General R elativity and Gravitation, Vol. 9, No. 10 (1978), pp. 857-877

On a New Variational Principie in General Relativityand the Energy oCthe Gravitational Field

JERZY KIJOWSKI

lnstitute for Mathematical Methods in Physics, University ot Warsaw,ul. Hoza 74, 00-682 Warsaw,Poland

Received December 15, 1977

Abstract

A new variational principie based on the affine connection in space-time is proposed.This leads to a new formulation of general relativity. The gravitational field is a field of in­ertial frames in space-time. The metric g appears as a momentum canonically conjugate tothe gravitational field. In the case of simple matter fields, e.g., scalar fields, electromagneticfields, Proca fields, or hydrodynamical matter, the new formulation is equivalent to thetraditional one. A new formulation of conservation laws is proposed.

§(1): lntroduction

The present paper contains a new formulation of the general relativitytheory. It is based on a variational principle that differs considerably from theone proposed by Hilbert [5] and also from the so-called Hilbert-Palatini varia­tional principle. The fundamental field in our approach is the symmetric affine

connection r in space-time M (in terms of local coordinates we have r~v= r~J.JThis field plays the role of the gravitational potential. The physical interpreta­tion of such a potential follows from the discussion below.

Given an affine connection r in space-time M we define at each point x EMa class of local coordinate systems. This class is composed of those coordinate

systems for which the coefficients r~lJ at x vanish. If (x /.I) and (y lJ) are two co­ordinate systems belonging to this class, then

(1.1)

8570001-7701/78/1000-0857$05.00/0 @ 1978 Plenum Publishing Corporation

858 KIJOWSKI

Such a class of coordinate systems will be called a local inertial frame at thepoint x. Conversely, if a local inertial frame at each point x EM is given we may

define an affine connection r putting r~v = O in any coordinate system belong­ing to the local inertial frame. We see that the field r in M may be treated as afield oflocal inertial frames defined at each point x EM. Thus the gravitationalpotential in our approach is a field of local inertial frames in four-dimensionalspace- time. The gravitational field strength is a curvature tensor (defined by firstderivatives of the potential) which measures the nonf1atness of the field of in­

ertial frames. Of course the metricg/lv also appears in our theory. It is a mo­men tum canonically conjugate with respect to r. Given a field rtv we maycalculate g/lV by differentiation of rtv provided the properties of the matter(sources of gravity) are known (e.g., we know the Lagrangian ofthe matterfield). Distances in space-time are thus determined by the properties of thematter.

For empty space (no matter) and for several simple matter fields (e.g.,electrodynamical fields, Proca fields, hydrodynamical matter) our theory iscompletely equivalent to the standard formulation of general relativity proposedby Einstein [3] and Hilbert [5]. For more complicated matter fields our formu­lation differs from Hilbert's. We give stron g physical argument s for such a modi­fication and hope that this modification is in the spirit of early papers in generalrelativity.

The main result of this paper is the definition of the energy of the gravita­tional field (more generally, a formulation of conservation laws in general reIa­tivity). Because of various difficulties in defining the energy of the gravitationalfield (pseudotensors, super-Hamiltonians, etc.), many authors do not believethat this energy can be localized. Many artificial explanations of these difficul­ties have been proposed (see, e.g., [8], chap. 20). In our approach the energy ofthe gravitational field is no more mysterious than the energy of any other field(although it has its own particular properties). The second important result ofour theory is the appropriate formulation of the Cauchy problem (or boundary­value problem) and also a new definition of gravitational radiation. Solving theso-called "constraint equations" for boundary data or Cauchy data we foundthat the energy density is one among four independent data. This result will bepresented in a subsequent paper.

aur theory of gravity differs from the standard one, especially in the case ofhighly condensed matter. The Penrose theorem coneerning singularities does notapply in this ease. Whether our theory eontains a serious remedy against singu­larities is not, however, obvious at the moment.

The present paper may be eonsidered a simple applieation of two importantphysieal ideas. The first one is Tulezyjew's idea that all dynamie al variabIes inany field theory should be defined as generating funetions of the dynamies withrespeet to an appropriate eanonieal strueture. This idea unifies Hami1tonian,

NEW V ARIATIONAL PRINCIPLE 859

Lagrangian, and all other formulations of the field theory. This idea was firstpublished in [15] and is probably one of the most important physical ideas theauthor ever learned. Further elaboration of this idea may be found in [6]. Someresults of [6] that are necessary for our purposes are briefly sketched here (see

Section 5). The second fundamental idea, which says that r~v and not g/lV

should be the primary quantity in general relativity, was suggested in the beauti­ful and important paper of Szczyrba [13]. This idea has already been proposedin the context of unified theory (see, e.g., [2,4, 12]). None of these approaches,however, succeeded in proving that general relativity itself can be formulatedthis way.

