这里翻译一篇由栗夫席兹与哈拉特尼科夫对于在共动参考系内伪奇点产生的几何学机理解释，作为二卷《场论》的 97的P 323的第2个注释给出。

引力场方程的宇宙学解的奇异性

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

苏联科学院物理问题研究所，莫斯科，美国。(1961年2月28日)

 

在广义相对论的宇宙学应用中，广泛使用了众所周知的爱因斯坦引力场方程的（弗里德曼模型）解，它是基于物质空间分布的完全均匀性和各向同性的假设。这一假设在数学方面的准确度有限，更不用说它在实际宇宙中的实现最多只能是近似的。因此，自然就有疑问：所得模型的重要属性 —— 时空奇点的存在，在多大程度上取决于这个特定的假设？解决这个问题，对整个宇宙学来说是最重要的，需要对物质和引力场在空间的相当任意的分布所产生的情况进行考察。这里对这种研究的结果做了一个简短的总结。

 

在处理这个问题时，参考系的自然选择是一个系统，受制于条件, , （我们将称这样的系统为同步参考系/共动参考系，因为它允许沿整个空间的时钟同步）。很久以前，朗道就指出，由于引力场方程之一（方程），度规行列式一定在有限时间内不可避免地变成零。然而，这个结果（最近也被其他作者独立发现[1]）绝不是证明 ——（与其他文献中表达的观点相反）—— 度规中真正的（物理）奇点的存在是不可避免的，不能被参考系的任何变换所排除。这个奇点可能是虚构的、非物理的，仅仅与所选参考系的具体性质有关。

 

这个问题的答案来自于对同步参考系中时空特性的几何分析。

 

很容易看出，在一个同步参考系中，时间线是四维时空的测地线。这一特性可用于在任何时空中对这种参考系进行几何构造。我们选择一个任意的空间超曲面，并构建一组对该超曲面法线的测地线。如果我们现在把时间坐标定义为一个给定的世界点和超曲面的交点之间的测地线的长度，那么，我们就可以很容易地看到，我们构造了一个同步参考系。

 

但任意一簇测地线一般在一些包络超曲面上相互交错 —— 类比与几何光学的四维超曲面。因此，出现一个奇点存在的几何原因，是由于同步参考系的特定属性，因此显然具有非物理性质。然而，需要强调的是，一般来说，四维时空的任意度量也允许存在不相交的时间状的测地线集。但是，引力场方程的上述特性意味着它们所承认的性质排除了这种测地线集存在的可能性，因此，在任何同步参考系中，时间线必然相互交错。

 

这意味着，从分析的角度来看，在同步参考系中，爱因斯坦场方程有一个相对于时间的虚构奇点的一般解。

 

因此，与这个一般解一起存在的另一个一般解的任何伪奇点都被消除了，这个一般解也将是一个一般解，但将有一个真正的奇点。解的一般性的标准是它所包含的（空间坐标的）任意函数的数量。在这些函数中，一般也有这样的函数，其任意性仅仅是由于方程所允许的参考系的自由选择所造成的。重要的只是"物理上不同的"任意函数的数量，它不能因为参考系的任何具体选择而减少。对于一般的解决方案，这个数字必须是8个；这些函数必须规定有可能提出任意的初始条件，确定物质的密度和三个速度分量的初始空间分布，以及确定自由引力场的四个数量。(后者的数量可以通过考虑弱引力波而得出；因为这些波是横向的，它们的场由两个服从二阶微分方程的量来描述，因此，这个场的初始条件必须由四个空间函数给出）。

 

当然，上述几何学方面的考虑并不排除存在具有真正奇点的严格宇宙学解的可能性。事实上，对这种解的广泛搜索（由我们两人进行[2]）表明，其中最宽松的解只包含七个物理上不同的任意函数，即比一般解所要求的少一个；因此，即使这个解尽管宽松，也只是一个特例。换句话说，这个解是不稳定的；存在着导致其耗散的小扰动。由于在同步参考系中，奇点不可能完全消失，这就意味着，作为扰动的结果，它必须转变成一个虚构的奇点。

 

因此，我们得出的基本结论是，物理时间奇点的存在不是广义相对论的宇宙学模型的强制性属性。在物质和引力场任意分布的一般情况下，会导致没有这样的奇点。

 

这个结果在形式上对朝向两个时间方向的奇点同样有效。然而，在物理上，这些方向当然是不等价的，在问题本身的陈述中，这两种情况就有本质的区别。未来的奇点只有在它被先前任何时刻所给的相当任意的条件所承认时才有物理意义。另一方面，完全清楚的是，在宇宙演化过程中达到的物质和场的分布，根本没有理由符合实现具有物理奇点的特殊解决方案所必需的具体条件。即使人们承认在某个时间点上实现了这种特定的分布，但由于不可避免的波动，在接下来的时间里，它将不可避免地被违反。因此，上述结果排除了未来存在奇点的可能性；这意味着宇宙的收缩（如果它真的实现的话）之后必须再次转变为扩张。至于过去的奇点，仅基于引力场方程的研究，只能对初始条件的可接受性施加某些限制，在现有理论的框架内不可能完全阐明。

 

这项工作的详细说明将发表在《实验和理论物理学杂志》（美国）。

 

我们衷心感谢朗道教授对我们工作的持续关注和多次讨论。

[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) 待发表]. 

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].