VOLUME 2, NUMBER 8 PHYSICAL RKVIKW LKTTKRS APRIL 15, 1959

served values of the charge and magnetic-mo-ment radii indicate that this resonance shouldoccur at v -8.5m (square of the pion momen-tum in the s-s barycentric system). In Fig. 1the function ~F (s) I' is ylotted for this value of

vr and several values of thewldthr I Fig 2the pion form factor is plotted for g & a. Since itis less than one over most of this region, its ap-pearance in the denominator of the integral ofEq. (9) will produce an additional enhancement.

In conclusion, our Eq. (9) for the weight func-tions together with the approximation given inEq. (10) for the pion-pion scattering amplitudesuggests that a p-p resonance of suitable positionand width could lead to agreement between dis-persion theory and many aspects of nucleon elec-tromagnetic structure. Detailed calculations arein progress.

We are indebted to Professor Geoffrey F. Chewfor his advice throughout this work, and for ad-vance communication of some of the results on

pion-pion scattering. We also acknowledge thehelp of James S. Ball and Peter Cziffra in obtain-ing Eq. (4).

This work was done under the auspices of the U. S.Atomic Energy Commission.

~Visitor from the Argentine Army.~Chew, Karplus, Gasiorowicz, and Zachariasen, Phys.

Rev. 110, 265 (1958).2Federbush, Goldberger, and Treiman, Phys. Hev.

»2, 642 (1958).3S. D. Drell, 1958 Annual International Conference

on High-Energy Physics at CERN, edited by B. Ferretti(CERN, Geneva, 1958).

4If the integral in Eq. (1) fails to converge, one canuse the usual subtracted form, as discussed in refer-ence l.

~S. Mandelstam, Phys. Rev. 112, 1344 (1958).SR. Omnes, Nuovo cimento 8, 316 (1958).~See, for example, Appendix II of Fubini, Nambu, and

Wataghin, Phys. Rev. 111, 329 (1958).SG. F. Chew, Lawrence Radiation Laboratory (private

communication, 1959).

ENERGY OF THE GRAVITATIONAL FIELD

P. A. M. DiracInstitute for Advanced Study, Princeton, New Jersey

(Received March 20, 1959)

Einstein's equations for the gravitational fieldare valid for any system of coordinates and makeit difficult for one to distinguish physical effectsfrom effects of the curvature of the coordinatesystem. In consequence there is no obvious de-finition for the energy of the gravitational field.The usual definition, in terms of a componentt,o of the stress pseudotensor, makes the energydepend very much on the system of coordinatesand is thus not satisfactory.

For physical problems one can restrict thegravitational field to be weak, of the order of y,the gravitational constant. One can then use asystem of coordinates for which the g differfrom their values in special relativity by quanti-ties of the order y. Even if one restricts oneselfto such coordinate systems (and renounces, forexample, the use of polar coordinates), one canstill make arbitrary changes of order y in thecoordinate system and such changes producechanges in the energy of the same order as theenergy itself, so the difficulty persists.

In discussing the question people have usually

lost sight of the primary requirement for theenergy, that it shall be a useful integral of theequations of motion. The development of theHamiltonian form of gravitational theory' enablesone to make a new attack on the problem, takingthis utility requirement into account.

In the Hamiltonian form one deals with the stateat a certain time x, which state is described bydynamical variables for all values of z', z, z'for this one value of xo. It is found that the onlyvariables needed to describe the gravitationalfield are the six g (r, s=1, 2, 3) and their con-jugate moments p&~. The four g 0 do not enterinto the description of the state at a certain time.They are needed only to provide a connection be-tween the state at one time and the state at aneighboring time.

We are thus led to the condition that the energyat a certain time should involve only the gzs, p~s,and the nongravitational variables and should notdepend on the g&0. The usual definition in termsof to does not satisfy this condition, nor does thedefinition recently yroyosed by Mgtller

368

p~z. UMz 2, NUMezR 8 P HYSICAL REVI E W LETTERS APRiI. 15, 1959

We shall take over the Hamiltonian worked outin reference 1, keeping the same notation exceptthat we shall change the sign of aB the g» tomake g«negative (which entails changing alsothe sign of the prs). Thus we have the Hamil-tonian

where a lower suffix added to a field quantity de-notes an ordinary derivative. We have also theconstraints

K~=O, K =G.

In an ordinary Hamiltonian theory one takesthe Hamiltonian itself to be the total energy, asit is always an integral of the motion. This willnot do in the present theory, because, on accountof the constraints (4), the Hamiltonian (1) equalszero. So long as one considers the exact equa-tions of motion there does not seem to be anyuseful integral that one could take to be the totalenergy.

