UDC 531.51

PHYSICS

P. P. GRISHCHUK

ON SPATIALLY HOMOGENEOUS GRAVITATIONAL FIELDS

(Presented by Academician Ya. B. Zel’dovich on 28 V 1969)

We shall call a metric spatially homogeneous (s.h.) if it admits a simply or multiply transitive group of motions \(\check{G}_r\) \((r \ge 3)\), acting on spacelike hypersurfaces. Such a definition is often encountered in the literature \((^{1-3})\), although it is not entirely justified from the physical point of view \((^3)\). Let us consider the case \(r = 3\). Metrics satisfying the stated requirements are known. Usually one chooses a semigeodesic (synchronous) coordinate system in which the hypersurfaces of transitivity have equations \(x^0=\mathrm{const}\), and each component of the Killing vectors \(\xi^i\) is at most a function of \(x^1, x^2, x^3\). Then the metrics obtained as a result of integrating the Killing equations can be written in the form \((^{1,2})\)*

\[ ds^2 = dx^{0\,2} + g_{ik}\,dx^i dx^k = c^2 dt^2 + \alpha_{ab} e_i^a e_k^b dx^i dx^k . \]

Here \(\alpha_{ab}\) are arbitrary functions of \(t\), and \(e_i^a\) are Killing vectors of the group \(\check{G}_3\), reciprocal to the given ones \((^4)\). The quantities \(e_i^a\) are defined by the relations \(e_i^a e_a^k=\delta_i^k\), \(e_i^a e_b^i=\delta_b^a\). For each of the 9 types of real nonisomorphic structures \((^{5,7})\), \(e_i^a\) can be chosen so that the structure constants \(\bar{C}_{bc}^{\,a}\) from the relations

\[ e_{i,k}^a - e_{k,i}^a = \bar{C}_{bc}^{\,a} e_i^b e_k^c \tag{1} \]

coincide with the structure constants \(C_{bc}^{\,a}\) of the given \(G_3\) \((\xi_{b,k}^i \xi_c^k - \xi_{c,k}^i \xi_b^k = C_{bc}^{\,a}\xi_a^i)\). The form of the operators \(\xi_a^i\) and \(e_i^a\) is given in the Appendix.

S.h. metrics satisfying the Einstein equations \(R_{\mu\nu} = -\varkappa (T_{\mu\nu} - {}^1/_2 g_{\mu\nu}T)\) are being studied intensively in connection with their cosmological applications. As the energy–momentum tensor of the medium one most often takes

\[ T_{\mu\nu} = (\rho c^2 + p)u_\mu u_\nu - p g_{\mu\nu}, \tag{2} \]

where \(\rho c^2\) and \(p\) are the energy density and pressure; \(u_\mu\) are the components of the velocity of the medium, which in the present case have the form \((^2)\) \(u_0=n_0(t)\), \(u_i=n_a(t)e_i^a\). Generally speaking, \(u^i\) are not equal to zero, i.e. the synchronous coordinate system used is not comoving with the medium. However, in cosmological problems we are primarily interested in the behavior of the medium—its density, deformation, rotation, etc.; for this it is convenient to use a comoving reference system (c.r.s.). And in those cases where one seeks solutions satisfying previously imposed requirements (for example, the requirement of a specified dependence of the density on the proper time \(\tau\) of an element of the medium—

* Greek indices \(\mu, \nu, \ldots\) take the values 0, 1, 2, 3; Latin \(i, j, k, \ldots\), 1, 2, 3. “Frame” indices \(a, b, c, d, \ldots\) take the values 1, 2, 3. A comma with an index denotes partial differentiation.

which is important in calculating the chemical composition of matter), it is practically impossible to do without an s.s.r. The aim of the article is to derive the Einstein equations in an s.s.r. for all types of spatially homogeneous metrics.

