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Some considerations concerning newtonian charts Gegenberg, Jack David 1972

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SOME CONSIDERATIONS CONCERNING NEWTONIAN CHARTS by JACK DAVID GEGENBERG B.A., University of Colorado, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in the Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF April, BRITISH COLUMBIA 19 7 2 In presenting t h i s thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department The University of B r i t i s h Columbia Vancouver 8, Canada A B S T R A C T In Part I, spherically symmetric solutions of Rastall's 1971 gravitational f i e l d equations for empty space-time are examined. One static solution is found to be just a static spherically symmetric Newtonian metric; i.e., the metric of Rastall's 1968 scalar theory of gravity. However, there are other solutions which satisfy the same boundary conditions at spatial i n f i n i t y . It is observed that the time-like vector f i e l d n y appearing in the f i e l d equations is not uniquely defined when the metric is assumed to be spherically symmetric. Part I con-cludes with a discussion of the effects of this ambiguity upon the solutions of the f i e l d equations. Part II is a discussion of an alternative procedure for generalizing Rastall's 1968 theory of gravity. The new, gener-alized Newtonian metric is assumed to satisfy the linearized vacuum f i e l d equations of General Relativity in the weak-fi^ld limit. The quantities from which generalized Newtonian metrics are constructed are then found to exhibit wave-like behavior. i i TABLE OF CONTENTS Page INTRODUCTION 1 PART I 4 (a) Newtonian Metrics and A Scalar Theory of Gravity . . 4 (b) The Spherically Symmetric Field Equations 9 (c) The Solutions of the Field Equations 13 (d) Uniqueness 17 (e) Uniqueness of the Vector Field n y 21 PART I I . . . . ". 27 (a) Generalized Newtonian Metrics. 27 (b) The Linearized Einstein Equations and Generalized Newtonian Metrics , .30 CONCLUSION 34 REFERENCES 36 i i i • A C K N O W L E D G M E N T S I wish to thank Professor P. Rastall for suggesting the topic of this thesis and for his patient supervision of my research. iv I N T R O D U C T I O N A physical theory is said to admit a class of pre-ferred charts (or "co-ordinate systems," the differential geometric definition of which appears in Section I (a)) i f the members of the class can be distinguished from charts not in the class by a well-defined physical procedure. For example, the Special Theory of Relativity admits the class of " i n e r t i a l charts" as preferred charts. It has been suggested ( c f . Ein-stein, 1956) that the notion of preferred charts leads to philosophical ambiguities. » The General Theory of Relativity, as usually formu-lated, is without preferred charts. This lack, while perhaps a philosophical strength, often presents technical d i f f i c u l -ties and hinders our understanding of many aspects of the theory (for example, the d i f f i c u l t y in defining local gravi-tational energy stems in large measure from the lack of pre-ferred charts). Some gravitational theorists have attempted to surmount these d i f f i c u l t i e s by introducing "co-ordinate conditions," a procedure by which most of General Relativity remains intact, but in which preferred charts are introduced. Examples of this are Fock's "harmonic co-ordinates" (Fock, 1959), and the "canonical co-ordinates" of Arnowitt, Deser and Misner (Arnowitt, Deser, Misner, 1962). 1 2 However, i f one is willing to put aside the philoso-phical arguments against preferred charts, then i t seems that one should look more carefully at theories in which preferred charts enter on a less ad-hoc level than they do in the theories mentioned above. Examples of such theories are the numerous Lorentz covariant theories of gravitation (for a recent example, see Coleman, 1971) and Rastall's scalar theory of gravitation (Rastall, 1968). However, these attempts, to date, have not been as successful as General Relativity in satisfying certain experimental or mathematical c r i t e r i a . In this thesis, I shall examine two possible generali-zations of Rastall's 1968 scalar theory. One of these generali-zations is a generally covariant theory (Rastall, 1971) in which the preferred charts, called Newtonian charts, of the scalar theory are in general not present. In Part I, I w i l l examine spherically symmetric solutions of the f i e l d equations of the* 1971 theory and show that i t is conceivable that, in this case, Newtonian