Si w— mm mm W& BHBH MM MM HoffiflBoBN m huh m WJBI3uw .H ■1 I ■ ^Ml • ^A : 9 HI I I ■V ■I ffiSK 89* ■ v H& *< M W fflP W - ' " VU-V I ■iiillJllilll! • Warn LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 510.84 I£6r no. 100-110 cop. 3 no. 107 cop- 3 AN APPROXIMATE STRESS ENERGY TENSOR FOR GRAVITATIONAL FIELDS by A. H. Taub UNIVERSITY OF ILLINOIS DIGITAL COMPUTER LABORATORY URBANA, ILLINOIS Report No. 107 March 27, 1961 This work was supported in part by the National Science Foundation under grant G950j5« ABSTRACT An invariant formulation in Minkowski space-time of an approximation to the Einstein theory of gravitation is given. In this formulation a tensor is introduced which may he interpreted as the approximate stress energy tensor of the gravitational field. Conservation laws involving this tensor and the material stress energy tensor are formulated. The behavior of these tensors under "gauge transformations" of the weak gravitational fields is discussed. The classical limit of the conservation of energy equation is studied and the results are compared to some observations of H. Bondi on a possible analogue of the Poynting vector for a gravitational field. AN APPROXIMATE STRESS ENERGY TENSOR FOR GRAVITATIONAL FIELDS by A. H. Taub 1. Introduction It is the main purpose of this paper to formulate and discuss conservation laws in invariant form in Minkowski space-time for an approxi- mate version of the Einstein theory of gravitation. These laws will involve the approximate energy and momentum of the material and gravitational fields . The discussion will be mainly concerned with a first approximation to the Einstein theory but may be extended to higher approximations. We shall relate the results obtained to some observations of H. Bondi^ concerning an analogue to the Poynting vector for classical time -dependent gravitational fields . The Minkowski space-time will be used as the underlying space in which the discussion will take place. In principle any fixed Riemonnian space-time may be used. There are, however, two reasons for choosing the Minkowski one: (a) with this choice the Newtonian approximation is readily obtained from the first approximation given below by neglecting terms of the order of l/c and (b) the underlying space-time admits a ten parameter group of motions, the inhomogeneous Lorentz group. Use is made of the latter fact in formulating conserved quantities. the metric tensor g of space-time as defined over the Minkowski space by The approximate theory mentioned above is obtained by considering tensor g of space-time as & uv a convergent power series expansion in G -27 2~ = 1.864 x 10" cm gr * (l.l) k - 2S . 1.861, x 10 -2T cm _-l c where G is Newton's constant of gravitation and c is the velocity of light of the special theory of relativity. We assume that S uv = V + k V + 2- h (2)uv + "" (1.2) where t)^ is the metric tensor of the Minkowski space-time. The coordinate -1- system in which equation (1.2) holds may be an arbitrary one. The equations satisfied by the g . that is h , h/p\ ... will be derived from the Einstein field equations: G = -kc 2 T (1.3) UV UV \ y/ where G = R - ijj g R , (1.4) nv uv 2 °uv N ' R is the Ricci tensor and R the scalor curvature tensor formed from the g uv uv The tensor T is the stress energy tensor of the matter "creating" the uv gravitational field. Both the tensors G and T may be considered as functions of k and uv [xv written as G = G, v +kG /n N + 75- G/_s + .... (1.5) uv (0)nv (l)jiv 2 (2)uv v ,2 T = T/ A % +kT/,x + #~ T/ v + (1.6) uv (O)uv (l)uv 2 (2)uv It is evident from equation (1.2) that G (oW