There exist coordinate transformations (containing 4 arbitrary functions of \(t\)) that leave invariant the form of \(\xi_i^{a}\) and the equation \(x^0=\mathrm{const}\). Having found these transformations, it is easy to show that, by a choice of the arbitrary functions, all \(u^i\) can be made equal to zero, i.e., an s.s.r. can be introduced. The synchronicity of the coordinate system is thereby violated. Suppose that the choice of s.s.r. has been made, and integrate Killing’s equations. As a result we obtain

\[ g_{00}=\alpha_{00},\qquad g_{0i}=\alpha_{0a}e_i^{a},\qquad g_{ik}=\alpha_{ab}e_i^{a}e_k^{b}, \tag{3} \]

where the \(\alpha\)’s are arbitrary functions of \(t\). In view of the possibility of direct physical interpretation of the quantities entering the Einstein equations, we write these equations in the chronometrically invariant (ch.i.) form proposed by Zelmanov \((^6)\):

\[ {}^{*}\dot D+D_{ik}D^{ik}+A_{ik}A^{ki}+{}^{*}\nabla_iF^i-F_iF^i=-\frac12(\rho+U), \tag{4} \]

\[ {}^{*}\dot D_{ik}-(D_{ij}+A_{ij})(D_k^{\,j}+A_k^{\,j})+DD_{ik}-D_{ij}D_k^{\,j}+3A_{ij}A_k^{\,j}+ \]

\[ +\frac12({}^{*}\nabla_iF_k+{}^{*}\nabla_kF_i)-F_iF_k-C_{ik} =\frac12(\rho h_{ik}+2U_{ik}-Uh_{ik}), \]

\[ {}^{*}\nabla_k(h^{ik}D-D^{ik}-A^{ik})+2F_kA^{ik}=J^i, \]

The constants \(c\) and \(\varkappa\) are equal to 1; a dot denotes \(\dfrac{\partial}{\partial t}\), while a dot with an asterisk denotes \({}^{*}\dfrac{\partial}{\partial t}\). Ch.i. quantities and operators (marked by \(*\)) are expressed in terms of the ordinary operators and the metric \(g_{\mu\nu}\) as follows:

\[ {}^{*}\frac{\partial}{\partial t}=\frac{1}{\sqrt{g_{00}}}\frac{\partial}{\partial t}; \qquad {}^{*}\frac{\partial}{\partial x^i}= \frac{\partial}{\partial x^i}-\frac{g_{0i}}{g_{00}}\frac{\partial}{\partial t}; \]

the metric tensor

\[ h_{ik}=-g_{ik}+\frac{g_{0i}g_{0k}}{g_{00}}; \]

the tensor of deformation velocities

\[ D_{ik}=\frac12{}^{*}\dot h_{ik}; \]

the vector of gravitational-inertial force

\[ F_i=(1-w)^{-1}(w_{,i}-\dot V_i); \]

the tensor of angular velocity of rotation

\[ A_{ik}=\frac12(V_{k,i}-V_{i,k})+\frac12(F_iV_k-F_kV_i), \]

where the auxiliary quantities \(w,V_i\) are defined by \(g_{00}=(1-w)^2,\ g_{0i}=-V_i(1-w)\). Further, the Ricci tensor of space is

\[ C_{lk}=H_{lk}-A_{ki}D_l^{\,i}-A_{li}D_k^{\,i}-A_{kl}D, \quad\text{where } H_{lk}=H_{lki}^{\ \ \ i}; \]

\[ H_{lki}^{\ \ \ j}Q_j=({}^{*}\nabla_{ik}-{}^{*}\nabla_{ki})Q_l-2A_{ik}{}^{*}\dot Q_l; \tag{5} \]

\(Q_l\) is an arbitrary vector; \({}^{*}\nabla_i\) is the operator of covariant differentiation, constructed according to the usual rules from the metric \(h_{ik}\) and \({}^{*}\dfrac{\partial}{\partial x^i}\). Finally,

\[ \rho=T_{00}(g_{00})^{-1},\qquad J^i=T_0^{\,i}(g_{00})^{-1/2},\qquad U^{ik}=T^{ik}. \]

