By ilikeafrica in arXiv — 19 Nov 2025

A Proposal of Positive-Deﬁnite Local Gravitational

이 논문은 일반 상대성 이론의 새로운 방법론을 제안한다. 연구진은 espacio - 시간을 2 차원에서 구분하여 4 차원 칼루차 - 클라인 접근법을 사용하여 일반 상대성 이론을 다시 설명한다. 그들은 double null 가우스를 사용하여 Lagrangian 부피를 도출하고, 이것이 E-H Lagrangian 부피와 동일한 필드 방정식으로 이어진다고 주장한다.

논문은 다음과 같은 중요 결과를 제시한다:

* 일반 상대성 이론을 1+1 차원 양자장 이론로 재설명할 수 있다.
* 그들은 double null 가우스에서 Lagrangian 부피가 E-H Lagrangian 부피와 동일한 필드 방정식을 이끄는다고 주장한다.
* 이 Lagrangian 부피를 사용하여 일반 상대성 이론의 지역 Hamiltonian 부피와 지역 중력 에너지 밀도를 계산할 수 있다.
* 이는 음이 아닌 양적 변수로 표현되며, Bondi 질량과 ADM 질량에 대한 볼륨 적분을 통해 확인되었다.

결과적으로, 논문은 일반 상대성 이론의 새로운 방법론을 제안하고, 중력 에너지 밀도를 지역적으로 정의할 수 있는 방법을 제공한다.

A Proposal of Positive-Deﬁnite Local Gravitational

arXiv:gr-qc/9401012v1 15 Jan 1993SNUTP 93-78A Proposal of Positive-Deﬁnite Local GravitationalEnergy Density in General RelativityJ.H. Yoon∗Center for Theoretical Physics,Seoul National University, Seoul 151-742, Korea(November 2, 2018)AbstractWe propose a 4-dimensional Kaluza-Klein approach to general relativity inthe (2,2)-splitting of space-time using the double null gauge.

The associatedLagrangian density, implemented with the auxiliary equations associated withthe double null gauge, is equivalent to the Einstein-Hilbert Lagrangian density,since it yields the same ﬁeld equations as the E-H Lagrangian density does. Itis describable as a (1+1)-dimensional Yang-Mills type gauge theory coupled to(1+1)-dimensional matter ﬁelds, where the minimal coupling associated withthe inﬁnite dimensional diﬀeomorphism group of the 2-dimensional spacelikeﬁbre space automatically appears.

The physical degrees of freedom of gravi-tational ﬁeld show up as a (1+1)-dimensional non-linear sigma model in ourLagrangian density. Written in the ﬁrst-order formalism, our Lagrangian den-sity directly yields a non-zero local Hamiltonian density, where the associatedtime function is the retarded time.

From this Hamiltonian density, we obtaina positive-deﬁnite local gravitational energy density. In the asymptotically∗e-mail address: snu00162@krsnucc1.bitnet1

ﬂat space-times, the volume integrals of the proposed local gravitational en-ergy density over suitable 3-dimensional hypersurfaces correctly reproducethe Bondi mass and the ADM mass expressed as surface integrals at nulland spatial inﬁnity, respectively, supporting our proposal. We also obtain theBondi mass-loss formula as a negative-deﬁnite ﬂux integral of a bilinear in thegravitational currents at null inﬁnity.PACS numbers: 04.20.-q, 04.20.Cv, 04.20.Fy, 04.30.+xTypeset using REVTEX2

I. INTRODUCTIONThe exact correspondence of the Euclidean self-dual Einstein’s equations to the equa-tions of motion of 2-dimensional non-linear sigma models with the target space as the area-preserving diﬀeomorphism of 2-surface [1,2] has inspired us to look further into the intriguingquestion whether the full-ﬂedged general relativity of the 4-dimensional space-time can bealso formulated as a certain (1+1)-dimensional ﬁeld theory. Recently we have shown thatsuch a description is indeed possible, and constructed the action principle [3–6] in the frame-work of the 4-dimensional Kaluza-Klein theory in the (2,2)-splitting.

In this approach, the4-dimensional space-time, at least for a ﬁnite range of space-time, is viewed as a ﬁbred man-ifold that consists of the (1+1)-dimensional “space-time” and the 2-dimensional “auxiliary”ﬁbre space.There are certain advantages of this 4-dimensional KK approach to general relativityin the above splitting, which led us to develop this formalism.We list a few of them.First of all, in (1+1)-dimensions, there exist a number of ﬁeld theoretic methods recentlydeveloped thanks to the string-related theories. Hopefully the rich mathematical methods in(1+1)-dimensions might also prove useful in studying general relativity in the (2,2)-splitting,classical and quantum.

Moreover, in this KK formulation, general relativity can be viewed asa (1+1)-dimensional gauge theory with the prescribed interactions and auxiliary equations.Since the major advantage of gauge theory formulation is that gauge invariant quantitiesautomatically solve Gauss-law equations associated with the gauge invariance, the problemof solving constraints of general relativity could be made even trivial, at least for some ofthem. Furthermore, this formulation allows us to forget about the space-time picture ofgeneral relativity; instead, it enables us to study general relativity much the way as we dofor Yang-Mills theories coupled to matter ﬁelds in (1+1)-dimensions, putting the space-timephysics into a new perspective.This (1+1)-dimensional method, however, is not entirely new since it was virtually usedin analyzing gravitational waves [7,8], and was further developed in the spin-coeﬃcient3

formalism [9–12] and the null hypersurface formalism [13–19]. In these formalisms, a specialgauge which we may call the double null gauge is chosen such that two real dual nullvector ﬁelds whose congruences span the (1+1)-dimensional submanifold are singled out,and the Einstein’s equations are spelled out in that gauge.

A characteristic feature of theseformalisms, among others, lies in that the true physical degrees of freedom of gravitationalﬁeld show up in the conformal 2-geometry of the transverse 2-surface [8,13–17,20]. Thisfeature, that has been particularly useful for studying the propagation of gravitational wavesin the asymptotically ﬂat space-times, further motivated the canonical analysis [18,21–26]of the null hypersurface formalism, in the hope of getting quantum theory of gravity byquantizing the true physical degrees of freedom of gravitational ﬁeld.In view of these advantages of the KK formalism and the null hypersurface formalism, ittherefore seems worth combining both formalisms to see what could be learnt more aboutgeneral relativity.In this article, we shall present such a formalism.In this approach,it is the (1+1)-dimensional submanifold spanned by two real dual null vector ﬁelds thatwe imagine as “space-time” and the remaining transverse 2-surface as the “auxiliary” ﬁbrespace.

As a by-product of our KK approach in the double null gauge, we obtain a new resultwhich we report in this article. Namely, we propose a positive-deﬁnite local gravitationalenergy density in general relativity without referring to the boundary conditions, and showthat the volume integrals of the proposed energy density over suitably chosen 3-dimensionalhypersurfaces correctly yield the Bondi and the ADM surface integral at null and spatialinﬁnity in the asymptotically ﬂat space-times, respectively.

We also obtain the Bondi mass-loss formula as a negative-deﬁnite ﬂux integral of the gravitational degrees of freedom atnull inﬁnity [7,8,11]. The proposed local gravitational energy density comes directly fromthe local Hamiltonian density of general relativity described as the 4-dimensional KK theoryin the double null gauge.

The associated time function is the retarded time, and has thephysical interpretation [27] as the phase of the local gravitational radiation in situationswhere gravitational waves are present.This article is organized as follows. In section II, we present the 4-dimensional KK theory4

in the (2,2)-splitting, using the double null gauge. It will be seen that, even when the gaugesymmetry is an inﬁnite dimensional symmetry such as the group of diﬀeomorphisms, theKK idea is still useful by showing that the KK variables transform properly as gauge ﬁeldsand tensor ﬁelds under the corresponding gauge transformations [3–6,28].

