Abstract. An expression for quasi -local energy is sugges ted.The calculation looks locally at an Alexandrov
neighborhood whose boundary is defined by the intersection of the future and past cones from nearby time
like separated points. The calculation uses the canonical formalism on a null cone and defines the quasi-local
ene rgy as a two-surface integral overthe convergence ofthe past cone minus a similar integral over a surface
in Minkowski space. This definition is suggested by the results of an analysis at null infinity.
1. INTRODUCTION
A local energy density cannot be defined in general relativity. However, there havebeen a number of attempts to define a quasi-local energy. That is, one does not ask foran energy density, but rather one asks for the energy content within a volume of spacedefined by a closed spatial two-surface . This notion was introduced by Roger Penrose in connection with his twistor program (1982). In Minkowski space, the energyand momentum are contained in the 10 component angular momentum twistor (Penrose and Rind1er 1986). Penrose's idea is to construct a corresponding quantity forthe gravitational field using solutions of the twistor equation V (AwB) = O. For mostcases, this does lead to a satisfactory result for energy-momentum, but not for angularmomentum. However, the twistor equation does not have an easily visualizab1emeaning and therefore there have been a number ofother proposals (Brown and York 1993;Dougan and Mason 1991; Hayward 1993). Perhaps the formulation closest to that ofPenrose is that of Dougan and Mason (1991). But there the spinors which are definedto be holomorphic are equally elusive in their interpretation.
Here I wish to propose a definition which clearly depends on the light bendingproperties of mass. It is a definition which is derived directly from the Einsteinequations and mirrors the calculations at null infinity which leads to the TrautmanBondi-Sachs (TBS) energy. The particular path taken follows the reexamination ofthe gravitational radiation field at null infinity (Goldberg and Soteriou 1995). In thereexamination, the analysis was done with the null surface canonical formalism usingan anti-self-dual connection and densitized triad as phase space variables (Goldberg ,Robinson, and Soteriou 1992) as in the Ashtekar formalism on a space-like surface(Ashtekar 1991). In this approach one follows the Bondi-Sachs (Bondi , Van Der Berg,and Metzner 1962; Sachs 1962) and Newman-Penrose (1962) calculations . Thus, weimpose data on an outgoing null surface, integrate the hypersurface equations, and
then examine the time development off of future null infinity. One of the time development equations describes the rate of change of mass (energy) and leads to adefinition of the global energy. It turns out that the mass is defined by the 1/r 2 part ofthe convergence f.L of the generators of I +.
The idea , then , is to redo this calculation in a small finite domain bounded byoutgoing and incoming null cones - an Alexandrov neighborhood. Again one imaginesgiving data on the outgoing null cone , carrying out the integrations of the hypersurfaceequations, and examining the time development of the convergence of the null rays onthe incoming surface. A problem arises here which does not appear at null infinity. Atnull infinity, the Minkowski value of f.L which is r-v l / r drops out of the calculationand therefore out of the definition of the mass. It is not clear that the same occurs ona finite null boundary.
In the following section, the general formalism is set up with a discussion of theresults at null infinity. The Alexandrov neighborhood will be set up and the conservation equations will be discussed in section 3. The problem with an appropriate zero forthe quasi-local energy will also be discussed in section 3. Finally, in section 4 there isa brief discussion of the results.
2. FORMALISM
In the analysis at null infinity we introduced a null tetrad based on a foliation of nullsurfaces t = constant:
eO = (N + av\Ni)dt + av\dxi
ea = v\(N idt + dx i)
eo = N - 1(Dt - N iDi)
ea = (vai + aaN-1 N i)Di - aaN-1Dt ,
where aa = aDla and Vai Vb i = J ab . (All indices have the range 1-3 and repeatedindices sum . Bold face indices refer to the one forms and tetrads. The signature is - 2and - iN v is positive, where v is the determinent ofV Oli' ) The connection coefficientsare defined by
deOi = e{3 1\ wOi {3
The self-dual components of the connection are
1 .+wOi {3 = _( wOi {3 _ !:-rp {3 wlW)
2 2" IW
and are represented by rA , (A ,B , .. . = 1 - 3):
1r 1 := 2( w0 1
+ w23
) , r2 := w2 1
, r3 := w 0
3.