For the tensor (2), and in an s.s.r.,

\[ J^i=0,\qquad U^{ik}=p h^{ik}. \]

For spatially homogeneous metrics (3), all the vector and tensor quantities considered can be “expanded” with respect to the frame vectors \(e_i^{a}\), with coefficients depending only on \(t\). For example,

\[ h_{ik}=\left(-\alpha_{ab}+\frac{\alpha_{0a}\alpha_{0b}}{\alpha_{00}}\right)e_i^{a}e_k^{b} =\gamma_{ab}(t)e_i^{a}e_k^{b}. \]

With the help of \(\gamma_{ab}\), frame indices are shifted. We denote the frame components of the quantities \(D_{ik},V_i,F_i,A_{ik},C_{lk}\) by the corresponding lowercase letters; then

\[ d_{ab}=\frac12{}^{*}\dot\gamma_{ab},\qquad v_a=-\frac{\alpha_{0a}}{\sqrt{\alpha_{00}}},\qquad f_a=-{}^{*}\dot v_a, \]

and, taking (1) into account,

\[ a_{ab}=\frac12 v_c C^c{}_{ba}+\frac12(f_av_b-f_bv_a). \]

We find how \(c_{ab}\) is expressed in terms of \(\gamma_{ab}, v_a, C^a{}_{bc}\). In (5), as \(Q_i\) we substitute \(e_i^{d}\). We introduce the notation \(-\Gamma^d{}_{ac}=\)

\[ = e_a^i e_c^k {}^{*}\nabla_k e_i^d. \]
Then
\[ e_a^i e_b^j e_c^k {}^{*}\nabla_j e_{ki}^d = -\Gamma^d_{a c\oplus b}+\Gamma^d_{gc}\Gamma^g_{ab}+\Gamma^d_{ag}\Gamma^g_{cb}, \]
where
\[ \Gamma^d_{ac\oplus b}=e_b^j {}^{*}\nabla_j \Gamma^d_{ac}. \]
Since
\[ {}^{*}\!\left(e_l^d\right)^{\bullet}=0, \]
from (5) we obtain
\[ H_{ab}=H_{abc}{}^c=-2\Gamma^c_{a[c\oplus b]}+2\Gamma^c_{d[b}\Gamma^d_{a|c]}+2\Gamma^c_{ad}\Gamma^d_{[bc]}, \tag{6} \]
where \(X_{[a|b|c]}=\frac12(X_{abc}-X_{cba})\). The required \(c_{ab}\) are related to \(H_{ab}\) by
\[ c_{ab}=H_{ab}-a_{bc}d_a{}^c-a_{ac}d_b{}^c-a_{ba}d. \tag{7} \]

Let us compute the quantities \(\Gamma^c_{ab}\). Applying the operator \({}^{*}\nabla_k\) to \(\gamma_{ab}=e_a^i e_{bi}\), we get
\[ \Gamma_{abc}+\Gamma_{bac}=v_c^{*}\gamma_{ab}, \]
and from the definition of \(\Gamma^d_{ac}\) and (1) it follows that
\[ \Gamma^d_{bc}-\Gamma^d_{cb}=C^d{}_{cb}. \]
If one uses the equalities obtained from these by a cyclic interchange of indices, one can find
\[ -\Gamma_{abc}=\frac12(C_{abc}+C_{bca}-C_{cab}-v_c^{*}\gamma_{ab}-v_b^{*}\gamma_{ac}+v_a^{*}\gamma_{bc}). \]
After minor transformations,
\[ -\Gamma^c_{ab}=\frac12\left[C^c{}_{ab}+\gamma^{cd}(\gamma_{ag}C^g{}_{bd}+\gamma_{bg}C^g{}_{ad})\right] -v_a d_b{}^c-v_b d_a{}^c+v^c d_{ab}. \tag{8} \]