§(2): The Variational Principle Based on Affine Connection

According to the philosophy given in the Introduction, we consider two in­

teracting fields: the gravitational field r~v and a matter field '-PA. The nature ofthe index A is not specified. In the present section we formulate the dynamicsfor such a theory in terms of a variational principIe. The Lagrangian is a scalardensity on M and depends as usual on both fields and their first derivatives:

where

r~VK = aKr~V

'-PAK = aK '-PA

The field equations of our theory are the Euler-Lagrange equations:

(2.1)

(2.2)

(2.3)

We introduce also the momenta rr-!VK and PAK canonically conjugate to thegravitational field and to the matter field, respectively. The corresponding dy­namical equations for the moment a are

(2.4)

(2.5)

860 KIJOWSKI

(2.6)

(2.7)

The momentum 1ft}K is symmetric in indices J.l.and ZJ because of the symmetryr~,.tVK = r~VfLK' Equations (2.3) can thus be rewritten in the following form:

o fLVK _ otK1f~ -or~fLV

K ot

°KPA = o~

It will be shown in §(4) that equations (2.5) and (2.7) are standard field equa­tions for the matter field. In order to be able to interpret the remaining equa­tions (2.4) and (2.6) as equations for the gravitational field we limit ourselves to

the case when t depends on derivatives rtvK only via the curvature tensor:

(2.8)

Otherwise, the theory could not be coordinate independent. The only algebraicidentities satisfied a priori by the curvature tensor are the antisymmetricity inindices "1 and o :

ROl(3'YÓ = -ROl(3ó'Y

and the so-called "second identity":

ROl(3'Yó + ROl'YÓ(3 + ROló(3'Y = O

(2.9)

(2.10)

(2.13)

A priori, the number of independent components of ROI (3'YÓ is thus 80. Theother identities will follow from field equations.

The above assumption about the Lagrangian reduces strong1y the number ofindependent components of the momentum 1f-/,VK because of the identity

1f/,VK = ~ aR!'Yó (2.11)aR (3'Yó ar /J-VK

For the sake of simplicity we assume that t has been extended to a function de­

pending on all possible components ROl(3'Yó, not necessarily satisfying the ident i­ties (2.9) and (2.1 O). The auxiliary quantity

(3'VÓ atZOl' =~ -41:::1.)(, (2.12)

(3'YÓ

is thus an arbitrary tensor density with 44 independent components. The aboveconvention is the unique exception from the general rule that we always dif­ferentiate with respect to independent components of tensors. Following thisrule we obtain from (2.8) the equation

aR Ol

~ = 001 [OfLOKOV - OfLOV OK + OVOKOfL - OVOfLOK]"r~ ~ (3 'Y Ó (3 'Y Ó (3 'Y Ó (3 'Y ÓU fLVK

NEW VARIATIONAL PRINCIPLE

and

1T!VK = -2(Z![VK] + Z;\,[,uK])

These square brackets denote antisymmetrization. The quantity

W!VK = -4Z![VK]

861

(2.14)

(2.15)

is an arbitrary tensor den sity which is antisymmetric in indices v and K. Thus

(2.16)

The immediate consequence of (2.15) and (2.16) is

(2.17)

(2.18)

(2.20)

The reader may easily check that the number of independent components of1T!VK is also 80.

Using our assumption about t we may calculate the right-hand side of (2.6)as follows:

at at aR Ol. a*t--=---~+-­ar,uil;, aR 0i.(J'Y o ar ,ul\.v ar ,ul\.v

where the last term denotes the derivative of t with respect to those gammasthat are not contained in the curvature tensor. We have also the identity

+ ó~ó~rffo + r~'Yó~ó~ - ó~ó~q'Y - r~o ó~ó~ (2.19)

Applying (2.19) to (2.18) and using (2.14) we obtain

atl\. = a* ~ + 2(zf[vo] r rft + zf[,uo ] r;oar,uv ar,uv

+z,u[Ov]rOl. +zv[o,u]rOl.)Ol. 1\.0 Ol. A.6

= a*~ _ (zf[ov] + Zl\.0 [(J'Y]) rrftar ,uv

- (zf[o,u] + Zl\.° [(J,u]) r;o - 2(zcf[ vo] + z:l,uo]) rl\.~

a*t 1 1= -- + -1T (JOVr.H + -1T (Jo,ur v + 1T ,uvor Ol.ar l\. 2 x po 2 X (Ja Oi. 1\.0,uv

(2.21)

862 KIJOWSKI

This result enables us to rewrite equation (2.6) in the covariant form:

- a*ta 1TJ.J.VK+1T{3v8rJl: +1T{3J.l8rv -1TJ.lv8r'" =--

K A A!J8 A {38 '" A8 ar AJ.lV

or

a*tJ.lVK =-A

'VK1TA a rlJ, V

where 'Vdenotes the covariant derivative.

