On the Curv ature of Space²

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On the Curv ature of Space²

By A. Friedman in Petersburg^*

With one ® gure. Received on 29. June 1922

§1. 1. In their well-known works on general cosmologic al questions, Einstein¹ and de Sitter² arrive at two possible typ es of the universe; Ein- stein obtains the so-called cylindrical world, in which space³ has constant, time-indep endent curvature, where the curvature radius is connected to the total mass of matter present in space; de Sitter obtains a spherical world in which not only space, but in a certain sense also the world can be addressed as a world of constant curvature.⁴ In doing so both Einstein and de Sitter make certain presupp ositions about the matter tensor, which correspond to the incoherence of matter and its relativ e rest, i.e. the velocity of matter will be supp osed to be su ciently small in comparison to the fundamental velocity⁵ — the velocity of light.

² Originally published in Zeitschrift f Èur Physik10, 377-386 (1922), with the title ª ÈUber die Kr Èumm ung des Raumes”. Both papers are printed with the kind permission of Springer-V erlag Gmb H & Co. KG, the cu rren t co pyrigh t owner, and translated by G.

F. R. Ellis and H . van Elst, Dep artmen t of M athemat ics and Applied M athematics, Un iversit y of Cap e Town, Rondebo sch 7701, South Africa. Some obvious typ os have b een corrected in this translation.

1 Einstein, Cosmological co nsiderations relating to the general theory of relativit y, Sitzungsberichte Berl. Akad. 1917.

2 de Sitter, On Einstein’ s theory of gravitation and its astro nomical consequences.

M onthly Notices of the R. Astronom. Soc. 1916-1917 .

3 By ª space” we understand here a space that is describ ed by a manifold of three dimensions; the ª world” corresp onds to a manifold of four dimensions.

4 Klein, On the integral form of the con servation theorems and the theory of the spatially closed world. G Èotting. Nachr. 1918.

5 See this name by Eddington given in his book: Espace, Temps et Gravitation , 2 P artie, S. 10. P aris 1921.

1991

0001-7701/99/1200-1991$16.00/0 °c1999 Plen um Publishing Corporation

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The goal of this Notice is, ® rstly the derivation of the cylindrical and spherical worlds (as special cases) from some general assumption s, and secondly the proof of the possibility of a world whose space curvature is constant with resp ect to three coordinates that serve as space coordinates, and dependent on the time, i.e. on the fourth— the time coordinate; this new typ e is, as concerning its other properties, an analogue of the Einstein cylindrical world.

2. The assumption s on which we base our consideratio ns divide into two classes. To the ® rst class belong assumptions which coincide with Ein- stein’ s and de Sitter’ s assumption s; they relate to the equations which the gravitational potentials obey, and to the state and the motion of matter.

To the second class belong assumption s on the general, so to speak geo- metrical character of the world; from our hyp othesis follows as a special case Einstein’ s cylindrical world and also de Sitter’ s spherical world.

The assumptions of the ® rst class are the following:

1. The gravitationa l potentials obey the Einstein equation system with the cosmological term, which may also be set to zero:

Rik ¡ 1/ 2 gikR + l gik = _¡ k Tik (i, k = 1, 2, 3, 4), (A ) here gik are the gravitational potentials, Tik the matter tensor, k — a constant, R = g^ikRik; Rik is determined by the equations

Rik = ¶ ²lgp_g

¶ xi¶ xk ¡ ¶ lgp_g

¶ xs

{

^ik^s

}

^¡ ^¶^¶ ^x^s

{

^ik^s

}

⁺

{

^ia^s

}{

^{k s}^a

}

^, ^{(B )}

where xi(i = 1, 2, 3, 4) are the world coordinates, and

{

^ik^l

}

the ChristoŒel sym bols of the second kind.⁶

2. Matter is incoherent and at relativ e rest; or, less strongly expressed, the relativ e velocities of matter are vanishingly small in comparison to the velocit y of light. As a consequenc e of these assumption s the matter tensor is given by the equations:

Tik = 0 for i and k not = 4,

T44 = c²r g44, (C )

where r is the densit y of matter and c the fundamental velocity; further- more the world coordinates are divided into three space coordinates x1, x2, x3 and the time coordinate x4.