Since \(\Gamma^c_{ab}\) depend only on \(t\), we have
\[ \Gamma^c_{ab\oplus d}=v_d^{*}(\Gamma^c_{ab})^{\bullet}, \]
and, taking this fact into account, formulas (6)—(8) solve the problem posed. In the Appendix are given \(\bar c_{ab}\), computed by (6)—(8) under the condition \(a_{00}=1\), \(a_{0a}=0\). Let us also introduce the notation
\[ {}^{*}\nabla_i=e_i^c \Box_c. \]
It is not difficult to prove that for any quantity
\[ X_{i\ldots}^{\ldots k}=x_{a\ldots}^{\ldots b}(t)e_i^a\cdots e_b^k \]
one has
\[ {}^{*}\nabla_j X_{i\ldots}^{\ldots k} =e_j^c e_i^a\cdots e_b^k\,\Box_c x_{a\ldots}^{\ldots b}, \]
where
\[ \Box_c x_{a\ldots}^{\ldots b} =v_c^{*}\bigl(x_{a\ldots}^{\ldots b}\bigr)^{\bullet} -x_{d\ldots}^{\ldots b}\Gamma^d_{ac}-\cdots +x_{a\ldots}^{\ldots d}\Gamma^b_{dc}. \]

Finally, let us pass to proper time by means of
\[ d\tau=\sqrt{a_{00}}\,dt \]
(which is equivalent to the choice \(a_{00}=1\)) and write equations (4) in the s.s.o. in the form of a system of ordinary differential equations for functions of \(\tau\) ( \(d/d\tau\) is denoted by a dot):
\[ \begin{gathered} \dot d+d_{ab}d^{ba}+a_{ab}a^{ba}+\Box_a f^a-f_a f^a=-\frac12(\rho+3p),\\ \Box_b(\gamma^{ab}d-d^{ab}-a^{ab})+2f_b a^{ab}=0,\\ \dot d_{ab}-(d_{ac}+a_{ac})(d_b{}^c+a_b{}^c)+d d_{ab}-d_{ac}d_b{}^c+3a_{ac}a_b{}^c \\ +\frac12(\Box_a f_b+\Box_b f_a)-f_a f_b-c_{ab} =\frac12(\rho-p)\gamma_{ab}. \end{gathered} \tag{9} \]

Let us note that from the hydrodynamic equations
\[ \dot\rho+d(\rho+p)=0,\qquad v_a\dot p=(\rho+p)f_a, \]
which are a consequence of (9), it follows that
\[ f_a v_b-f_b v_a=0, \]
i.e.
\[ a_{ab}=\frac12 v_c C^c{}_{ba}. \]
Moreover,
\[ v_a=-a_{0a}=k_a\exp\int \frac{dp}{\rho+p}, \]
where \(k_a\) are arbitrary constants.

The author thanks A. L. Zelmanov for discussion.

Appendix

Introduce the notation
\[ X_a=\xi_a^i\frac{\partial}{\partial x^i}=\xi_a^i p_i; \]
denote the determinant of the matrix \(\gamma_{ab}\) by \(\gamma\); then:

Type I.
\[ X_1=p_1,\quad X_2=p_2,\quad X_3=p_3,\quad e_i^1=(1,0,0),\quad e_i^2=(0,1,0),\quad e_i^3=(0,0,1),\quad \bar c_{ab}=0. \]

Type II.
\[ X_1=p_1,\quad X_2=p_2,\quad X_3=x^2p_1-p_3,\quad e_i^1=(1,x^3,0),\quad e_i^2=(0,1,0), \]
\[ e_i^3=(0,0,1),\quad \bar c_1^2=\bar c_1^3=\bar c_2^3=0,\quad -\bar c_1^1=\bar c_2^2=\bar c_3^3=\frac{1}{2\gamma}\gamma_{11}^2. \]

