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UDC 530.12:531.51
MATHEMATICAL PHYSICS
D. D. SOKOLOV
ON THE TOPOLOGY OF UNIVERSE MODELS
(Presented by Academician Ya. B. Zel’dovich, 24 IV 1970)
1°. An important question in cosmology is the problem of singling out those classes of manifolds that can serve as models of the Universe. For a correct consideration of Einstein’s equations it is necessary first, proceeding from physical considerations, to choose the manifold on which they will be defined. On the basis of the most general physical laws, a number of principles have been proposed that restrict the topology of Universe models independently of their concrete form*.
\((\mathfrak{B})\) \(\mathfrak{M}^4\) is a smooth manifold on which one can define an indefinite metric of signature \((1,3)\).
\((\mathfrak{G}_1)\) All timelike and isotropic vectors can be divided into two classes \(\{+\}\) and \(\{-\}\) in such a way that, if \(\xi \in \{+\}\), then \(-\xi \in \{-\}\), and conversely, and if \(\xi_n \in \{+\}\,(\{-\})\), then \(\lim \xi_n \in \{+\}\,(\{-\})\).
\((\mathfrak{G}_2)\) \(\mathfrak{M}^4\) contains no closed timelike lines.
\((\mathfrak{G}_3)\) On \(\mathfrak{M}^4\) there exists a smooth point function \(f(P)=t\) such that in some local coordinates \(t\) coincides with the time coordinate.
\((ZN)\) \(\mathfrak{M}^4\) contains no nonorientable spaces \(\mathfrak{M}^3\).
2°. The class \(\mathfrak{B}\) is narrower than the class of all smooth manifolds, since an indefinite metric can be defined not on all manifolds (a necessary condition for this is equality to 0 of the Euler characteristic \((^3)\)). The principle \(\mathfrak{B}\) is the most fundamental of all these principles. The principles \(\mathfrak{G}_1\)—\(\mathfrak{G}_3\) were proposed by K. Gödel \((^7)\) as different formulations of the principle of causality. The principle \(ZN\) was proposed by Ya. B. Zel’dovich and I. D. Novikov as a restriction following from \(CP\)-noninvariance \((^{1,2})\).
3°. Repeating literally the reasoning of \((^{1,2})\), one can obtain a restriction on the topology of Universe models following from \(CT\)-noninvariance. If \(CP\)-noninvariance makes it possible to distinguish the two possible orientations of \(\mathfrak{M}^3\), then \(CT\)-noninvariance distinguishes the two possible directions of time at points of \(\mathfrak{M}^3\), i.e., the two possible orientations of the timelike vectors defined on \(\mathfrak{M}^3\). That is, upon transport along a closed contour on \(\mathfrak{M}^3\), a timelike vector must not change its orientation, which coincides with the definition of two-sidedness of \(\mathfrak{M}^3\) \((^3)\). Thus, the following restriction on the topology of Universe models is proposed:
\((\mathfrak{MB})\) \(\mathfrak{M}^4\) cannot contain one-sidedly embedded \(\mathfrak{M}^3\)’s.
Let us note that, if \(ZN\) is an internal restriction with respect to \(\mathfrak{M}^3\), then \(\mathfrak{MB}\) is an external one.
4°. Let us give an illustration of the use of the principles \(ZN\) and \(\mathfrak{MB}\): we construct the manifold \(\mathfrak{M}^4\)—the direct product of the Klein bottle with the plane \((p;r)\). Recall that the direct product of two manifolds with coordinates \((u;v)\) and \((p;r)\) is a manifold whose points have coordinates \((u;v;p;r)\). The Klein bottle is obtained by gluing, in a square in the plane \((u;v)\), two sides with preservation of orientation and two with reversal of it, as a result of which a coordinat—
* The following notation is used below: \(\mathfrak{M}^4\) is space-time; \(\mathfrak{M}^3\) is space (concerning this concept see §8); \(R^1\) is the real line. The letters numbering the principles also denote the classes of manifolds singled out by them.
ian grid with “periodic” coordinates:
\[ (u; v+2\pi)=(u; v);\qquad (u+2\pi; 2\pi-v)=(u; v). \tag{1} \]
Let us specify on this \(\mathfrak M^4\) two metrics:
\[ ds^2=dp^2-du^2-dv^2-dr^2, \tag{2} \]
\[ ds^2=du^2-dv^2-dp^2-dr^2. \tag{3} \]
In (2), \(\mathfrak M^3\) is the direct product of a Klein bottle with \(R^1\)—nonorientable—and (2) does not satisfy \(ZN\); in (3), \(\mathfrak M^3\) is the product of a cylinder and a line—one-sidedly embedded in \(\mathfrak M^4\) (for the proof see \((^3)\)), and (3) does not satisfy \(\mathfrak{MB}\). The idea of the examples is borrowed from \((^6,^8)\). The fact that (3) does not satisfy \(\mathscr G_2\) is immaterial \((^9)\).
