On the Curv ature of Space ²

Transcription

1 Gener al Relativity and Gravitation, Vol. 31, No. 1, 1999 On the Curv ature of Space ² By A. Friedman in Peterburg * With one gure. Received on 9. June In their well-known work on general comologic al quetion, Eintein 1 and de Sitter arrive at two poible typ e of the univere; Eintein obtain the o-called cylindrical world, in which pace 3 ha contant, time-indep endent curvature, where the curvature radiu i connected to the total ma of matter preent in pace; de Sitter obtain a pherical world in which not only pace, but in a certain ene alo the world can be addreed a a world of contant curvature. 4 In doing o both Eintein and de Sitter make certain preupp oition about the matter tenor, which correpond to the incoherence of matter and it relativ e ret, i.e. the velocity of matter will be uppoed to be u ciently mall in comparion to the fundamental velocity 5 the velocity of light. ² È Originally publihed in Zeitchrift f Èur Phyik 10, (19), with the title ª Uber die Kr Èumm ung de Raume. Both pap er are printed with the kind permiion of Springer-V erlag GmbH & Co. KG, the curren t copyrigh t owner, and tranlated by G. F. R. Elli and H. van Elt, Departmen t of M athematic and Applied Mathematic, Univerit y of Cap e Town, Rondeboch 7701, South Africa. Some obviou typ o have 1 been corrected in thi tranlation. Eintein, Comological conideration relating to the general theory of relativit y, Sitzungberichte Berl. Akad de Sitter, On Eintein theory of gravitation and it atronomical conequence. Monthly Notice of the R. Atronom. Soc By ª pace we undertand here a pace that i decrib ed by a manifold of three dimenion; the ª world correp ond to a manifold of four dimenion. 4 Klein, On the integral form of the conervation theorem and the theory of the patially cloed world. G Èotting. Nachr See thi name by Eddington given in hi book: Epace, Temp et Gravitation, P artie, S. 10. P ari /99/ $16.00/0 c1999 Plen um Publihing Corporation

2 199 Friedman The goal of thi Notice i, rtly the derivation of the cylindrical and pherical world (a pecial cae) from ome general aumption, and econdly the proof of the poibility of a world whoe pace curvature i contant with rep ect to three coordinate that erve a pace coordinate, and dependent on the time, i.e. on the fourth the time coordinate; thi new type i, a concerning it other propertie, an analogue of the Eintein cylindrical world.. The aumption on which we bae our conideratio n divide into two clae. To the rt cla belong aumption which coincide with Eintein and de Sitter aumption ; they relate to the equation which the gravitational potential obey, and to the tate and the motion of matter. To the econd cla belong aumption on the general, o to peak geometrical character of the world; from our hyp othei follow a a pecial cae Eintein cylindrical world and alo de Sitter pherical world. The aumption of the rt cla are the following: 1. The gravitationa l potential obey the Eintein equation ytem with the comological term, which may alo be et to zero: R ik 1/ g ik R + l g ik = k T ik (i, k = 1,, 3, 4), (A) here g ik are the gravitational potential, T ik the matter tenor, k a contant, R = g ik R ik ; R ik i determined by the equation R ik = lg p g lg p g x i x k x {ik } x {ik } + {ia }{k a }, (B ) where x i (i = 1,, 3, 4) are the world coordinate, and {ik l } the ChritoŒel ymbol of the econd kind. 6. Matter i incoherent and at relativ e ret; or, le trongly expreed, the relativ e velocitie of matter are vanihingly mall in comparion to the velocit y of light. A a conequenc e of thee aumption the matter tenor i given by the equation: T ik = 0 for i and k not = 4, T 44 = c r g 44, (C ) where r i the denit y of matter and c the fundamental velocity; furthermore the world coordinate are divided into three pace coordinate x 1, x, x 3 and the time coordinate x 4. 6 The ign of R ik and of R i diœeren t in our cae from the uual one.