Type III.
\[ X_1=p_1,\quad X_2=p_2,\quad X_3=x^1p_1-p_3,\quad e_i^1=(e^{x^3},0,0),\quad e_i^2=(0,1,0), \]
\[ e_i^3=(0,0,1),\quad \bar c_1^3=\bar c_2^3=0,\quad \bar c_1^2=-\frac1\gamma\gamma_{11}\gamma_{12},\quad \bar c_1^1=\bar c_3^3=\gamma^{33}+\frac{1}{2\gamma}\gamma_{12}^2,\quad \bar c_2^2=-\frac{1}{2\gamma}\gamma_{12}^2. \]

Type IV. \(X_1=p_1,\ X_2=p_2,\ X_3=(x^1+x^2)p_1+x^2p_2-p_3,\ e_i^{1}=(e^{x^3}, x^3e^{x^3},0),\ e_i^{2}=(0,e^{x^3},0),\ e_i^{3}=(0,0,1),\ \bar c_1^{3}=\bar c_2^{3}=0,\ \bar c_1^{2}=\dfrac{1}{\gamma}\gamma_{11}^{2},\ \bar c_1^{1}=2\gamma^{33}-\dfrac{1}{2\gamma}\times \gamma_{11}(\gamma_{11}+2\gamma_{12}),\ \bar c_2^{2}=2\gamma^{33}+\dfrac{1}{2\gamma}\gamma_{11}(\gamma_{11}+2\gamma_{12}),\ \bar c_3^{3}=2\gamma^{33}+\dfrac{1}{2\gamma}\gamma_{11}^{2}.\)

Type V. \(X_1=p_1,\ X_2=p_2,\ X_3=x^1p_1+x^2p_2-p_3,\ e_i^{1}=(e^{x^3},0,0),\ e_i^{2}=(0,e^{x^3},0),\ e_i^{3}=(0,0,1),\ \bar c_{ab}=2\gamma^{33}\gamma_{ab}.\)

Type VI. \(X_1=p_1,\ X_2=p_2,\ X_3=x^1p_1+qx^2p_2-p_3,\ e_i^{1}=(e^{x^3},0,0),\ e_i^{2}=(0,e^{qx^3},0),\ e_i^{3}=(0,0,1),\ \bar c_1^{3}=\bar c_2^{3}=0,\ \bar c_1^{2}=\dfrac{1}{\gamma}\gamma_{11}\gamma_{12}(q-1),\ \bar c_1^{1}=\dfrac{1}{2\gamma}\gamma_{12}^{2}\times (1-q^2)+(1+q)\gamma^{33},\ \bar c_2^{2}=-\dfrac{1}{2\gamma}\gamma_{12}^{2}(1-q)^2+q(1+q)\gamma^{33},\ \bar c_3^{3}=\dfrac{1}{2\gamma}\gamma_{12}^{2}(1-q)^2+(1+q^2)\gamma^{33},\) where \(q\ne 0,1\).

Type VII. \(X_1=p_1,\ X_2=p_2,\ X_3=-x^2p_1+(qx^2+x^1)p_2+p_3,\ e_i^{1}=\bigl[e^{-\frac12 qx^3}\bigl(q\sin \frac{p}{2}x^3+p\cos \frac{p}{2}x^3\bigr),\ 2e^{-\frac12 qx^3}\sin \frac{p}{2}x^3,\ 0\bigr],\ e_i^{2}=\bigl[-2e^{-\frac12 qx^3}\times \sin \frac{p}{2}x^3,\ e^{-\frac12 qx^3}\bigl(-q\sin \frac{p}{2}x^3+p\cos \frac{p}{2}x^3\bigr),\ 0\bigr],\ e_i^{3}=(0,0,1),\ q^2<4,\ p=\sqrt{4-q^2},\ \bar c_1^{1}=\dfrac{1}{2\gamma}\bigl[\gamma_{22}^{2}-(\gamma_{11}-q\gamma_{12})^2\bigr],\ \bar c_2^{2}=-\dfrac{1}{2\gamma}\bigl[\gamma_{22}^{2}+(\gamma_{11}+q\gamma_{12})^2-2\gamma_{11}(\gamma_{11}+q\gamma_{22})\bigr],\ \bar c_3^{3}=\dfrac{1}{2\gamma}\bigl[-(\gamma_{11}+\gamma_{22}+q\gamma_{12})^2+2(\gamma_{11}^{2}+\gamma_{22}^{2}+2\gamma_{12}^{2})+2q^2\gamma_{11}\gamma_{22}\bigr],\ \bar c_1^{3}=\bar c_2^{3}=0,\ \bar c_1^{2}=-\dfrac{1}{\gamma}\bigl[\gamma_{12}(\gamma_{11}+\gamma_{22})-q\gamma_{11}\gamma_{22}\bigr].\)