charts reappear, in a sense, as preferred charts. To establish this rigorously, however, entails proving an analogue in Rastall's 1971 theory, of Birkhoff's theorem in General Rela-t i v i t y , which states that a l l spherically symmetric solutions of the vacuum f i e l d equations differ from the Schwarzschild solution only by a co-ordinate transformation (Bonner, 1962). In Rastall's theory, the establishment of this theorem depends upon overcoming two problems, neither of which I was able to solve: 3 (i) what is the nature of the boundary conditions on the f i e l d equations? (ii) do a l l the mathematical objects appearing in the f i e l d equations have unambiguous definitions? The second possible generalization of the scalar theory involves generalizations of the Newtonian charts. At this writing, these sorts of considerations (Rastall, unpub-lished notes, 1972) have not attained the status of a f u l l theory of gravitation. However, the "generalized Newtonian charts" should constitute a class of preferred charts in some new theory in a manner roughly analogous to the way Newtonian charts are preferred charts in the scalar theory. In Part II, I w i l l define Generalized Newtonian charts, and show that the metric tensor in a space-time admitting these charts as pre-ferred charts, constitutes a rather interesting solution of the linearized vacuum equations of General Relativity. I. (a) Newtonian Metrics and A Scalar Theory of Gravitation We assume, as in General Relativity, that space-time, M, is a smooth, pseudo-Riemannian manifold of dimension 4 and metric signature +2. We depart from General Relativity, however, in assuming that space-time is also endowed with pre-ferred charts, called Newtonian charts: at each point in the manifold there exists a chart (u,x) in which the components of the metric have the form - 2 * * Sab = e 6ab §ou = ho = " e 2* 6ou ( 1 ) where M + R i s smooth. Our convention on the range of in-dices is that lower-case Latin indices have range 1, 2, 3, while lower-case Greek indices have range 0, 1, 2, 3. The summation convention applies t° repeated lower-case indices. We can promote these considerations to a theory of gravitation by (i) identifying the orbits of test-particles and photons with the geodesies of the manifold (more properly: the geodesies of the metric connection on M); and ( i i ) by postu-lating f i e l d equations. In particular, we choose (Rastall, 1968) 7 4.TT.Gp V Z * = + e G ) (2) CE 2 where v is the ordinary spatial Laplacian operator, G^  and Cg (u,x) is a chart at p in M i f and only i f u is an open set containing p and x : u R is a homeomorphism. 5 are the classical gravitational constant and speed of light, respectively; e is the energy-density of non-gravitational matter and fields, and is the energy-density of the gravi-tational f i e l d . For weak fields (e^ ~ 0) and slow speeds, 2 equation (2) is just Poisson's equation and tyC-^ is the classi-cal gravitational potential. In vacuum, when e = 0 , i t is postulated (Rastall, 1968) that 7 V <f> = %V<j)' A(J> (3) Newtonian charts differ from the i n e r t i a l charts of "Special Relativity in a very important aspect; namely Newton-ian charts, in most cases, define a state of absolute rest. Thus a particle at rest in one Newtonian chart, is at rest in a l l Newtonian charts whose domains include the location of the particle (Rastall, 1968, Appendix). So when one wishes to calculate the gravitational f i e l d produced by a given source, one must specify the motion of the source with respect to the local Newtonian chart. This leads to d i f f i c u l t i e s in at least one important problem -- the calculation of the paths of pla-nets in the spherically symmetric f i e l d of the sun. In particu-lar, the following situation occurs (Rastall, 1968, 1969): (i) Assume the sun is at rest in a local Newtonian chart. Then the predictions of the theory are in .reasonable agreement with experiment (perhelion advance is 92% of that predicted in General Relativity). ( i i ) Assume the sun has some non-zero speed in a local Newtonian chart. Then 6 the paths of planets calculated do not agree with observation except for implaus-ibly small speeds. Since there is no reason for assuming that the sun is at rest in the local Newtonian chart, i t is clear that the scalar theory of gravity is incomplete. As a theory of static gravitation, i t is acceptable, but in non-static cases one needs a more generaly theory. In principle, at least, a solution to this d i f f i c u l t y consists in postulating a set of generally covariant f i e l d