LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 510o84 U6r no. 111-130 cop .3 The person charging this material is re- sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. To renew call Telephone Center, 333-8400 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN BUILDING SEP 21 1^80 SEP 2 1930 UiiE ONLY L161— O-10Q6 Digitized by the Internet Archive in 2013 http://archive.org/details/conservationlaws114taub ho. 114 cop. 3 CONSERVATION LAWS AND VARIATIONAL PRINCIPLES IN GENERAL RELATIVITY* by A. H. Taut UNIVERSITY OF ILLINOIS DIGITAL COMPUTER LABORATORY URBANA, ILLINOIS Report No. 11^ April 23, 1962 This work was supported in part by the National Science Foundation under Grant Gl6^89 . ABSTRACT Special relativistic field theories involving tensor fields and derivable from variational principles are generalized to general relativistic theories. It is assumed that the Lagrangian is a function of the tensor field components and their first derivatives. The Lagrangian is first written in a general coordinate system in Minkowski space-time and then the restriction that the underlying space-time is flat is removed. The Einstein field equations for the gravitational field may be derived as was shown by Hjlbert, and the matter energy tensor is obtained as a function of the tensor field. The technique of E. Noether is used to derive conservation laws and relations be- tween the various matter energy tensors which arise. CONSERVATION IAWS AND VARIATIONAL PRINCIPLES IN GENERAL RELATIVITY 1. Introduction In this paper we examine a method of formulating in general rela- tivity a field theory described in special relativity "by a variational principle involving a Lagrangian function which is a scalar function, in 1 n Minkowski space time, and which in turn depends on a tensor field cp V-, . • -v A 1 m (written as cp.) and its first derivatives. The method we shall employ is derived from a form of the equivalence principle and is as follows: We shall use a general curvalinear coordinate system in Minkowski space-time in which to express the Lagrangian function aC. We shall then con- sider the g occurring in X. as the metric tensor of a general Riemannian space and determine the nature of this space by deriving the Einstein field equations from the variational principle 5 1=0 (1.1) g with i = / (RtK# ^g aSc, (1.2) J v R the scalar curvature of the Riemannian space, and 6 I the variation in I g produced by a variation of the quantities g . The equation determining the tensor field cp will be taken to be the Euler equation of 6 1=0 (1.3) cp v ' where 5 I is the variation in I, given by equation (1.2), produced by a varia- tion of the tensor field cp V-, • • -v 1 m It will be shown that there exists a symmetric tensor T formed from the components of cp and g which as a consequence of equation (l-3) [J. v satisfies T^ V = (1.10 ;v where the semi-colon denotes the covariant derivative and as a consequence of (1.2) satisfies G^ V - R^ V - |g^ V R = -KT^ V (1.5) where R is the Ricci tensor of the space-time with metric tensor g . The °uv tensor T may thus "be said to represent the "gravitational matter tensor" which "creates" the gravitational field represented by the metric tensor g [J, v uV Another tensor t , a nonsymmetric one, which is a generalization of the special relativity " inert ial" stress -energy tensor of the field cp will also be derived. It will be shown that t pu + 2T pu + n pm-X = (1>6) where N puX = _ N p\u (1>?) Equation (1.6) thus relates the inertial and gravitational matter tensors. 