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We present the (1+1)-dimensional method for studying general relativity of

arXiv:gr-qc/9212015v1 29 Dec 1992(1+1)-Dimensional Methods forGeneral RelativityJong-Hyuk Yoon ∗AbstractWe present the (1+1)-dimensional method for studying general relativity of4-dimensions. We first discuss the general formalism, and subsequently drawattention to the algebraically special class of space-times, following the Petrovclassification.

It is shown that this class of space-times can be described by the(1+1)-dimensional Yang-Mills action interacting with matter fields, with thespacial diffeomorphisms of the 2-surface as the gauge symmetry. The constraintappears polynomial in part, whereas the non-polynomial part is a non-linearsigma model type in (1+1)-dimensions.

It is also shown that the representationsof w∞-gravity appear naturally as special cases of this description, and wediscuss briefly the w∞-geometry in term of the fibre bundle.1.IntroductionFor past years many 2-dimensional field theories have been intensively studied aslaboratories for many theoretical issues, due to great mathematical simplicities thatoften exist in 2-dimensional systems.Recently these 2-dimensional field theorieshave received considerable attention, for different reasons, in connection with generalrelativistic systems of 4-dimensions, such as self-dual spaces [1] and the black-holespace-times [2, 3]. These 2-dimensional formulations of self-dual spaces and black-hole space-times of allow, in principle, many 2-dimensional field theoretic methodsdeveloped in the past relevant for the description of the physics of 4-dimensions.This raises an intriguing question as to whether it is also possible to describe gen-eral relativity itself as a 2-dimensional field theory.

Recently we have shown thatsuch a description is indeed possible, and obtained, at least formally, the correspond-ing (1+1)-dimensional action principle based on the (2+2)-decomposition of gen-eral space-times1[4]. In particular, the algebraically special class of space-times (the∗Center for Theoretical Physics and Department of Physics, Seoul National University, Seoul 151-742, Korea.

e-mail address: SNU00162@KRSNUCC1.Bitnet. This work was partially supported bythe Ministry of Education and by the Korea Science and Engineering Foundation.1Here we are viewing space-times of 4-dimensions as locally fibrated, M1+1 × N2, with M1+1 asthe base manifold of signature (−, +) and N2 as the 2-dimensional fibre space of signature (+, +).1

J.H. Yoon :(1+1)-Dimensional MethodsPetrov type II), following the Petrov classification [5], was studied as an illustrationfrom this perspective, and the (1+1)-dimensional action principle and the constraintsfor this class were identified [6].

In this (2+2)-decomposition general relativity showsup as a (1+1)-dimensional gauge theory interacting with (1+1)-dimensional matterfields, with the minimal coupling to the gauge fields, where the gauge symmetry isthe diffeomorphisms of the fibre of 2-spacial dimensions. In this article we shall re-view our recent attempts of the (1+1)-dimensional formulation of general relativity.This article is organized as follows.

In section 2, we present the general formalismof (2+2)-decomposition of general relativity, and establish the corresponding (1+1)-dimensional action principle.In section 3, we draw attention to the algebraically special class of space-times [5, 7],following the Petrov classification, and present the (1+1)-dimensional action principlefor this entire class of space-times. We shall show that the spacial diffeomorphisms ofthe 2-surface becomes the gauge fixing condition in this description.

The constraintis polynomial in part, whereas the non-polynomial term is a non-linear sigma modeltype in (1+1)-dimensions. As such, this formulation might render the problem of theconstraints of general relativity manageable, at least formally.In section 4, we discuss the realizations of the so-called w∞-gravity as special cases ofthis description.

We find the fibre bundle as the natural framework for the geometricdescription of w∞-gravity, whose geometric understanding was lacking so far [8, 9].In this picture the local gauge fields for w∞-gravity are identified as the connectionsvalued in the infinite dimensional Lie algebra associated with the area-preservingdiffeomorphisms of the 2-dimensional fibre. Due to this picture of w∞-geometry, weare able to construct field theoretic realizations of w∞-gravity in a straightforwardway.

In section 5, we summarize this review and discuss a few problems for the futureinvestigations.2. (2 + 2)-decomposition of general relativityConsider a 4-dimensional manifold P4 ≃M1+1 × N2, equipped with a metric gAB(A, B, · · · = 0, 1, 2, 3)2.