6 Th e sign ofRik and ofR is diŒeren t in our case from the usual one.

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3. The assumption s of the second class are the following:

I. After distributio n of the three space coordinates x1, x2, x3 we have a space of constant curvature, that however may depend on x4 — the time coordinate. The interval⁷ ds, determined by ds² = gikdxidxk, can be brought into the following form through intro duction of suitable space coordinates:

ds² = R²(dx²₁+ sin²x1dx²₂+ sin²x1sin²x2dx²₃)

+ 2 g14dx1dx4+ 2 g24dx2dx4 + 2 g34dx3dx4+ g44dx²₄. Here R depends only on x4; R is proportiona l to the curvature radius of space, which therefore may change with time.

2. In the expression for ds², g14, g24, g34 can be made to vanish by corresponding choice of the time coordinate, or, shortly said, time is orthogonal to space. It seems to me that no physical or philosophi cal reasons can be given for this second assumption; it serves exclusively to simplify the calculation s. One must also remark that Einstein’ s and de Sitter’ s world are contained as special cases in our assumptions.

In consequen ce of assumptions 1 and 2, ds² can be brought into the form

ds² = R²(dx²₁+ sin²x1dx²₂+ sin²x1sin²x2dx²₃) + M²dx²₄, (D ) where R is a function of x4 and M in the general case depends on all four world coordinates. The Einstein universe is obtained if one replaces in (D) R² by ¡ ^R_c2² and furthermore sets M equal to 1, whereby R means the constant (indep endent of x4) curvature radius of space. De Sitter’ s universe is obtained if one replaces in (D) R² by ¡ ^R_c2² and M by cos x1:

dt²= ¡ R²

c² (dx²₁+ sin²x1dx²₂+ sin²x1sin²x2dx²₃) + dx²₄, (D1) dt²= _¡ R²

c² (dx²₁+ sin²x1dx²₂+ sin²x1sin²x2dx²₃) + cos²x1dx²₄.⁸ (D2)

7 See e.g. Eddington, Espace, Temps et Gravitation , 2 P artie. P aris 1921.

8 The ds, which is assumed to have the dimension of time, we denote by dt ; then the co nstan t k has the dimension Length

M ass and in CGS-units is equal to 1, 87.10^± ²⁷. See Lau e, Die Relativit Èatstheorie , Bd. II, S. 185. Braun sc hweig 1921.

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4. Now we must make an agreement over the limits within which the world coordinates are con® ned, i.e. over which points in the four- dimensiona l manifold we will address as being diŒerent; without engaging in a further justi® cation, we will supp ose that the space coordinates are con® ned within the following intervals: x1 in the interv al (0, p); x2 in the interv al (0, p) and x3 in the interval (0, 2p); with resp ect to the time co- ordinate we make no preliminary restrictiv e assumption, but will consider this question further below.

§2. 1. It follows from the assumptions (C) and (D), if one sets in equations (A) i = 1, 2, 3 and k = 4, that:

R 9 (x4) ¶ M

¶ x1

= R 9 (x4)¶ M

¶ x2

= R 9 (x4)¶ M

¶ x3

= 0 ;

from this there arise the two cases: (1.) R9 (x4) = 0, R is independent of x4, we wish to designate this world as a stationary world; (2.) R 9 (x4) not

= 0, M depends only on x4; this shall be called the non-statio nary world.

We consider ® rst the stationary world and write down the equations (A) for i, k = 1, 2, 3 and further i not = k, thus we obtain the following system of formulae:

¶ ²M

¶ x1¶ x2 ¡ cotg x1 ¶ M

¶ x2

= 0,

¶ ²M

¶ x1¶ x3 ¡ cotg x1 ¶ M

¶ x3

= 0,

¶ ²M

¶ x2¶ x3 ¡ cotg x2 ¶ M

¶ x3

= 0 .

Integration of these equations yields the following expression for M : M = A(x3, x4) sin x1sin x2+ B (x2, x4) sin x1+ C (x1, x4), (1) where A, B , C are arbitrary functions of their arguments. If we solve the equations (A) for Rik and eliminate the unknown density r⁹ from the still unused equations, we obtain, if we substitute for M the expression (1), after somewhat lengthy, but completely elementary calculation s the following two possibilitie s for M :

M = M0 = const, (2)

9 The densit yr is in our case an unknown function of the world coordinates x1,x2,x3, x4.

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M = (A0x4+ B0) cos x1, (3) where M0, A0, B0 denote constants.

If M is equal to a constant, the stationary world is the cylindrical world. Here it is advantageous to operate with the gravitational potentials of formula (D1); if we determine the density and the quantit y l, Einstein’ s well-known result will be obtained:

l = c²

R² , r = 2

k R² , M = 4 p² k R , where M means the total mass of space.