equations which have the property that they reduce to equation (2) in the case of gravitostatics. Rastall proposed such a theory (Rastall, 1971), the f i e l d equation of which are Q . + n n. R, = S (4) x y [ v ; i r ] y [ v 'IT ] y [ v i r ] The symbols appearing here have the following definition: [ y , v ] denotes anti-symmetrization in y , v ; R are the components of the Riemann tensor in a y v r r p r given chart. In particular, i f in this chart the metric has components g y v> then a 8 R = £ " S. + S [ r r - r r i y v r r p 6 y [ p , T r ] v tov[p,7r]y b y p , a VTT,8 y T r , a v p , 8 where the r = g rp and r° are the components of the y v , TT ° t t p | i v y v metric connection; R^ ^ are components of the Ricci-tensor, R = girpR ; R is the curvature scalar, R = g y vR ; n y is y v 6 T r y v p ' ' to y v 7 a time-like vector f i e l d which we w i l l discuss in detail below; Q = R + n , rn p R ; uv yv iryvp' S r . is a tensor giving the distribution of sources." For an y [ v TT ] ° ° ideal f l u i d , Randall chose Sy[vrr] = 2 T y [ v ; T r ] + ^ eE " P E ^ f v ^ J y + 2 ( > E + Pfi) ' [ v11* ] n„ (5) where is the stress-energy-momentum tensor of an ideal f l u i d , eg is the (non-gravitational) energy density, and the pressure density of the f l u i d . In general, S tay be con-structed from (5) by defining E^ and p^ by • e P = -n Mn VT h yv P P = " T (g + n n ) T Y V  rE 3 6 y v y vJ Now i f we write equation (4) and (5) in a static New-tonian chart (the metric has the form given by equation (1) with <f> independent of time) , and define n y = 6 e"* (6) y O v J then equation (4) reduces to equation ( 3 ). So we seek a geo-metric definition of the time-like vector f i e l d n y which reduces to equation (6) in the case of a static Newtonian chart. Such a definition is provided by the following consideration: 8 Consider an orthonormal tetrad (W ) = (W , Wn, W0, ^ o r v o' 1' 2 * Wj). The tetrad or p h y s i c a l components of the Riemann tensor with respect to (W ) are R . = Wy W* Wp R One can show (Landau and L i f s h i t z , pp. 305-306) that the Rie-mann tensor is uniquely determined by a pair of complex 3 x 3 matrices, S and H, defined by Sab = ^ [ " R ( a ) (b) + R(a+3) (b + 3) ~ i ( R ( a ) (b + 3) + R(a+3) ( b ) ) ] Hab = % [ R(a) (b) + R(a+3) (b + 3 ) + i (" R(a) (b + 3) + R(a+3) (b) } 1 ( 7 ) where Latin indices enclosed by parentheses denote pairs of indices with the convention: 12=(3), 23=(1), 31=(2), 10=(4), 20=(5), 30=(6). If S is non-degenerate and of Petrov type I (i.e., can be diagonalized), then there exists a unique ortho-normal tetrad ( w ), called the P r i n c i p a l tetrad, such that S is diagonal. Rastall defines the time-like vector f i e l d n y in equation (4) to be the time-like vector f i e l d of the principal tetrad; i.e., n y = W . Now i f we construct the Riemann tensor ' ' o from the metric (1), then we find that, except for certain t r i -v i a l functions <$>, the corresponding matrix S is non-degenerate and of Petrov type I, so we are able to calculate the principal tetrad (W ) and in particular, we find that a 9 o pO as desired. In general, n y w i l l be a function of the metric and i t s f i r s t and second derivatives. In many cases of physical interest the matrix S and hence nMmay not be uniquely defined. In such cases, one hopes that the physics w i l l not be affected by the choice of any par-ticular mathematically permissible n y. I. (b) The Spherically Symmetric Field Equations Let us consider the case of the metric having the form g u - e a §22 = r 2 2 - 2 g 3 3 = r sin e Y Soo = " e gpv = 0 i £ ^ v < 8 ) with respect to so-called "curvature co-ordinates" (r, 6 , $, t ) . The functions a and y depend on r, t only. This is one form of a spherically symmetric metric. We can now write down the con-nection components rj^v and find the only non-vanishing ones are: il = Ha' • = ha ea^ '12 13 r 10 x10 hi r 3 '23 r 1 T T  !33 2 = cote = -re 33 1 00 r o •r s in 2ee" a = - sine cose = *2Y ( 9 ) where ' denotes 3/3r and • denotes 3/3t. The only non-vanishing components of the Riemann ten-sor are: R1212 %ra 1R1313 s i n2 e R l n 2 R2323 r 2 s i n 2 e ( l - e ~ a ) R1220 -hx'a R1330 s i n2 e R 1 2 2 Q R1010 e aA(a,Y) + e YB(a, Y) R2020 i Y " a i hre1 Y R3030 . 2 s i n 9 R2020 (10) where A ( C X , Y ) = ~ha - %(a) + vay B(a, Y) = ky" + % ( Y ' ) " %a'Y' ( I D 11 The Ricci tensor and the curvature scalar are R l l • = - I a' + e a " Y A ( a , Y ) R22 : = h re a(y'-a') + e ' a - l R33 = . 