§(3): Gravitationallnteraction

(2.22)

(2.23)

The framework developed in Section 2 is much too large for the purposes ofthe theory of gravitation. We do not know at the moment whether the interac­tion of the matter field with the full curvature tensor corresponds to any inter­action really existing in nature . However, to describe the gravitational field weneed only a smalI part of this interaction corresponding to the symmetric partof the Ricci tensor:

(3.1)

where

(3.2)

Identities (2.9) and (2.1 O) do not imply any a priori symmetry of the Riccitensor. We thus restrict ourselves to the case when t does not depend on the full

curvature tensor but only on KJ.lv, In this case we obtain further reduction ofindependent components of 1T!VK:

l at aK{381T!VK = _

2 aK{38 arAJ.lVK

(We differentiate over ten independent components of K~v')The equation

implies

aK{38 _~K~J.l~V ~V~J.l~K ~K~V~J.l ~J.l~V~K--A- - UA U({3u8) - UAU({3u8) + UA U({3u8) - UA U({3u8)ar J.lVK

We introduce the following notation:

1T{38 = ~aK{38

(3.3)

(3.4)

(3.5)

(3.6)

NEW VARIATIONAL PRINCIPLE

Applying (3.5) to (3.3) we obtain

ILVK _ "K /-LV d/-L V)Kn" -u"n - u" n

Now equation (2.6), or (2.23), reads

'IJ n/-LV - o (/-L 'IJ nV)K = ~" "K ar"

/-LV

This implies

3--2 'lJKn/-LK = o" a*lV -ar"

/-LV

and

863

(3.7)

(3.8)

(3.9)

(3.1 O)

AU the information about the momentum nfVK is contained in the sym­metric tensor density 1T/-Lv. We shall show in Section 4 that our theory providesthe correct description of gravity if we interpret the den sity n/-LV as the contra­

variant density of the pseudo-Riemannian metric tensor g/-LV (up to a multiplica­tive constant). We define the metric by the formula

nlLV = - (lIk) (_g)1/2 g/-LV (3.11 )

where k is the gravitational constant and g = det(g/-Lv). We adopt the convention(+, - , - , - ) for the sign of the metric. In the simplest case when l depends on

gammas only via K/-Lv we obtain

(3.12)

and equation (3.10) reads

(3.13)

This is equivalent to the equation

(3.14)

and further to the equation

r/-L"V == {,.~v} (3.15)

(we remember that rtv is always symmetric !), where {tv} are the Christoffelsymbols built up of the metric g/-LV:

{/-L"v} = lg"K(avgK/-L + a~KV - aKg/-Lv) (3.16)

864 KIJOWSKI

In order to show that our variational principie provides the eomplete theory ofgravity, we must prove that equations (2.4) or equivalently (3.6) are Einsteinequations. This will be done in Seetion 4.

§(4): Partial Legendre Transfonnation and the Matter Lagrangian

In the present seetion we keep all the assumptions about the Lagrangian .fthat we made earlier. Thus

(A A A).f=.f r}.Lv,Kllv,'-P ,'-P K

The eomplete differential of.f is equal to

d f' - l l }.LVd A l }.LvdK l d A K d A,L-z A r}.Lv+z1T }.LV+ A '-P +PA '-P K

where we denote

and

(4.1)

(4.2)

(4.3)

(4.5)

(4.4)lA = aKPAK

Equation (4.2) is equivalent to the system of equations

}.LV a.flA =--

ar;v

1T}.Lv=~

aKllv

l _ atA - a'-PA

K at

PA = a-Y<{! K

The faetor t appears in (4.2) sinee we always take into aeeount the symmetryof KJlv and differentiate over independent eomponents. The strueture of thesystem (4.5) is similar to the strueture of the system

. aL

P = aq

aLP = ---;-

aq

(4.6)

in elassical mechanies. The equations of many other field theories mayaiso bewritten in a simi1ar way. We give in rabIe I the list of objeets that are analogousin different dynamie al theories. For a deeper diseussion of this analogy see [6].

NEW VARIATIONAL PRINCIPLE

Table I. Objects analogous in different dynamical theories

865

Mecharucs

Position q

Velocity q

MomentumaL

p =--:­aq

Force p

dL = [idq + pdq

Field <pA Theory

Field <pA

Partial deriva tives

,~AK = aK<pA

MomentumK at

PA =-:;-:Aa<p K

Correspondingpart of for­mula (4.2)

Electrodynamics

Electrodynamical poten­tial A f-L (a connectionin a principal fibrebundle over space­time)

Electrodynamical fieldf= curvA (the curva­ture of the connec­tionA)

Electrodynamicalinduction

f-LV ath =~aff-Lv

Electric current

/ = af-Lhf-Lv

General Relativity

Gravita tional poten tial

r~v (a connection inthe tangent bundleover space-time)

Gravitational fieldthat one could ex­pect to be the wholecurvature of r. Atthe moment we un­derstand only itspart, namely, K f-LV'

Momentum

f-LVK _ at1Tr... -~

ar f-LVKor

~V=~_

aKf-LvThis is the metric inspace-time.

The current

Jfv = VK1Tr...f-LVK

the interpretation ofwhich will be givenlater.