Next, we presentthe Lagrangian density for general relativity in the double null gauge, which is implementedby the auxiliary equations associated with the double null gauge using the Lagrange mul-tipliers. This Lagrangian density is equivalent to the Einstein-Hilbert Lagrangian density,since it yields the same ﬁeld equations as the E-H Lagrangian density does.

We shall presentthe 10 Einstein’s equations in the double null gauge. Moreover, we shall ﬁnd that 2 of these10 equations are in a form of the Schr¨odinger equation, reminiscent of the Brill wave equa-tion [29–31], i.e.

the initial value equation of the axi-symmetric gravitational waves at themoment of time symmetry.In section III, we shall present this Lagrangian density in the ﬁrst-order formalism,with the retarded time identiﬁed as our clock variable. This immediately leads to the localHamiltonian density and thus to the local gravitational energy density that we are interestedin.

The proposed local gravitational energy density is positive-deﬁnite. We shall further showthat the volume integral of the proposed local gravitational energy density over a suitablychosen 3-dimensional hypersurface become a surface integral, using the vacuum Einstein’sequations and the Bianchi identities.

In the asymptotically ﬂat space-times, this surfaceintegral becomes the Bondi and the ADM surface integral deﬁned at null and spatial inﬁnity,respectively. We also derive the Bondi mass-loss formula from the proposed gravitationalenergy density.In Appendix A, we shall describe the general (2,2)-splitting of space-time, and presentthe resulting E-H Lagrangian density without picking up a special gauge, as we need it whenwe wish to obtain the ﬁeld equations from the variational principle.

In Appendix B, we shallintroduce the covariant null tetrads, as we shall use them in Appendix C where we showthat the proposed volume integrals in section III can be expressed as surface integrals. InAppendix D, we shall show that κ2±, which appears when we discuss the Bondi mass-loss, is5

positive-deﬁnite.II. THE LAGRANGIAN DENSITY IN THE DOUBLE NULL GAUGEIn this section we combine the null hypersurface formalism with the 4-dimensionalKaluza-Klein approach where space-time is viewed as a ﬁbred manifold, i.e.

a local product ofthe (1+1)-dimensional base manifold and the 2-dimensional ﬁbre space. Let the vector ﬁelds∂/∂XA = (∂/∂u, ∂/∂v, ∂/∂ya) (a = 2, 3) span the 4-dimensional space-time.

In a Lorentzianspace-time we consider here, there always exist two real null vector ﬁelds, which we maychoose orthogonal to the 2-dimensional spacelike surface N2 spanned by ∂/∂ya. Followingthe KK idea [32], the two null vector ﬁelds can be represented as the linear combinations ofthese basis vector ﬁelds∂∂u −A a+∂∂ya,and∂∂v −A a−∂∂ya,(2.1)for some functions A a± (u, v, y).

Since these null vector ﬁelds are assumed to be normal toN2, the line element may be written in a manifestly symmetric way as follows;ds2 = −2dudv + φab(A a+ du + A a−dv + dya)(A b+du + A b−dv + dyb),(2.2)where φab(u, v, y) is the 2-dimensional metric on N2. Notice that, as a consequence of pickingup two null vector ﬁelds normal to N2, 2 out of the 10 metric coeﬃcient functions weregauged away in (2.2).

In addition, one more function was removed from (2.2) by choosingthe coordinate v such that Cdv′ = dv for some function C, i.e. by choosing C = 1.

Theelimination of these 3 functions may be viewed as a partial gauge-ﬁxing of the space-timediﬀeomorphism, and may be better understood in terms of the dual metric, which we maywriteg++ = g−−= 0,g+−= g−+ = −1,g+a = A a−,g−a = A a+ ,gab = φab −2A a+ A b−. (2.3)6

That g++ = g−−= 0 means that du and dv are dual null vector ﬁelds, and that g+−= −1is a normalization condition for v, given an arbitrary function u. We shall call this gaugeas the double null gauge1, and general relativity formulated in this gauge is referred to asthe double null formalism [13–17].

The 3-dimensional hypersurface deﬁned by u = constantis a null hypersurface since it is metrically degenerate; within each null hypersurface the2-dimensional spacelike space N2 deﬁned by v = constant is transverse to both du and dv.In order to see whether this 4-dimensional KK program is justiﬁable in the absence of anyKilling symmetry, as is the case here, we have to ﬁrst examine the transformation propertiesof φab and A a± in (2.2) under the action of some group of transformations associated withN2. The most natural group of transformations associated with N2 is the diﬀeomorphismsof N2, i.e.

diﬀN2. Under the diﬀN2 transformationy′a = y′a(u, v, y),u′ = u,v′ = v,(2.4)these ﬁelds must transform asφ′ab(u, v, y′) = ∂yc∂y′a∂yd∂y′bφcd(u, v, y),(2.5a)A′a± (u, v, y′) = ∂y′a∂yc A c±(u, v, y) −∂±y′a,(2.5b)so that the line element ds2 remains invariant [3–6].

Under the corresponding inﬁnitesimaltransformationδya = ξa(u, v, y),δu = δv = 0,(2.6)where ξa is an arbitrary function, we ﬁndδφab = −[ξ, φ]ab,(2.7a)δA a± = −D±ξa = −∂±ξa + [A±, ξ]a,(2.7b)1We notice that this double null gauge is valid only for a ﬁnite range of space-time.See forinstance [13,14,30].7

where A± := A a± ∂a and ξ := ξa∂a. Here the brackets are the Lie derivatives associated withdiﬀN2,[ξ, φ]ab = ξc∂cφab + (∂aξc)φcb + (∂bξc)φac,(2.8a)[A±, ξ]a = A c±∂cξa −ξc∂cA a± .

(2.8b)This observation tells us two things. First, diﬀN2 is the residual symmetry [11] of the lineelement (2.2) which survives even after the double null gauge was chosen.

Second, diﬀN2should be viewed as a local2 gauge symmetry of the Yang-Mills type, since A a± and φabtransform as a gauge ﬁeld and a tensor ﬁeld under the diﬀN2 transformations, respectively.This feature is rather surprising, since the KK variables were often thought to be useful forhigher dimensional gravity theories where the degrees of freedom associated with the extradimensions are suppressed in one way or another3. In our 4-dimensional KK approach togeneral relativity, we leave all the “internal” degrees of freedom intact so that all the ﬁelds in(2.2) depend on all of the coordinates (u, v, ya).

In spite of these generalities, the Lagrangiandensity associated with the metric (2.2) can be still identiﬁed, `a la Kaluza-Klein, as a gaugetheory Lagrangian density deﬁned on the (1+1)-dimensional “space-time”, with diﬀN2 asthe associated local gauge symmetry, as we shall see shortly.The metric (2.2) in the double null gauge can be obtained from the general KK lineelementds2 = γµνdxµdxν + φab(A aµ dxµ + dya)(A bν dxν + dyb),(2.9)where µ, ν = 0, 1 and a, b = 2, 3. From this we obtain (2.2) by introducing the retarded andadvanced coordinate (u, v) and the ﬁelds A a± ,u = 1√2(x0 −x1),v = 1√2(x0 + x1),(2.10a)A a± = 1√2(A a0 ∓A a1 ),(2.10b)2Here local means local in the (1+1)-dimensional “space-time”.3See however [28,33].8

assuming the (1+1)-dimensional “space-time” metric γµν to beγ+−= −1,γ++ = γ−−= 0,(2.11)in the (u, v)-coordinates, where +(−) represents u(v). In Appendix A, the general E-HLagrangian density [3–5] for the metric (2.9) and the prescription how to obtain it arepresented.

Using the “ansatz” (2.11), we can easily show that the general E-H Lagrangiandensity (A32) reduces to the following expression corresponding to the metric (2.2). If weneglect the auxiliary equations associated with the double null gauge (2.11) for the moment,it is given by [5]L0 =qφh12φabFa+−Fb+−+ 12φabφcdn(D+φac)(D−φbd) −(D+φab)(D−φcd)oi,(2.12)where we ignored the surface terms.