\(5^\circ\). Let us note that \(ZN\) and \(\mathfrak{MB}\) are physically proved by direct experiments.
\(6^\circ\). What is the relation between the principles considered above? We shall show that, by strengthening \(\mathscr G_3\), one can obtain a principle from which \(\mathfrak B, \mathscr G_1, \mathscr G_2\), and \(\mathfrak{MB}\) follow.
\((\mathfrak{KB})\) On \(\mathfrak M^4\) one can define a smooth function of a point \(f(P)=t\) such that, in some local coordinates, \(t\) coincides with the time coordinate, \(t\) ranges over the interval \((a,b)\), and \(f^{-1}(a)=\mathfrak N_1^3\)* and \(f^{-1}(b)=\mathfrak N_2^3\) are three-dimensional closed smooth manifolds.
Then \(f(P)\) is a Morse function of the manifold \(\mathfrak M^4\) without critical points and \(\mathfrak N_1^3\) \((^4)\) is diffeomorphic to \(\mathfrak N_2^3\), and
\[ \mathfrak M^4=\mathfrak N^3\times R^1. \tag{4} \]
As is easy to verify, the principles \(\mathfrak B, \mathscr G_1, \mathscr G_2, \mathfrak{MB}\) are in this case trivially fulfilled, while \(ZN\) forbids nonorientability of \(\mathfrak N^3\). That is, \(\mathfrak{KB}\) reduces the problem of the topology of models of the Universe to the three-dimensional one, where it is already posed as the problem of different “gluings” of space, i.e., as the problem of Heegaard diagrams \((^3)\). Let us note that all cosmological models known to the author that have physical meaning, as well as solutions of the equations of the general theory of relativity, satisfy \(\mathfrak{KB}\). It may be supposed that \(\mathfrak{KB}\) follows from \(\mathscr G_1-\mathscr G_3, \mathfrak B\), and \(\mathfrak{MB}\). It is also interesting to find out precisely which manifolds enter into the classes considered above.
\(7^\circ\). The principles formulated above admit a natural generalization to the case of arbitrary dimension and signature; here classes \(\mathfrak B_p^i\) arise (i.e., the class of \(n\)-dimensional smooth manifolds on which one can specify a metric of signature \((p;q)\)), etc. However, their intersection is nonempty only for some \(n,p,q\), so that, perhaps, \(n,p\), and \(q\) can be obtained from these principles. Let us note that if a violation of \(CPT\) were observed, it would not impose restrictions analogous to \(ZN\) and \(\mathfrak{MB}\), since \(PT\) does not change the orientation of \(\mathfrak M^4\).
\(8^\circ\). In conclusion we give various definitions of \(\mathfrak M^3\):
\((\mathfrak C)\) \(\mathfrak M^3\) is any spacelike three-dimensional submanifold embedded in \(\mathfrak M^4\) \((^7)\).
\((\mathfrak{GM})\) \(\mathfrak M^3\) is a spacelike three-dimensional submanifold that every timelike geodesic intersects once and only once \((^5,^7)\).
In some works of Ya. B. Zel’dovich and A. L. Zel’manov, for \(\mathfrak M^4\) representable in the form (4), the following definition of space is meant:
\((ZZ)\) \(\mathfrak M^3\) is a spacelike submanifold diffeomorphic to \(\mathfrak N^3\).
* \(f^{-1}(a)=\mathfrak N_1^3\) means that for all \(\tau\) sufficiently close to \(a\), \(f^{-1}(\tau)\) is diffeomorphic to \(\mathfrak N_1^3\).
For the definition \(G\) it has not been proved that at least one \(M^4\) satisfies \(ZN\); the difficulties of \(GM\) are discussed in \((^5)\). In the case of justification of \(KP\), the definition \(ZZ\) appears preferable to the author.
The author is grateful to Academician Ya. B. Zel’dovich and Prof. E. G. Poznyak for valuable discussion and attention to the work.
Moscow State University
named after M. V. Lomonosov
Received
3 IV 1970
CITED LITERATURE
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