2 - s i n 8 R22 R00 = - -e Y" a[B(a,Y)+^Y] - A (a R01 = - Rir> - - —a 10 r (12) a l l other R vanish. R = 2[e~ u(B(a,y)+ $(y'-a') + K) + e" YA(a, Y) - ±7] r r (13) It is easy to check that a principal tetrad of the Rie-mann tensor of equation (10) is ( w a)> given by XY E wy = s t e " a / 2 1 ul y y = Wy = ±6 9 2 r y2 - z M s Wy - 1 * 3 rsinO u3 n y = Wy = 6 e " Y / 2 (14) o uo If we now construct the physical components of the Rie-mann tensor with respect to (W ) and thence construct the matrix S, we find that S is diagonal but degenerate 12 0 0 0 0 s 2 0 0 (15) So the tetrad of equation (4) is not a unique principal tetrad for the system. Nevertheless, for the present we shall assume that the vector f i e l d n y is given by n y = &uQe • Finally, we construct the tensor Q and find r ^11 Q22 = " hre a' + e -1 Q 3 3 - sin 8 Q 2 2 Qoo = Qoi = Qio e Y °B(a , Y) - A (a , y) 1-- —a r (16) a l l other Q vanish. Now we can write out explicitly the left-hand sides of the f i e l d equation (4). The only components which do not identically vanish are: Q1[1;0] + n l n [ l R , 0 ] = ~k'a (% +B(a, Y)+e a" YA(a, Y) )(17) Q0 [0;1] + n 0 n [ 0 R , l ] r e Y _ C i [ % B ' ( a , Y ) - % Y ' B ( a , Y ) - % a ' B ( a , Y ) + h ^ Y " - h ^ ( Y ' ) 2 - | | « ' Y ' - h V' - -^+(^ )2-+ W ( « , Y ) - | Y ' A ( a , Y ) - k | ( a - Y ) a + ^ e Y (18) Q 2 [ 2 ; l ] + n 2 n [ 2 R ' l ] = ^e"G(-^a'^r(a')24) + 1^ ( 1 9 ) ^2[2;0}+ n2 n[2 R'0] = % r e ~ ( _ a ' + a a , ) (20) 13 In the next section, we w i l l find solutions for the vacuum f i e l d equation, Q . . + n n r R, = 0 xy [V;TT] y [v 'TT] In particular we w i l l solve for a and y in the following four equations, which are merely equations (17) - (20) with the right-hand sides set to zero (and common, constant, non-zero factors divided out): a(±j + B(a,y) + e a _ YA(a, Y) ) = 0 (21) r e - <V 2B' ( a , Y ) - %Y*B (a , Y ) - %a'B(a,Y) + k\v" ~ %^(V) 2 - 4 - a ' Y * - k-j Y* " F « + ? ( a ) - -3] + -3-+ e ~ Y [ W ( a , Y ) " ^ Y'A(a , Y ) . " ^ ( a - Y ) a ] = 0 (22) e" a(-%ra" + % r ( a ' ) 2 - I) + I = 0 (23) e" a(-a' + o a » ) = 0 (24) I. (c) The Solutions of the Field Equations Consider equation (24). It is equivalent to ^ - 0 The general solution of this is well known to be e" a = f(r) + g(t) (25) 14 where £ and g are at least c in their respective arguments. Now we may re-write equation (23) as: % r ( e ' a ) " - i ( e " a ) + ±- = 0 or r 2 ( e ~ a ) M - 2(e" a) - -2 (26) So for equations (23) and (24) to be consistent, we must have [substitute (25) into (26)] r 2 f " ( r ) - 2 f ( r f = -2(1 - g(t)) (27) But this is nonsense unless g(t) is a constant, say g(t) = g Q. Equation (27) is an inhomogeneous Cauchy differential equation and the general solution i s : f(r) = ^ + c 2 r 2 + 1 - g Q (28) So we f i n a l l y have: e" a = ^ + c 2 r 2 + 1 - g Q + g Q = p- * c 2 r 2 + 1 (29) We shall now impose the boundary condition that the metric by asymptotically f l a t ; i.e., that e Y+l, e a-^l as r->°°. This implies that we must choose the constant of integration, c 2 = 0. Thus, e = CI + — ) or c, a(r) = - l n ( l + ^ (30) 15 Now note that a = 0 is a solution of equation (21), so we are l e f t with equation (22), which becomes: e" a[%B' (a, Y) - % Y'B(a, Y) - %a'B(a,Y) + k\y" - k\{y')2 " £ V Y * " k±z?' ~ |a" + i ( a ' )2 - ^j] + = 0 r r r This can be further simplified by noticing that the last four terms in the above equation are identically zero by equation (23). Now use equation (11) and simplify to get: Y"' " % ( Y * ) 3 + %Y'Y" + (-fa» + | ) Y " " (%<*' + ^ ) ( Y ' ) 2 + C-Jsa" + %(a')2 - |a' - lj) y • = 0 (31) r It would be very d i f f i c u l t to solve (31) directly. But one solution caibe found by the following trick, and in the next section I shall prove that this solution (for which y is a function of r alone) is the only analytic solution given the appropriate boundary conditions. The metric (8) is only one example of a spherically symmetric metric. . By transforming from the "curvature co-ordi-nates" (r, e , ip, t) to "isotropic co-ordinates" ( p , 0 , \j>, t) we obtain another form for the spherically symmetric metric. The transformation from (r, 0 , i\>, t) to ( p , 0 , [l, t) is given by (Landau and Lifschitz, p. 331): c 1,2 P ' t = t ' (32) r = (1 - j-) P <J> = * 16 The metric g in isotropic co-ordinates has the 'y v form: §11 = e S 3 2 §22 = 6 p 3 2 . 