Here φ = det φab, and Fa+−is the diﬀN2-valued ﬁeldstrength, and D±φab is the diﬀN2-covariant derivative deﬁned asFa+−= ∂+A a−−∂−A a+ −[A+, A−]a,(2.13a)D±φab = ∂±φab −[A±, φ]ab,(2.13b)where [A+, A−]a and [A±, φ]ab are the Lie derivatives deﬁned as[A+, A−]a = A c+∂cA a−−A c−∂cA a+ ,(2.14a)[A±, φ]ab = A c±∂cφab + (∂aA c±)φcb + (∂bA c±)φac,(2.14b)respectively (see Appendix A). Recall that, in the double null formalism of general relativity,the transverse 2-metric with a unit determinant, i.e.the conformal 2-geometry, is thetwo physical degrees of freedom of gravitational ﬁeld [8,13–17].

Since we are essentiallyreformulating the double null formalism from the KK point of view in this article, we wouldlike to see ﬁrst of all what the Lagrangian density looks like when written in terms ofthe conformal 2-geometry. Let us therefore decompose the 2-metric φab into the conformalclassesφab = Ωρab,(Ω> 0anddet ρab = 1),(2.15)9

where ρab is the conformal 2-geometry4 of the transverse 2-surface N2. If we deﬁne σ byσ := ln Ω, the second term in (2.12) becomesK := 12qφ φabφcdn(D+φac)(D−φbd) −(D+φab)(D−φcd)o= −Ω−1(D+Ω)(D−Ω) + 12Ωρabρcd(D+ρac)(D−ρbd)= −eσ(D+σ)(D−σ) + 12eσρabρcd(D+ρac)(D−ρbd),(2.16)where D±Ω, D±σ, and D±ρab are the diﬀN2-covariant derivativesD±Ω= ∂±Ω−[A±, Ω],(2.17a)D±σ = ∂±σ −[A±, σ],(2.17b)D±ρab = ∂±ρab −[A±, ρ]ab,(2.17c)and [A±, Ω], [A±, σ], and [A±, ρ]ab are given by[A±, Ω] = A a± ∂aΩ+ (∂aA a± )Ω,(2.18a)[A±, σ] = A a± ∂aσ + ∂aA a± ,(2.18b)[A±, ρ]ab = A c±∂cρab + (∂aA c±)ρcb + (∂bA c±)ρac −(∂cA c±)ρab,(2.18c)respectively.

Here ∂aA aµ -terms are included in the Lie derivatives, since Ωand ρab are tensordensities of weight −1 and +1 under diﬀN2, respectively. Thus (2.12) becomesL0 = 12e2σρabFa+−Fb+−−eσ(D+σ)(D−σ) + 12eσρabρcd(D+ρac)(D−ρbd).

(2.19)In order to ﬁnd the correct variational principle that yields all the 10 Einstein’s equations,however, we must implement (2.19) with the auxiliary equations associated with the double4This may be viewed as a ﬁnite analogue of the physical transverse traceless degrees of freedom ofthe spin-2 ﬁelds propagating in the ﬂat space-time [14]. The double null gauge may be also viewedas an analogue of the Coulomb gauge in Maxwell’s theory, where the physical degrees of freedomare the transverse traceless vector potentials [34].10

null gauge.These equations can be found by ﬁrst varying the general E-H Lagrangiandensity (A32) with respect to γ++,γ−−, and γ+−, and then plugging the double null gauge(2.11) into the resulting equations. They are found to beC± := D2±σ + 12(D±σ)2 + 14ρabρcd(D±ρac)(D±ρbd) = 0,(2.20a)C0 := −12eσρabFa+−Fb+−+ D+D−σ + D−D+σ + 2(D+σ)(D−σ) + R2= 0,(2.20b)respectively, where R2 := φacRac is the scalar curvature of N2.

Remarkably, the equationsC± = 0 in (2.20a) may be written in a form of the Schr¨odinger equation. Using the identityeσ/2D2±σ + 12(D±σ)2= 2 D2±eσ/2,(2.21)the equations C± = 0 becomeD2±eσ/2 + κ2±eσ/2 = 0,whereκ2± := 18ρabρcd(D±ρac)(D±ρbd) ≥0.

(2.22)That κ2±, a bilinear in the currents of the gravitational degrees of freedom, is positive-deﬁnite can be shown easily (see Appendix D). The equations (2.22) are the analoguesof the Brill wave equation5 [29–31], as they are of the Schr¨odinger equation type for awave function corresponding to a state of zero energy in the potential −κ2±, coupled tothe external gauge ﬁelds A a± !

Thus, viewed as a scattering problem, the scattering dataeσ/2 in (2.22) is an auxiliary ﬁeld that can be determined by the potential −κ2± up to5Our Schr¨odinger equations look like one-dimensional wave equations coupled to gauge ﬁelds, butactually they are 3-dimensional partial diﬀerential equations like the Brill wave equation, due tothe Lie derivatives along A± = A a± ∂a in (2.22).The wave function in (2.22) is related to theconformal factor of the 2-dimensional wavefront, the spatial projection of the null hypersurfaceu = constant, rather than that of spacelike hypersurface. But it should be also mentioned that,for the metric (2.2), the area measure of the 2-dimensional wavefront and the volume measure of3-dimensional null hypersurface u = constant are the same [35].11

some integral “constant” functions. The generic behaviors of solutions of the 2 Einstein’sequations C± = 0 are therefore expected either of the scattering type, or of the bound-stateor resonance type, corresponding to the asymptotically ﬂat space-times or spatially closeduniverses, respectively, on a par with the Brill wave equation.The correct variational principle is now given byL = 12e2σρabFa+−Fb+−−eσ(D+σ)(D−σ) + 12eσρabρcd(D+ρac)(D−ρbd) +Xα=±,0λαCα,(2.23)where λα’s are the Lagrange multipliers which should be put to zero after variation.

Theequations of motions for A a± , σ, and ρab (subject to det ρab = 1) can be obtained by varying(2.23), with λα = 0. Here we present the results only;(a) D−e2σρabFb+−+ eσ(D−σ)(∂aσ) −∂a(eσD−σ) −12eσρbcρde(D−ρbd)(∂aρce)+∂beσρbcD−ρac= 0,(2.24a)(b) D+e2σρabFb+−−eσ(D+σ)(∂aσ) + ∂a(eσD+σ) + 12eσρbcρde(D+ρbd)(∂aρce)−∂beσρbcD+ρac= 0,(2.24b)(c) (D+σ)(D−σ) + 2D(+D−)σ + 12ρabρcd(D+ρac)(D−ρbd) + eσρabFa+−Fb+−= 0,(2.24c)(d) D(+eσρacD−)ρbc−12e2σρbcFa+−Fc+−−12δabρcdFc+−Fd+−= 0,(2.24d)where the symmetric symbol is normalized such that (αβ) := (αβ + βα)/2.Togetherwith the 3 equations C± = 0, C0 = 0 that we obtain by varying (2.23) with respect toλ±, λ0, these ﬁeld equations are identical to the 10 Einstein’s equations spelled out in thedouble null gauge, which we obtain by ﬁrst varying the general E-H Lagrangian density(A32) in Appendix A, and then imposing the double null gauge (2.11).Therefore theLagrangian density (2.23) is equivalent to the general E-H Lagrangian density (A32), withthe understanding that λα’s are to be set to zero after variation.The Lagrangian density (2.23) may be naturally interpreted as the Yang-Mills typeLagrangian density on the (1+1)-dimensional “space-time”, interacting with the (1+1)-dimensional “matter” ﬁelds σ and ρab.The corresponding local gauge symmetry is the12

built-in diﬀN2, and the “matter” ﬁelds couple to the diﬀN2-valued gauge ﬁelds throughthe minimal couplings. In addition, each term in (2.23), including the auxiliary equations,is manifestly invariant under the diﬀN2 transformations.