Corresponding part offormula (4.2)

Equations (4.5) enable us to calculate "moment a" and "forces" in terms of"positions" and "velocities." Now we would like to make a Legendre transfor­mation and replace some velocities by some momenta. This procedure is analo­gous to passing from equations (4.6) to the equations

. 'OHq=-'Op

in classical mechanies. We apply our Legendre transformation only to gravita­

tional degrees of freedom, Le., we exchange 1ff-LV with Kf-Lv. In order to do thatwe define the function

'OH

P = - 'Oq(4.7)

(4.8)

(4.9)

(4.11)

866 KIJOWSKI

where K/.Lv is now treated as a function of variabies (r;v, 11/.Lv, tpA , tpAK). Thisfunction is calculated by solving (4.5) with respect to K/.Lv. Thus

dL =dt- !d(1T/.LvK/.Lv)

- 1 J/.LVdr A 1 K d /.LV d A Kd A-z A /.LV-Z /.LV 1T +JA tp +PA tpK

The above formula is equivalent to the following system of equations:

J:V=~ arA/.LV

(4.10)aL

JA = a",A

aLK __

PA - a",AK

The function L is called the matter Lagrangian. It is a function of matter field

",A, its derivatives, and the geometry of space-time (the affine connection rtvand the metric g/.Lv represented by its covariant den sity 1T/.LV).The third and thefourth equations of the system (4.10) are precisely the Euler-Lagrange equa­tions for the field tpA with the Lagrangian L. We show that the second equationis precisely the Einstein equation. The formula (3.11) implies

a k (a 1 a a )- a1T/.LV = (_ g)1/2 ag/.Lv - "2 g/.Lvg (3 aga(3

Hence

and

_ /.LV = /.LV k__ a(3 ~

R - g R/.Lv g K/.LV - (_ g)1/2 g aga(3

Substituting (4.13) for the corresponding term in (4.12) we obtain

1 1 aL

K/.Lv - "2 Rg/.LV = k (_ g)1/2 ag/.Lv

The symmetric tensor

G/.Lv=K/.Lv- !RgJ.lv=R(/.Lv)- !Rg/.LV

(4.12)

(4.13)

(4.14)

(4.15)

NEW V ARIATIONAL PRINCIPLE

is the Einstein tensor. The tensor

867

(4.16)

is usually called "the symmetric energy-momentum tensor of the matter field."We prefer to call this quantity the stress tensor (this nomenclature comes from[6] and will be explained in the Section 5). Thus

G/l.V = kT/l.v (4.17)

In the case when oC (or L) does not depend explicitly on r/l.\' the first equationofthe system (4.10) is equivalent to (3.15). In this case our theory is completelyequivalent to the Einstein theory. This is the case of the scalar field theory(Klein-Gordon-Einstein equations), electrodynamics (Maxwell-Einstein equa­tions), Proca-Einstein field, and the theory of hydrodynamical matter (includingdust). For instance, the scalar field corresponds to the Lagrangian

(4.18)

(4.19)

The reader may easily check that equations (4.5) are Klein-Gordon-Einsteinequations. The corresponding matter Lagrangian is as usual

L = l (-g)I/2 (g/l.vr.p/l. r.pv - m2tp2)

= - (k/2) {n/l.V r.p/l. r.pv + k [- det (n/l.V)p/2 m2 r.p2}

since

(4.20)

We could start the description of the dynamics of our theory by means of the

function L instead of t.Equations (4.9) ar (4.10) give Jfv ,K/l.v,JA , and p 1in terms of r~v,n/l.v, ~, and r.pAI{ and are equivalent to (4.2) or (4.5). In orderto pass from one description to another we make the Legendre transformation:

(4.21)

where the variable n/l.V has been eliminated with the help of equations (4.10).From our point of view L is not a Lagrangian but a mixed quantity that playsthe role of the Lagrangian for the matter field (it depends on "pasitions" r.pA

and "velacity" r.pAI{) and the role af the Hamiltonian for the gravitational field

(it depends on "position" rtv and "momentum" n/l.V). The true Lagrangian isthe function t.Our interpretation of the formula (4.21) differs from the stan­dard one. The right-hand side of(4.21) is not a sum ofthe "matter Lagrangian"plus the "gravitational Lagrangian" but simply the Legendre transformationfrom one to another mathematical descriptian af the same physical objeet,

which consists of interacting matter and gravitation. The term l n/l.V K/l.v on the

868 KIJOWSKI

right-hand side of (4.21) is analogous to the term pq in the formula

L =pq- H(p,q) (4.22)

which gives the Lagrangian if you eliminate p from the right-hand side. We see

that the so-called Hilbert-Palatini Lagrangian, which depends on r!;V, K/lv, rr/lV,

<.pA, and <.pAK'is similar to the function L (q, q, p) given by above formula if youforget to eliminate the momentum.

Generally, however, Jtv does not vanish and equation (3.15) has additionalterms on the right-hand side. The Einstein equation (4.17) always holds, but thefulI theory is not equivalent to Einstein's theory because of the nonmetricity ofthe connection r.The structure of space-time differs from the structure of thefiat space in two different aspects: the curvature and the nonmetricity. Thesource ofthe curvature is the stress tensor (4.16). The source ofthe nonmetric·

ity is the tensor aL/ar /ll>-v.We give in Section 5 the physical interpretation ofboth sources.