In the second possible case, when M is given by (3), we arrive by means of a well-behaved transformation of x410 at de Sitter’ s spherical world, in which M = cos x1; with the help of (D2) we obtain de Sitter’ s relations:

l = 3 c²

R² , r = 0, M = 0 .

Thus we have the following result: the stationary world is either the Einstein cylindrical world or the de Sitter spherical world.

2. We now want to consider the non-statio nary world. M is now a function of x4; by suitable choice of x4 (without harming the generalit y of the consideration), one can obtain that M = 1; to relate this to our usual picture, we give ds² a form that is analogous to (D1) and (D2):

dt² = ¡ R²(x4)

c² (dx²₁+ sin²x1dx²₂+ sin²x1sin²x2dx²₃) + dx²₄. (D3) Our task is now the determination of R and r from the equations (A).

It is clear that the equations (A) with diŒering indices give nothing; the equations (A) for i = k = 1, 2, 3 give one relation:

R 9²

R² + 2 R R9 9 R² + c²

R² ^¡ l = 0, (4)

the equation (A) with i = k = 4 gives the relation:

3 R 9² R² + 3 c²

R² ^¡ l = k c²r, (5)

10 Th is transformation is given by the formuladx4 = p _A

0x4+ B0dx4.

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with

R 9 = dR dx4

and R 9 9 = d²R dx²₄ .

Since R9 is not = 0, the integration of equation (4), when we also write t for x4, gives the following equation:

1

c²

(

^dR^dt

)

²⁼ ^A^¡ ^{R +}^R³^l^c² ^R³ ^, ⁽⁶⁾

where A is an arbitrary constant. From this equation we obtain R by inversion of an elliptic integral, i.e. by solving for R the equation

t = 1

c

s

^a^R^s^A¡ x +^x ₃^l_c2 x³ dx + B , (7) in which B and a are constants, and where also care must be taken of the usual condition s of the change of sign of the square root. From the equation (5), r can be determined to be

r = 3 A

k R³ ; (8)

the constant A is expressed in terms of the total mass of space M according to:

A = k M

6 p² . (9)

If M is positive, then also A will be positive.

3. We must base the consideration of the non-statio nary world on equations (6) and (7); here the quantit y l is not determined; we will assume that it can have arbitrary values. We determine now those values of the variable x at which the square root of formula (7) can change its sign. Restricting our considerations to positive curvature radii, it su ces to consider for x the interv al (0,

¥

) and within this interv al those values of x that make the quantit y under the square root equal to 0 or

¥

^{. One}

value of x for which the square root in (7) is equal to zero, is x = 0; the remaining values of x, at which the square root in (7) changes its sign, are given by the positive roots of the equation A¡ x + ₃^l_c2x³ = 0. We denote ₃^l_c2 by y and consider in the (x, y)-plane the family of curves of third degree:

y x³¡ x + A = 0 . (10)

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A is here the parameter of the family, that varies in the interval (0,

¥

^).

The curves of the family (s. Fig.) cut the x-axis at the point x = A, y = 0 and have a maxim um at the point

x = 3 A

2 , y = 4

27 A² .

From the ® gure it is apparent that for negative l the equation A_¡ x +

l

3c²x³ = 0 has one positive root x0 in the interv al (0, A). Considering x0

as a function of l and A:

x0= H( l, A),

one ® nds that H is an increasing function of l and an increasing function of A. If l is located in the interv al

(

^0,⁴⁹^A^c²²

)

, the equation has two positive roots x0 = H( l, A) and x9₀= q (l, A), where x0falls in the interval

(

^A,³²^A

)

and x9₀ in the interval

(

³2^A,

¥ )

; H( l, A) is an increasing function both of l and of A,q (l, A) a decreasing function of l and A. Finally if l is greater than ⁴₉_A^c²2, the equation has no positive roots.

We now go over to the consideration of formula (7) and precede this consideration by the following remark: let the curvature radius be equal

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to R0 for t = t0 ; the sign of the square root in (7) for t = t0 is positive or negative according to whether for t = t0 the curvature radius is increasing or decreasing; by replacing t by ¡ t if necessary, we can always make the square root positive, i.e. by choice of the time we can always achieve that the curvature radius for t = t0 increases with increasing time.

4. We consider ® rst the case l > ⁴₉_A^c²2, i.e. the case when the equation A¡ x + ₃^l_c2x³ = 0 has no positive roots. The equation (7) can then be written in the form

t¡ t0 = 1

c

s

^R^R⁰ ^s^A¡ x +^x ₃^l_c2x³ dx, (11) where according to our remark, the square root is always positive. From this it follows that R is an increasing function of t; the positive initial value R0 is free of any restriction.

