hidekazu nariai

G en eral Relativ ity an d G ravi tation , Vol. 31, No. 6 , 1999

On a New Cosmological Solution of Einstein’s Field

Equations of Grav itation ²

Hidekazu Nariai*

Received April 10, 1951

Communicated by Prof. Z. Hitotuyan agi

1. INTRODUCTION

In the previous paper,1 the author obtained the following solut ion of Ein-

stein’ s ® eld equat ions for the homogeneous static universe exhibit in spher-

ical symmetry.

ds2=

1

L [ (A cos log r + B sin log r)2 dt2

±1

r2(dr2

+ r2dh2

+ r2sin

2hdw

2) ], (1)

where L is the cosmological constant and A and B are arbit rary constants.

We have shown that this solut ion corresponds to the empty universe and,

moreover, this line element can not be transformed into the standard form.

The ® eld equat ions in the empty universe which this solut ion must

satisfy are, however, reduced to

Rmn = Lgmn , (2)

² Originally published in T he Scien ce Repor ts of the Toho ku Un iversity Series I, vol.

XXXV , No. 1 (1951) , p. 46-57. Reprinted with the kind permission of the Astro-

nom ical Inst itute of the Tohoku University in Sendai (J apan) , the current copyright

owner.* Institute of Ast ronomy, Faculty of Science, Tohoku University1 H. Nariai, Sc i. Rep . Tohoku Un iv. Ser. 1, vol. X X X IV , No. 3, p. 160 (1950) .

9 6 3

0 0 0 1 - 7 7 0 1 / 9 9 / 0 6 0 0 - 0 9 6 3 $ 1 6 . 0 0 / 0 ° c 1 9 9 9 P l e n u m P u b l i s h i n g C o r p o r a t i o n

9 6 4 N ar ia i

where Rmn is the contracted Riemann± ChristoŒel tensor. In this paper, at

® rst, we rederive the line element (1) as the solut ion of the ® eld equat ions

(2) starting from the isot ropic static line element and then discuss in detail

the geometrical and physical natures of the solut ion under considerat ion.

2. FIELD EQUATIONS FOR STATIC EMPTY UNIVERSE EXHIBIT-

ING SPHERICAL SYMMETRY

The static isotropic line element exhibit ing spherical symmetry is writ-

ten as follows.

ds2= en dt2 ± em

(dr2+ r2 dh

2+ r2

sin2

hdw2), (3)

with m = m(r) and n = n(r). And its ® eld equat ions (2) for the empty

matter-distribut ion are reduced to the following system of diŒerential equa-

tions.

m 9 9 +n 9 9

2+

n 9 2

4±

m 9 n 9

4+

m 9

r= ± Lem

, (4)

m 9 9

2+

m 9 2

4+

m 9 n 9

4+

n 9 + 3m 9

2r= ± Lem

, (5)

n 9 9

2+

n 9 2

4+

m 9 n 9

4+

n 9

r= ± Lem

, (6)

where primes denote diŒerentiat ion with respect to r. Taking the diŒerence

of (4) and (6), we get.

m 9 9 ±m 9 n 9

2+

m 9 ± n 9

r= 0 . (7)

The integral is

m 9 = 2(Aen / 2 ± 1)/ r, (8)

where A is an integrat ion constant. Subst ituting (8) into (5), we get after

suitable arrangements

2j 9 +j2 ± 1

r+

L

rexp ( 2 s j

rdr) = 0, (9)

or

j 9 9 +j 9 (1 ± j)

r+

j(1 ± j2 )

r2= 0, (9) 9

O n a N e w C osm olog ic a l S olu t ion 9 6 5

where j = Aen / 2 . It is usually very di� cult to obtain the general solut ion

of (9) 9 on account of its non-linear character, but we can ® nd several

part icular solut ions. For example, the solut ions j1 = 1 (A > 0), j2 = ± 1

(A < 0) and j3 = 0 (A = 0) are easily found from the formal nature of

(9) 9 . Moreover, (9) 9 is also satis® ed by j4 =1 - a/ r1+ a/ r (A = 1, L = 0) and

j5 = a2 - r 2

a2 + r 2 (A = 1).

The solut ions j1 and j2 represent the ¯ at space-t ime, and j4 corre-

sponds to the Schwarzschild exterior solut ion in the isotropic form of line

element (3). From j5 we obtain, put ting a = 2 without any loss of gener-

ality,

ds2= ( 1 ± r2 / 4

1 + r2 / 4 )2

dt2 ± R 2(dr2

+ r2 dh2

+ r2sin

2hdw

2) (1 + r2

/ 4) - 2, (10)

where R2 = 3/ L. This is the static isotropic form of the de Sitter universe.

These last two expressions correspond to the solut ions in an isotropic form

representing the special cases (L = 0 or m = 0) of the general standard-

form solut ion

ds2= c dt2 ± c - 1 dr2 ± r2dh

2 ± r2sin

2hdw

2, (11)

where c = 1 ± 2m/ r ± Lr2 / 3.