§(S): Energy-Momentum Tensors and Stress Tensors

In the present section we give a brief survey of some results presented in[6]. These results concern the energy-momentum tensors and stress tensors inthe general field theory. AlI these results are based on a new definition of energyin the field theory. Energy is the so-called generating function of the dynamics.We consider here a matter field <.pA in a fixed given space-time geometry. The

Lagrangian is the function depending on <.pA and <.pAK(Le., r!;v and g/lv arefixed). We thus have

dL =p1d<.p~ + JA d <.pA

JA = aKP1

(5.1)

(5.2)

The energy of the matter field is assigned to a three-dimensional surface elementdal>- and to a vector field X/l which we interpret as an infinitesimal transforma­tion of space-time. We denote this energy by Emat(X, da). It is defined by apartial Legendre transformation when we replace velocities in the direction ofX/l (Le., the Ue derivative t<.pA) with the projection p Idol>- of the momentumxpI onto the surface da>.. (for further details see [6]). The result of this Legendretransformation is linear with respect to dal>-:

Emat(X,da)=E~at(X)dal>- (5.3)

The proportionality coefficient E~at(X) is a vector density which we call theenergy density of the matter field corresponding to the infinitesimal transforma­tion X. The quantity E~at(X) is defined uniquely. No "complete divergence"

NEW VARIATIONAL PRINCIPLE 869

(5.4)

may be added or subtracted. The final formula proved in [6] is

E~at(X) =p}(f,<pA)- XALx

The above quantity depends linearlyon X and its derivatives up to some finiteorder. This order depends on the nature of the field <pA . In the simplest case ofthe scalar field we have

f, <p= XIl all<p = Xlltpll (5.5)xLe., (5.4) does not contain any derivative af X. If tpA is a tensor field then theLie derivative f, <pA depends on XIl and its first derivatives. If the space-time isxequipped with a connection rtv, the quantity (5.4) may thus be spanned as alinear combination of XIl and VvX/l:

E~at(X) = t~XIl + tAl}.LVvXIl (5.6)

The coefficients tAIl and tA~ are called, respectively, the first and the secondenergy-momentum tensor. This defmition applies to aU field theories known tothe author. (In electrodynamics, for example, there is no problem with gauge

invariance. The quantity tAIl is in this case what is usually called "the symmetricenergy-momentum tensor" and the quantity tAVIl vanishes.) The main result ofthis approach is that there are two energy-momentum tensors. An attempt tohave the energy proportional to XIl only is not justified and fails in a generalsituation. The quantity E~at(X) is a conserved current if and ouly if the fieldX is the so-caUed symmetry field of the theory. For the general field X we de­fine a scalar density

(5.7)

which we cali virtual action. This quantity measures the nonconservation of the ( 'j

energy, Le., how much action is spent in the dragging of the field <pA along X. ­For symmetry fields we have W(X) = O. The virtual action depends linearlyonXIl and its derivatives up to a finite order. If <pA is a tensor field then W(X) de-pends on first and secand derivatives ouly:

W(X) = TIlXIl + TVIlVvXIl + TA~ V(A Vv)XIl (5.8)

We took into account only the symmetric part V(A Vv)XIl of the second deriva­tive, which is independent of X and its first derivatives. The antisymmetric partis not an independent quantity:

V[A Vv]XIl = lXuRlluAV (5.9)

The coefficient TIl in the expansion (5.8) is called the force. The coefficientsT~ and TA~ are called, respectively, the first and the second stress tensor. Thisterminology is taken from elastastatics, where the virtual work is a sum of

870 KIJOWSKI

terms proportional to the shift vector (analogous to XM) and to the strain tensor(analogous to 'VvXM), respectiv~ly. Corresponding proportionality coefficientsare force and the stress tensor. Comparing (5.6) with (5.8) and using the defini­tion (5.7) of the virtual action, we obtain the following relation between energy­momentum tensors and stress tensors:

and

TV = tV +" t"AVM M V"A M

T"Av = 2~("Av)M l M

(5.10)

(5.11)

(5.12)

Now we limit ourselves to the case of Lagrangians that are invariant with respectto general coordinate transformations. The general Hamilton-Jacobi theoremproved in [6] enables us to calculate stress tensors by differentiation of theLagrangian with respect to the geometry. The following formulas are proved:

(5.13)

where

(5.14)

(5.15)

and

T"Av =_~M arfv

Stress tensors describe the rigidity of the matter field, i.e., the answer of thematter for space-time deformation.