2n §33 = e p s i n g 0 0 = - e < S (33) where 3 and 6 are functions of p and t only. In particular B(p) = 4 l n ( l - ^ -) (34) Now notice that i f we demand 3 = -6 ^-2^, <j> independent of time, then (33) looks like the metric in a spherically symmet-ri c Newtonian chart. In the case we know that the f i e l d equa-tions reduce to just 2 , V (J) = %V (j) • V <J) Rastall has already found a spherically symmetric solution for this equation (Rastall 1968), and, in the appropriate notation i t is just equation (34). So we suspect that Y ( r ) = 6(p(r)) = - 3 ( P ( r ) ) = - 4 l n ( l - ^ r ^ , ) (35) is a solution of equation (31). Transforming back to curvature co-ordinates, we obtain Y ( r ) = 4 ln[%(l±/l + p-)} and i f we demand eY->l as r->°°, we have Y ( r ) = 4 In A + p-)} (36) 17 Indeed, when we put (36) and (30) in (31) , we see that we have found a solution. Unfortunately, there may be at least two more solu-tions of the f i e l d equations, namely a = - i n ( l + p.) Y = constant (37) and a = 0 Y = Y ( r ) (38) a solution of (31) with a = 0, i f such a solution exists. It is obvious that neither solution agrees with experiment (e.g., (38) gives no perhelion advance). Presumably, to obtain a unique solution of the f i e l d equations, we would have to impose other boundary conditions. Since asymptotic flatness requires the vanishing at spatial i n f i n i t y of the par-t i a l derivatives of a l l orders of the metric, these yet-to-be specified boundary conditions would relate in some way to the sources of the gravitational f i e l d . I. (d) Uniqueness The general uniqueness theorems for solutions of dif-ferential equations (Birkhoff and Rota, 1962) are very d i f f i c u l t to apply in equation (31). However, i t can be shown by a straight-forward procedure that i f we demand YCX)-* 0 as r-*°°, then the non-t r i v i a l solution of (31) near i n f i n i t y is unique to analyticity, 18 i.e., there is only one analytic non-zero solution near in-f i n i t y . c l In equation (31) set a = -ln(l+—) and change the in-c l r dependent variable to x = — to obtain Tt ! - U Y J + - * Y Y + 9v./-, + Y\Y + / i v r n v i u J Y -< Y  +  + 2x(l+x) Y + 4TT1T7)+XTTWY' = 0 (39^ This can be written as a second order differential equation by writing f(x) = Y'(x) (40) Note that f(x) is analytic at x = 0. f" +-j.ff + Cllx+8), f, r_^f3 (9x+8 ) f2 1 n _ f ^ C f + x(l+x) ) f C ^ f 4x(l+x) f + x(l+x) £ : ) " (41) let us examine analytic solutions of (41) in the neighbourhood of x = 0; i.e., set f(x) = Z a X n (42) n=0 R Using the result that ( Z a Xn) ( z b x11) = Z c X n n=0 n n=0 n n=0 n where c = Z a, b , n k = Q k n-k (See Fulks, p. 398) for x in the intersection of the circles of OO 0 0 convergence of Z _ Yn and Z b Xn, we obtain Oa A r> n n n=0 19 f 2(x) = E b Xn; b = E a, a , (43. a) n=0 n n k=0 K n K x 00 n f~(x) = E c Xn; c = E a,b , (43.b) n=0 n n k=0 K n K £ , ( X ) = nlQ ( n n ) a n + 1 X n (43. c) f(x)f'(x) = E , xn. d = E fn-v+lla f43 dl n=0  dn ' n k = 0 V n  K i J an-k+l <.45.aj £"(X) = nl0 (n+1) (n+2) ^ 4 3 ' ^ Substitute this into equation (41), simplify, and group like powers of x to get: n f 0 ( C „ + l ) ( n + 2 ) a n + 2 • ,» • 1* « n • 4 g n - £ * 1^ + 2J„ • kn>Xn  + + 2 b 0 * V JTTW = 0 ( 4 4 ) where n e = E r, (n-k+1) a , .-n k = Q k^ n-k+1 n Q = E r, (n-k+2) a , . 0 k=0 n-k+2 n la = E r, b , C 4 5) n k = Q k n-k n j = E r. b . , Jn k = Q k n-k+1 n K = E r a n k=Orkan-k+l 20 and 1 = E r X n = 1 - x + . . . and 1+x n n n=0 So we see that Za X is a solution i f n=0 n 4a r + 2bQ + a Q = 0 (46) d c (n+1) (n+2)a + *1 + *Ie +4g - + fh +2j + k = 0 ^ J ^ J n+2 2 2 n 6n 4 4 n Jn n (47) I claim that (46) and (47) imply that given a Q(i.e.,given f(0) = Y'(0)) the coefficients a^, i > 0 are unique. Thus there is only one analytic solution, given f(0). Proof: From (43.a) and (46), we have & 1 = -a Q(2a 0 + l)/4 (48) Thus given , a^ is unique. Now use equations (43) and (45) in (47) to get [(n+1)(n+2) + 4r Q(n+2)]a n + 2 = (terms in a^, m<n+2) (49) i.e., (49) is linear in a-n+2' This proves the uniqueness of a , given a . n' & o Does this imply that y ( x) is unique near x = 0? Suppose 8(x) is also a solution near x = 0 and that B'(0) = y'(0) Then 6'(x) = y'(x) near x = 0, or p(x) = y(x) + c(t). But i f we demand that the metric be asymptotically f l a t , then we also have 21 f3(0) = y(0) = 0, which implies c(t) = 0. Thus, in terms of r, y(r) given by (36) is the only analytic solution of (31) i f the metric is assumed to be asymptotically f l a t . Note that both a and y are time - independent. I. (e) Uniqueness of the Vector Field n y Instead of the orthonormal tetrad defined by equation (14), consider a new tetrad defined by W.a " (50) where L yv is a 4 x 4 real matrix which has the form of a Lorentz transformation in the (r,t)-plane, in particular, at a given point in space-time (L a6) " - 0 0 -o 0 0 1 0 0 0 0 0 1 0 •6o 0 0 (51) where a = (1-8') 