H ° d-T) The following discussion will center about the discussion of the equations [ which are consequences of equations (1.3)- In equation (1.8) the semi-colon denotes the covariant derivative with respect to the metric tensor g Because of the Bianchi identities , we have (G g VP ) A E . (1.9) If equations (1.5) and 1.6) are substituted into equations (1.3) and the resulting equations are regarded as identities in k we obtain G/ x = - nc 2 T, . v . (1.10) (n)uv (n-l)uv These equations may be regarded as differential equations for the determination -2- of the h, x in terms of h, N and T/ n (m = 1,2, ... n-l) . The T, % (n)uv (m)|iv (m)nv % ' (m)nv must be such that m = (1.11) k=o r d m ,. 2 _ vpx (kc T g ) _ Ldk m ^ v > p t s\ VP where T is given by equation (1.6), and the g are functions of k which satisfy -, vP dc dk s dk g u.J-^; We also require that where the comma denotes the covariant derivative with respect to the tensor Equations (1.12) and (1.13) are equations for the determination of the T/ \ in terms of h/ \ and T f \ with r = 1,2, ... n and s = 0,2, . . ,n-l, (n)uv (r)^iv (s)|av ' ' ' ' 2. The Calculation of G^ (Dv We begin our discussion by considering the expansion of the Christoffel symbols as power series in k. Thus v c} = I s ^ ( V a + gp ^ v " s «fy ] (2 ' 1} where we have used the notation >p - = g «i (2.2) We may write u. 1 + where w w ■ ( t { -i -3- Thus the Christoffel symbol calculated from the "T s . In a galilean coordinate system L V ^(o) = (2.5) (o) It may he verified that < a - {v"o} (l) - I "^V* + W " h av,P> < 2 - 6 > where, as above the comma denotes the covariant derivative with respect to the tensor rj . It follows from equation (2.6) that where h = r n (2.8) The Ricci tensor is defined by the equation v=-i°vL + {-L-WW + WW <«> Hence and (0)uv R m =-| a l +l°l (2.10) (l)uv V v J(i), V°/(i), v On substituting from equation (2.6) and (2.7) into equation (2.10) we obtain r (t\ = ~ k n P0 (b rt + h rt - h n - n h ) (2.11) (l)uv 2 ' v Pv,u Pu,v uv,P 'Pv ,u ,a v ' Since R, = we have R/.v = rf v R. . = - n pa (h _ ,<* - h J (2.12) (1) ! (l)uv ' v pa,£ ' ,p y ,o s J -k- Thus we may write G (l)uv = R (l)uv ~ 2 V R (l) 2 ' v Pv,u nP,v uv,P 'uv Pa,£ ' ',0 • - 7? 1 a ( k « - k i OS , aBx 2 ' v Pv,u uv,P -11 k« « n K + n k Q ii H '* 'uv Pa,p ' 'pv ua,p ' ,a (2.13) where k = h -ii) h (2.14) uv m-v 2 uv Because the Minkowski space is a flat space it follows that u G/.x = T) TU G (2.15) (1) V TV v *" and the order of covairant differentiation is immaterial, that is 1 * * * 'ap = t * * * * 'pa ' It may be verified by using equations (2.l4) and (2.15) that G/./ = (2.16) (1) v,u v U V 3. The Calculation of Gv^x In this section we shall evaluate the above tensor in terms of h h/ \ and their first and second derivatives. We shall show that it uv, (2)uv may be written as a sum of two tensors. One of these contains the second derivatives of h and h/ ? x and has a vanishing divergence. The other is a function of the h its first derivatives and G /n ^ uv (ljuv It follows from equations (2.1) that to w ■ ».". • H <. <>■» where B va = ^ I (h (2)vP,a + h (2)ap,v " h (2)va,P ) (5<2) and A is defined by equations (2.6) -5- By differentiating equation (2-9) twice with respect to k and setting k = we obtain R »«-W«/t P ')w/" t ' v,cl at T>\ ax pt (3-3) Since it follows that _/, U0 vt , vt ua 1 , uv cjt 1 , ax uv N „ - 2(1T t] + h n - 2 h ^ -2 hT V ) R (i) aT In view of equations (2.10) and (3.3) we may write the above equation as G ^ = (g ^ g vx . 