§(6): Stress Tensors as Sources ofthe Gravitational Field. A Comparisonof Hilbert's and Dur Description of Gravity

In the case when L does not depend on r but only on the metric g, the sec­

ond stress tensar T"A~ vanishes. This happens when the second energy­momentum tensor is antisymmetric: t"A~ = -tV~. The source of gravity is inthis case the first stress tensar alone. Both aur approach and Hilbert's lead tothe equations

(6.1)

(6.2)

NEW V ARIATIONAL PRINCIPLE

equivalent to the equations

871

A_{A' (rlL" - IL"I 6.3)

GIL" = kTIL" (6.4)

However, in a general case there are two stress tensors TIL" and TA"w It is reason­able to expect that both are sources of gravity. It follows from (4.10) and (5.15)that our approach to the theory of gravity consists in keeping the Einstein

equation (6.2) and taking the second stress tensor TA~ as the source of non­metricity of the connection:

(6.5)

(6.6)

or

" IL" - 2 ,,(ILT")K - TIL"VA1T -3"uA K A

by virtue of (3 .10). Hilbert's approach consists in keeping the equation (6.3) andputting on the right-hand side of the Einstein equation (6.4) the quantity TIfr",which we calI the Hilbert tensor. In order to define this tensor we treat L as afunction of the metric and its first derivatives contained in the definition of

rILA" = {IL;;'}' The Hilbert tensor is defined as the so-called Lagrange derivative:

(6.7)

It was proved in [6] that

TIfr" = TIL" - 'VA(T\IL") - t TIL"A) (6.8)

This formula is equivalent to the so-called Belinfante-Rosenfeld theorem (cf.[1,6,11]). From the point ofview ofthe matter field 'PA the Hilbert tensorhas no direct physical interpretation, in contrast to the stress tensors defined bythe expansion (5.8). The elear interpretation of the sources of gravity is the [irstimportant advantage of our theory.

We draw the attention of the reader to the fact that TIL" is the stress tensor,which in general does not reduce to the energy-momentum tensor tILV' It fol­lows from (5.10) that tAIL= T~ happens only for tAVIL= O, Le., in the case ofscalar field or electromagnetic field (and of course in the case of no matter whenboth stress tensors and energy-momentum tensors vanish). Apart from thesecases, sources of gravity correspond to nonconservation of the energy and mo­men tum rather than to the energy momentum itself.

§(7): Energy Conservation Laws

We apply the general definition of the energy density to the full system con­sisting of both the matter field and the gravitational field. The formula (5.4)

872

now reads

KIJOWSKI

(7.1)EA(X) = p}(ftpA) + !1TrA( fr;v) - XAf,X X

The factor l appears since we always take into account the symmetry of gam­mas. The Lie derivative of the affine connection contains derivatives of the field

X up to the second order:

No expansion similar to (5.6) can thus be expected. However, we may expand

EA(X) with respect to three independent quantitiesXK, 'VvXK, and 'V(IJ.'VV)XK,

which gives rise to three energy-momentum tensors. To calcu1ate them we usethe formulas (4.21), (5.4), (5.6), and (7.2). The result is

EA(X) = Xa {tAa + [(- g)/2kP/2 R(j~ - l 1TjLVAR ~va}

+ t AVIJ.'V vXIJ. + -! 1TfvA'V (IJ.'V v)XK (7.3)

The above formula gives us the interpretation of the quantity 1TfVA as the thirdenergy-momentum tensor of the gravitational field (or of the full system). Thesecond energy-momentum tensor of the full system equals that of the matterfield. The first energy-momentum tensor of the fuli system is the sum of tAa

(matter energy-momentum tensor) plus the term

7Aa = [(-g)1/2/2k]R(j~- l1TfvARKlJ.va (7.4)

which one could calI the first energy-momentum tensor of the gravitational field.Many attempts have been made in general relativity to express the energy

den sity in the form proportional to the field Xa alone and not to its derivatives(see [9,14]). Formula (7.3) shows that this program cannot be carried out, atleast if we want to deal with coordinate-invariant objects. One can, however,fulfill this program in a coordinate-dependent way if we integrate by parts aUterms in (7.3) that contain partial derivatives of X. For example,

tAVIJ.'VvXIJ. = tA~(avxlJ. + rtr,Xa) = Xa(r tr,tA~ - avtAVa) + av(tAVIJ.XIJ.) (7.5)

Similarly,

I1T IJ.VAD D XK = I1T IJ.VAD D XK = l a (1T IJ.VAD XK)2 K V (IJ. V v) 2 K V IJ.v v 2 IJ. K V V

+ l a [XK(1T IJ.vpr A - D 1T IJ.VA)]2 v K PIJ. v IJ. K

- IXKa (1T IJ.vPr A _ D 1T IJ.VA)2 v K PIJ. v IJ. K

+ Ixar K (1T IJ.vPr A - D 1TIJ.VA) (76)2 av K PIJ. v IJ. K •

We combine the above two formulas and obtain

EA(X)=Xa(tAa +j"Aa)+ alJ.AIJ.A (7.7)

873NEW VARIATIONAL PRINCIPLE

where the quantities 5'\ and AJlA are defined as follows:

<lA = rA + r Jl tAV _ a tAV +.l r K (1T JlvPr A - \7 1T IWA).J a a va Jl v a 2 av K PJl v Jl K

- t av(1TaJlVprA - VJl1TaJlVA) (7.8)

and

The energy density EA(X) can thus be represented as Xa (t\ + 5'\) plus a fulidivergence. The quantity 5'Aa is a "pseudotensor." This procedure, however, iscbmpletely arbitrary andartificial and has no physical meaning.