Finally, from j3 = 0, (8) and (5), we get

em= 1/ Lr2

, (L > 0). (12)

Subst itut ing (12) into (6) , we get

g9 9 +g9

r+

g

r2= 0, (en

= g2). (13)

The general solut ion is

g = A cos log r + B sin log r, (14)

where A and B are arbit rary constants. Hence we get the line element

ds2+ (A cos log r + B sin log r)

2 dt2 ±1

Lr2(dr2

+ r2dh2

+ r2sin

2hdw

2) , (15)

which, by adjust ing the scale of time-variable t, becomes to (1) obtained

in the previous paper. We are thus led to the conclusion that the solut ion

(1), which is also the same as the case j3 = 0, is one of possible solut ions

other than the well-known one for the empty static universe.2

2 We can prove more generally that the solut ion (15) is the only one which has spherical

symm et ry and sat is ® es the condition. r2 em = funct ion of t only.

9 6 6 N ar ia i

3. GEOMETRICAL PROPERTIES OF THE LINE ELEMENT

Performing the following transformation of coordinat es in the line

element (1).

t = t( L

A2 + B 2 )1/ 2

(16)

and r = exp(±x + tan - 1 BA ) or r1 = R sin x (R2 = 1

L ), we get

ds2= cos

2xdt

2 ± R2(dx

2+ dS 2

), (17)

or

ds2= ( 1 ±

r21

R 2 ) dt2 ±

dr21

1 ± r21 / R 2 ± R 2 dS 2

, (18)

where dS 2 = dh2 + sin2

hdw 2 . If we compare (17) or (18) with the corre-

sponding line elements of the de Sitter universe.

ds2= cos

2xdt

2 ± R 2(dx

2+ sin

2xdS 2

) , (19)

or

ds2= ( 1 ±

r21

R 2 ) dt2 ±

dr21

1 ± r21 / R 2 ± r2

1 dS 2, (20)

where R 2 = 3/ L, we can then ® nd their similarit ies. Really the diŒerent

point s are concerned only with the coe� cients of dS 2 and the values of

R 2L, and the other parts are entirely the same. Therefore, as far as we

consider the purely radial case dS = 0 such as the propagat ion of light in

the radial direct ion, both solut ions give the same results.

Moreover, the de Sitter universe can be represented as a hypersurface

of imaginary radius of curvature iR embedded in the ® ve-dimensional Eu-

clidean space. We can also obtain a similar representation of our solut ion

by the following transformat ion of coordinat es.


Z1 = R sin h cos w ,

Z2 = R sin h sin w ,

Z3 = R cos h ,

Z4 = r1 , (21)

Z5 = R( cosht

R )r

1 ±r2

1

R 2,

Z6 = R( sinht

R )r

1 ±r2

1

R 2.

Subst itut ing (21) into (18) , we get, after suitable arrangements,

ds2= ± dZ 2

1 ± dZ 22 ± dZ 2

3 ± dZ 24 ± dZ 2

5 + dZ 26 , (22)

where

Z 21 + Z 2

2 + Z 23 = R2

and Z 24 + Z 2

5 ± Z 26 = R2

. (23)

We can, therefore, interpret the solut ion considered here as a four-dimen-

sional hypersurface, which is a section of two ® ve-dimensional hypersur-

faces (23)3 embedded in the six-dimensional pseudo-Euclidean space (22) .

Finally, if we write the spat ial part of the original line element (1) as

ds2 we get

ds2

= R2(dx2

+ dy2+ dz2

)/ r2, (24)

where x = r sin h cos w , y = r sin h sin w , z = r cos h. Taking the following

transformation of coordinat es.

x/ r2= x*

, y/ r2= y*

, z/ r2= z*

(i.e., rr*= 1), (25)

we get

ds2

= R2(dx*2

+ dy*2+ dz*2

)/ r*2. (26)

Hence we can see that the spat ial part of (1) is invariant under the inversion

given by (25) .

3 We can also take, in place of (23) , the following two expressions.

z12 + z2

2 + z32 + z4

2 + z52 - z6

2 = 2R 2 , z12 + z2

2 + z32 - z4

2 - z52 + z6

2 = 0 .

T he former is a hyper-hyperboloid of one sheet and the latter a hypercone.

9 6 8 N ar ia i

4. MOTION OF A FREE PARTICLE

Let us consider the geodesic equat ions of motion of a free part icle in

the space-t ime considered here. Start ing from (22) , (23) and applying the

variat ional principle under the constraint condit ions, we get

d2zi

ds2= azi (i = 1, 2, 3),

d2 zj

ds2= bzj (j = 4, 5, 6),

ü ý þ(27)

where a+ b = 1/ R 2 and a, b are parameters, of which general solut ion may

be obtained without di� culty. But let us here start from the equat ion (17)

direct ly. Among forty¶ gm n

¶ xl ’ s in (17) , only two are non-vanishing quant ities,

i.e.,¶ g3 3

¶ hand

¶ g4 4

¶ x , hence as the non-vanishing ChristoŒel symbols Clmn

remain.