3 On the Curv ature of Space The aumption of the econd cla are the following: I. After ditribution of the three pace coordinate x 1, x, x 3 we have a pace of contant curvature, that however may depend on x 4 the time coordinate. The interval 7 d, determined by d = g ik dx idx k, can be brought into the following form through intro duction of uitable pace coordinate: d = R (dx 1 + in x 1 dx + in x 1 in x dx 3 ) + g 14 dx 1 dx 4 + g 4 dx dx 4 + g 34 dx 3 dx 4 + g 44 dx 4. Here R depend only on x 4 ; R i proportiona l to the curvature radiu of pace, which therefore may change with time.. In the expreion for d, g 14, g 4, g 34 can be made to vanih by correponding choice of the time coordinate, or, hortly aid, time i orthogonal to pace. It eem to me that no phyical or philoophi cal reaon can be given for thi econd aumption; it erve excluively to implify the calculation. One mut alo remark that Eintein and de Sitter world are contained a pecial cae in our aumption. In conequence of aumption 1 and, d can be brought into the form d = R (dx 1 + in x 1 dx + in x 1 in x dx 3) + M dx 4, (D ) where R i a function of x 4 and M in the general cae depend on all four world coordinate. The Eintein univere i obtained if one replace in (D) R by R c and furthermore et M equal to 1, whereby R mean the contant (indep endent of x 4) curvature radiu of pace. De Sitter univere i obtained if one replace in (D) R R by c and M by co x1 : dt = dt = R c (dx 1 + in x 1 dx + in x 1 in x dx 3) + dx 4, (D 1 ) R c (dx 1 + in x 1 dx + in x 1 in x dx 3) + co x 1 dx 4. 8 (D ) 7 See e.g. Eddington, Epace, Temp et Gravitation, P artie. P ari The d, which i aumed to have the dimenion of time, we denote by dt ; then the contan t k ha the dimenion Length and in CGS-unit i equal to 1, ± M a 7. See Laue, Die Relativit Èattheorie, Bd. II, S Braunc hweig 191.

4 1994 Friedman 4. Now we mut make an agreement over the limit within which the world coordinate are con ned, i.e. over which point in the fourdimeniona l manifold we will addre a being diœerent; without engaging in a further juti cation, we will upp oe that the pace coordinate are con ned within the following interval: x 1 in the interval (0, p); x in the interval (0, p) and x 3 in the interval (0, p); with repect to the time coordinate we make no preliminary retrictiv e aumption, but will conider thi quetion further below.. 1. It follow from the aumption (C) and (D), if one et in equation (A) i = 1,, 3 and k = 4, that: R9 (x 4) M x 1 = R9 (x 4 ) M x = R9 (x 4) M x 3 = 0 ; from thi there arie the two cae: (1.) R9 (x 4) = 0, R i independent of x 4, we wih to deignate thi world a a tationary world; (.) R9 (x 4 ) not = 0, M depend only on x 4; thi hall be called the non-tationary world. We conider rt the tationary world and write down the equation (A) for i, k = 1,, 3 and further i not = k, thu we obtain the following ytem of formulae: M M cotg x 1 = 0, x 1 x x M M cotg x 1 = 0, x 1 x 3 x 3 M M cotg x = 0. x x 3 x 3 Integration of thee equation yield the following expreion for M : M = A(x 3, x 4 ) in x 1 in x + B (x, x 4 ) in x 1 + C(x 1, x 4 ), (1) where A, B, C are arbitrary function of their argument. If we olve the equation (A) for R ik and eliminate the unknown denity r 9 from the till unued equation, we obtain, if we ubtitute for M the expreion (1), after omewhat lengthy, but completely elementary calculation the following two poibilitie for M : M = M 0 = cont, () 9 The denit y r i in our cae an unknown function of the world coordinate x1, x, x3, x4.