For the weak-field approximation it is reason-able to divide the Hamiltonian into two parts, apart that gives the main motion corresponding tog G--5

Gand apart that gives the correcting

terms due to small deviation of the g 0 from theG p, G

values -5&G. These two parts are

H ~ = '(w+XMI) d x~ (5)

wherej. 1 f

=prsprs 'prrpss+ 'grsugrsu grrugssu)

~ grsr guus gurPusr~'

and

+cor =J ~( +g00)(grsrs grrss ML)

- g~{2P„-X~„(}

(PAL

The constraints in this approximation are

&xsr s &riess ML = 2P -3'Mr=0. (8)

I +~~o

where $CI and SC are functions only of the grs,prs, and nongravitational variables, namely

-1( rs, r s), -1+2 rs) uv

2 Q5x(ag -g„„)+tc (rc e )„}„+xmi,(2)

QSs =P guvs (P gus)v+ Ms&

The constraints now cause H to have thevalue zero, but not Hmajn The difference hasarisen because of the neglect of a surface termat infinity in the derivation of (5), such neglectbeing justifiable because a term of this naturein the Hamiltonian does not influence the equa-tions of motion.

Let us now consider an example for which, atlarge distances y from the origin, there is nomatter present and g „andp are of order

Such examples often occur in practice. W'enow have so of order y ~ at large distances, soamain converges. It is a constant of the motion,provided we take values for the g&O which pre-serve its convergence Hmain is now a usefulintegral of the motion, because its constancy isnot a consequence merely of the constraints (8).Vfe may thus reasonably define Hmain to be thetotal energy.

For an example in which there is continuousemission of gravitational waves, g and pat great dist nces are of order r 1. Hmaln nowdoes not converge, corresponding to the totalenergy of the gravitational waves being infinite.The energy of physical importance is now thetotal energy within a large region R. To be ableto obtain such an energy we need an expressionfor the energy density, at. any rate for largevalues of y.

The expression (5) for the total energy in thecase of convergence suggests that we look uponw+XMI, in general as the energy density, sothat m is the energy density of the gravitationalfield. Let us examine whether this is permissibletaking first the special case when our coordinatesystem is such that

gpO- poThe conservation of energy would require that

&(w+ XMf )/&x' = err,where 4 is some 3-vector, which can be inter-preted as the energy fiux.

Now the first of the constraints (4), if evaluatedfrom (2) to the second order in y, gives

zo+XMI = - K e

This leads immediately to (10) with

k =- X e p.

But this kz cannot be interpreted as the energyflux because it is of order y, whereas the energy

VOLUME 2~ NUMBER 8 P HYSICAI, RKVIK W I, KTYKRS APRIL 15, 1959

density zo in the absence of matter is of ordery'.

We have, with the conditions (9),

8(ge+XMI )/Sx'=J[w+X~f, w'+X'~L ]d'x'. (12)

We see from (6) that w involves only undifferen-tiated momentum variables and dynamical coor-dinates differentiated not more than once, andwe may assume that XII. is similar. It followsthat [so+XMI, , w'+X'ML, ] cannot involve 6{@-x')differentiated more than once. Since it is anti-symmetrical between the two points x andz',we can infer that it must be of the form

6, (u + XM~) = fa~'[so+ X-MI, 2p'„»i

- K'M„ld'x .If the hypersurface x'= constant is changed byeach point of it being shifted normally through adistance a, where a is a function of g~ zs, ~s oforder y, the change in so+3~L is

(14)

6&(g1 +K~I ) =f8 [tU+ XMI, g'&&&isa - g &&&&&i

(15)

The energy density, defined as ce+XML, is sub-ject to these two uncertainties.

It may easily be verified that, when the condi-tions (9) hold, the component too of the stresspseudotensor is just equal to m, so the conserva-tion law proved above is not a new result. Whenthe conditions (9) do not hold, too differs from zo

by terms involving derivatives of the g 0. If the

[u+x~~, m'+x'ML]= (k„+k~')6~(o-x'), (13)

for some 3-vector k~. Substituting this into (12),we get just the result (10). The gravitationalpart of k~, which comes from [w, m'], is linearhomogeneous in the p~z and linear homogeneousin the g, so it is of order y, the same as m.

So this kz can be interpreted as the energy fluxand the conservation law is verified.

When the conditions (9) do not hold, the abovededuction gets spoilt by the extra change in

cv+XMI produced by Hcor, given by (7). Thesource of the trouble is that ge+XMI gets alteredif one makes a change in the coordinate systemof order y. The alteration consists of two parts,corresponding to the two terms in the integrandin (7). If the coordinates z~ are changed byx"-a~+ $~, where the $~ are functions of x', x',

of order y, the change in gr+3MI is

energy density is defined in terms of too, it issubject to the above two uncertainties and a fur-ther uncertainty because of its dependence onthe g&0. So the use of zo instead of t~ makessome improvement.