2 to (W ) a Now form the physical components of R with respect r c y \JTT p R „ * = Wy Y!Va W71 Wp R a g Y ° a 8 Y o y v r r p If we now construct the 3 x 3 matrix S [equation (7)] from Ra6 Y6' w e f i n d t h a t S=S, where S is' constructed from R a 6 a ( 5 [equa-tion (6)]. Thus a l l orthonormal tetrads of the form 22 (50) are also principal tetrads. In particular, our spheri-cally symmetric, system admits an in f i n i t e number of time-like vector fields of the form (n y) = (-Be"Y/2, 0, 0, e " Y / 2 ) a (52) where 3 is a smooth function of (r,t) and a = (1-8 ) 2 We have shown that for the choice of a particular n y namely that for which 8=0 in equation (52), there exists a physically acceptable solution of the f i e l d equations. If our theory is to be physically acceptable, other choices of n y must yield no distinct physically meaningful solutions of the f i e l d equations. So far we have not been able to prove this; indeed the question is not even well-posed until we say what we mean by "physically acceptable solutions." We shall, however, give a partial answer to the question of whether there are solutions of the f i e l d equations that are independent of the choice of 8 (i.e., in the choice of principal tetrad). We demand that the f i e l d equations (vacuum) hold for the following two choices of the time-like vector f i e l d n y: n y = 6 e " y / 2 (53) uo v J n y = L y n v (54) v where L y vis a rotation in the rt-plane. Thus Q . + n n r R , =0 (55) Q . . + n n r R , . =o . (56) 23 where 6 =• R + i i p n a R yv yv pyva = R + L paL aBn an B R . yv pyva = R + M p a R yv pyva (57) and where upa _ Tp T a „ a 6 M = L FaL 3n n Equation (56) becomes R r . - (MpaR ). + M R , .=0 (58) y[v;ir] pyatv - ' jTr] y [v 'IT] J where M = g g M p a yv &py°av and we have used R = -R pyva pyav But from (55) , R . = (n pn aR n n. R, . y v ; IT pya [v; ; ir ] y [v ' TT ] Using this in (58) and defining K p a = n pn a - M p a (59) we obtain (K p aR . ) . - K r R, = 0 (60) pya[v J ;7r ] y[v'ir] This is a set of f i r s t order linear equations in the quantities K p a ( f i r s t order non-linear equations in L yv). If the only solutions of equations (60) were L yv = 6yv, then i t 24 would follow that there are no solutions of the f i e l d equations that are independent of the choice of 8. In general, equations (60) are quite intractable, so we consider a special case, name-ly that L y y is an infinitesimal Lorentz transformation, i.e., n y = n v + X l y n v (61) where l y v = - l v y and l 0 y = 6 l y l ( r , t ) Then to f i r s t order in X, we have K p a = - X ( n p l A B + n a l p 8 ) n B = -X(6 +6 )e(r,t) (62) v po l a oo Ip-^  ' J v J where E ( r , t ) = e " Y l ( r , t ) . Using (62) in (60) and evaluating covariant derivatives, we obtain (to f i r s t order in X) : - X{(R , + R, )e - (R , + R, )e v o y l v l y O v , TT Oy 1 TT l y o r r }\j + e [R i + R-i ' R i " R i « o y l v , T r l y o v , T r o y l T r , v l y O T r , v - r a (R , + R, ) + r a (R , + R, ) y j T o a l v l a o v y v O a l i T l a o i r + C f i lyWVl[v> R,,r e a +Y]} = 0 (63)' In particular, (63) contains four non-trival equations: 25 R1220 e»l + e( R1220,l • ri2 R1220 } = 0 ( 6 4^ R1220e,0 + e ( R1220,0 + ^ ^ ^ l o i O ^ = 0 ( 6 5 ) R i o i o S i + £ ( R i o i o , i + ( r o i " r i ^ R i o i o " e * + Y R , i ) = 0 (66) R i o i o £ > o + G . ( R i o i o , o + ( r o i r o o ) R i o i o " e a Y R»O : ) = 0 (67) Now R^220 = ~liTa> a n <^ w e b- a v e shown that in order for (55) to hold we must have ct = 0. Thus R-^ 220 = ^ a n c* (^4^ comes t r i v i a l . However, (65) becomes h e r22R1010 = 0 or 7 -hre~a Y ( % Y " + % ( Y * ) " %a ' Y ' ) e = 0 or ( Y " + % ( Y ' J 2 " %a»Y ')e = 0 If ( Y" + % ( Y ' ) 2 - %a'y') f 0, then we must have e = 0. This then gives us the desired result n y = n u. It' is worth verifying that 2 CY" + h(y') • ^CC'Y 1) = 0 is not consistent with any solution of the vacuum f i e l d equations (21) - (24). 