Therefore (2.23) should be dulyregarded as a gauge theory formulation of the vacuum general relativity [4–6].That ρab is the physical degrees of freedom can be also seen as follows. In this (1+1)-dimensional interpretation, the diﬀN2-valued gauge ﬁelds A a± are auxiliary ﬁelds since theyhave no propagating (i.e.

no transverse traceless) degrees of freedom. Moreover, as we haveseen already, σ is also an auxiliary ﬁeld that is determined by ρab through the equationsC± = 0.This conﬁrms that the two physical degrees of freedom of gravitational ﬁeldare indeed contained in ρab.

It seems appropriate to notice here that, in the propagatingequations of motion (2.24d) for ρab, the source term is given by12e2σρbcFa+−Fc+−,(2.25)whose trace is precisely the local gravitational energy density, as we shall see in the nextsection. This indicates that the local energy density in general relativity indeed plays theanalogous role as the local charge density does in Maxwell’s theory.III.

THE LOCAL GRAVITATIONAL ENERGY DENSITYIn this section, we shall ﬁnd the local Hamiltonian density of general relativity. Thiscan be obtained simply by writing the local Lagrangian density (2.23) in the ﬁrst-orderform using a suitable time coordinate.

The most natural time in this formulation seems theretarded time u [11,27]. With the retarded time as our clock, the ﬁrst term in (2.23) maybe written asLYM := 12e2σρabFa+−Fb+−= e2σρabFb+−∂+A a−−∂−A a+ −[A+, A−]a −12Fa+−.

(3.1)In terms of the phase space variables (Πa, A a−), where Πa is deﬁned as13

Πa = e2σρabFb+−,(3.2)this can be written asLYM = Πa∂+A a−−12e−2σρabΠaΠb + A a+ D−Πa,(3.3)ignoring the surface terms. Here D−Πa is the diﬀN2-covariant derivative of the density Πadeﬁned asD−Πa = ∂−Πa −[A−, Π]a,(3.4)where [A−, Π]a is the Lie derivative of Πa,[A−, Π]a = A c−∂cΠa + (∂aA c−)Πc + (∂cA c−)Πa.

(3.5)The second and third term in (2.23) are already in the ﬁrst-order form, apart from theterms proportional to A a+ whose variation yields the Gauss-law equations associated withthe residual diﬀN2 invariance. Putting these all together, the Lagrangian density (2.23) canbe written in the following Hamiltonian form6L = Πa∂+A a−−eσ(D−σ)(∂+σ) + 12eσρabρcd(D−ρbd)(∂+ρac)−12e−2σρabΠaΠb + A a+ Ca +Xα=±,0λαCα,(3.6)where Ca is given byCa = D−Πa + eσ(D−σ)(∂aσ) −∂a(eσD−σ) −12eσρbcρde(D−ρbd)(∂aρce)+∂beσρbcD−ρac,(3.7)which is the same as (2.24a) if we use (3.2).

From (3.6) the local gravitational Hamiltoniandensity H is given by6Notice that the Hessian of this Lagrangian density is zero, so that the usual method of Hamil-tonization does not work. This is the reason that we did not introduce the momenta conjugate toσ and ρab.

See for instance [36–39].14

H = 12e−2σρabΠaΠb −A a+ Ca −Xα=±,0λαCα. (3.8)Using the 5 equations Ca = 0, C± = 0, and C0 = 0, the local Hamiltonian density (3.8)becomesE = 12e−2σρabΠaΠb = 12e2σρabFa+−Fb+−≥0,(3.9)which is positive-deﬁnite for any σ and Πa, since the conformal 2-metric ρab has a positive-deﬁnite signature.

Thus, at least formally, we have obtained, in the double null gauge, apositive-deﬁnite local gravitational energy density for the vacuum general relativity! Thetime function associated with this non-zero local energy density is the retarded time u thatwe may choose at will7.

This is our proposal of the positive-deﬁnite local gravitational energydensity in this article. This seems to be against the usual argument that, in general relativity,local gravitational energies can not be deﬁned because they can be always “transformed”away due to the equivalence principle, let alone the positive-deﬁniteness.

In our deﬁnitionof the local gravitational energy density, however, the “ﬁeld strength” Fa+−in (2.13a) is thecoeﬃcient of the commutator of the two null vector ﬁelds ∂± −A a± ∂a, which measures thetwist of the parallelogram made of two successive parallel transports of these null vector ﬁeldsalong each other. Certainly the twist of this null parallelogram can not be “transformed”away even in a local Lorentz frame, and thus can serve as a measure of gravitational energyassociated with the parallelogram surrounding the space-time point under consideration.The proposed local gravitational energy density is just the square of this local “ﬁeld strength”multiplied by the canonical integration measure.In order to appreciate what this really means, however, we have to ﬁrst deﬁne the volumeintegral of the local gravitational energy density E over a 3-dimensional hypersurface deﬁned7For the asymptotically ﬂat space-times, the number of possible choices of the retarded time uis equal to the number of an arbitrary, monotonically increasing function of three variables (u, ya)[11].

This may be true for other space-times as well.15

by u = constant, and evaluate it for the asymptotically ﬂat space-times, since the totalgravitational energy is well-deﬁned only for the asymptotically ﬂat space-times. As we nowshow, for the asymptotically ﬂat space-times, the volume integrals of the proposed localenergy density over suitably chosen 3-dimensional hypersurfaces can be re-expressed as theBondi and ADM surface integral at null and spatial inﬁnity, respectively.

We shall alsoderive the Bondi mass loss-formula as a negative-deﬁnite ﬂux integral of a bilinear in thegravitational currents at null inﬁnity.A. The Bondi MassIn this subsection we wish to show that, for the asymptotically ﬂat space-times, thevolume integral of (3.9) over the u = constant null hypersurface is precisely the Bondi massas measured at null inﬁnity.

Let us ﬁrst notice that the volume integral E, whereE = 12Zdvd2y e2σρabFa+−Fb+−≥0,(3.10)is positive-deﬁnite for any topology of N2 [10]. In order to express (3.10) as a surface integral,it is necessary to write it in a slightly diﬀerent form using the ﬁeld equations.

For this letus consider the following identityD+D−σ −D−D+σ = −Fa+−∂aσ −∂aFa+−. (3.11)Using (3.11), the integral of the equation C0 = 0 in (2.20b) over N2 with the integrationmeasure eσ may be written as12Zd2y e2σρabFa+−Fb+−=Zd2y eσnR2 + 2(D+σ)(D−σ) + 2D+D−σo+Zd2y ∂a(eσFa+−).

(3.12)The last term in (3.12) is zero for any 2-surface N2 that we assume compact without bound-ary. Thus (3.10) becomesE =Zdvd2y eσnR2 + 2(D+σ)(D−σ) + 2D+D−σo.

(3.13)16

Let us also integrate the equation (2.24c) over N2, using (3.11), to obtainZd2y eσn(D+σ)(D−σ) + 2D+D−σo= −Zd2y e2σρabFa+−Fb+−−12Zd2y eσρabρcd(D+ρac)(D−ρbd)−Zd2y ∂a(eσFa+−),(3.14)where the last term may be also dropped. Thus the volume integral (3.13) becomesE =Zdvd2y eσnR2 + (D+σ)(D−σ) −12ρabρcd(D+ρac)(D−ρbd)o−Zdvd2y e2σρabFa+−Fb+−,(3.15)or,E = 12Zdvd2y e2σρabFa+−Fb+−= 13Zdvd2y eσnR2 + (D+σ)(D−σ) −12ρabρcd(D+ρac)(D−ρbd)o.

(3.16)To show that this8 can be expressed as a surface integral, the covariant null tetrad notationthat we described in Appendix B is useful. Let us notice that the Gauss equation in the(2,2)-splitting of space-time is given by9 [40–43]R2 + (D+σ)(D−σ) −12ρabρcd(D+ρac)(D−ρbd) = hAChBDCABCD,(3.17)where CABCD(A, B, · · · = 0, 1, 2, 3) is the conformal curvature tensor, and hAB is the 2-metric on the transverse surface N2, i.e.

the covariant form of φab. Then the volume integralE may be written as8Notice that this energy integral is diﬀerent from the one in Hayward’s paper [40].