1 g uv g a T) f UV . ,110 VT 1 UV 0Tv G (2) " ^ ^ " 2 n n J R (2)ax (3^) .UV r (2) " where ua ^vt 1 uv aTi r T /(2) > P i(2) V . ua vt . vt \io 1 , uv aT 1 , aT uv \ / .a A P c a [ h K n + h if - g ^ 1 " 2 h n A a T " A ap 5 t J, a /,ua vt , vt ua 1 ,uv „aT l ,aT \xv\ (.a A P R a\ - 2 br T) + h t)^ - — h K T] - — h vr A - A ,& | ^ ' ' 2 ' 2 ' ),a\ ox op t J It may be verified that .uv (2) ^v + 2c E uv k H^ V =0 , v (3.5) (3.6) and -6- r= r^-|f r)|-B%5>;) + i ,Ma I i^j j* + j i^ - 1 i, w j ^ T -p v - 1 *r) * ra . n ^v^ar _ 1 ln at) with , ua .tv . |iv , ax + h h - h h iT ,X , ux , av .pa ,vx ,cl ,a ,x ,a ,x - [h(h% av + h av n PlJ - h^n* - n"" h pa ) + [ h h >P ( .. ,Pa Pu av «nv Pa i^ v = h^ Ph pv . (5-7) (3-8) 2c _,uv -/ ua vx 1 uv axs / .p A .p .\ s -^ E^ = 2(tf t, - ^ if 1 )( AflX A px - A 0t A J »0 »© t a> - 20T1T + h^iT - f ^V l - I h wl T,^ (A^ T - A» p 6") + [h( k p ^ T, av + k av ,/* - k ^ v n <* - ^ k pa )^ a (3«9) , . ux , av . pa , vx + h _ h - h _ h ,a where as before pv , uv 1 uv . k h = h r ■ | T Equation (3«9) may be shown to be equivalent to 2 2c ^uv , u . xp av . v . xP cru — — E^=-k H „k n - k „ k Ti K k x,P ,0 ' t,P >0 ' + k^ k vp n TX + k p I k^ v P,X ,\ ! ,P ,T - C ^ + 1 k * w ^ - t v.. w -- -K) 1 UV + 2^ ax pa 1 aa px 1 W k ,P n - 2 k aa,T k ,P n U h ,pV -7- Pa (3.10) It follows from equations (3.5) and (3-6) that (2),v -~ E ,v (3.11) In view of the field equations, that is equations (1.10), and equation (2.l6) this equation may "be written as T£ V + m£ + E^ = (3.12) Equation (3»H) holds for arbitrary h and h/p\ , that is it is an identity in these quantities. Equations (3. 12) are the approximate equations of motion of the matter represented by the tensor T (cf equation (1.6)), correct to terms 2 involving k . They are written in invariant form in the Minkowski space. If h is interpreted as a tensor in this space (approximately) representing the gravitational field created by the matter, then we may regard the last term in equations (3«12) as the "stress-energy"* tensor of the gravitational field. It is the object of this paper to evaluate the right hand side of equations (3-9) for a particular choice of the tensor T/_n , that is, for a particular choice of h . We first make some remarks concerning some con- sequences of these equations. ho Conservation Equations We may write equations (3»12) as (# v + E^ v ) t = (k.l) where # a . T^ a . + k T^. (4.2) (0) (1) Multiplying equation (h.l) by an arbitrary vector field of Minkowski space- time, x and summing we obtain v (x(M^ + E^)) -i(r + E^)U +X 5=0, since both M^ v and E^ v are symmetric. -8- When \ is a Killing vector of Minkowski space-time , that is when \ + X =0 (4.3) H,V V,H N •" we obtain the conservation equation (x v (^ ♦ Bf") ) >(i - . (4.4) As is well known the general solution of equations (4.3) is given by \ = F X v + a (4.5) in a galilean coordinate system in Minkowski space-time where F = - F p and are independent of x and the a are constants. There are thus ten linearly independent \ and associated with each of these there is a conserva- tion theorem of the form of equation (4.4). The four vectors which in a galilean coordinate system have the coordinate ^ = ^ a - 1,2,3,4 (4.6) will be said to be associated with the conservation of energy and momentum. Equation (4.4) implies that / X (# v + E^ V )n d 5 v = (4.7) V (J. where the integral is taken over a closed three dimensional hypersurface in Minkowski space-time and n d? v = J~^h -sr- "sr - -sr - du dv dw (4.8) (J. y UV0T ou ov 6V x ' if u, v and w are variables giving a parameterization of the hypersurface. For use in later sections we drive an equation based on the Bianchi identity: }^ v = G^ v + G Pv {* \ + G^ { v \ = ;v jv {p [xj |u pj -9- If this equation is differentiated twice with respect to k and k is set equal to zero we obtain (2),v (1) > W(i) WlvpJd) That is °(2)n V ,v = °(1) P h vP,H " h ,v G (l)n " ffl (l) V,P In view of equations (1.10), (2.l6) and (4.2) this equation may be written as I <„ ■ I fe v. - v «?.