2 So we assume Y " + % ( Y ' ) " ho.' y' = 0. This implies that the quantity 3(a,y) appearing in equation (22) is zero. Using this and equation (23) brings (22) into the form: %re" a(y" - % ( y ' ) 2 - f a V " | Y ' ) = 0 or ? ? i %re" a(y." + h(y') - % a ' Y ' -CY') - CX'Y' - ^ Y ' ) = 0 26 or Y ' (y1 + «' + i ) = 0 Clearly (y * + a' + i) = 0 is incompatible with any of the solutions discussed in section 1(c). However y 1 = 0 is compatible with the solution (37). But i f we reject solutions (37) and (38) on physical grounds, then we have established that ( Y " + % ( Y * ) 2 - W Y ' ) f 0 so that we must have e = 0. We have thus shown that infinitesimal changes of the form (63) in the definition of n u [equation (53)] lead to f i e l d equations with t r i v i a l solutions only. It remains a problem to examine the effect of more general changes in n y [equation (54)] on the solution of the resulting f i e l d equations. II. (a) Generalized Newtonian Metrics We shall now examine space-times that admit a class of preferred charts, called generalized Newtonian charts, which include Newtonian charts as a special case. Let M be a smooth, four-dimensional manifold. A pseudo-Riemannian metric g on M is a 2-covariant, non-degenerate, sym-metric, smooth tensor f i e l d on M with signature +2. Given a metric g on M, define a new metric g by g = e" 2 < ! >(g + f n x n) . (1) where f and <j> are smooth real-valued functions on M and n is a smooth covariant vector f i e l d on M. It is easy to see that g satisfies the definition above of a pseudo-Riemannian metric on-M. If (u,x) is a chart at p in M, then the covariant com-ponents of g, g, n at p with respect to (u, x) are g a g , g , n a, while the contravariant components of these objects are, respectively g a B , g a^, n a. It is straightforward to show that (Rastall, unpublished notes 1972) 2<j) -f = e ( g a 3 - £ ( ! + £ j ) _ 1 n a n p ) (3) where n a defined by n a = g a^n 0 and where ga^n n = j . Now let us P a p require n to be a unit time-like vector f i e l d with respect to g, i.e 27 g n n. = j = -1 Let {e } be a basis of the tangent space to M at p, and let {eP } be the dual basis of the cotangent space (i.e., e y(e ) = 6 ). Then we can choose (e } such that the follow v yv y ing are true: (i) {e^ } is an orthonormal basis with respect to the metric g, i.e., g(e , e ) = n ° y ' v y v where nab = 6ab n = n ' = - 6 Oy yO yo C i i ) e° = n It follows that In particular, = e~2(f)[n + f 6 6 ] yv oy ov S(ey> Sa^ = e " 2 * % a ; a = 1, 2, 3 i(e , e ) = e"2<l>(n + ffi ) 5 y ' ov v yO yO Let a be in the tangent space of M at p. Define the length a with respect to the metric g and g by 29 A Let a be any tangent parallel to n (p) and 3 any tangent orthogonal to n (p), where n (p) is the contravariant vector f i e l d corresponding to n. Then L = e " * ( l - f ) % L a a If we choose (l-f)'= e 4* (5) then L = e*L a a L = e"*LQ a 3 Roughly speaking, g "stretches" tangents parallel to n by a factor and tangents orthogonal to n by a factor e ^ in com-parison to g. If we then define j and f appearing in equations (2) and (3) by (4) and (5), then we obtain: 30 g o  = e 2*g o  + ( e 2* " e 2*)n n D -a8 2<J> aB /• -2* 2*, a 3 g = e y g - (e - e y)n n p (6) We can now define a generalized Newtonian metric; namely, let g be a f l a t metric, then g, defined by equation (1), is a generalised Newtonian metric i f n is time-like and (1-f) = e ^ A chart in which |gag| = 5 a g i n equation (6) is called a gen-eralized Newtonian chart. Newtonian charts are a special case of generalized Newtonian charts, defined by the condition n = &' p po which i m p l i e s t h a t g reduce to F 6a3 -2<j> ^ab & ab g = -e 6 6Oa oa as in equation (1) of Part I. II. (b) The Linearized Einstein Equations and Generalized Newtonian Metrics The eventual goal of these considerations is the con-struction of a geometric theory of gravitation based upon gen-eralized Newtonian metrics. However, in the" remainder of this essay I shall undertake a less ambitious task; namely, I shall show that generalized Newtonian metrics which differ i n f i n i t e s -imally from fl a t metrics are not uninteresting solutions of the linearized vacuum f i e l d equations of General Relativity. 31 In equation (6), expand e 9 in a parameter \: • 2* 2 e = l + x<$> + 0 ( x ) a s x - > o (7) (So we must have X<j> = 2$). Then equations (6)have the form § a B = § a B + X h a B + 0 ( x 2 ) g a B = g a S - Ah a B + 0(A 2) (8) where g is a fl a t metric in Minkowski co-ordinates, i.e., g . = n . and where &a3 aB h = -(g + 2n n j $ (9) aB ° a B a 3 and v a B a y 36, h = g 'gp h Y<5 The components (in a Minkowski chart) of the Ricci tensor defined by g are then R a B = h X C h'a3 " ^*>** ^ * > a i T + D ^ g ) + °(^) (10) where a comma preceding an index denotes the partial derivative with respect to the co-ordinate labelled by that index and h = h a O = g a 3 — a —g ( i i ) c X 3 X 32 Also, R, the curvature scalar i s , in the linear approximation: R = g a \ g = J s U D h - 2(h a S -%g a 6h), ag] + 0 ( x 2 ) (12) As usual, when deaing with the linearized Einstein equations great simplification results i f the "Hilbert gauge" is imposed, i.e., i f we demand that h a B satisfies ( h a 3 - %g a Bh) ,a = 0 (13) In this case R a g = ^ D \ g + . ( 1 4 > R = %XQh + 0(X 2) (15) Notice that h „ - kg „ h = -2n n.