — the four-dimensional analogs of the caustic surfaces of geometrical optics. Thus there exists a geometrical reason for the appearance of a singularity, which is due to specific properties of the synchronous reference system and is therefore obviously of a nonphysical nature. It is to be emphasized, however, that an arbitrary metric of a 4-space in general allows also for the existence of nonintersecting sets of time-like geodesics. But the above-mentioned property of the gravitational equations means that the metric admitted by them excludes the possibility of the existence of such sets, so that the lines of time necessarily intersect each other in any synchronous reference system.

This means, from the analytical point of view, that in a synchronous system of reference the Einstein equations have a general solution with a fictitious singularity with respect to time.

Thus any foundation is removed for the existence, along with this general solution, of yet another, which would also be a general one but would have a real singularity. The criterion of the generality of the solution is the number of arbitrary functions (of the space coordinates) it contains. Among these functions there are in general also such, whose arbitrariness is due merely to the freedom in the choice of the reference system admitted by the equations. What is essential is only the number of the "physically different" arbitrary functions, which cannot be decreased by any specific choice of the reference system. For the general solution this number must be eight; these functions must provide for the possibility to put arbitrary initial conditions, determining the initial space distributions of the density and the three velocity components of the matter, and of the four quantities which determine the free gravitational field. (The latter number can be arrived at, e.g., by considering weak gravitational waves; since these waves are transverse, their field is characterized by two quantities which obey differential equations of the second order, and therefor e the initial conditions for this field must be given by four space functions.)

The above geometrical considerations do not exclude, of course, the possibility of the existence of narrower classes of cosmological solutions with a real singularity. Indeed an extensive search (carried out by two of us[2]) for such solutions has shown that the widest of them contain only seven physically different arbitrary functions, i.e., one less than it is required for a general solution; hence even this solution in spite of its wideness is only a special case. In other words, this solution is unstable; there exist small perturbations which lead to its dissipation. Since in the synchronous reference system the singularity cannot disappear entirely, this means that it must go over, as a result of the perturbation, into a fictitious one.

Thus we are led to the fundamental conclusion that the existence of a physical time singularity is not an obligatory property of the cosmological models of the general relativity theory. The general case of an arbitrary distribution of matter and gravitational field leads to an absence of such a singularity.

This result is formally equally valid for the singularities towards both directions of time. However, physically these directions are of course not equivalent and there is an essential difference between both cases already in the statement of the problem itself. The singularity in the future can have a physical meaning only if it is admitted by quite arbitrary conditions given at any previous moment of time. On the other hand, it is perfectly clear that there are no reasons at all for the distribution of matter and field attained in the course of the evolution of the universe to comply with the specific conditions which are necessary for realization of the special solution with a physical singularity. Even if one admits the realization of such a specific distribution at some moment of time, it will inevitably be violated in the following time already as a result of the unavoidable fluctuations. Therefore the above results exclude the possibility of the existence of a singularity in the future; this means that the contraction of the universe (if it is at all to come) must afterwards change again to an expansion. As to the singularity in the past, an investigation based only on the gravitational equations, can only impose certain restrictions on the admissible character of the initial conditions, the complete elucidation of which is impossible in the framework of the existing theory.

A detailed account of this work will be published in the Journal of Experimental and Theoretical Physics (U.S.S.R.).

Our sincere thanks are due to Professor L. D. Landau for his constant interest in our work and for numerous discussions.

[1]A. Komar, Phys. Rev. 104, 544 (1956).

[2]E. M. Lifshitz and I. M. Khalatnikov, J. Exptl.

Theoret. Phys. {U.S.S.R.) 39, 149 and 800 (1960) [translations: Soviet Phys. —JETP 12, 108 (1961) and to be published].