From these we obtain the self-dual components of the Riemann tensor as
~RA = drA + 7)A BC r B 1\ r C .2
(2)
(3)
(4)
Q UASI-LOCAL ENERGY
In a 3 + 1 decomposition , we have
and
RAij = 2A
A[i ,j ] + 27)ABCAB UAC i]'
RAOi = D iBA - A A i O',
The derivative operator D acts on the index A as
D d A := adA + 27)A BcABdc .
The SO(3) invariant metric
377
(5)
(6)
and its inverse,
°°- 1
°°- 1
are used to raise and lower the uppercase Latin self-dual, triad indices.When written in terms of the self-dual Riemann tensor, the Lagrang ian leads to
(AA i , ~A i ) as canonical variables where
~l i = - VV li
, ~2 i = -a:vi, ~3i = - V V3
i, V i = V V 2
i. (7)
In addition, there are a numb er of functions which act as Lagrange multilpiers : B A ,
v i , a: and /l i. From the Lagrangian , we get the following constraints:
H o. (1 . 2 .) 0,.- vt R ij ~3J + R ij ~ l J =
H iA .
0,.- - R ij ~AJ =g A Di ~A
i = 0, (8a)
cPi1 . 2 .
°.- R i j ~3J + R ij ~ l J
Ci .- ~2 i + a:Vi = 0.
(note that cPivi = H o and cPi ~3 i = Hi~ l i ) and conditions on the Lagrange multipliers:
0,
(8b)
(8c)
Q A B := 8:81 + 8N1.The last equation, Xi, results from the propagation of the constraint Ci = °whichdefines the propagation of~2i = 0.
378 JOSHUA N. GOLDBERG
In addition, the Hamiltonian equations of motion for the dynamical variables are1 l ' 1 . 2A i, O = DiB +NJ R ij - l;!v JR ij ,
2 2 ' 2A i ,O = DiB + NJ R ij - Ili '
A3· 0 = D ·B 3 + Nj R 3 - Nvl R l .t , t t J _ tJ'
In (Goldberg and Soteriou 1995) we assumed the space-time to be asymptoticallyMinkowskian. But, to solve the equations, we must impose coordinate and gaugeconditions. The surfaces t = constant form a foliation of null surfaces near nullinfinity and give us one coordinate. Space-like cuts of these surfaces are assumed tohave the topology 8 2 . Hence, the generators can be labeled by the angular coordinates(e, ¢). We follow the convention of Bondi and Sachs (Bondi, Van Der Berg, andMetzner 1962; Sachs 1962) in choosing the coordinate r along the null generators tobe the luminosity distance, so that
. 2 . e _.1" k 2 (10)-zr sm TJ1 jk tr LJ3 = V ,
and we can setL;1 i = - ir2 sin e8i
1. (I I)
Then the null rotations generated by 9 1 and 9 2 allow us to fix L;3i tangent to the surfaces r = constant, that is, L;31 = 0, and then to set A 1 1 = 0. The latter is equivalentto setting t = °in the Newman-Penrose formalism (1962). Reality conditions thentell us that Vi must be set equal to E3 i at the end of the calculation.
For the initial data we give (A32, L;32) and then the integration proceeds in thesame order as in (Goldberg, Robinson, and Soteriou 1992). In the course of integrating the constraint equations on u = 0, a number of r -independent functions areintroduced. After applying the reality conditions and consistency with the Minkowskispace limit, only three complex functions remain: C 1a associated with A1a and M associated with A2 a . The time development of these functions is then determined by thepropagation equations for A 1a and vaA2 a, the conservation equations. And becauseof the Bianchi identities, that is the only information coming from those equations(Bondi, Van Der Berg, and Metzner 1962; Sachs 1962). The C 1a are associated withangular momentum and M is the mass aspect. Thus, the conservation equations giveus information about the rate of change of angular momentum and mass. Note thatL;3iA1 i = vo:, vi A \ = - v{3 , and A 2
a va = v u , Here 0:, {3, and 11 are theNP coefficients defining the connection on the sphere and the convergence of the nullrays. However, at I + it is only the I /r3 part of 0:, {3, and the I / r2 part of 11 whichare relevant. Here we will be concerned only with the mass which is defined as theintegral over the two- surface u = constan t on I +. Its time derivative is non-positive(ibid.).