Since the curvature radius cannot be less than zero, so it must, de- creasing from R0 with decreasing time t, reach the value zero at the instan t t9 . The time of growth of R from 0 to R0 we want to call the time since the creation of the world;¹¹ this time t9 is given by:

t9 = 1

c

s

⁰^R⁰ ^s^A¡ x +^x ₃^l_c2x³ dx. (12) We designate the world considered as a monotonic world of the ® rst kind.

The time since the creation of the (monoton ic) world (of the ® rst kind), considered as a function of R0, A, l, has the following properties:

1. it grows with growing R0; 2. it decreases, when A increases, i.e. the mass in space increases; 3. it decreases, when l increases. If A > ²₃R0, then for an arbitrary l the time that has ¯ owed since the creation of the world is ® nite; if A

£

²3R0, then there can always be found a value of l = l1 = ₉⁴_A^c²2 such that as l approaches this value, the time since the creation of the world increases without limit.

5. Now l shall lie in the interv al

(

^0,⁹⁴^A^c²²

)

; then the initial value of the curvature radius can lie in the intervals: (0, x0), (x0, x9₀), (x9₀,

¥

^{). If}

R0 falls into the interval (x0, x9₀), then the square root in formula (7) is imaginary; a space with this initial curvature is impossible.

11 The time since the creatio n of the world is the time that has ¯ owed from that instan t when the space was one p oint (R = 0) until the presen t state (R = R0); this time may also b e in® nite.

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We will dedicate the next section to the case when R0 lies in the interv al (0, x0); here we still consider the third case: R0 > x9₀ or R0 >

q (l, A). By considerations which are analogous to the preceding ones, it can be shown that R is an increasing function of time, where R can begin with the value x9₀ = q (l, A). The time that has elapsed from the instant when R = x9₀ until the instant which corresponds to R = R0, we again call the time since the creation of the world. Let this be t9 , then it is

t9 = 1

c

s

^x9^R⁰⁰ ^s^A¡ x +^x ₃^l_c2x³ dx. (13) This world we call a monotonic world of the second kind.

6. We consider now the case when l falls between the limits (¡

¥

^{, 0).}

In this case if R0 > x0 = H( l, A), so the square root in (7) becomes imaginary, the space with this R0 is impossible. If R0 < x0, then the considered case is identical with the one that we left aside in the previous section. We thus suppose that l lies in the interv al

(

^¡

^¥

^,⁹⁴^A^c²²

)

^{and that}

R0 <x0. Through known considerations¹² one can now show that R is a periodic function of t, with period tp, we call this the world period; tp is given through the formula:

tp = 2

c

s

⁰^x⁰^s^A¡ x +^x ₃^l_c2x³ dx. (14) The curvature radius varies thereby between 0 and x0. We want to call this world the periodic world. The period of the periodic world increases, when we increase l, and tends towards in® nity, when l tends towards the value l1 = ₉⁴_A^c²2.

For small l the period is represented by the approximation formula tp = p A

c . (15)

With resp ect to the periodic world, two viewp oints are possible: if we regard two events to be coincident if their space coordinates coincide and

12 See e.g. W eierstrass, On a class of real periodic functions. Monatsber. d. K Èonigl.

Akad. d. W issensch. 1866, an d H orn, On the theory of sm all ® nite oscillations.

ZS. f. Math. und Physik 47, 400, 1902. In our case, the con siderations of these au thors m ust be altered ap propriately; the periodicit y in our case can be determin ed by elemen tary con siderations.

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the diŒerence of the time coordinates is an integer multiple of the period, then the curvature radius increases from 0 to x0 and then decreases to the value 0; the time of the world’ s existence is ® nite; on the other hand, if the time varies between ¡

¥

^{and +}

¥

(i.e. we consider two events to be coincident only when not only their space coordinates but also their world coordinates coincide), then we arrive at a true periodicit y of the space curvature.

7. Our knowledge is completely insu cient to carry out numerical cal- culations and to decide, which world our universe is; it is possible that the causality problem and the problem of the centrifugal force will illuminate these questions. It is left to remark that the ª cosmologica l” quantit y l remains undetermin ed in our formulae, since it is an extra constant in the problem; possibly electrodynamical considerations can lead to its evalua- tion. If we set l = 0 and M = 5.10²¹ solar masses, then the world period becomes of the order of 10 billion years. But these ® gures can surely only serve as an illustration for our calculation s.

Petrograd, 29. May 1922.