>„ - 2 T (o) w] <*•« 5. Gauge Invariance In this section we shall discuss the effects of a transformation of coordinates in the Reimannian space- time on the tensors h and h/,_\ in the Minkowski space-time. We recall that in any coordinate system h = ^L g ] . (5.1) (n)nv \ dk nV/ k=0 Under the transformation of coordinates in the Reimannian space-time defined by the equations y^ = y^(x) (5,2) the tensor g transforms as V (x) = ^T (y(x)) y^y| v (5-3) It then follows that if the functions y are independent of k that the quantities h/ \ transforms as tensors in the Minkowski space-time under the transformation given by equations (5«2) where these are now interpreted as a transformation of coordinates in the Minkowski space-time. -10- If the functions y depend on k, it follows from equations (5«2) and (5.3) that (n)uv - B ^n g uvj k=o does not transform as a tensor in the Minkowski space-time. In this case equations (5°2) which may be written as y^ = /(x;k) (5.10 may he interpreted either as a transformation of coordinates for fixed k or as a congruence of curves for variable k. It is sufficient to discuss the case where equations (5*^0 are such that That is, y^ = x^ + kf^xjk) (5.6) For a general transformation of the form of (5°^) is obtained from (5*6) by following it by a transformation independent of k. Let us write (2L-*. and set a^ } - ^ . (5.8) The functions a are the components of a contravariant vector field, the vector field tangent to the congruence of curves (5°*0 at the point x . In fact under the transformation of coordinates x = g (x) with the definition 'a a, y. y = g (y) -ii- ILLINOIS LIBRARY we have \ /k=o \oy yk=o ox However the functions a, . (x) do not have a vector transformation (2) law. Indeed we have (2) \ dk 2 L ax T ( 2 ) o* P dx T Note that the quantity i >" - £. + Q- .V (5-9) does obey the transformation law of a vector. It follows from equations (5.3) end the definition (5-l) that under the transformation (5«*0 subject to equation (5*5) h* = h - a - a (5-10) [IV (IV \i,v v,u * (J T h =h -b -b - 2ti a a (2)uv (2)nv ^> v v ^ 0T »»■ > v / * 0*CJ *(J\ - 2(h „ &+ h a + h a° ) uv,P ua ,v vcr ,[x = h/_> - 2(h _ a P + h m & + h m a? ) \2)\XV UV,P |KJ ,V V0 ,\3i - (b - 2a n a a ) - ft - 2a' a ) (5.11) where the vector b is given by equations (5-9) an a we have made use of the fact that T) is the metric tensor of a flat space. In case h = h, x = uv (2)nv h* = - (a + a ) (5.12) uv u,v v,u -12- L (2) uv (b - 2a „ u n,a a a ) - (b - 2a „ a*) v,a M (5.13) That is, even when the Eeinannian space-time is flat hut a non-galilean coordinate system is used which arises from a galilean one by a transformation of the type given by equations (5°^)> "the quantities h and h/_v need not vanish. However, they are of the form given by equations (5. 