QUASI-LOCAL ENERGY
3. QUASI-LOCAL ENERGY
379
We wish to mirror the calculation at null infinity in an Alexandrov neighborhood .Thus we consider two close timelike separated points , p and q with q to the future ofp. Thus the neighborhood, A, is defined as the intersection of the causal future of pand the causal past of q. We further assume that p and q are sufficiently close thatthere is a unique timelike geodesic connecting them. Let t be the proper time alongthe geodesic with p the point at t = 0 and q the point at t = to. Label the future nullcones with origin on the geodesic by t. The past cone N+ from q cuts these conesand together with t = 0,N-, defines A. Now consider the past null cones from thegeodesic. They intersect the future cones in a folliation of two-surfaces. Label thesesurfaces by the coordinate r , which we take to be the luminosity distance: the area ofthe two-surface equals 471-r2 . Label the null rays on N + by the angles ((), ¢). Thus,we have (t, r , () , 1;) as coordinates in A. With these coordinate conditions , the tetradin (1) becomes (a , b , a, b, '" = 2, 3)
()o = dt ,
()l = v 1l (N 1dt + dr ),
()a = va a(dx a + Nadt)
eo = Ot - u'»;el = Vl 10r ,
ea = Va -o;
(12)
In fixing the above tetrad, we have used the r -dependent null rotations to align Va a
tangent to the two-surfaces. With these coordinate and tetrad conditions, eo and el aretangent to the generators ofN + and N - respectively. The only remaining freedom isan r -independent phase of the ea.
From-ir2si n () 'T/ljk v j L,ak = v 2
.
and and our fixing VI i tangent to the outgoing null rays, we find
(13a)
(13b)
Our tetrad choice is such that A 1 1 = 0 and from the guage constraint 92 we getA 3 1 = O. We further assume that the strong energy condition holds so that for anarbitrary null vector we have
If we are to integrate the equations, we give as initial data A 32 and L,32. Then wecan proceed to solve the equations as in (Goldberg and Soteriou 1995). However, forour present interests it is unnecessary to do so. We are interested in the informationcontained in the conservation equation (9b). Therefore, we may assume that we havea known space-time. Thus, we know all the tetrad and connection coefficients in somecoordinate system. Then, given p and q, we form the needed transformation xl' ----*
380 JOSHUA N. GOLDBERG
(t, r, B, ¢) in the neighborhood A. The Bianchi identities tell us that the equations forA 1a and vaA2a need be satisfied only at one value of r . Therefore, in the following ,these equations are assumed to be evaluated on the surface S = N - n N + .
In the above we have made use of the condition ViJ-l i = O. Also , since the calculationis local, there may be matter present and that is indicated by the matter tensor on theright hand sides.
At I +, Bondi and Sachs identify the equations for Al a with conservation of angular momentum with the help of the conformal Killing vectors of the sphere. It isstill open how one can extend this definition locally. Furthermore, this definition hasdifficulties even at I +. As noted in section 2, the A l a are related to the N-P spincoefficients a and {3.