Type VIII. \(X_1=p_2,\ X_2=x^2p_2+p_3,\ X_3=e^{x^3}p_1+x^{2\,2}p_2+2x^2p_3,\ e_i^{1}=(1,x^1e^{-x^3},-x^1),\ e_i^{2}=(0,-2x^1e^{-x^3},1),\ e_i^{3}=(0,e^{-x^3},0),\ \bar c_1^{2}=-2\bar c_2^{3}=-4\left(\gamma^{23}+\dfrac{1}{\gamma}\gamma_{12}A\right),\ \bar c_1^{3}=-2\left(2\gamma^{33}-\dfrac{1}{\gamma}\gamma_{11}A\right),\ \bar c_1^{1}=\bar c_3^{3}=-2\left(2\gamma^{13}-\dfrac{1}{\gamma}\gamma_{22}A\right),\ \bar c_2^{2}=2\left[\gamma^{22}-\dfrac{1}{\gamma}(\gamma_{22}+\gamma_{13})A\right],\) where \(A=\gamma_{22}-\gamma_{13}\).

Type IX. \(X_1=p_2,\ X_2=\cos x^2p_1-\operatorname{ctg}x^1\sin x^2p_2+\dfrac{\sin x^2}{\sin x^1}p_3,\ X_3=-\sin x^2p_1-\operatorname{ctg}x^1\cos x^2p_2+\dfrac{\cos x^2}{\sin x^1}p_3,\ e_i^{1}=(\cos x^3,\sin x^3\sin x^1,0),\ e_i^{2}=(-\sin x^3,\cos x^3\sin x^1,0),\ e_i^{3}=(0,\cos x^1,1),\ \bar c_1^{2}=-2\gamma^{12}-\dfrac{1}{\gamma}\gamma_{12}A,\ \bar c_1^{1}=\dfrac{1}{2\gamma}A^2-B-\gamma^{11}-\dfrac{1}{\gamma}(\gamma_{11}^{2}+\gamma_{12}^{2}+\gamma_{13}^{2}),\) where \(A=\gamma_{11}+\gamma_{22}+\gamma_{33},\ B=\gamma^{11}+\gamma^{22}+\gamma^{33};\) the remaining components \(\bar c_a^{b}\) are obtained by a cyclic replacement of the indices \(1,2,3\).

For some geometrical properties of the metrics considered, see \((^3,^8)\).

Sternberg State Astronomical Institute

Received
26 V 1969

CITED LITERATURE

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\(^{2}\) E. Schücking, In: Gravitation, 1963.
\(^{3}\) Л. П. Грищук, Astr. Zh., 44, 5, 1097 (1967).
\(^{4}\) Л. П. Эйзенхарт, Continuous Groups of Transformations, IL, 1947.
\(^{5}\) L. Bianchi, Lezioni sulla teoria dei gruppi continui finiti di trasformazioni sperri, Pisa, 1918.
\(^{6}\) А. Л. Зельманов, DAN, 107, 6, 815 (1956); Proceedings of the VI Conference on Cosmogony, 1959.
\(^{7}\) А. З. Петров, New Methods in General Relativity, “Nauka,” 1966.
\(^{8}\) G. F. R. Ellis, M. A. H. MacCallum, Comm. Math. Phys., 12, 108 (1969).