The energydensity he proposed is the minus of L0 in (2.19), modulo the Euler density, and is not positive-deﬁnite. This diﬀerence may be traced back to the fact that in his paper both u and v are treatedas the time variables.9Here we used the vacuum Einstein’s equations.17

E = 13Zdvd2y eσhAChBDCABCD. (3.18)In Appendix C, we have shown that, using the Bianchi identity∇[MCAB]CD = 0,(3.19)this can be expressed as the surface integralE = 13Zdvd2y eσhAChBDCABCD = 13 lim vZd2y eσhAChBDCABCD,(3.20)where lim means that the integral over N2 is to be evaluated at the limiting boundaryvalue(s) of v. This expression picks up the coeﬃcient of 1/v-term, and becomes preciselythe Bondi mass10 [40–43] in the limit as v approaches to inﬁnity!

Notice that the parameterv/√2 becomes the area radius in the limit v →∞(keeping u = u0 = constant), as ourmetric (2.2) approaches tods2 →−2dudv + 12(v −u)2(dϑ2 + sin2ϑdϕ2)(3.21)at null inﬁnity.B. The Bondi Mass-Loss FormulaIn this subsection we shall continue to obtain the Bondi mass-loss formula in the presenceof the gravitational radiation in the asymptotically ﬂat space-times.For this, we maysimply take a u-derivative of the integral (3.20), using suitable vacuum Einstein’s equations.However, since we wish to account for the mass-loss in terms of the physical degrees offreedom ρab, we shall work with the volume integral (3.13).Recalling that diﬀN2 is the residual gauge symmetry of the metric (2.2), we may ﬁx thissymmetry by choosing A a+ = 011 by a suitable coordinate transformation on N2.

Then the10We have a factor of 1/3, which could be taken care of by a suitable normalization.11This is equivalent to the assumption that ∂/∂u is a twist-free null vector ﬁeld [4,5,44]. However,there could be some topological obstructions against globalizing this choice.18

equation C+ = 0 in (2.22) reduces to the following Schr¨odinger equation∂2+eσ/2 + κ′2+eσ/2 = 0,whereκ′2+ := 18ρabρcd(∂+ρac)(∂+ρbd) ≥0. (3.22)In the following we shall use this equation when we examine the rate of change in E as theretarded time u advances.

Let us further notice that the metric (2.2) becomes, in the gaugeA a+ = 0,ds2 = −2dudv + eσρab(A a−dv + dya)(A b−dv + dyb). (3.23)With ya = (ϑ, ϕ), where ϑ, ϕ are the angles of S2, we ﬁnd by comparing (3.23) with (3.21)the following asymptotic behaviors of the metric as v approaches to inﬁnity,eσ = O(v2),ρab = O(1),A a−= O(1/v2).

(3.24)In the gauge A a+ = 012, the volume integral (3.13) becomesE =Zdvd2y eσnR2 + 2(∂+σ)(D−σ) + 2∂+D−σo=Zdvd2yneσR2 + 2∂+D−eσo,(3.25)where we used the identity∂+D−eσ = eσ(∂+σ)(D−σ) + eσ∂+D−σ. (3.26)The ﬁrst term in (3.25) is the v-integration of the Euler number χ, whereχ = 14πZd2y eσR2 = 2, 0, −2(g −1),(3.27)for N2 = S2, T2, and the 2-surface Σg of genus g, respectively.

Since the u-derivative of theEuler integral is zero, the rate of change in E as u increases comes from the second term in(3.25). For the asymptotically ﬂat space-times, we may assume N2 = S2 so that12This gauge choice is only for convenience, since we already obtained the covariant expression ofthe Bondi mass in the previous subsection.19

dEdu = 2ZS2dvd2y ∂2+(D−eσ)= 2Zv=∞,S2d2y (∂2+eσ) −2Zv=v0,S2d2y (∂2+eσ)1 + O(v−10 ),(3.28)where the domain of the v-integration was chosen from v0 to ∞, and we used the asymptoticbehaviors (3.24). Here v0 is some point that lies suﬃciently far away from the sources ofgravitational waves along the out-going null direction such that the gravitational waves arecontained entirely in the range v0 < v ≤∞at the instant u = constant.

In this asymptoticregion, we may also assume the out-going null condition [7,8,11],eσ = 12v2sinϑn1 + f(u, ϑ, ϕ)v+ O( 1v2)o,(3.29)for some function f(u, ϑ, ϕ). From this, it is found that∂2+eσ = 2eσ/2∂2+eσ/2 1 + O(1/v)= −14eσρabρcd(∂+ρac)(∂+ρbd)1 + O(1/v),(3.30)where in the second line we used the Schr¨odinger equation (3.22).

Thus dE/du becomesdEdu = −12 limv→∞ZS2d2y eσρabρcd(∂+ρac)(∂+ρbd)+12Zv=v0,S2d2y eσρabρcd(∂+ρac)(∂+ρbd)1 + O(v−10 ). (3.31)Since there are no propagating gravitational degrees of freedom in the region v ≤v0 thecurrents of gravitational waves must vanish in this region, so thatρab∂+ρac = 0forv ≤v0.

(3.32)Thus the second term in (3.31) vanishes, and we ﬁnally havedEdu = −12 limv→∞ZS2d2y eσρabρcd(∂+ρac)(∂+ρbd) ≤0. (3.33)Apart from the integration measure eσ, this ﬂux integral over S2 at null inﬁnity is expressedentirely in terms of the physical degrees of freedom, and is negative-deﬁnite.

This is preciselythe Bondi mass-loss formula!It must be stressed that the gravitational energy carried20

away to null inﬁnity by the gravitational radiation is given in a bilinear combination of thegravitational currents, in excellent accordance with our experience that observables are veryoften expressed in bilinears of the physical ﬁelds. This strongly supports the view held bythe geometric quantization school [36–38] that, in general relativity, as in other non-linearﬁeld theories, it is the current rather than the conformal 2-geometry that should be regardedas the fundamental physical ﬁeld.The total radiated gravitational energy to null inﬁnity between the null time interval u0and u can be obtained by integrating (3.33), and is given byE(u) −E(u0) = −12 limv→∞Z uu0duZS2 d2y eσρabρcd(∂+ρac)(∂+ρbd).(3.34)C.

The ADM MassNow we shall show that the volume integral of the proposed local gravitational energydensity (3.10) over a spacelike hypersurface reproduces the ADM surface integral. Let usmake the following coordinate transformationr = −12u + v.(3.35)In the new coordinates (u, r, ya), the metric (2.2) becomesds2 = −du2 −2dudr + eσρabn(A a+ + 12A a−)du + A a−dr + dyaon(A b+ + 12A b−)du + A b−dr + dybo.

(3.36)Since we still have the residual gauge symmetry associated with the diﬀN2 invariance in(3.36), we may well ﬁx this residual symmetry by choosingA a+ + 12A a−= 0. (3.37)Then (3.36) reduces tods2 = −du2 −2dudr + eσρabA a−dr + dyaA b−dr + dyb.

(3.38)21

In the limit as r →∞, both (3.36) and (3.38) approach over to the ﬂat space-time metricds2 →−du2 −2dudr + 12r2(dϑ2 + sin2ϑdϕ2),(3.39)showing that r/√2 becomes the area radius in this limit. Moreover the u-coordinate is theproper time at each point on the u = constant hypersurface, suggesting that the volumeintegral be deﬁned over the spacelike hypersurface u = constant in the new coordinates,since the ADM surface integral is associated with a unit time translation at spatial inﬁnity.To ﬁnd the relevant Hamiltonian for the ADM mass we need to write the local Lagrangiandensity in the new coordinates.

