Reports of the Academy of Sciences of the USSR
1967. Volume 176, No. 5

UDC 530.12:531.51

PHYSICS

I. S. SHIKIN

A HOMOGENEOUS AXISYMMETRIC COSMOLOGICAL MODEL IN THE ULTRARELATIVISTIC CASE

(Presented by Academician L. I. Sedov on 29 XII 1966)

The present paper is a continuation of (¹), in which a homogeneous axisymmetric solution of the Einstein equations and Maxwell equations was considered under the condition that space is filled with a perfect fluid and, in the comoving system, there is a homogeneous magnetic field directed along the axis of symmetry \(z\). In the comoving system, which is at the same time synchronous, the metric has the form (¹)

\[ -ds^2=-c^2d\tau^2+a^2(\tau)\,[d\chi^2+f^2(\chi)d\varphi^2]+b^2(\tau)dz^2 . \tag{1} \]

In the closed model \(f(\chi)=\sin\chi\), in the open model \(f(\chi)=\operatorname{sh}\chi\), in the flat model \(f(\chi)=\chi\).

For dustlike matter \(a(\tau)\) and \(b(\tau)\) are given in (¹). Below we consider the case of the ultrarelativistic equation of state \(e=3p\).

The equation \(T^k_{0;k}=0\) for \(e=3p\) gives (¹) \((2\dot a/a)+(\dot b/b)=-3\dot e/4e\) (the dot denotes differentiation with respect to proper time \(\tau\)), i.e.

\[ e=\frac{3D}{8\pi k/c^4}\,(a^2b)^{-4/3},\qquad D=\text{const}. \tag{2} \]

In the case under consideration, the gravitational equations reduce to three equations ((¹), formulas (3)—(5)), of which we shall use the following two together with equation (2):

\[ \frac{\dot a^2}{c^2a^2}+2\frac{\dot a\dot b}{c^2ab}\pm\frac{\alpha^2}{a^2} = \frac{8\pi k}{c^4}(e+W) = \frac{3D}{(a^2b)^{4/3}}+\frac{B_1^2}{a^4}; \tag{3} \]

\[ \frac{2\ddot a}{c^2a}+\frac{\dot a^2}{c^2a^2}\pm\frac{\alpha^2}{a^2} = \frac{8\pi k}{c^4}(-p+W) = -\frac{D}{(a^2b)^{4/3}}+\frac{B_1^2}{a^4}. \tag{4} \]

The closed model corresponds in (3)—(4) to the upper sign and \(\alpha^2=1\), the open model to the lower sign and \(\alpha^2=1\); for the flat model \(\alpha^2=0\).

1. Let us consider the solution for the case when there is no magnetic field \((B_1^2=0)\). This question was considered earlier in (², ³).

For \(B_1^2=0\), from (3) and (4) we obtain for \(a\) and \(b\) the equations

\[ 3Y\frac{d^2Y}{da^2}-6\left(\frac{dY}{da}\right)^2\mp14\alpha^2\frac{dY}{da}-8\alpha^4=0,\qquad Y\equiv\frac{a\dot a^2}{c^2}. \tag{5} \]

\[ \frac{D}{b^{4/3}}=a^{2/3}\left[\mp\alpha^2-\frac{dY}{da}\right]. \tag{6} \]

It is convenient to introduce the parameter \(\xi\) according to

\[ \frac{1}{\xi^2}=\mp\alpha^2-\frac{d(a\dot a^2/c^2)}{da}\equiv\mp\alpha^2-\frac{dY}{da}. \tag{7} \]

The possibility of introducing the parameter in this way (for physically real conditions, when \(e=3p>0\) and the constant \(D>0\)) follows from formula (6), which then assumes the form

\[ b^{4/3}=\frac{D\xi^2}{a^{2/3}}. \tag{8} \]

Equation (5) is transformed with the aid of (7) into the form

\[ \frac{d(a\dot a^2/c^2)}{(a\dot a^2/c^2)} = 3\,\frac{(\mp\alpha^2\xi^2-1)\,d\xi}{\xi(3\mp\alpha^2\xi^2)} . \tag{9} \]

From formulas (7) and (9) we obtain

\[ da=(a\dot a^{\,2}/c^2)\,3\zeta\,d\zeta/(3+a^2\zeta^2). \tag{10} \]

Integrating (9) and (10), we obtain \(a\) and \(\dot a\) as functions of \(\zeta\), after which \(\tau(\zeta)\) is also determined. The dependence \(b(\zeta)\) is expressed by (8).