2. Notation In order to avoid an excessive use of indices, we employ the following notation. A 1 n /_ _ N V ~ 9 _ T (2.1) 1" m The symbol ~ is to be read as "stands for." A 1 n In q> ... ' r . ,.t = 9 '1' ' ' m;u "1" "m.,[i iV- T T*' T «.... T n-" T . a. . . .a. _pa. _ . . .a a. 1 i-r l+l n + > cp i-J. "* n pi ( } 1 I DLX 1 m ff_ . . .a In p cp rr H T . . .T . pT . . . .1 i T .11 1 J-1 K J+l m J^ -3- where the comma denotes the ordinary derivative and ,P ~P X ] Tax =g 2 ( W + S ^a " ^t^" (2 ° } Ll We also write ty., and ty ^ where t . . .T 1 n T . . • T [i I'm \i 1 n and t . . .t cr. .. .a A . 1 m In /_ _v ♦"A* ~* a,.. .a * r ...t (2 - 5) In 1 m ll A 1 m ll 1 n /_ r \ In 1 m That is, the quantity given in (2.5) is a scalar and that in (2.6) is a vector. We consider ^ as a function of three sets of variables: g . cp and cp and define % - &■ (2.8) dcp pa =-^ ' (2 - 9) dcp ;n In each case the remaining two sets of variables are kept constant in the partial differentiations. We also define _.pA.p. \ 1 m ll 1 l-l l+l n 1 ^ 1 n 1 m (2.10) T n . . .T . n \T . . . . .T O^ . . .0 1 j-1 j+1 m ll 1 n ap P 9 g H In 1 m-1 j m J ■It- It follows from equations (2.3) that < T = 8 P 4 (6g oMT ♦ 8g XT;o - 8g aT . x ) (2.11) and from (2.2) that a a v - 1 a n . . .a. , pa. -. . . .a a . e/ A N /c Av \ 1 i-l H l+l n c n ! •<*,„>-<*>,„ -£ » v^ 1 ^ i -) cp x m 5 r p Y T r-- T j-i p Vi--- T m \f where the variations are produced by varying the cp and the g . Hence Pa 6 ^ A J " Pa ^ A )., = \ pXp,i ^-u + 6 s^-n - 5g uo-X } A ;u A ;u d A.PJM- H±sP JJ-p , A. = M^ 8g x where M A. P U = J. ^ p A. P U + pP AU + p ^U P + pP U^ _ p U P A. _ p UA. P ^ (2.13> I X P^ ._ i (p^-P^ + p P^ + p^P + d 3 ^ t^P^- tJ^P k It follows from this equation that (2.12) ^Xpu _ p \pu , 1 ( p p\u _ p \pu + p ^up + p pu\ _ pUpX _ p u\p^ (2.14) 3- The Euler Equations The relation between the energy -momentum tensor for the cp-field which appears in the Einstien field equations and the variation of <5*f with respect to rii the g was first pointed out by Hilbert J as is noted in Pauli's classical discussion of the theory of relativity [2] where additional references may be found. In this section we shall derive the Einstein field equations and the equations that must be satisfied by the cp-field. -5- The total variation of the integral I given "by equation (1.2), that is the change in I produced by the changes 6g and Sep is given by 51 = 8 I + 8 I g 9 A L J* Kf A + (G^ V + K'O 8g |V + K-(q A Bcp A + P^ B(q>.„)) Vv ■A {(rr- g p v ff )V; P L]^ a (3-D where G is defined in equations (1.5), q A in (2.8) and p^ in equation (2.9) A 'A It follows from equation (2.12) that equation (3-l) may be writter as 61 =f [(G^ V + ^ V ) B g + R(q A Scp A + P^ (&cp A ) ;| V " + M^° 8g ) ( pa u-v pp. vefs e (g g - g g ) &£ M-v;pJ ;a g d x On integrating by parts this in turn may be written as 81 = -^{[g-^^K^-p^b/}^ A V ■Crf"" Bg+Kp?V - (g p0 g^- g P V )5 giiv . f (3.2) g d x v^ff where the symmetric tensor (3-3) and M^ va is defined by equations (2.10) and (2.I3). By requiring 61 to vanish for arbitrary variations which vanish on the boundary of the region V we obtain the Euj_er equations -6- G ^v +|a nv =0 {3k) and F A E %" »L " ° (5-5) Equations i^.h) are the Einstein field equations with a matter tensor given "by equation (3'3)- Because this tensor arises from the variation of I with respect to the gravitational field g we call T the gravitational matter tensor. uv Note that even in the Minkowski space-time T^ is different from 0^ in a general coordinate system. Equations (3«5) ar e the equations for the field cp . They may be obtained from the special relativity equations in a galilean coordinate system by replacing every ordinary derivative by a covariant one. They obviously reduce to the equations of special relativity in case the tensor g is the metric tensor of Minkowski space-time. k. Infinitesimal Coordinate Transformations In this section we shall use a technique similar to that of E. Noether [ 3] to derive conservation laws from the invariance properties of the Lagrangian . Under the infinitesimal coordinate