12) and (5.I3). We shall call the transformation \xv \xv n (2)nv -» n (2)^v -X- * where the h and h/ p \ are given by equations (5«10) and (5.11) a gauge transformation. It is the transformation induced on these tensors by the ■x- ft coordinate transformation (5 •**■)• When h and h/_\ are substituted into equations (2.13) and (j>.k) we will obtain quantities we shall denote as G* 11 and G^ v . These are the coefficients of the first and second powers (1) (2) of k in the expansion of the tensor *\iv 1 nV " 2 g R = G *Uv which may be obtained from the tensor G^ v by using the fact that G ^ v arises from G^ v by means of the transformation (5.4) and the transformation law G*^(x) = G aT (x) ^ y v la |t (5-lM It follows from this equation by setting k=o, by differentiating with respect to k and setting k=o, and by differentiating twice with respect to k and setting k=o that (0) (0) ft '(I) G^ v =G uv (1) (5.15) and '(2) (2) - 2 G kiv L (i),p a P - G^ T a v - G TV B? (1) (1) (5.16) -13- 4t -* These equations may also be derived by substituting h and h/ p N into the equations defining G . and G. . as functions of these quantities. Since equations (5.10) hold identically in h and h/^v we have I G *°1 = 2c! E *ar m 2c£ E ar ^aP & t ^Pt & a (2),T k ,T k ,T (l) ,TP ,-q ,TP m'c . ..ar ax P tP a \ 21 ^ + G a ,P + G a ; P ),T (5.17) as follows from equation (5«l6) and (2.l6) . It is a consequence of equations (5«15), (5«l6) and (l«10) that !^V + M nv* = T uv + kT uv _ k (0) (1) (0) (1) T^ v a P - T^ T a v -(o),p (0) >-< T TV a^ (0) >' We now define the vector X- = \ ~ k <» P \,e + N, V> (5.18) (5.19) where X is one of the Killing vectors of Minkowski space-time, that is, X satisfies equations (4.3) • It may be verified as a consequence of equations (5-l8) and (5-19) that X* M*^ = X M^ V - k(a P (x T^ V ) „ - X T^ a v n ) (5-20) U I* \, H (0) > p H (0) > p / where terms in k have been neglected ; and W and M are defined by means of equations (4.2) and the corresponding equation for the starred quantities. If we multiply equations (5*20) by (l - ka ) we then obtain to the same accuracy (1 - ka a J X* M^* = X M uv - k ,0 u [ (aP \ %\ pP V P " \ (0) S ;P and hence [ (1 - ka^) x; ^ - (X M^ V ) - k[a V (x Tf .) 1 ;V u ',v L H (0)SPj / (X M^ V ) U ,v (5.21) -14- The first form of equation (5°2l) holds for an arbitrary vector \ » The second form of this equation follows from the first form by virtue of the uv fact that \ is a Killing vector and T has a vanishing divergence. " (0) It follows from equation (5.21), by integration over a region of Minkowski space-time bounded by a closed three-dimensional hypersurface, that f(l - ka a ) X* *T* n d 5 v = fx # v n d 5 v = - J ^E^ v % 6? v. (5.22) It is of course a consequence of the first of equations (5*17) that f - E |iv * n d 3 v . (5.23) /^^■■/\ The difference between the surface integral of E and that of uv E is due to the fact that the hypersurface in Minkowski space-time into which the hypersurface defined by the equations x = x \u, v, w; transforms under the transformation defined by equations (5»^) differs from the former one. Thus we see that although the gravitational energy tensor E is not gauge invariant, the conserved quantities computed from it are related by equations (5«22) which take into account the fact that the gauge transformations arise from coordinate transformations in the Riemannian space-time . 6. Bondi's Relation In this section we shall compare equation (hok) with \ = \v > where the r\ are evaluated in a galilean coordinate system, to a set of equations first derived by Bondi ( l) from classical arguements. We shall derive his relations by studying the classical limit of the Einstein field -15- U.V equations for weak fields for the case where T7 . is the stress energy tensor of a perfect fluid. In forming the classical limit we shall neglect terms involving l/c . McVittie (2) has given the h associated via equations (1.10) with such a T .In the notation used above McVittie 's results may be (0) written as follows: In a galilean coordinate system let 2g, 5 * sK (, + !!M) h = -2P o o + w + — *«-j T] (iV M. V Jd '[XV (6.1) then where h = 2(cp + A) c T|r = Z 8 ^ H=l (6.2) (6.3) and 1+4 2 uv |a v 2 ^nJ Vv (6.4) with 2f, v = f - 2g/ v (6.5) hence X %> = * (i=l It may be verified from equations (2.13) that (6^6) -16- (1) AJ f (*0,ij Z, (')*" U) i M = c 2 /V,.x +♦#-*').