5 On the Curv ature of Space 1995 M = (A 0 x 4 + B 0 ) co x 1, (3) where M 0, A 0, B 0 denote contant. If M i equal to a contant, the tationary world i the cylindrical world. Here it i advantageou to operate with the gravitational potential of formula (D 1 ); if we determine the denity and the quantit y l, Eintein well-known reult will be obtained: l = c R, r = k R, M = 4 p k R, where M mean the total ma of pace. In the econd poible cae, when M i given by (3), we arrive by mean of a well-behaved tranformation of x 10 4 at de Sitter pherical world, in which M = co x 1 ; with the help of (D ) we obtain de Sitter relation: l = 3 c R, r = 0, M = 0. Thu we have the following reult: the tationary world i either the Eintein cylindrical world or the de Sitter pherical world.. We now want to conider the non-tationary world. M i now a function of x 4 ; by uitable choice of x 4 (without harming the generalit y of the conideration), one can obtain that M = 1; to relate thi to our uual picture, we give d a form that i analogou to (D 1) and (D ): dt = R (x 4) c (dx 1 + in x 1 dx + in x 1 in x dx 3) + dx 4. (D 3 ) Our tak i now the determination of R and r from the equation (A). It i clear that the equation (A) with diœering indice give nothing; the equation (A) for i = k = 1,, 3 give one relation: R9 R + R R9 9 R the equation (A) with i = k = 4 give the relation: 3 R9 R + c l = 0, (4) R + 3 c R l = k c r, (5) 10 Thi tranformation i given by the formula dx4 = p A 0 x4 + B 0 dx4.

6 1996 Friedman with R9 = dr and R9 9 = d R dx 4 dx. 4 Since R9 i not = 0, the integration of equation (4), when we alo write t for x 4, give the following equation: 1 c ( dr dt ) = A R + l 3 c R 3, (6) R where A i an arbitrary contant. From thi equation we obtain R by inverion of an elliptic integral, i.e. by olving for R the equation t = 1 c a R x A x + l dx + B, (7) 3 c x3 in which B and a are contant, and where alo care mut be taken of the uual condition of the change of ign of the quare root. From the equation (5), r can be determined to be r = 3 A k R 3 ; (8) the contant A i expreed in term of the total ma of pace M according to: A = k M 6 p. (9) If M i poitive, then alo A will be poitive. 3. We mut bae the conideration of the non-tationary world on equation (6) and (7); here the quantit y l i not determined; we will aume that it can have arbitrary value. We determine now thoe value of the variable x at which the quare root of formula (7) can change it ign. Retricting our conideration to poitive curvature radii, it u ce to conider for x the interval (0, ) and within thi interval thoe value of x that make the quantit y under the quare root equal to 0 or. One value of x for which the quare root in (7) i equal to zero, i x = 0; the remaining value of x, at which the quare root in (7) change it ign, are given by the poitive root of the equation A x + l 3 c x3 = 0. We l denote 3 c by y and conider in the (x, y)-plane the family of curve of third degree: y x 3 x + A = 0. (10)

7 On the Curv ature of Space 1997 A i here the parameter of the family, that varie in the interv al (0, ). The curve of the family (. Fig.) cut the x-axi at the point x = A, y = 0 and have a maximum at the point x = 3 A, y = 4 7 A. From the gure it i apparent that for negative l the equation A x + l 3 c x3 = 0 ha one poitive root x 0 in the interval (0, A). Conidering x 0 a a function of l and A: x 0 = H( l, A), one nd that H i an increaing function of l and an increaing function of A. If l i located in the interval ( 0, 4 c 9 A ), the equation ha two poitive root x 0 = H( l, A) and x9 0 = q (l, A), where x 0 fall in the interval ( A, 3 A and x9 0 in the interval ( 3 A, ) ; H( l, A) i an increaing function both of l and of A, q (l, A) a decreaing function of l and A. Finally if l i greater c than 4 9 A, the equation ha no poitive root. We now go over to the conideration of formula (7) and precede thi conideration by the following remark: let the curvature radiu be equal )