Our theory is invariant under the group of all differomorphisms of space­time M. This means that any vector field X in M genera te s a one-parameter sym­metry group of the theory. Following the general Hamilton-Jacobi thearemgiven in [6] we conclude that the energy density (7.3) is a conserved quantity:

(7.1 O)

for any vector field X since any vector field X is a symmetry field of the theory.(The reader should distinguish between the symmetry fields of the theory andsymmetry fields of a particular geometry in M; the latter may be very few areven absent.) The conservation law (7.10) mayaiso be proved directly from thefollowing formula:

(7.11)

where the antisymmetric tensar den sity HAJl(X) equals

HAJl(X) = [(_g)1/2(k] VI A X/.L] + Xa tl A/.L] a + t (XJlt(Aa) a - XAt(Jla) a)

(7.12)

and

lndeed, formula (7.11) implies

aAEA(X) = aA aJlHAJl (X) = O

(7.13)

(7.14)

The proof of energy form ulas (7.11) and (7.12) is given in Section 9.The physical interpretation of our energy den sity EA(X) is given infollow­

ing sections. Let us notice that in the case of empty space-time (no matter)E\X) reduces to the quantity proposed already by Komar in [7] (see also[14]):

(7.15)

874 KIJOWSKI

§(8): Energy in Large-Scale Problems. Mach PrincipIe

(8.3)

We give the interpretation of EA.(X) as the fulI energy density of a system.This energy is earried by both the matter field and the gravitational field. Oneeould be afraid that our definition of energy leads to embarras de richesse sineewe have as many different energie s EA.(X) as different veetor fields X in M. Wewill show below how to interpret this situation.

Suppose we have a family X of observers (world lines) whieh move (freely ornot) in spaee-time M. As a veetor field X we take the four-velocity field definedby this family of world-lines. The quantity EA.(X) is ealIed energy density takenwith respeet to the fami1y of observers X (eL [10]).

Let L be a three-dimensional hypersufraee in M and let l:) C L be a boundeddomain in L. The integral

r E\X) d0A. = 1ap,HA.P,(X)doA. =1HA.p,dhp, (8.1)Ja c) afJ

is the amount of energy (taken with respeet to X) eontained in tl. Here atl de­

notes the two-dimensional boundary of tl and dfA.p,is the two-dimensionalsurfaee element on al:). The formula(8.l) shows that the energy does not dependon l:) and X but depends on1y on al:) and the value of X over al:). Henee we donot need the three-dimensional family X of observers but only the two-dimen­sional family X eomposed of those that pass through the boundary of l:) . Thisfamily spans a three-dimensional tube in M. The surfaee al:) is an interseetion ofthis tube with the hypersurfaee L (see Figure 1). We denote the quantity (8.1)by E(X, atl) and ealI it the energy eontained inside the tube, taken with respeetto the fami1y X of external observers and ealculated on the seetion al:) of thetube. ut al:)' be another seetion of the tube eorresponding to a three­dimensional surfaee l:)'. We denote by S the three-dimensional surfaee of thetube eontained between al:) and al:)' and by V the four-dimensional volumeinside the tube and eontained between tl and l:)'. The eonservation law (7.14)implies

0= 1aA.EA.(x) dV = J EA.(X) d0A.= E(X, al:)') - E(X, al:))v av

+ rad (X, S) (8.2)

where

rad (X, S) =1EA.(X) doA.S

is the amount of energy radiated from the tube through S. In the simplest easeof no radiation through Swe have the exaet eonservation of the energy

E(X, al:)') =E(X, al:)) (8.4)

NEW V ARIATIONAL PRINCIPLE

Fig. l.

875

In this case the quantity EGO independent of a section may be defined. It isthe energy of matter and gravitation contained in the tube.

The above definition of energy may be considered to be a formulation ofthe Mach principIe. The energy is always defined with respect to a family X ofdistant observers (dis tan t stars). The energy is not created. Any increase ordecrease of the energy is always due to incoming or outgoing radiation.

One of the important products of our theory is the definition (8.3) of radia­tion. It turns out that this defmition is extremely useful in solving the so-called"constraint equation" for Cauchy data or boundary data in general relativity.The discussion of this problem will be given in a subsequent paper.

The energy may be defined with respect to any crazy family of observers.Usually, however, the physical situation distinguishes a smalI class of suchfamilies. This enables us to reduce the number of conserved quantities to thosethat can be really measured. In the subsequent paper we analyze situationswhere ten independent conserved quantities (four momenta and six angularmomenta) are distinguished. We will also show there the connection betweenthe energy and the inertial mass.