Closed model. Let us first consider the case \(\zeta^2=3\) (\(\zeta\) then cannot serve as a parameter). From (7), introducing \(\eta\) according to \((2/3^{1/2})c\,d\tau=a\,d\eta\), we obtain

\[ a=A(1-\cos\eta),\qquad (2/3^{1/2})c(\tau-\tau_0)=A(\eta-\sin\eta), \qquad A=\text{const}. \tag{11} \]

The curve (11) is a cycloid. \(b^4=(3D)^3/a^2\). For \(a=0\), \(b=\infty\) and \(e=3p=\infty\).

Excluding the case just considered, from (9) and (10) we obtain

\[ a\dot a/c^2=A(\zeta^2-3)^2/\zeta,\qquad A=\text{const}; \tag{12} \]

\[ a(\zeta)/A=A_1+9\zeta-\zeta^3,\qquad A_1=\text{const}; \tag{13} \]

\[ c(\tau-\text{const})/A=\pm 3\int^{\zeta} [\zeta a(\zeta)/A]^{1/2}\,d\zeta . \tag{14} \]

The range of variation of \(\zeta\) is determined by the requirement that the expression under the radical, \(\zeta a(\zeta)/A\), not be negative. In this case various cases are possible, depending on the magnitude of the constant \(A_1\).

If \(A_1>6\cdot 3^{1/2}\), then the cubic (13) has one real root \(\zeta=\zeta_1>0\); \(\zeta a(\zeta)/A\ge 0\) for \(0\le \zeta\le \zeta_1\). According to (13), (12), (8), and (2) we have: for \(\zeta=\zeta_1\), \(a=0\), \(\dot a=\infty\), \(b=\infty\), and \(e=3p=\infty\); for \(\zeta=0\), \(a/A=A_1\ne 0\), \(\dot a=\infty\), \(b=0\), and \(e=\infty\). Near the value \(\zeta=0\), taking \(\tau=\tau_0\) at \(\zeta=0\), we have
\[ a/A\approx A_1+9(4A_1)^{-1/3}[c(\tau-\tau_0)/A]^{2/3}. \]
It is seen that this formula is valid for \(\tau>\tau_0\) and for \(\tau<\tau_0\) (on both sides of the value \(a/A=A_1,\ e=\infty\)). Near \(\zeta=\zeta_1\), taking \(\tau=\tau_1\) at \(\zeta=\zeta_1\), we obtain
\[ a/A\approx [27(\zeta_1^2-3)^2/12\zeta_1]^{1/3}[c(\tau-\tau_0)/A]^{2/3}. \]
This formula is also valid for \(\tau>\tau_1\) and for \(\tau<\tau_1\) (on both sides of the value \(a=0,\ e=\infty\)). In the interval \(2\tau_0-\tau_1\le \tau\le \tau_1\), the solution is given by (14) with \(\text{const}=\tau_0\) and with \(\zeta=0\) as the lower limit of integration. To continue the solution through the value \(\tau=\tau_1\), in (14) one must take the corresponding additive constant. Thus we obtain a periodic dependence \(a(\tau)\); over one period \(a(\tau)/A\) increases from the value \(a=0,\ \dot a=\infty\) (\(b=\infty,\ e=\infty,\ \zeta=\zeta_1>3^{1/2}\)), passes through a maximum (\(\zeta=3^{1/2}\)), and decreases to the value \(a/A=A_1,\ \dot a=\infty\) (\(b=0,\ e=\infty\)), after which it changes in the reverse order down to the value \(a=0\).