8 1998 Friedman to R 0 for t = t 0 ; the ign of the quare root in (7) for t = t 0 i poitive or negative according to whether for t = t 0 the curvature radiu i increaing or decreaing; by replacing t by t if neceary, we can alway make the quare root poitive, i.e. by choice of the time we can alway achieve that the curvature radiu for t = t 0 increae with increaing time. 4. We conider rt the cae l > 4 c 9 A, i.e. the cae when the equation A x + l 3 c x3 = 0 ha no poitive root. The equation (7) can then be written in the form t t 0 = 1 R x c R 0 A x + l dx, (11) 3 c x3 where according to our remark, the quare root i alway poitive. From thi it follow that R i an increaing function of t; the poitive initial value R 0 i free of any retriction. Since the curvature radiu cannot be le than zero, o it mut, decreaing from R 0 with decreaing time t, reach the value zero at the intan t t9. The time of growth of R from 0 to R 0 we want to call the time ince the creation of the world; 11 thi time t9 i given by: t9 = 1 c 0 R 0 x A x + l 3 c x3 dx. (1) We deignate the world conidered a a monotonic world of the rt kind. The time ince the creation of the (monoton ic) world (of the rt kind), conidered a a function of R 0, A, l, ha the following propertie: 1. it grow with growing R 0;. it decreae, when A increae, i.e. the ma in pace increae; 3. it decreae, when l increae. If A > 3 R 0, then for an arbitrary l the time that ha owed ince the creation of the world i nite; if A R 3 0, then there can alway be found a value of l = l 1 = 4 c uch that a l approache thi value, the time ince the 9 A creation of the world increae without limit. 5. Now l hall lie in the interval ( 0, 4 c 9 A ) ; then the initial value of the curvature radiu can lie in the interval: (0, x 0), (x 0, x9 0 ), (x9 0, ). If R 0 fall into the interval (x 0, x9 0 ), then the quare root in formula (7) i imaginary; a pace with thi initial curvature i impoible. 11 The time ince the creation of the world i the time that ha owed from that intan t when the pace wa one point (R = 0) until the preen t tate ( R = R 0 ); thi time may alo be in nite.

9 On the Curv ature of Space 1999 We will dedicate the next ection to the cae when R 0 lie in the interval (0, x 0 ); here we till conider the third cae: R 0 > x9 0 or R 0 > q (l, A). By conideratio n which are analogou to the preceding one, it can be hown that R i an increaing function of time, where R can begin with the value x9 0 = q (l, A). The time that ha elaped from the intan t when R = x9 0 until the intant which correpond to R = R 0, we again call the time ince the creation of the world. Let thi be t9, then it i t9 = 1 c x9 0 R 0 x A x + l dx. (13) 3 c x3 Thi world we call a monotonic world of the econd kind. 6. We conider now the cae when l fall between the limit (, 0). In thi cae if R 0 > x 0 = H( l, A), o the quare root in (7) become imaginary, the pace with thi R 0 i impoible. If R 0 < x 0, then the conidered cae i identical with the one that we left aide in the previou ection. We thu uppoe that l lie in the interval (, 4 c 9 A ) and that R 0 < x 0. Through known conideration 1 one can now how that R i a periodic function of t, with period t p, we call thi the world period; t p i given through the formula: t p = c 0 x 0 x A x + l dx. (14) 3 c x3 The curvature radiu varie thereby between 0 and x 0. We want to call thi world the periodic world. The period of the periodic world increae, when we increae l, and tend toward in nity, when l tend toward the value l 1 = 4 c 9 A. For mall l the period i repreented by the approximation formula t p = p A c. (15) With repect to the periodic world, two viewp oint are poible: if we regard two event to be coincident if their pace coordinate coincide and 1 See e.g. Weiertra, On a cla of real periodic function. Monatber. d. K Èonigl. Akad. d. Wiench. 1866, and Horn, On the theory of mall nite ocillation. ZS. f. Math. und Phyik 47, 400, 190. In our cae, the conideration of thee author mut be altered appropriately; the periodicity in our cae can be determined by elemen tary conideration.

10 000 Friedman the diœerence of the time coordinate i an integer multiple of the period, then the curvature radiu increae from 0 to x 0 and then decreae to the value 0; the time of the world exitence i nite; on the other hand, if the time varie between and + (i.e. we conider two event to be coincident only when not only their pace coordinate but alo their world coordinate coincide), then we arrive at a true periodicit y of the pace curvature. 7. Our knowledge i completely inu cient to carry out numerical calculation and to decide, which world our univere i; it i poible that the cauality problem and the problem of the centrifugal force will illuminate thee quetion. It i left to remark that the ª comologica l quantit y l remain undetermined in our formulae, ince it i an extra contant in the problem; poibly electrodynamical conideration can lead to it evaluation. If we et l = 0 and M = olar mae, then the world period become of the order of 10 billion year. But thee gure can urely only erve a an illutration for our calculation. Petrograd, 9. May 19.