The local Lagrangian density can be found directly from(2.23)L = 12e2σρabFa+−Fb+−−eσ(D+σ)(D−σ) + 12eσρabρcd(D+ρac)(D−ρbd) +Xα=±,0λαCα |∂−=∂r,(3.40)with ∂−replaced by ∂r everywhere. Thus the relevant volume integral is given byE = 12Zdrd2y e2σρabFa+−Fb+−|∂−=∂r= 13Zdrd2y eσnR2 + (D+σ)(D−σ) −12ρabρcd(D+ρac)(D−ρbd)o|∂−=∂r= 13Zdrd2y eσhAChBDCABCD.

(3.41)Repeating the same reasoning as in Appendix C, with ˜Ω−1 = r, we ﬁnd that the volumeintegral (3.41) becomes the surface integral,E = 13 lim rZd2y eσhAChBDCABCD,(3.42)which becomes precisely the covariant expression of the ADM mass of the asymptoticallyﬂat space-times in the limit as r approaches to inﬁnity! [40–43]IV.

DISCUSSIONSIn this article, we combined the double null formalism of general relativity with the KKformalism in the (2,2)-splitting, and proposed a local gravitational energy density of general22

relativity. As we have seen so far, there are a number of notable features of this descriptionwhich deserve further remarks.

First of all, this formalism explicitly brings out the gaugetheory aspects of general relativity of the 4-dimensional space-times. Although it has beenrealized for a long time that the local diﬀeomorphism invariance in general relativity is ona par with the local gauge symmetry in gauge theories, it seems fair to say that the full-ﬂedged gauge theory formulation of general relativity is still lacking.

Our 4-dimensional KKapproach to general relativity in the (2,2)-splitting, using the double null gauge, seems toprovide such a formulation, as we have described in this article. Thus we may well take careof the Gauss-law equations associated with the diﬀN2 invariance by considering the diﬀN2invariant quantities only.Moreover, this formalism shows that, in the double null gauge, local gravitational energydensity of general relativity can be well-deﬁned, and moreover, is positive-deﬁnite.Thevolume integral of this local gravitational energy density over the 3-dimensional null andspacelike hypersurface correctly reproduces the Bondi and ADM surface integral at null andspatial inﬁnity, respectively.

The Bondi mass-loss due to the gravitational radiation in theasymptotically ﬂat space-times is given by a negative-deﬁnite ﬂux integral of the bilinearin the gravitational currents at null inﬁnity.It should be mentioned that the proposedgravitational energy density can be also used to deﬁne quasi-local gravitational energies fora ﬁnite region of a given 3-dimensional hypersurface in a straightforward way [40,45–49].The non-zero local Hamiltonian density proposed in this article also has a direct bearingto the problem of time [50–54]. The time associated with the non-zero Hamiltonian is theretarded time u, which has the physical meaning as the phase of the gravitational radiationwhen gravitational waves are present.The canonical analysis of our formalism is underprogress [55], which will shed further light on this important issue.In addition, this formalism does seem to indicate the intriguing possibility that quantumgeneral relativity of the 4-dimensional space-time may be regarded as a (1+1)-dimensionalquantum ﬁeld theory.

For instance, one might even speculate that quantum gravity mightbe a ﬁnite theory, given that the renormalizability depends critically on the dimensions of23

“space-time”. However, it must be addressed that this formalism is for the vacuum generalrelativity only.

It certainly is an interesting question to see whether this formalism can beextended to include matter ﬁelds. We leave this problem for the future investigation.ACKNOWLEDGMENTSIt is a great pleasure to thank G.T.

Horowitz for the hospitality during the Space-Time93 program at ITP, Santa Barbara, where part of this work was conceived. The author alsothanks S. Carlip, C.W.

Misner, V. Moncrief, E.T. Newman and others for their interest inthis approach and for encouragements, and K. Kuchaˇr for suggesting him to look into thenull hypersurface formalism that was of invaluable help to this work.

He also thanks D.Brill for a number of valuable suggestions on the preliminary version of this article, and forkindly informing him the related work of S.A. Hayward, and for encouragements. This workis supported in part by the Ministry of Education and by the Korea Science and EngineeringFoundation through the SRC program.APPENDIX A: THE E-H LAGRANGIAN DENSITY IN THE GENERAL(2,2)-SPLITTINGIn this appendix, we shall make a general (2,2)-decomposition of space-time, and obtainthe corresponding E-H Lagrangian density [3–5] without picking up a particular gauge.

Byexamining the transformation properties of the metric in the (2,2)-splitting under the diﬀN2transformations, we shall ﬁnd that each ﬁeld can be identiﬁed either as a scalar, a tensor,or a gauge ﬁeld with respect to the diﬀN2 transformations, respectively, suggesting that theKK program works even in the absence of any Killing vector ﬁelds. Then we simplify thegeneral E-H Lagrangian density by introducing the double null gauge to obtain L0 in (2.12)[5], which also has the diﬀN2 symmetry as the residual symmetry.The 4-dimensional space-time may be regarded as a ﬁbred manifold, i.e.

a local productof two 2-dimensional submanifolds M1+1 ×N2, for which we introduce two pairs of the basis24

vector ﬁelds ∂µ = ∂/∂xµ(µ = 0, 1) and ∂a = ∂/∂ya(a = 2, 3), respectively. The correspondingmetrics on M1+1 and N2 will be denoted as γµν and φab, respectively.

Then the general lineelement of the 4-dimensional space-time can be written asds2 = γµνdxµdxν + φab(A aµ dxµ + dya)(A bν dxν + dyb). (A1)Formally this is quite similar to the “dimensional reduction” in KK theory, where M1+1 isregarded as the (1+1)-dimensional “space-time” and N2 as the “internal” ﬁbre space.

Inthe standard KK reduction one assumes a restriction on the metric, namely, an isometrycondition, to make A aµ a gauge ﬁeld associated with the isometry group. Here, however,we do not assume such isometry conditions, and allow all the ﬁelds to depend on both xµand ya.

Nevertheless A aµ (x, y) can still be identiﬁed as a gauge ﬁeld, but now associatedwith an inﬁnite dimensional diﬀeomorphism group diﬀN2. To show this, let us consider thefollowing diﬀeomorphism of N2,y′a = y′a(x, y),x′µ = xµ.

(A2)Under this transformation, we ﬁndγ′µν(x, y′) = γµν(x, y),(A3)φ′ab(x, y′) = ∂yc∂y′a∂yd∂y′bφcd(x, y),(A4)A′aµ (x, y′) = ∂y′a∂yc A cµ (x, y) −∂µy′a,(A5)such that the line element (A1) is invariant. Under the corresponding inﬁnitesimal trans-formationδya = ξa(x, y),δxµ = 0,(A6)these becomeδγµν = −[ξ, γµν] = −ξc∂cγµν,(A7)δφab = −[ξ, φ]ab = −ξc∂cφab −(∂aξc)φcb −(∂bξc)φac,(A8)δA aµ = −∂µξa + [Aµ, ξ]a = −∂µξa + (Acµ∂cξa −ξc∂cA aµ ),(A9)25

where the bracket represents the Lie derivative that acts on the “internal” indices a, b, etc,only.Notice that the Lie derivative, an inﬁnite dimensional generalization of the ﬁnitedimensional matrix commutators, appears naturally. Associated with this diﬀN2 transfor-mation, the diﬀN2-covariant derivative Dµ is deﬁned byDµ = ∂µ −[Aµ,],(A10)where the bracket is again the Lie derivative along Aµ = A aµ ∂a.

With this deﬁnition, wehaveδA aµ = −Dµξa,(A11)which clearly indicates that A aµ is the gauge ﬁeld valued in the inﬁnite dimensional Liealgebra associated with diﬀN2. Moreover the transformation properties (A7) and (A8) showthat γµν and φab are a scalar and a tensor ﬁeld, respectively, under diﬀN2.

The ﬁeld strengthFaµν corresponding to A aµ can now be deﬁned as[Dµ, Dν] = −Faµν ∂a = −{∂µA aν −∂νA aµ −[Aµ, Aν]a}∂a,(A12)which transforms covariantly under the inﬁnitesimal transformation (A6),δFaµν = −[ξ, Fµν]a. (A13)To obtain the E-H Lagrangian density, we have to ﬁrst compute connections and curvaturetensors.

