VOLUME 20, NUMBER 2 PHYSICAL REVIEW LETTERS 8 JANUARY 1968 where A. is a constant, our experimental result Atomic Energy Commission. limits the total branching ratio of the K+ into ~N. Cabibbo and R. Gatto, Phys. Rev. Letters 5, this channel to be less than l.lx10 382 (1960). The vector-meson-dominant model'~' and Y. Hara and Y. Nambu, Phys. Rev. Letters 16, 875 (1966). the g-pole model" both predict branching ra- 3Y. Fujii, Phys. Rev. Letters 17, 613 (1966). K+ —m++y+y that much than tios for are lower 4S. Okubo, R. E. Marshak, and V. S. Mathur, Phys. the upper limits which we have been able to Rev. Letters 19, 407 (1967). set in this experiment. 5A. H. Rosenfeld, A. Barbaro-Galtieri, W. J. Podol- We wish to thank E. Segre for advice and sky, L. R. Price, P. Soding, C. G. Wohl, M. Roos, encouragement, W. Hartsough and the Beva- and W. J. Willis, Rev. Mod. Phys. 39, 1 (1967). tron staff for an efficiently running machine, 6I. Lapidus, Nuovo Cimento 46A, 668 (1966). ~S. Oneda, Rev. 1541 and E. McLeish for her effects at the scanning Phys. 158, (1967). 86. Oppo and S. Oneda, Phys. Rev. 160, 1397 (1967). table. Y. Fujii, private communication. ~ G. Faldt, B. Petersson, and H. Pilkuhn, Nucl. Phys. )Work performed under the auspices of the U. S. B3, 234 (1967).

POSITIVE DEFINITENESS OF GRAVITATIONAL FIELD ENERGY* D. Brill Sloane Physics Laboratory, Yale University, New Haven, Connecticut

and S. Deser Department of Physics, Brandeis University, Waltham, Massachusetts (Received 20 September 1967)

The total gravitational field energy functional is shown to have only one extremum un- der variation of the metric field variables. At the extremum the energy vanishes and space is Qat; second variation shows that the vacuum state is also a local minimum.

It has been increasingly recognized in recent infinite series in the metric field variables. years that the gravitational field, as described On the other hand, an affirmative answer is by general relativity, shares many of the ba- clearly essential for a satisfactory physical sic physical properties of Lorentz-covariant interpretation of the theory. field theories, particularly of massless gauge In this Letter we regard the energy as a func- systems. Thus, asymptotically flat spaces, tional of the gravitational field variables and which describe isolated physical systems, can consider its variational properties under change be assigned a well-defined total energy-momen- of geometry. We thereby establish that the tum P~, satisfying the physical requirement functional has only one extremum, flat space. that PW is covariant under Lorentz transfor- Further, the second variation about this "point" mations and invariant under interior coordinate —the energy of a weak field —is shown to be transformations. ' The fact that the field is positive. From these results, and to the ex- self-coupled, i.e., that gravitational field en- tent that our functional behaves like a function ergy is itself a source of the gravitational field, of a finite number of variables, 4 the positive- is reflected in the nonlinear nature of the Ein- ness problem is resolved in the affirmative: stein equations, particularly of the constraints The vacuum (absence of any field excitations) that determine the energy. This is also the is the lowest energy state, and, as a corollary, reason that the following fundamental problem vanishing energy implies flatness. W'e restrict had resisted solution until now: Is the total ourselves here to the source-free field, but energy of the gravitational field positive def- the proof holds also in the presence of normal initeY' The difficulty of this question is due matter sources that have positive energy and to the implicit nature of the Hamiltonian of the are minimally coupled to gravitation. We con- theory —it can be given explicitly only as an sider here only "nonpathological" systems, VOLUME 20, NUMBER 2 PHYSICAL REVIEW LETTERS 8 JwNvxRv 1968