transformation a a* a .a ,, _ x x — > x = x + | (^-l) where £ is an arbitrary vector , the products of whose component may be neglected, the integral I undergoes the variation 51 = -J (R - K^H -g A (k.2) since the boundary of the region of integration is transformed and since the integrand is a scalar function of coordinates. -7- Since the g and cp are tensors they are transformed under (^.l) in accordance with the equations T g uv (x } = g aT ( x ( x )) * . dx dx ox ox and a corresponding equation for cp . These produce variations in the fields g and cp given by uv 6g = - 5 - 'I M-v u;v v;u (h.3) . o^...o v— i o n ...a. pa. . . .a a. e A In .p \ 1 l-l 1 ^ l+l n , l *" ~ ** T....T ' + /L * T....T 5 ;p 1 m:p . 1 m ' K l (+-M 9 cr_ .. .a 1 n 1 j-l p j+1 m ' J obtain On substituting equations (4-3) an d (^-^) into equation (3.2) we 5i = f { (2je^ v + k^ v ) i . + k^ a cp a A^i A J v n;v p ' ;o v (R -K5C) l' _h a v -g d x (^•5) where i T . . .T 1 m 1 l-l l+l n a, . . .o . . va. .... .a au. 1 l-l l+l n 5 ^cp 1 m 1 j-1 j+1 m a . . .0 1 n a. . . .o 1 n cp ,P^ T n . . .T . n pT . . . .T 1 J-1 J+1 m and a a A c a ^ t = p cp - 8 X; p A ;p p ( + •6) ■8- in deriving (4.5) we have made use of the fact that 2G^ V | - (g P V V -g P V a )U ^ ) = - (R*°) n u;v u;v v;u ;pa ;a Note that iT = when equations (3-5) ar e satisfied. Since the tensor N is antisymmetric in v and a we have (I N^ Va ) = u ' ;vo That is, (g N^ va + | # W ) = or r uva- T uav (g N^ vu ) = (| N^ v ) u;v ja u ;v ;o In view of this equation and equation (4.2) we may write equation (4. 5) a s V n^V , T UV A e P (2T^ V + O | + F A cp" | ; b u;v A Y ;p b / > "v -g d x V (t ° + N CTV ) g p p p;v jj -g d x = (^.8) On integrating this equation by parts we obtain f (F fl cp A - 2T ° -L° ) l P S^A + f (2T °V t *+ H ^ ) l P ] ^d^x = J v I. P P P ;v " J;a (*.9) When the tensor field cp is a solution of the Euler equations (3-5) the above equations become T ^v^ n;v ■sT-e 4 g d x + V (t ° + N uv )| p av P ;v V-g d X (4.10) and -9- J v ;v u V (21^ t- + N ° v )|P p p p ;v j,' r d X = (4.1l) Both equations must hold for arbitrary volumes and arbitrary vectors £ If E is a Killing vector, that is, satisfies u M-; v v;u then equations (4.10) implies for such vectors that H (t °; N av )*' p p ;v' J;° t reduces to t pa = in the case of special relativity. It should be noted that as a consequence of equation (^.13), equation (4.1l) may be written as / — k v -g d x =0 V (2T tf + t ^N ° V ) ^ P P p p ;v j> which in turn may be written as an integral over the hypersurface S boundary the volume V, namely 2T °; t <^N ° V p p P ;v r n dS = a (h.±6) where n dS is the element of volume in S, and £ is an arbitrary vector. Equation (1+.13) may be used to relate the time rate of change of the three- demensional volume integrals of T and t by choosing the hypersurface surface S to consist of the hyperplanes t = constant and t + dt = constant. 5. Concluding Remarks The results obtained above may be readily generalized to the case where there are a number of cp-fields present. In such a case for each such field there will be an associated T and a corresponding t . The right-hand side of the Einstein field equations will contain the sum of the T V-V -11- and this tensor will be related to the sum of the v "by equations analogous to equations (k.lk) . In case the cp-field is a spinor field a simular discussion to that given above can be made. The special relativistic Lagrangian must first be generalized by replacing ordinary derivatives of the spinor field by covariant ones . The variations in the metric tensor may be performed by varying the generalized Dirac matrices which satisfy the relation 7 7 + 77 = 2g 1. U V V (i. U-V The details of this process will be described in a subsequent paper. It should be pointed out that if we define the scalar y- 4