C144 = ± sin x cos x / R2

, C233 = ± sin h cos h,

C323 = cot h, C

414 = ± tan x .

(28)

Subst itut ing (28) into the geodesic equat ions

d2 xl

ds2+ C

lmn

dxm

dsdxn

ds= 0, (29)

we get the following system of diŒerential equat ions.

d2 x

ds2 ±sin x cos x

R2 ( dt

ds )2

= 0, (30)

d2h

ds2 ± sin h cos h( dw

ds )2

= 0, (31)

d2w

ds2+ 2 cot h

dh

dsdw

ds= 0, (32)

d2 t

ds2 ± 2 tan xdx

dsdt

ds= 0 . (33)

If we put h = p / 2 and dh/ ds = 0 in (31) , we get d2 h/ ds2 = 0, hence the

plane mot ion is possible. In the following we restrict ourselves only to this

case. The integral of (32) reduces to

dw

ds= h, (34)


where h is an integrat ion constant. Integrat ing (33) , we get

dt / ds = c/ cos2

x , (35)

where c is an integrat ion constant and c > 0. We may use (17) in place of

(30) , and we get, from (17) , (34) , (35) and the condit ion h = p / 2,

dx

ds= ±

pc2 ± (1 + h2R 2 ) cos2 x

R cos x. (36)

Eliminat ing ds from (34) , (35) and (36) , and using r1 = R sin x in place

of x , we get

dr1

dw= ± a

h(r2

1 + b)1 / 2

, (37)

dr1

dt= ± a

c ( 1 ±r2

1

R 2 ) (r21 + b)

1 / 2, (38)

where a = (h2 + 1R 2 )1/ 2 > 0 and b =

c2 - ( 1+ h 2 R 2)

a2 = R2 ( c2

1+ h 2 R 2 ± 1).

The equat ions (37) and (38) are our fundamental equat ions of motion.

Integrat ing (37) , we get as an orbital expression

r1 = (e F ± be - F) / 2, (39)

where F º ± ah (w ± w0 ) and w 0 is an integrat ion constant. We can regard

F in (39) as a new angle variable. As the typical special cases of (39) , we

haver1 = sinh F (b = + 1),

r1 = 12 exp F (b = 0),

r1 = cosh F (b = ± 1).

ü ý þ(40)

The spat ial part of (18) is entirely diŒerent from the euclidean character,

hence it seems to be impossible to treat (r1 , F ) as ordinary polar coordi-

nates. But regarding (r1 , F ) as an orthogonal net of coordinat es, we can

see the features of the orbit s. (See Fig. 1.)

When b > 0, the range 0 £ r1 <p

± b must be excluded from the

domain of part icle-motion, but when b ³ 0, there exist no restriction. And

then, from (38) , we get

T º ± a

c(t ± t0 ) = s dr1

(1 ±r 2

1

R 2 )p

r21 + b

, (41)

9 7 0 N ar ia i

F ig u r e 1 . I. r1 = sinh F , II. r1 = 1

2exp F , III. r1 = cosh F .4

where t0 is an integrat ion constant, and we may regard T as a new time

variable. Integrat ing the right -hand side of (41) and after suitable arrange-

ments, we get

r1 =ì í î

R[ m2 ± 1

m2 (e 2 m T + 1e 2 m T - 1 )2 ± 1]1 / 2

(b > 0, m2> 1),

R[ 1

1 + R2 e - 2T ]1/ 2

(b = 0, m2= 1),

R[ 1 ± m2

1 ± m2 ( e2 m T - 1e 2 m T + 1 )2 ]1 / 2

(b < 0, m2 < 1),

(42)

where m2 = 1 +b

R 2 = c2

1+ h 2 R 2 . Eliminat ing r1 from (39) and (42) , we get

for the diŒerent cases respectively

F = log[ R f [ ]1/ 2

+ ( [ ] + m2 ± 1)1 / 2 g ], (43)

where [ ] represents the expression in the bracket on the right -hand side

of (42) , and F = F (T ) is equivalent to w = w (t ). Thus we can obtain

the rigorous solut ion of the geodesic equat ions of motion. In a part icular

case of radial mot ion (h = 0), it becomes a = 1/ R , b = R 2 (c2 ± 1) and

m = c, and the expressions (38) and (42) keep their forms. The behavior

of part icle-motion is the same as that in the de Sitter universe, hence the

so-called ª velocity-distanceº relat ion is established.

4 W hen b = + 1, w < 0 seem s to be impossible, but this does not m ean any loss of

generality, as we can see from its de® nition.


5. CONCLUSION

The solut ion considered here is a new cosmological static solut ion

of Einstein’ s ® eld equat ions of gravitat ion, which has an empty matter-

distribut ion. Its line element can not be transformed into the standard

one, and it is shown that this is the only one having such a charact er.

The geometrical propert ies of this solut ion are very similar to those

of de Sitter universe and in part icular, as far as we consider situat ions in

the radial direct ion, both universes permit the ident ical result s.

Feb. 13th 1951

Astronomical Institute

Tohoku University

hidekazu nariai

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