For this purpose it is convenient to introduce the following horizontal lift basisˆ∂A = (ˆ∂µ, ˆ∂a) where [32]ˆ∂µ := ∂µ −A aµ ∂a,ˆ∂a := ∂a . (A14)From the following commutation relations[ˆ∂A, ˆ∂B] = fCAB ˆ∂C,(A15)we ﬁnd the structure functions fCAB (x, y)26

faµν = −Faµν ,fbµa = −fbaµ = ∂aA bµ ,fCAB= 0,otherwise. (A16)The virtue of this basis is that it brings the metric (A1) into a block diagonal formˆgAB =γµν00φab,(A17)which drastically simpliﬁes the computation of the scalar curvature.The connection coeﬃcients and the curvature tensors in this basis are given by [32,56]ˆΓCAB = 12ˆgCDˆ∂AˆgBD + ˆ∂BˆgAD −ˆ∂DˆgAB+ 12ˆgCDfABD −fBDA −fADB,ˆRDABC= ˆ∂BˆΓDAC−ˆ∂AˆΓDBC + ˆΓDBE ˆΓEAC −ˆΓDAE ˆΓEBC + fEAB ˆΓDEC ,ˆRAC = ˆgBD ˆRABCD,R = ˆgAC ˆRAC,(A18)where fABC := ˆgCDfDAB .

In components the connection coeﬃcients are given byˆΓαµν = 12γαβˆ∂µγνβ + ˆ∂νγµβ −ˆ∂βγµν,ˆΓaµν = −12φab∂bγµν −12Faµν ,ˆΓνµa = ˆΓνaµ = 12γνα∂aγµα + 12γναφabFbµα ,ˆΓbµa = 12φbc ˆ∂µφac + 12∂aA bµ −12φbcφae∂cA eµ ,ˆΓbaµ = 12φbc ˆ∂µφac −12∂aA bµ −12φbcφae∂cA eµ ,ˆΓµab = −12γµν ˆ∂νφab + 12γµνφac∂bA cν + 12γµνφbc∂aA cν ,ˆΓcab = 12φcd∂aφbd + ∂bφad −∂dφab,(A19)The following identities are also useful;ˆΓµµν = 12γαβ ˆ∂νγαβ,ˆΓββa = 12γαβ∂aγαβ,ˆΓaaν = 12φab ˆ∂νφab −∂aA aν ,ˆΓaνa = 12φab ˆ∂νφab,ˆΓaab = 12φac∂bφac. (A20)27

The Ricci tensors and the scalar curvature are given byˆRµν = ˆRαµαν+ ˆRaµaν ,ˆRac = ˆRbabc+ ˆRαaαc ,R = γµν ˆRµν + φac ˆRac. (A21)Thus, in order to obtain the E-H Lagrangian density, we need to calculate γµν ˆRµν and φac ˆRaconly.

Let us ﬁrst deﬁne R′µν and Rac as follows;R′µν = ˆ∂αˆΓαµν −ˆ∂µˆΓααν + ˆΓββα ˆΓαµν −ˆΓαµβ ˆΓβαν ,Rac = ∂bˆΓbac −∂aˆΓbbc + ˆΓddb ˆΓbac −ˆΓbad ˆΓdbc . (A22)Notice that formally R′µν is identical to the Ricci tensor of M1+1, except that ˆ∂µ was usedinstead of ∂µ.

For this reason it might be called the “gauged” Ricci tensor of M1+1, whereasRac is the usual Ricci tensor of N2. After a long computation we obtain γµν ˆRµν and φac ˆRacas follows;γµν ˆRµν = γµνR′µν −12γµνγαβφabFaµα Fbνβ −14γµνφabφcd(ˆ∂µφac)(ˆ∂νφbd)+14γµνφabφcd(ˆ∂µφab)(ˆ∂νφcd) + γµνφbc(ˆ∂µφac)(∂bA aν )−γµνφab(ˆ∂µφab)(∂cA cν ) + γµν(∂aA aµ )(∂bA bν ) −12γµν(∂aA bµ )(∂bA aν )−12γµνφabφcd(∂aA cµ )(∂bA dν )−( ˆ∇µ + ˆΓccµ )12γµνφab ˆ∂νφab −γµν∂aA aν−( ˆ∇a + ˆΓααa )12φabγµν∂bγµν,(A23)φac ˆRac = φacRac + 14γµνγαβφabFaµα Fbνβ −14φabγµνγαβ(∂aγµα)(∂bγνβ)+14φabγµνγαβ(∂aγµν)(∂bγαβ)−( ˆ∇µ + ˆΓccµ )12γµνφab ˆ∂νφab −γµν∂aA aν−( ˆ∇a + ˆΓααa )12φabγµν∂bγµν.

(A24)Here the derivatives ˆ∇µ and ˆ∇a are compatible with the metrics γαβ and φbc in the horizontallift basis, respectively, such thatˆ∇µγαβ = ˆ∂µγαβ −ˆΓδµα γδβ −ˆΓδµβ γαδ = 0,ˆ∇aφbc = ∂aφbc −ˆΓdab φdc −ˆΓdac φbd = 0. (A25)28

For an object of the mixed-type such as Xb···β···a···α··· in this basis, these derivatives act as follows;ˆ∇µXc···β···b···α···= ˆ∂µXc···β···b···α··· −ˆΓδµα Xc···β···b···δ···+ ˆΓβµδ Xc···δ···b···α··· + · · · ,ˆ∇aXc···β···b···α···= ∂aXc···β···b···α··· −ˆΓdab Xc···β···d···α··· + ˆΓcad Xd···β···b···α···+ · · · . (A26)For instance, the followings are true,ˆ∇µ(φab ˆ∂νφab) = ˆ∂µ(φab ˆ∂νφab) −ˆΓαµν (φab ˆ∂αφab),ˆ∇a(γµν∂bγµν) = ∂a(γµν∂bγµν) −ˆΓcab (γµν∂cγµν),ˆ∇µ(∂aA aν ) = ˆ∂µ(∂aA aν ) −ˆΓαµν ∂aA aα ,(A27)which we used in (A23) and (A24).

The scalar curvature R of the metric (A1) becomes,using (A23) and (A24),R = γµνR′µν + φacRac −14γµνγαβφabFaµα Fbνβ−14γµνφabφcdn(Dµφac)(Dνφbd) −(Dµφab)(Dνφcd)o−14φabγµνγαβn(∂aγµα)(∂bγνβ) −(∂aγµν)(∂bγαβ)o−( ˆ∇µ + ˆΓccµ )jµ −( ˆ∇a + ˆΓααa )ja. (A28)Here jµ and ja are deﬁned asjµ = γµνφab ˆ∂νφab −2γµν∂aA aν ,ja = φabγµν∂bγµν,(A29)and Dµφab is the diﬀN2-covariant derivativeDµφac = ˆ∂µφac −(∂aA eµ )φec −(∂cA eµ )φae= ∂µφac −[Aµ, φ]ac,(A30)where [Aµ, φ]ac is the Lie derivative of φac along Aµ = A eµ ∂e,[Aµ, φ]ac = A eµ ∂eφac + (∂aA eµ )φec + (∂cA eµ )φae.

(A31)29

Thus the E-H Lagrangian density in this (2,2)-splitting ﬁnally becomesL′ = √−γqφ R= √−γqφhγµνR′µν + φacRac −14γµνγαβφabFaµα Fbνβ−14γµνφabφcdn(Dµφac)(Dνφbd) −(Dµφab)(Dνφcd)o−14φabγµνγαβn(∂aγµα)(∂bγνβ) −(∂aγµν)(∂bγαβ)oi−√−γqφn( ˆ∇µ + ˆΓccµ )jµ + ( ˆ∇a + ˆΓααa )jao,(A32)where γ = det γµν and φ = det φab. It can be shown that the last two terms in (A32) aretotal divergences [3–5],√−γqφ ( ˆ∇µ + ˆΓccµ )jµ = ∂µ√−γqφ jµ−∂a√−γqφ A aµ jµ,(A33)√−γqφ ( ˆ∇a + ˆΓααa )ja = ∂a√−γqφ ja,(A34)using (A20) and the following identitiesˆ∇µjµ = ˆ∂µjµ + ˆΓααµ jµ,ˆ∇aja = ∂aja + ˆΓbba ja.