Here we will restrict our considerations to the energy equation, (9b), which wewrite as (va = VV2a)
a(v2aA2a) = iJ aA2 + Vi {B 2 . _ 2A 1 B 2 + 2A2.B 1at 2 a , t t t
+ Nj A\j - A2j,i - 2A 1j A2i + 2A 2jA\ ]}
- 41TA:vT 10 '
(15)
(16)
Now, V2aA2a = u, where J-l is the spin coefficient for the convergence of the nullrays on N +. At I +, it is the 1/ r2 part of J-l which is identified with the mass aspect.The Minkowski space contribution is of order 1/ r and drop s out of the calculation in(Goldberg, Robinson, and Soteriou 1992). Clearly, the difference in the convergencefrom its Minkowski space value is a measure of the mass enclo sed. At I +, the energyis defined by the integral of the 1/r2 part of J-l over a two-surface cut. For our choiceof coordinates and tetrad vectors, the Bianchi identities tell us that when all the otherequations have been satisfied, (14) need be satisfied only at one value ofr. Therefore,we would like to define the quasi-local mass by
4mo.VI = Re{ i [v2aA2a]r2sin B dB d¢} ,
where S is the intersection of N- n N +. Note that any matter present whi ch flowsinto the surface contributes to the rate of change of the convergence. That is containedin (15).
While the above expression is finite , presumably, it still contains the convergencewhich would be present for such a sphere and incoming null surface in Minkowski
Q UASI-LOCAL ENERGY 381
space. To determine an appropriate subtraction, one can embed the surface S isometrically in Minkowski space (d'Inverno and Vickers 1995). Since p and q are connected by a geodesic, the embedding surface may be chosen to be a space-like plane withtime-like normal equal to the tangent vector to the geodesic at p. So doing takes intoaccount the distortion of the rays due to a Lorentz transformation. Then one calculates the convergence of the incoming rays normal to the embedded surface. Since thesurface will not in general be a sphere, the rays will not form a cone. But , in the neighborho od of S, the convergence will be well defined. Therefore, the above expressionfor the quasi-local mass should be modified by subtracting the corresponding integralin Minkowski space.
4. CONCLUSION
What we have presented above is a suggestion of an approach to quasi-local masswhich is explicitly derived from the field equations . We have used the formalism developed in (Goldberg and Soteriou 1995; Goldberg , Robinson, and Soteriou 1992),but the resulting idea is not tied to the formalism. The important point is to recognizethe importance of the convergence of incoming rays in defining the energy or massconfined in a finite domain . This is precisely what is done by astronomers for estimating the mass of intervening matter by measuring - or estimating - the bending of lightby distant galaxies. There are other suggestions in the literature, but their derivationsare either ad hoc (Hayward 1993) or contain some choice which does not have a directphysical meaning (Dougan and Mason 1991). Recent work (Brown, Lau, and York)mentioned earlier, comes closest to the point of view expressed here.
It is a pleasure to dedicate this paper to John Stachel. We have had many opportunities to discuss energy and conservation laws. At one point I considered using his2+2 formalism for the discussion in this paper. It is certainly similar, particularly asdeveloped by Ray d'Inverno and James Vickers (1995).
ACKNOWLEDGEMENTS
I want to thank David Robinson and John Madore for discussions and to thank LionelMason for his critical remarks.
Syracuse University
REFERENCES
Ashtekar, A. 1991. Lectures on Non-perturbative Canonical Gravi ty (notes prepared in collabora tion withR. Tate) . Singapore: World Scientific.
Bondi , H., M. Van Der Berg, and A. Metz ner. 1962. Proc. Roy. Soc. A 269 :21.Brown , D., and J. York. 1993. Phys. Rev. D 47:1407.Brown, J. D., S. R. Lau, and J. York. preprint gr-qc/98 10003.d' Inverno , R., and J. Vickers. 1995. Class. Quantum Grav. 12:753.Dougan, A. J., and L. Mason . 1991. Phys. Rev. Lett. 67 :2 119.
382 J OSHUA N. GOLDBERG
Go ldberg , J. N., D. C. Robin son , and C. Soteriou. 1992. Class. Quantum Grav. 9:1309.Go ldberg , J. N., and C. Soteriou . 1995. Class. Quant um Grav. 12:2779.Hayward, G. 1993. Phys . Rev. 047:3275.Newman, E. T., and R. Penrose. 1962. J. Math . Phys . 3:566.Penrose, R. 1982. Proc. Roy. Soc. A 38 1:53.Penrose, R., and W. Rindl er. 1986. Spinors and Space-Time. Ca mbridge: Ca mbridge University Press.Sachs, R. 1962. Proc. Roy. Soc. A 270: 103 .