For \(A_1=6\cdot 3^{1/2}\), from (13) and (14) we obtain, putting \(\zeta=2\cdot 3^{1/2}\cos^2\lambda\):

\[ a/A=6\cdot 3^{1/2}\sin^2\lambda\,(2\cos^2\lambda+1)^2, \]
\[ c\tau/A=(27/4)^{1/2}(12\lambda+2\sin^3 2\lambda-3\sin 4\lambda). \tag{15} \]

For \(0<A_1<6\cdot 3^{1/2}\), the cubic (13) has three real roots \(\zeta_1>0\), \(\zeta_2<0\), and \(\zeta_3<\zeta_2<0\); \(\zeta a(\zeta)/A\ge 0\) for \(\zeta_3\le \zeta\le \zeta_2\) and for \(0\le \zeta\le \zeta_1\), so that two types of solutions are possible. Solutions with \(0\le \zeta\le \zeta_1\) are completely analogous to those considered above. The type of solutions for which \(\zeta_3\le \zeta\le \zeta_2\) turns out to be different: for \(\zeta=\zeta_3\), \(a=0\), \(\dot a=\infty\), \(b=\infty\), and \(e=\infty\); for \(\zeta=\zeta_2\) likewise \(a=0\), \(\dot a=\infty\), \(b=\infty\), and \(e=\infty\). The function \(b\) in this case does not vanish. The asymptotic formulas for \(a(\tau)\) near \(\zeta=\zeta_3\) and \(\zeta=\zeta_2\) are analogous to the one given above for \(\zeta=\zeta_1\); the dependence \(a(\tau)\) is analytically continued on both sides of the value \(a=0,\ e=\infty\) and has a periodic character.

For \(A_1=0\), evaluating the integral (14), we obtain

\[ (a/A)^2=\zeta^2(9-\zeta^2)^2,\qquad c(\tau-\text{const})/A=\mp(9-\zeta^2)^{3/2}. \tag{16} \]

For \(\zeta=0\), \(a=0\), \(\dot a=\infty\), \(b=0\), and \(e=\infty\); for \(\zeta^2=9\), \(a=0\), \(\dot a=\infty\), \(b=\infty\), and \(e=\infty\).

Negative values of the constant \(A_1\) do not lead to new types of solutions, since the corresponding formulas reduce to the formulas with positive \(A_1\) by changing the sign of \(\xi\) and \(a/A\).

Open model. Integrating (9) and (10), we obtain

\[ \frac{a\dot a^{\,2}}{c^2}=A\frac{(\xi^2+3)^2}{\xi},\qquad A=\mathrm{const}; \tag{17} \]

\[ a/A=A_1+9\xi+\xi^3,\qquad A_1=\mathrm{const}; \tag{18} \]

\[ c(\tau-\tau_0)/A=\pm 3\int^{\xi}[\xi a/A]^{1/2}\,d\xi . \tag{19} \]

Again the range of variation of \(\xi\) is determined by the requirement that \(\xi a/A\) be \(\ge 0\). The trinomial (18), for all values of the constant \(A_1\), has one real root \(\xi=\xi_1\). For \(A_1>0\) this root is \(\xi_1<0\); \(\xi a/A\ge 0\) for \(\xi\le \xi_1\) and for \(\xi\ge 0\), so that two types of solutions are possible. The first type corresponds to \(\xi\ge 0\). At \(\xi=0\) \((\tau=\tau_0)\), \(a/A=A_1\ne 0\), \(\dot a=\infty\), \(b=0\), and \(e=\infty\). The asymptotics near the value \(\xi=0\) \((\tau=\tau_0)\) has the same form as that given above for the closed model, and is valid for \(\tau>\tau_0\) and \(\tau<\tau_0\). As \(\xi\) grows from 0 (i.e., as \(|\tau-\tau_0|\) increases), \(a/A\) increases from the value \(A_1\) to \(\infty\), and \(b^4\) increases from 0 to \(D^3/A^2\).

The second type of solutions corresponds to \(\xi\le \xi_1\). At \(\xi=\xi_1\), \(a=0\), \(\dot a=\infty\), \(b=\infty\), and \(e=\infty\). Near \(\xi=\xi_1\) \((\tau=\tau_0)\),

\[ a/A\approx [27(\xi_1^2+3)^2/12\xi_1]^{1/3}\,[c(\tau-\tau_0)/A]^{1/3}. \]

As \(|\xi|\) increases from the value \(|\xi_1|\) (i.e., as \(|\tau-\tau_0|\) increases), \(a^2\) grows from zero, while \(b^4\) decreases, and as \(|\xi|\to\infty\),

\[ a^2\to\infty,\qquad b^4\to D^3/A^2. \]

In this case \(b\) nowhere vanishes.