(A35)In the (u, v) coordinatesu = 1√2(x0 −x1),v = 1√2(x0 + x1),(A36)the following substitutionγ+−= −1,γ++ = γ−−= 0,(A37)together withA a± = 1√2(A a0 ∓A a1 ),(A38)leads to the double null gauge (2.2) and enormously simpliﬁes the E-H Lagrangian density(A32). The resulting expression is L0 in (2.12).30

APPENDIX B: THE COVARIANT NULL TETRADSIn this appendix, we describe the kinematics of a Lorentzian space-time of 4-dimensionsusing the covariant null tetrads [9–12,19]. This allows us to compare the variables in thetraditional double null hypersurface formulation and our KK variables directly.

Moreover,in order to express the volume integral (3.18) as a surface integral, it is better to use thecovariant null tetrad notation. Let the two real dual null tetrads lA and nA (A=0,1,2,3) bethe gradient ﬁelds for some scalar functions u and v,lA = ∇Au,nA = ∇Av,(B1)so that ∇[BlA] = ∇[BnA] = 0.

The dual vector ﬁelds du and dv are related to the dual nulltetrads bydu = lAdXA,dv = nAdXA. (B2)We also have the vector ﬁelds ∂/∂u, ∂/∂v, and ∂/∂ya (a = 2, 3) which we may write∂∂u = uA∂∂XA,∂∂v = vA∂∂XA,∂∂ya = y Aa∂∂XA .

(B3)If we choose the basis vector ﬁelds of space-time so that ∇A = (∂/∂u, ∂/∂v, ∂/∂ya), thecomponents of the dual null tetrads and vector ﬁelds are given bylA = (1, 0, 0, 0)nA = (0, 1, 0, 0),uA = (1, 0, 0, 0),vA = (0, 1, 0, 0),y Aa= δ Aa . (B4)From this it follows thatlAuA = nAvA = 1,lAvA = nAuA = 0.

(B5)Since lAlA = nAnA = 0, the null tetrad lA and nA may be chosen asl = lA∂∂XA = −∂∂v + A a−∂∂ya,n = nA∂∂XA = −∂∂u + A a+∂∂ya,(B6)i.e.31

lA = (0, −1, A a−),nA = (−1, 0, A a+ ),(B7)such thatlAnA = −1. (B8)(n and l are the minus of the horizontal lift vector ﬁelds ˆ∂± = ∂± −A a± ∂a in the (u, v)-coordinates, respectively.) The condition (B8) is equivalent to the previous normalizationcondition g+−= −1 in (2.3), and means that, given an arbitrary function u, the functionv must be chosen in such a way that the normalization condition (B8) is satisﬁed.

We stillhave the freedom to orient lA and nA in space-time, and we ﬁx this freedom by demandingthat lA and nA are normal to the 2-dimensional spacelike surface N2 whose tangent vectorﬁelds are ∂/∂ya,hABlB = hABnB = 0,(B9)where hAB is the metric on N2 (hAB is the covariant expression of φab). Using hAB and lA,nA, the space-time metric gAB may be written asgAB = hAB −(lAnB + nAlB).

(B10)From these relations, we easily ﬁnd that (B10) is identical to the metric (2.2)ds2 = −2dudv + φab(A a+ du + A a−dv + dya)(A b+du + A b−dv + dyb). (B11)APPENDIX C: THE BONDI SURFACE INTEGRALWe now show that the volume integral (3.18)E = 13Zdvd2y eσhAChBDCABCD(C1)can be expressed as a surface integral, using the Bianchi identity of the conformal curvaturetensor32

∇[MCAB]CD = 0. (C2)Let us notice that, due to the Bianchi identity, the following is true for any scalar function˜Ω,∇[M˜Ω−1CAB]CD=∇[M ˜Ω−1CAB]CD.

(C3)If we contract (C3) by hAChBD, it becomeshAChBD∇[M˜Ω−1CAB]CD= hAChBD∇[M ˜Ω−1CAB]CD. (C4)Let us choose ˜Ω−1 as a function of (u, v) only, and let ∇M = ∇−.

Then the r.h.s. of (C4)becomeshAChBD∇[−˜Ω−1CAB]CD = 13hAChBD∇−˜Ω−1CABCD,(C5)since hAB∇B ˜Ω−1 = 0.

The l.h.s. of (C4) becomeshAChBD∇[−˜Ω−1CAB]CD= ∇[−hAChBD ˜Ω−1CAB]CD−∇[−hAChBD˜Ω−1CAB]CD.

(C6)Using hAC = gAC + lAnC + nAlC and the properties of the conformal curvature tensorgACCABCD = gBDCABCD = 0,CABCD = C[AB][CD],(C7)the second term in the r.h.s. of (C6) may be written as∇[−hAChBD˜Ω−1CAB]CD = −2∇[−lAnB]lCnD˜Ω−1CABCD.

(C8)Since lA and nA are non-zero only when A, B are + or −, we have∇[−lAnB]lCnD= 0,(C9)due to the repeated indices in the anti-symmetric symbol.Therefore the l.h.s.of (C4)becomes33

hAChBD∇[−˜Ω−1CAB]CD= ∇[−hAChBD ˜Ω−1CAB]CD. (C10)Thus (C4) becomes, using (C5) and (C10),hAChBD∇−˜Ω−1CABCD = ∇−hAChBD ˜Ω−1CABCD+ ∇AhAChBD ˜Ω−1CB−CD+∇BhAChBD ˜Ω−1C−ACD.

(C11)Integrating (C11) over the u = constant hypersurface with the canonical measure√h [35],it becomes, using √g =√h for the metric (B10),Zdvd2y√h hAChBD∇−˜Ω−1CABCD =Zdvd2y√h ∇−hAChBD ˜Ω−1CABCD= limZd2y√hhAChBD ˜Ω−1CABCD,(C12)since the surface integrals coming from the last two terms in the r.h.s. of (C11) vanish forany 2-surface N2 compact without boundary.

Here lim means that the integral over N2 mustbe evaluated at the limiting boundary value(s) of v. Now let us choose ˜Ωsuch that ˜Ω−1 = v.Then the identity (C12) becomesZdvd2y√h hAChBDCABCD = lim vZd2y√h hAChBDCABCD. (C13)Since√h = eσ in our previous notation, the volume integral (3.18) becomesE = 13Zdvd2y eσhAChBDCABCD = 13 lim vZd2y eσhAChBDCABCD,(C14)which is the desired expression for the Bondi surface integral as v approaches to inﬁnity.APPENDIX D: THE POSITIVE-DEFINITENESS OF κ2±Here we show that κ2± is positive-deﬁnite.

Let us introduce super indices A′, B′ for thesymmetric combinations (ac) and (bd), respectively, so thatρA′ := ρac,ρB′ := ρbd. (D1)Deﬁne the supermetric GA′B′ and its inverse GA′B′ by34

GA′B′ := 12(ρabρcd + ρadρcb),GA′B′ := 12(ρabρcd + ρadρcb),(D2)such thatGA′E′GE′B′ = δA′B′,whereδA′B′ = 12(δabδcd + δadδcb). (D3)This supermetric raises and lowers the super indicesGA′B′ρB′ = ρA′,GA′B′ρB′ = ρA′,(D4)and has a positive-deﬁnite signature since it becomesGA′B′ = diag(+1, +1/2, +1)forρab = δab.

(D5)Therefore it follows thatκ2± = 18ρabρcd(D±ρac)(D±ρbd)= 18GA′B′(D±ρA′)(D±ρB′) ≥0. (D6)35

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A Proposal of Positive-Deﬁnite Local Gravitational

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