i.e., systems with Euclidean topology which evant metric components. This is in complete can be reached from flat space by continuous analogy with electrodynamics, where the total deformation. Also, we assume that in such charge Q is defined by $E d5, while its value space at least one "minimal" hypersurface can for any system is obtained from the volume be introduced. integral fd'r p(r), because of the constraint We begin with the definition of the usual mass V ~ E = p. The Einstein constraints read' energy. ' For our purpose, it is adequate to 0 -p ij g 2 i ij take it as the flux integral, ~ z -=6t-g (~ ~ =.,.~ )=0, ~ =-v.~ =o, Zj = = lj 16wm fdS. (g. . .—g. . .) —)dS.(gg ) where S is the scalar curvature density of the Zjj2ZZ 2 hypersurface t=0, mlj is a three-tensor densi- over a closed two-surface at spatial infinity. ' ty related to the extrinsic curvature of this The value of nz for any physical system is ob- hypersurface (it is essentially Bpg,j, the mo- tained from its definition (1) by solving the four mentum conjugate to the field amplitude gij), Einstein constraint equations G& =0 for the rel- and v-=n . By (1) and (2) we may thus write the total mass energy

3 Q lj J 2 16~m=-fd r[(R-g (v ~.. ——,n )+(g.. . .—g, . ..)] 2j l2jj lj»

3 p ij l k l k — -1 'n 2 —- d rg [g (I' I'. . -I" . I' . ) g (7t ij,v, )]. . (3) kl Zj kZ lj Zj

Clearly we cannot simply vary m[gi, v'&] with respect to arbitrary changes fungi, 5&mil of the vari- ables, since the four nonlinear contraints of Eq. (2) must be respected under variation in order to compare only systems satisfying the Einstein equations. On the other hand, we cannot solve the R constraint in closed form for a single "constraint" metric component. Instead, we add to 5~g j a compensating 5~gij to guarantee that R =0 remains satisfied. The key to our procedure is the fact that a 6~gij can be found which actually leaves m unaffected. Consider the variations ijL g. . =5 g. . 6 m =5m +5m T ij n ij +4',ij' T where 5„gijis a normal, arbitrary variation vanishing faster than I/x at infinity, while e may (and indeed must, as we shall see) vanish as slowly as 1/r The sign. ificance of the decomposition of 5Tg;. is made apparent by writing lm T ij ij ij T lm so that Q; is clearly traceless and has five components, while e is essentially the trace of 5Tg;j. The hagi& are then unconstrained, but yield five equations upon varying. For simplicity in text, we = us e six Qz pij s instead, but the re suits are identical, with the redundant equation 8 0 entering the re- by. Similarly, we have divided DT7tlj into an arbitrary part 5mljT which is divergenceless, and a longitudinal part 5mzj~ which guarantees the maintenance of the constraints R& = 0 under the full vari- ation of Eq. (4). From the corresponding variations of the quantities occurring in (3),

—5 f@d~x= 2f@(Rd~—r+8$Ve d5, —5 f(g . . . -g . . .)d~~=-8/v~ dg, ii,jj ij, ij

lj y 2 p 3 lj y 2 p 3 & f(v 7t'. ~n )g d r=2 e(w v. . -~n' )g d y, lj Zj

it is easy to see that 5~m vanishes. Here the explicit metric dependence of the terms involv- straint Ra= 0, and shows incidentally that the ing mlj is fixed by the choice of the contravari- energy of a physical system in general relativ- ant tensor density m» as the basic variable. ity is a (three-space) conformal invariant. The vanishing of 6&m occurs only at the con- To assure that the R~ constraint remains sat- VOLUME 20, NUMBER 2 I HYSICAI. RZVIZW r. ZTYKRS 8 JANUARY 1968 isfied, we demand that 5TH =0. Using the for- ever Rij and re vanish on a spacelike surface, mulas and Einstein's equations hold. Finally, we must show that the extremum 6 R=8V 'e —4Re, 6vg =6m/g, g is in fact an absolute minimum; this corresponds to evaluating second variations of the energy we find that e must satisfy the Poisson equation9 about flat space (weak-field excitations). Here 2 — ij 12 one must take into account the first-order re- 8V e -6 [R g (n w. .-2w )] g (6&g) zj lations among the variations, particularly the