§(9): Proof ofthe Energy Fonnula

EA(X) = p1c;<pA) + t n!')'Ac;r{J~)- XA.f

= PAA(±:<pA) - XAL + XA(1/2k)(-g)1/2 Rx

+ t (xuRO!.{Ju')' + 'V')' 'V{JXO!.)(8~ n')'{3 - 8(~n')')A)

=E~at(X) + X\1/2k)y=g R + ! [XU(RA{3u')'n')'{3

- iRO!. n')'A_ 1RO!. n!3A) + n')'{3\7 \7 XA2 O!.u"! 2 {J U O!. v {3v ')'

- n')'A'V (O!. 'V')')XO!.] (9.1)

We use the formula

'V(O!.'V')')XO!. = 'VO!.'V')'XO!. - 'V[O!.'V')'] X O!.= 'VO!.'V')'XO!. - !XURO!.uO!.')' (9.2)

876 KIJOWSKI

and integrate by parts the last two terms of (9 .1). Then we apply the formula

1 RC< - R"2 c<u"{ - [u"{] (9.3)

(9.5)

The result is

EA.(X) = E~at(X) + XA.(1/2k) (_g) 1/2 R + l XU(RA.(3u"{1T(3"{ + R"{u1T"{A.)

+ \1"{ {[(_g) 1/2 /k] \1[A.X"{]} + l (\1"{1T(3A. - 5~\1c<1T(3C<)'V(3X"{ (9.4)

Now we use equations (5.6), (5.10), and the Einstein equations. We obtain

E~at(X) = xuru - XU\1 ).It!lA.u + tA.(3"{\1(3X'Y = - XU R("{u)1T"{A.

- XA.(I/2k)( _g)1/2 R + \1iXu t[ A."{] u) + t(A.(3)"{ \1(3X'Y - XU\1 ).It().IA.)u

The nonmetricity equation implies

t( A.(3) = 1 TA.(3 = _1 \1 1T A.(36 = _1 (\1 1TA.(3- 5 (A. \1 1T(3)c<)"{ 2 "{ 2 6 "{ 2 "{ "( C<

and

The above formulas imply

EA.(X) = \1).1{[(_g)1/2/k] \1[A.X).I] +Xu(t[A.).I]u + l 5~).I\1c<~]C<)}

+ XU( R "{A. 1 R "{A. + 1 RA. "((3 + "\l "\l A.(3)- ("{u)1T +"2 "{u1T "2 (3u"{1T V[(3vu]1T

But

(-R + 1 R ) "{A. + 1 RA. (3"{ - R "{A. 1 R "{A.("{u) "2 "{u 1T "2 (3a"{1T - ["(a] 1T -"2 "{a1T

+ 1 RA. 1T(3"{= - [(_g)1/2/k] (R g'YA. + R(A."{) )2 (3a"{ ba] a"{

Using an inertial system (r).l~= O) one can prove the formula

\1 [(3 \1a] 1TA.(3= [(_g)1/2 /k] (Rbu] g"{A. + R(A."{)u"{)

(9.6)

(9.7)

(9.8)

(9.9)

(9.10)

The last two formulas show that the second term in (9.8) vanishes. This com­pletes the proof.

Acknowledgments

The author is much indebted to Professor P. Mittelstaedt for his kind hospi­tality at the Institute of Theoretical Physics, University of KaIn, where thepresent paper was conceived. Special thanks are due to Professor F. Hehl formany discussions which, although not always very peaceful, were very stimulat­ing for the author; a1so, to Professor J. Ehlers for his important and valuableremarks about the energy problem.

1. Belinfante, F. J. (1940). Physica, 7,449.2. Eddington, A. S.(1963). The Mathematical Theory of R elativity (University Press,

Cambridge).3. Einstein, A. (1915). Preuss. Akad. Wiss. Berlin, Sitzber., 884.4. Einstein, A. (1925). Preuss. Akad. Wiss. Berlin, Sitzber., 414.5. Hilbert, D. (1915). Konigl. Gesel. Wiss. Gottingen, Nachr. Math.-Phys. Kl., 395.6. Kijowski, J., and Tu1czyjew, W. M., "A Symp1ectic Framework for Field Theories," to

appear in Springer Lecture Notes in Physics.7. Komar, A. (1959).Phys. Rev., 113,934.8. Misner, C. W., Thorne, K. S., and Whee1er, J. A. (1973). Gravitation (Freeman, San

Francisco).9. M~ller, C. (1958). Ann. Phys. (N.Y.),4, 347.

10. Pirani, F. A. E. (1962). Gauss's theorem and gravitational energy, in Les TheoriesRelativistes de la Gravitation-Royaumont Conference 1959 (CNRS, Paris).

11. Rosenfeld, L. (1940). A cad. Roy. Belg., 18, 1.12. Schr6dinger, E. (1954). Space-Time Structure (University Press, Cambridge).13. Szczyrba, W. (1976). Commun. Math. Phys., 51, 163.14. Trautman, A. (1962). "Conservation Laws in Genera1 Re1ativity," in Gravitation­

An lntroduction to Current Research, ed. Witten, L. (Wiley, New York).15. Tulczyjew, W. M. (1974). Symposia Math., 14, 247.

NEW VARIATIONAL PRINCIPLE

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