When the total energy is convergent it can beexpressed as a surface integral at infinity. Withthe present definition of energy this integral is,according to (11),

fZ-1(rC e"")„dS„. (16)

With the usual definition in terms of the pseudo-tensor, it is

The two expressions agree provided g 0—-5 0

+O(r ') at large distances.On account of the uncertainties (14), (15),

there does not seem to be any general definitionfor the energy density independent of the coor-dinate system. However, there is an importantspecial case when these uncertainties vanish,namely, when there is no matter present and thegravitational field, to the first order of accuracy,consists only of waves moving in one direction.

In the absence of matter, (14) and (15) give,with the help of (6),

62tv = -2p a (19)

s= 2psssms.

The second of the constraints (8) now gives p 3=0, so 6 IN also vanishes.

We can conclude that the energy density ofgravitational waves moving in a single directionis well-defined, independent of the coordinatesystem. It is only the interference energy den-sity of waves moving in different directions that

Let us suppose the gravitational field consistsonly of waves moving, say, in the direction ~3.Then the derivatives g and p will vanishunless I=8. 8 we now make a change in thecoordinate system so as to preserve the conditionthat the gravitational field consists only of wavesmoving in the direction g', we can introduce onlycoordinate waves moving in the direction gs.This requires that the derivatives 5 and ashall also vanish unless I=3. We now get 6,so

=0 and

SVG

VOLUME 2, NUMBER 8 PHYSICAL REVIEW LETTERS APRrL 15, 1959

is subject to uncertainties.Let, us go back to the problem of determining

the total energy within a large region R surround-ing some accelerating masses that are continu-ously emitting gravitational waves. Let us takea solution of the field equations in terms of re-tarded potentials without any ingoing waves.Then for large values of z, the main part of thegravitational field, of order z, consists ofwaves moving in only one direction at each point,namely radially outward. The energy density w

at large distances y is nom well-defined, in-dependent of any transformation of coordinatesthat preserves the character of the solution ofbeing expressible in terms of retarded potentials,and does not introduce any ingoing coordinatewaves.

The total energy within the region R, definedby

fR(K+ X~f ) d x

is now also mell-defined, because any transfor-mation of coordinates that affects only the cen-tral part of R will not change (20), on account of(11), while any permissible transformation ofcoordinates in the outer part does not affect gy

and so does not affect (20). Thus the uncertain-ties in the energy density defined by so+X~l donot affect calculations of the@mission of energyby gravitational waves.

The author's stay at Princeton mas supportedby the National Science Foundation.

This vrork formed the substance of an invited talkgiven at the New York Meeting of the American Physi-cal Society on January 30, 1959.

iP. A. M. Dirac, Proc, Roy. Soc. (London) A246„333 (1958).

tC. Mgller, Ann. Phys. 4, 347 (1958).

POSSIBLE DETERMINATION OF HYPERON PARITIES AND COUPLING STRENGTHS

Saul Barshay* and Sheldon L. GlashowInstitute for Theoretical Physics, University of Copenhagen, Copenhagen, Denmark

(Received February 26, 1959)

The notions of "global" or "universal" symme-tries, wherein the Z and A hyperons are treatedas members of the same multiplet structure incertain or all of their strong interactions, havebeen widely discussed, ' although a decisiveexperimental determination of the relative Z -A

parity is yet to be carried out. These detailedtheories require this relative parity to be even,but the possibility that it is odd has also beenmentioned. 4y ' Several rather difficult experi-ments have been suggested to determine the re-lative Z-A parity. ' ' Furthermore, analyses ofthe forward angle K-mesoa-proton dispersionrelations have been performed in order to deter-mine simultaneously the relative Z-A parity andthe relative K-A parity. '~" It was emphasizedsome time ago~ xx and again recentlyy &2 that thereare likely to be serious ambiguities in this pro-cedure. In particular, one must at present makethe somewhat subjective and certainly unjustifi-able assumption that, in the case of odd relativeZ -A parity, the renormalized pseudoscalarcoupling of E mesons is not an order of magni-tude greater than the renormalized scalar coup-

ling. In this note we shall discuss another ex-periment for determining the relative Z -A paritywhich may be feasible at this time.

Chew" has suggested a method of extractinginformation from differential cross sections bytheir extrapolation to nonphysical values of mo-mentum transfer. Thus an examination of the400-Mev neutron-proton angular distribution inthe backward directions has yielded a new evalua-tion of the pion-nucleon coupling constant. '4 Vfeshould like to suggest an analogous procedureap lied to the hyperon-nucleon data concernings ch processes as:

(b) Z +p-Z'+s,

(c) z +p-z +p,

(d) " +P-A+A,

(e) " +p=" +p.

Since at this time only processes (a) and (b)have been abundantly seen, me confine ourselvesto a discussion of what we may be able to learn