~ value ~ of 6e/6„g» at the extremum. The term + (w w ..-2m ). (6) 6'm/6Tg, &6Tgf~, which involves the curvature 6,~Tv j R, can be shown to be positive definite" '4 by Here V&' is the covariant Laplacian for the base explicit variation and use of Eq. (6). The mixed metric g,j. Note that the solution e is indeed variations 6'm/6Zg lj"6Tmf~ clearly vanish at necessarily O(1/x), and a linear functional of mij =0, since m is quadratic in mij. This leaves 2 6+g,&, 6m~&. [An alternate derivation of 6Tm 6 m/6Tw &6Tv, where m must include the proceeds from its definition (1), which gives variations 2fd'—r—g 'I'6m6m even though &= 0. directly 6m = -8fV'ed'r, with V'e given by (6).] However, since Vj57Tzj =0 at mij =0 we may The variations 57t» must also be treated care- fall back on the fact that at flat three-space must the R' constraints where rectangular coordinates = are fully as they respect (2). (g,& 6,&) For this purpose we may invoke a recent ex- allowed, the form plicit covariant solution~ of this transversal- 1 ~ ~ — ity equation. Qne may write mij =7tzjT+mzjL fd xg [6 m 6Tzj'm'. . 2(6w) ] where Vjnzj'T -=0. The variation of the trans- ~ ~ verse part 5mij T is thus free, while the three is positive if & 6Tw» =0. (This integral is sim- variations 5mzj~ are determined by the three ply the kinetic energy in the linearized approx- conditions 5Rz = 0 as solutions of Poisson-like imation. ) Thus the energy of weak systems equations in terms of 6gij. As was the case is strictly positive, "and in fact has the trans- for c, these need not be solved explicitly, since parent form

3 -z L 1 2 ~ 1 TT 2 TT 2jT —,'6 6 ~,m=(&n) d ~g m 6m. . I „=fd r[,'(V6 -g ) +(6n ) ]~0 (9) 6w ) 2j (orthogonality of transverse and longitudinal appropriate to a massless spin-2 particle, whose tensors) on the "minimal" surface m=0 whose variables are transverse traceless (TT) in the existence we are assuming (though of course flat-space sense. 6me 0). Qur considerations hold for isolated systems, The total variation of the mass energy is then i.e., for asympotically flat spaces [g&„-7I» effectively just +O(1/r), s~g„~-O(r ') at t=0] with Euclid- ean topology, and we have assumed that the ijT ij 6 m=(6 g, .6/6g, . +6n 6/6n )m, Poisson-type equation (6) (and a similar one T n zj ij for 6wL) have global solutions. Also, it is im- where the last set of variations can be treat- plicit in our variational process that we corn ed as entirely unconstrained. From Eq. (3) pare only spaces which lie on the same func- the Euler-Lagrange equations for the extremum, tional "branch" as flat space, and omit from Gym = 0, are" consideration any possible "pathological" spac- es which do not reduce smoothly to flat space lj 1 ij 2 ijT ij in some suitable limit, such as a limit of de- R -gg R+O(n)=0, w'=m =0. (8) creasing field intensity. (It is probably only Here O(~') indicates terms quadratic in m'~ which for such systems that the notion of energy is will vanish anyhow because of the other Euler at all applicable or relevant. ) equations, mij =0. Thus there is only one ex- If, further, our variational results on m can tremum which lies at the "point" mij =O=Rij be taken to imply that this functional is posi- (taking into account the constraints R0=0=Ri). tive, ~ they have the following implications: But this "point" is precisely flat space, ' since The "free-field" energy of the full classical the full curvature tensor R&~~p vanishes when- Einstein theory has zero as its lowest value. VOLUME 20, NUMBER 2 PHYSICAL REVIEW LETTERS 8 JANUARY 1968