For \(A_1=0\) we obtain

\[ a/A=\xi(9+\xi^2),\qquad c(\tau-\tau_0)/A=\pm\left[(9+\xi^2)^{3/2}-27\right], \]

\[ b^4=(D^3/A^2)\xi^4(9+\xi^2)^{-2}. \tag{20} \]

At \(\xi=0\), \(a=0\), \(\dot a=\infty\), \(b=0\), \(e=\infty\); as \(|\xi|\to\infty\), \(a^2\to\infty\), \(b^4\to D^3/A^2\).

Flat model (3). From equations (9) and (10) we have \((\alpha^2=0)\)

\[ \frac{a\dot a^{\,2}}{c^2}=\frac{9A}{\xi},\qquad A=\mathrm{const};\qquad \frac{a}{A}=9\xi+A_1,\qquad A_1=\mathrm{const}; \]

\[ \frac{c(\tau-\tau_0)}{A}=\pm 3\int^{\xi}[\xi a/A]^{1/2}\,d\xi . \tag{21} \]

For \(A_1>0\) the expression under the radical in the integral is nonnegative for \(\xi\ge 0\) and for \(\xi\le -A_1/9\). Again solutions of two types are possible.

For \(\xi\ge 0\), introducing \(\lambda\) according to \(\xi=(A_1/9)\operatorname{sh}^2\lambda\), from (21) and (8) we obtain

\[ a/A=A_1\operatorname{ch}^2\lambda,\qquad c(\tau-\tau_0)/A=(A_1^2/144)(\operatorname{sh}4\lambda-4\lambda), \]

\[ b^4=(D^3/9^6A^2)(a/A)^4\operatorname{th}^{12}\lambda . \tag{22} \]

At \(\tau=\tau_0\) the minimum value \(a/A=A_1\ne 0\) is attained; in this case \(\dot a=\infty\), \(b=0\), \(e=\infty\). Near \(\tau=\tau_0\),

\[ a/A\approx A_1+9(4A_1)^{-1/3}[c(\tau-\tau_0)/A]^{2/3} \]

(as in the closed and open models); as \(|\tau|\to\infty\), \(a/A\to\infty\), \(b^4\to (a/A)^4(D^3/9^6A^2)\).

For \(\xi\le -A_1/9\), introducing \(\lambda\) according to \(\xi=-(A_1/9)\operatorname{ch}^2\lambda\), we obtain

\[ (a/A)^2=A_1^2\operatorname{sh}^4\lambda,\qquad c(\tau-\tau_0)/A=(A_1^2/144)(\operatorname{sh}4\lambda-4\lambda), \]

\[ b^4=(D^3/9^6A^2)(a/A)^4\operatorname{cth}^{12}\lambda . \tag{23} \]

At \(\tau=\tau_0\) the minimum value \(a^2=0\) is attained; in this case \(\dot a=\infty\), \(b=\infty\), \(e=\infty\). Near \(\tau=\tau_0\),

\[ a/A\approx (81/4\xi_1)^{1/3}[c(\tau-\tau_0)/A]^{2/3}; \]

as \(|\tau|\to\infty\), \(a^2\to\infty\), \(b^4\to (a/A)^4(D^3/9^6A^2)\).

For \(A_1=0\), (21) becomes the Friedmann solution

\[ (a/A)^4=(9^6A^2/D^3)b^4=324c^2(\tau-\tau_0)^2/A^2, \]

which as \(\tau\to\tau_0\) \((e\to\infty)\) differs essentially from (22)—(23).

1. In the case where there is a magnetic field, one must consider the system (3)—(4) with constant \(B_1^2 \ne 0\). In this case it is not possible to obtain finite expressions.