- i ^ j (l) { (i) U)/AJ r The tensor T, . may now be calculated from Equation (1.10) where the (0) uv above quantities are used for G~ * If in the resulting equations we neglect / 2 ^ 1 ) the terms in 1/c we obtain for the classical limit (0) >iJ i; 1 =cp = pu. (6.7) (0) ,14 1 £, = - *, tt - 2 ^ ( i),ii ♦ ^d^ 5kl + x *«*« = pu * (I) t]\ = - (*,.* + *,,) . . = pu.u, (0) (i) v iJ y AJ i J The extreme right hand sides of equations (6.7) are obtained from the classical limit of the relativistic stress energy tensor of a fluid T"" - P(l + \ + -%) uV - \ f C PC c with 4 1 i U i 2 r" 1 TT 2 u - — , u = ~ , v = \ U V A/l - v 2 /c 2 ^1 " v 2 /c -17- The quantities f, . \ are not arbitrary but must be chosen so that a set of equations called consistency equations by McVittie must be satisfied. Lhese equations are determined from the requirement that the ten equations (5.7) determine the five quantities P, P and U. . When these are satisfied we find p ■ - •,« 8 ij PU i * + *,lM P = - v M * X 1,3 = 1,2,3 (6.8) where X is determined by the integrable equations ,j 1 = 1,2,3 (6.9 We next evaluate the right hand side of equation (4-9) ia the classical limit, that is by substituting from equation (6.7) for T^ v and from equations (6.1) and (6.2) with the terms in l/c omitted for h and h. Equation (4.9) then becomes j£ = ! T* Tf (-26 4 6^ + t, J - *> (^ - T* 6 k ) a,n 2 L ,a a) n P 'nP y ,u v t Q \ a ( \ a' _ If we set a = h in this equation we obtain on neglecting terms in l/c' inside the parentheses, ^■-IV*S)"iV BlJ, 1 • ij (6.10) (6.11) The first of these equations may be written as it = -M^T^ ), - i 6iJ + ^V i + 1 ( i+ #1 (0) d > 4 >J > 4 J '* (0) >* 2 (0) > 4 The scalar 9 is related to the Newtonian potential V by the equation V = 4tiG9 , as is evident from the first of equations (6.7)- Hence the above equation may be written as u i v ,j slJ ■&<*,* \i - ™ *>,j 5±J ♦ ( pvu i>,j 6« + i (pv) jU (6.11.) Equation (6.l4) has been derived by H. Bondi from purely classical argue- ments and has led him to suggest that the vector *1 * k < V ,4 V ,i " « *) is the gravitational analogue of the Poynting vector. That is, it represents the momentum of the gravitational radiation through unit area of a surface exterior to the moving matter. When equations (6.12) and (60I3) are added we obtain -19- or This equation relates the four dimensional divergence of the energy-momentum of the material field with the time rate of change of the potential energy of the mass distribution and the divergence of the vector p.. We shall compare equation (6.15) to the equation resulting from equation (k.k) by choosing X as mentioned above. To do this we substitute for k from equations (6.4) with the l/c terms omitted into equations (3. 10). We then obtain on neglecting l/c terms E kk = - I [+ Iq> . q> . 6 iJ + lflxp 6 iJ ] 2 L 2 ,1 , J ,ij J or E kk -■h[- 4VP + 5Fg v ,i v ,j 6 1J ] (6.16) or E E kl k |L3P >lt » >1 + WP >lt o 2 (lratj) 3* „ V ;i ♦ *W „ (6.17) On substituting equations (6.l6) and (6. 17) into equation {k.k) we obtain = -E 1 *,H (-^ + ^ V ,i V ,J 6iJ ),i» ij •w^ 7 ^^/ J (6.18) Note that equation (6.18) is similar to equation (6.15) in that its right hand member contains products of V and its first and second derivatives . It differs from the right hand member of equation (6.15) in the presence of the term V . V . 5 iJ A ,3 -20- X****" * "u^^,.^-^* 8 - ,-^r;^—- ■