value the V' This is reached only by vacuum state, the three-space t =0 with metric g~j Thus, denotes where there are no physical excitations (flat covariant differentiation in this metric, and g—=detZzj. = — space), and the single condition m =0 is equiv- The sign convention on Rz~ is Rz~ I zj y I'&qp j at alent to the vanishing of the full curvature ten- r =0. sor These properties coincide with ~We assume that the various Poisson equations we R»~p. need have global solutions; alternatively, we are re- those of all Lorentz-covariant systems consid- stricting attention to spaces for which this is the case. ered in the rest of physics, and have obvious No attempt is being made here to solve the unsolved relevance to quantization. mathematical problems of rigor connected with such equations in curved space. S. Deser, Compt. Rend. 264, 311 (1967), and Ann. Inst. Henri Poincare 7, 149 (1967). *Work supported in part by the National Aeronautics ~~The first part of Eq. (8) is obvious since it is the and Space Administration under Grant No. NSG 163-61 three-dimensional analog of the standard Einstein vari- and by the U. S. Air Force, Office of Aerospace Re- ational principle. The second part, 7t~& =0, results be- search, under Grant No. AF 368-67. cause the kinetic energy variation has only one term, R. Arnowitt, S. Deser, and C. W. Misner, Phys. fd3~g sr~ 6~ , "when m=0 and x~& =0 (due to Rev. 122, 997 (1961); R. Arnowitt and S. Deser, to be ~;~J= 0). The 'cross terms" fd3r g 4~JT6mgP van- published. ish since T and L tensors are orthogonal. 2Positiveness of the energy has been established for ~2For a detailed proof, see, for example, R. Arno- various special classes of solutions: H. Araki, Ann. witt and S. Deser, Ann. Phys. (N.Y.) 23, 319 (1963). Phys. (N. Y.) 7, 456 (1959); D. R. Brill, ibid. 7, 466 The existence of a flat surface of vanishing extrinsic (1959); R. Arnowitt, S. Deser, and C. W. Misner, ibid. curvature is a sufficient condition for flatness, but it 11, 116 (1960). The last paper also mentions some is not necessary, since curved surfaces can also be physical arguments for positiveness in general. imbedded in flat space. ~ & 3Unlike the situation in Lorentz-covariant theories, Positiveness of weak-field energy when 7L = 0 was it is well known that the zero point of energy is not ar- first demonstrated by Araki (Ref. 2). bitrary in general relativity; m must be zero for flat ~4In studying second variations of m, one must check space. that second variations of the constraints can also be 4For functionals, however, positiveness does not al- made to vanish. This is indeed the case, but details ways follow from knowledge that there is only one ex- of this are unnecessary in evaluating 6 m, just as de- tremum which is a local minimum. Existing mathe- tails of how 6R vanishes were unnecessary in evaluat- matical criteria do not yet seem directly applicable to ing BTrn. our problem. ~5If the "minimal" surface condition is lifted, it is 5We use the historical notation m for the energy known (Ref. 10) that the kinetic energy is not a positive (from the Schwarzschild solution mass). Alternatively, form (unless Rz~ =0). Physically, however, the the- I is also the invariant mass in c.m. frames, where p orem still holds, for the value of 7t just fixes the time =0 (which is probably always allowed for bounded sys- coordinate, and transformations leading from arbi- tems). The units are 16~=1=c; Latin indices range trary 7t to 7t = 0 exist (at least formally), leaving the 1, 2, 3; the signature is (+++—). value of m invariant. When n & 0, the method in text 6All the usual definitions of gravitational energy no longer leads to Eq. (8); one obtains an expression agree with Eq. (1) when g» and Bog~q approach their for 5~ depending explicitly on the constraint varia- Qat space values as 0(x ~) and 0(x ), respectively, tions. We have not been able to determine the corre- at infinity. For a more precise discussion of the defi- sponding Euler-Lagrange equations, but they are pre- nition and invariance of gravitational field energy, see sumably that generalization of (8) which expresses the Ref. 1. vanishing of R»zp(g, 7t) on an arbitrary spacelike sur- 70f course, dm/dt = 0 by the Einstein equations. face (the Gauss-Codazzi equations). We are indebted We follow the notation of the last paper of Ref. 2. to C. W. Misner for enlightening discussion of this All quantities ir. this Letter (except Rpqop) refer to question.