Let us consider the plane model for \(B_1^2 \ne 0\). Introduce the variables

\[ \omega = c^2/(a\dot a)^2,\qquad \xi = -(\omega^2/a)\,da/d\omega\,{}^* . \tag{24} \]

The system (3)—(4) \((\alpha^2=0)\) is then reduced to the form

\[ d\xi/d\omega = (\xi/3\omega^2)\,[\omega(8B_1^4\xi^2-14B_1^2\xi+6)+\xi(8B_1^2\xi-3)]; \tag{25} \]

\[ b^4 = D^3a^4(\xi\omega)^3/[B_1^2\xi\omega-(\omega-\xi)]^3. \tag{26} \]

The integral curves of equation (25) for \(B_1^2 \ne 0\) are shown in Fig. 1. In the sense of \(\omega\), one need consider only values \(\omega \ge 0\). According to (2), for nonnegative values of \(e=3p\) having physical meaning, the constant \(D\) must be \(>0\). It follows from (26) that for \(D>0\) the quantity \(b^4\) is negative for values of \(\omega\) and \(\xi\) inside the region bounded by the integral curves \(\xi=0\) and \(\xi=\omega/(1+B_1^2\omega)\), so that the inner part of this region (shaded in Fig. 1) need not be considered. In the remaining part of the half-plane \(\omega \ge 0\), equation (26) has singular points \(O(\omega=0,\xi=0)\) and \(A(\omega=0,\xi=3/(8B_1^2))\), lying on the \(\xi\)-axis, which is an integral curve. The singular points \(O\) and \(A\) have a structure more complicated than ordinary singular points.

Fig. 1.

The arrows indicate the direction of increase of \(a^2\) along an integral curve. The point \(O\) corresponds to the minimum of \(a^2\). When moving along an integral curve in the direction of increasing \(a^2\), one must pass from the lower half-plane to the upper one through an infinitely distant point \(\omega=\mathrm{const},\ \xi=\mp\infty\), corresponding to a finite value of \(a^2\). All curves arrive at the point \(A\), at which \(a^2=\infty,\ |\tau|=\infty\).

Near the point \(O\) (in the region \(\xi<0\)) \(\xi\approx -k_1^2\omega^2,\ k_1^2=\mathrm{const},\ a\approx \mathrm{const}\cdot\exp(k_1^2\omega)\); at the point \(O\), \(a\ne0,\ b=0,\ e=\infty\). It is essential that the minimum value of \(a^2\) attained at \(e=\infty\) is not equal to 0, and the energy density of the magnetic field \(W=B_1^2(c^4/8\pi k)/a^4\) is finite. The asymptotics near the singularity \(e=\infty\) in the presence of a magnetic field is expressed by formulas (22) for \(\lambda\to0\) \(^{**}\).

Let us note that for \(e=3p\) (in contrast to the case of dustlike matter) \(a^2(\tau)\) and \(b^2(\tau)\) are symmetric functions of \(\tau\) (with an appropriate choice of the time origin).

In the closed and open models, in the presence of a magnetic field, \(a^2\) likewise does not go to 0, since for small \(a\) in (3) and (4) the term \(B_1^2/a^4\) dominates over \(\pm \alpha^2/a^2\). The asymptotics near \(e=\infty\) in the closed and open models is also expressed by formulas (22) for \(\lambda\to0\).

Scientific Research Institute of Mechanics
Moscow State University
named after M. V. Lomonosov

Received
27 XII 1966

CITED LITERATURE

1. I. S. Shikin, DAN, 171, 73 (1966).
2. A. S. Kompaneets, A. S. Chernov, ZhETF, 47, 1939 (1964).
3. A. G. Doroshkevich, Astrofizika, 1, 255 (1965).

\[ \text{}^* \]
\(\omega \equiv \mathrm{const}\) corresponds to the isotropic Friedmann solution; for \(B_1^2\ne0\) the system (3)—(4) has no such solution.

\[ \text{}^{**} \]
For \(B_1^2=0\) the pattern of the integral curves
\[ \xi=(9A)^2\omega^2/[(9A)^2\omega-1] \]
differs substantially from Fig. 1. For \((\omega=1/(9A)^2,\ \xi=\mp\infty)\), \(a^2=\infty,\ |\tau|=\infty\). The curves with \(\xi<0\) correspond to (22), the curves with \(\xi>0\) to (23).