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On Isotropic Manifolds in the Theory of Relativity

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On Isotropic Manifolds in the Theory of Relativity ON ISOTROPIC MANIFOLDS IN THE THEORY OF RELATIVITY Bu D. l~. MOG~E ( R'.se:~rch Scholar, B~mbay University) Rece ived ffuly 24, 1939 (Communicated by Dr. G. S. Mahajani) THe condition of isotropy has played a great part in the general theory of relativlty in so far as it has considerably helped to obtain the solutions of Einstein's gravitational equations. It has been customary to take this condition as T, ~ =T91=T3' =--p (I) which is found to be only en approximate relation. The condition for ah isotropic manifold as understood in differential geometry is given by the equation B~,.~ = K (g,~ ~~ -- g,, 3',) (2) where the term on the left hand side is the Riemann-Christoffel tensor of the first kind for the fundamental tensor g:i, and 3t) is the KrSnecker delta. The relation (2) can be expressed in terms of Riemann-Christoffel tensor of the second kind as B,~~. ----g., B~,~ = K (g,~, g., -- g,, g..) (3) Here K is the Riemannian curvature which is, in general, supposed to vary from section to section of the space considered. It is the ob]ect of this paper to study the isotropy of relativity mani- folds from the point of view of differential geometry. Ir has been shown by Walker{ 11 that a Riemannian manifold V,t satisfying the condition of spherical symmetry about the geodesics of a Vn - 1 has the line-element of the form ds~ = dt ~ + g, dx" dx '~, (i, j = 1, 2. 9 n - 1) (4) In general, we can write (4) as ds 2 = g{j dx dx' dx', (i, j =0, 1, 2, 9 . 9., n -- 1) (5) where goo ----1, goi =gio =0, Si/ =g'ii, (i, j =1, 2 9 9 n --1). Ir will be understood that V,,- x is characterised by the line-element da ~ = g'ii dx 'i dx't (6) 275 276 D.N. Moghe The condition of isotropy for Vn- x can now be expressed as B ti,,.. ~ K t (g',. ajl - g',j 91 (7) It will be seen that for (i, j, l, m = 1, 2, 9 .., n -- 1) the relations (2) and (7) are the same, for go -g',%= ) (8) BŸ = B I,.,. Therefore, K = K'. (9) Hence, we deduce that the necessary condition that Vn is an isotropic manifold is that Vn- iis also another isotropic manifold with the same Riemannian curvature as that of Vn. Also from (2) we get, in addition B~i., = O, (i0) and Bi'o., ' Kg~,. l of, g~" B~o., = K I (::) of, I g~"~ B~%., = gjZ B~ a J [The retations (10) and (1l) can, otherwise, be proved as follows : We have B,j., -- ~ {im, a} b&'i bg~., _.:~ pg,,. _ ~g,i l . = . -~- +~{ij,~} ~t 'Y/Lax~ ~x~J = -- 89 [,ni,ii + .~g.~ ~t li.,.,~} + 89~t Ui, m] -- -.} g~,ra 3-t {ii, a} -- ~t ~ji, mi + 89~-t \ Ox' / (from the properties of the three-index symbols), = O, by the interchange of suffixes. .Stmilarly, B ooj = 88gap ~g_[ipbt-~--tbgM 89b2gii.bt ~' and o "" bS~i] ~t bt " ~t 2 Therefore, g'J B,'oj = g'~ B~0~.] Equations (10) & (11) supply the sufficient conditions for the isotropy of Vn- On 91 ~V[anifohis in/he Theory of ReIativity 277 The Riemannian eurvature K can now be expressed by the invariant relation of the forro K = 88 ~gie~r ~g~i3t -- 89g'! " ~'gubt ~ (12) Ir we write d~ 2 = U (t) fii dxidxi, (13) where gii = g'ii = U (t)fo' (i, j = 1, 2, 9 9 n -- 1), and g,i = g,ii = fu':' fil having the same significante as g0\ Ir is explicitly assumed that thefii's are not functions of t. For this case, therefore, (12) simplifies to 88 fii ff, p fip f~,i -- 89 fi, = K ; that is, • • = Z: (14) where a dot denotes a differentiafion with regard to t. The spaee being regarded as having a constant Riemannian curvature, K does not vary from point to point. The complete integral of (14) is of the form : u = A'- cos" (~ + V~t) (15) where A aada are constants of integration. Therefore, the line-element for an isotropic Riemannian manifold ha~-ing a spherical symmetry about the geodesics of a V'n- x can be written as ds 2 = dt z + A' cosz (a + V~t) f,i dxidxi (16) where V'n - t is itself an isotropic manifold characterised by da '~ =fii d xidx/. (17) We shall now apply these considerations to one of the most important line-elements in the theory of relativity satisfying the general conditions of spherical symmetry. This line-element is given by ds ~" = H (r, t) dt z -- F (r, t) {dr ~- + r~ 2 + r z sin 2 0dO2} (18) For line-elements of the forro (18), the condition of isotropy (2) reduces ultimately to relations of the type and B],. = -- Kg,, 3~, (l =v_j, i =/= m).~, (19) 2qow, ir we apply the process of corttraction of tensors to the relations (19) we get G=G, } and G# = -- Kg 6. (20) For cosmological considerations, Einstein has chosen c-.,; = ,~g,,,- (21) 278 D.N. Moghe as the modified law of gravitation, where ~ is the cosmical constant. Com- paring (21) with the second equation (20), we see that Riemannian curva- ture is given by K = -- ~. It will, therefore, be evident that the modified law Of gravitation is implied in the condition of isotropy as understood in Differential Geometry. K = 0 gives the original law of gravitation. For a perfect fluid (or, a disordered radiation which is the same asa perfect fluid), the material-energy tensor can be expressed as Ti/ = (Po + P) vi v] -- pgii (22) whieh gives the scalar T = gi; T;/= po - 3p = Poo (23) denoting the invariant density Poo. In the present case we must have ~o = 3p. Consequently, the invariant density vanishes, and, we deduce the following relation "rd =T2 ~ =Tg = --p (24) j ust as in the case of proper co-ordinates. Equations (24), therefore, gire us the usual (physical) condition of isotropy. Asa consequence of (24), it willbe found that v t =v 2 =v a ----0, and Tt ~ =0. This puts a sort of restriction of the field given by (18). Ir might be contended that this is a very severe restriction because the cross-component of the T') denoting the flow of matter from within or without the model vanishes. But the model in question is supposed to cover all the material limits of the universe at a particular epoch, so that, flow of matter Irom beyond or beyond these limits does not arise at all. Moreover, the internal motion is given by the geodesic equations, and not by Ti ~ Considering the tensor expression (22), we may say that the idea of isotropy is inherently present in it in so Ÿ as (22) is derived from the law of gravitation (21). For the line-element (18), the surviving cGmponents of the material-energy tensor satisfy the following relation : (T~O)'- + F (T:: -- T00) ('r:-" - T,') = 0(:) (25) which has been chosen by Walker as the condition of isotropy. This re- lation does not give isotropic pressure (i.e., the physical condition of iso- tropy) unless Tt~ 0. While, by a simple process and assumptions not inconsistent with physical ideas we have dednced the extra condition of isotropy (24). This goes to point out that (25) is not at all any real condi- tion of isotropy. REFERENCES 1. " On Riemannian sp~ces with sphericul symmetry about a liae, and the condi- tions for isotropy in General Rela~ivity, " Quart. Jour. Maths. (Oxford series), 6, 22, p. 87, Eq. 32. 2. Loc. cit., p. 89, Eq. 45. .
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