$ so the linearized Einstein a g toag a g tensor 6- D = R 0 - kg DR has a very simple form: ag a g ° a g r G a B = -*D(vy) + 0(X 2) (16) Now we impose the condition that the metric g be Riaci-flat, i.e., that R „ = 0 + 0(A 2) or equivalently G „ = 0 + 0(X 2) either of ag 1 J ag . which require P ( n a V ) = 0 , ( 1 7 ) If R = 0 + 0(A 2), then we also have R = 0 + 0(X 2), or a 8 • $ = 0 (18) 33 Thus i f we demand that g be R i c c i - f l a t , then the function $ must satisfy the wave-equation. We may also have • n a = .0 (19) i f n a and $ have the following form: $ = 4(s x a) a n = n (s x 6) (20) a a g where s^ is a constant n u l l vector with respect to the f l a t metric g. It is easy to check that (20) i s a solution of (17) by d i r e c t s u b s t i t u t i o n : P i (n n.$) = $(n n n . + n . n n ) + n n„ n $ + 2($Vn • vn. 1 — 1 a B a 1 — 1 B B 1 — ' a a B a B + n„Vn -v$ + n vn • v$) (21) B a a B where v A • VB = g y v ^ - M _ = t 3A 8B 8A 8B 3x y 8x y a=l 3x a 3x a 9x° 3x° Now i f (p and n„ satisfy (20) , then clearly (18) and (19) hold, so the f i r s t two terms of (21) vanish, and the last term becomes 2s s y($n' n' + n„n' $' + n n' $') u a B B a a B where 1 denotes differentiation with respect to (s^x01) . But since s is n u l l , i.e., s s y = 0, the above expression is zero, y y Notice that $ and n must both be advanced (s° > 0) or both re a tarted (s° < 0) solutions of the wave-equation. C O N C L U S I O N In Part I we examined spherically symmetric solutions of Rastall's 1971 gravitational f i e l d equations, which are third order, non-linear,partial differential equations. The imposition of the boundary condition that the metric by asymp-tot i c a l l y f l a t proved to be insufficient to guarantee a unique solution. One solution that is compatible with this boundary condition is equivalent, in appropriate co-ordinates, to a static Newtonian metric. We already know (Rastall, 1968) that such a metric gives experimentally satisfactory results for the case of planetary motion and the deflection of light by the sun. It is possible that by imposing other boundary conditions, or.:, by considering the relation of the f i e l d to its sources, one would be able to prove that this metric is the unique spheri-cally symmetric solution of the f i e l d equations. In addition, i f this were the case, we would have an analogue of "Birkhoff's Theorem" in Rastall's theory. Another d i f f i c u l t y is that the time-like vector f i e l d appearing in the f i e l d equations is not unambiguously defined in the case of spherical symmetry. This is clearly an unphysi-cal situation, unless i t can be shown that the ambiguity in n M does not lead to ambiguity in the solution of the f i e l d equa-tions. We found in section 1(e) that infinitesimal changes in n y do not lead to distinct solutions of the f i e l d equations. 34 35 In fact, i t was shown in that section that the structure of the f i e l d equations rules out infinitesimal changes in n y. Rastall's 1971 theory is a generalization of his 1968 scalar theory of gravity. Another generalization involves a larger class of metrics, called generalized Newtonian metrics. The effect of these metrics is to "stretch" the length of vec-tors parallel to a certain direction, and "compress" the length of vectors orthogonal to that direction. We have shown that there exist wave-like solutions of the linearized vacuum Ein-stein equations that have the form of a generalized Newtonian metric. R E F E R E N C E S Arnowitt, Deser, and Misner in Gravitation: An I n t r o d u c t i o n to Current Research, ed. L. Witten (Wiley, New York, 1962). Birkhoff and Rota, Ordinary D i f f e r e n t i a l Equations (Ginn and Co., New York,,196 2). Bonnor, in Recent Developments in General R e l a t i v i t y (Pergamon London, 1961) . ' Coleman, Journal of Physics A: General Physics, 4, 611 (1971). Einstein, The Meaning of Relativity, f i f t h ed., (Princeton University Press, Princeton, New Jersey, 1956). Fock, The Theory of Space, Time and Gravitation (Pergamon, London, 1959). Fulks, Advanced Calculus (Wiley, New York, 1961). Landau and Lifshitz, The Classical Theory of Fields, second ed., (Addison-Wesley, New York, 1961). Rastall, Canadian Journal of Physics, 46, 2155 (1968). Rastall, Canadian Journal of Physics, 47, 2161 (1969). Rastall, Canadian Journal of Physics, 49, 611 (1971). Rastall, Unpublished Notes, 1972. 36 

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