Let ∂µ = ∂/∂xµ (µ, ν, · · · = 0, 1) and ∂a = ∂/∂ya (a, b, · · · =2, 3) be a coordinate basis of M1+1 and N2, respectively, and choose ∂A = (∂µ, ∂a) asa coordinate basis of P4. In this basis the most general metric on P4 can be writtenas [10]ds2 = φabdyadyb +γµν + φabA aµ A bνdxµdxν + 2φabA bµ dxµdya.

(2.1)2From here on, we shall distinguish the two manifolds by their signatures to avoid confusion.Namely, M1+1 shall be referred to as the (1+1)-dimensional manifold and N2 as 2-dimensionalmanifold.2

J.H. Yoon :(1+1)-Dimensional MethodsFormally this is quite similar to the ‘dimensional reduction’ in Kaluza-Klein theory,where N2 is regarded as the ‘internal’ fibre 3 and M1+1 as the ‘space-time’.

In thestandard Kaluza-Klein reduction one assumes a restriction on the metric, namely, anisometry condition, to make A aµ a gauge field associated with the isometry group.Here, however, we do not assume any isometry condition, and allow all the fields todepend arbitrarily on both xµ and ya. Nevertheless A aµ (x, y) can still be identified asa connection, but now associated with an infinite dimensional diffeomorphism groupdiffN2.

To show this, let us consider the following diffeomorphism of N2,y′a = y′a(yb, xµ),x′µ = xµ. (2.2)Under these transformations, we findγ′µν(y′, x) = γµν(y, x),(2.3a)φ′ab(y′, x) = ∂yc∂y′a∂yd∂y′bφcd(y, x),(2.3b)A′aµ (y′, x) = ∂y′a∂yc A cµ (y, x) −∂µy′a.

(2.3c)For the corresponding infinitesimal variations such thatδya = ξa(yb, xµ),δxµ = 0,(2.4)(2.3) becomeδγµν=−[ξ, γµν] = −£ξγµν = −ξc∂cγµν,(2.5a)δφab=−[ξ, φ]ab = −£ξφab=−ξc∂cφab −(∂aξc)φcb −(∂bξc)φac,(2.5b)δA aµ=−∂µξa + [Aµ, ξ]a = −∂µξa + £Aµξa=−∂µξa + (Acµ∂cξa −ξc∂cA aµ ),(2.5c)where £ξ represents the Lie derivative along the vector fields ξ = ξa∂a, and actsonly on the ‘internal’ indices a, b, etc.Notice that the Lie derivative, an infinitedimensional generalization of the finite dimensional matrix commutators, appearsnaturally. Clearly (2.4) defines a gauge transformation which leaves the line element(2.1) invariant.

Associated with this gauge transformation, the covariant derivativeDµ is defined byDµ = ∂µ −£Aµ,(2.6)where the Lie derivative is taken along the vector field Aµ = A aµ ∂a.With thisdefinition, we haveδA aµ = −Dµξa,(2.7)3For the algebraically special class of space-times we shall consider in section 3, the fibre spaceN2 may be interpreted as the physical transverse wave-surface [5].3

J.H. Yoon :(1+1)-Dimensional Methodswhich clearly indicates that A aµ is the gauge field valued in the infinite dimensionalLie algebra associated with the diffeomorphisms of N2.

Moreover the transformationproperties (2.3a) and (2.3b) show that γµν and φab are a scalar and tensor field,respectively, under diffN2. The field strength Faµν corresponding to A aµ can now bedefined as[Dµ, Dν] = −Faµν ∂a = −{∂µA aν −∂νA aµ −[Aµ, Aν]a}∂a.

(2.8)Notice that the field strength transforms covariantly under the infinitesimal transfor-mation (2.4),δFaµν = −[ξ, Fµν]a = −£ξFaµν . (2.9)To find the (1+1)-dimensional action principle of general relativity, we must computethe scalar curvature of space-times in the (2+2)-decomposition.

For this purpose itis convenient to introduce the following non-coordinate basis ˆ∂A = (ˆ∂µ, ˆ∂a) where [11]ˆ∂µ ≡∂µ −A aµ (x, y)∂a,ˆ∂a ≡∂a . (2.10)From the definition we have[ˆ∂A, ˆ∂B] = fCAB (x, y)ˆ∂C,(2.11)where the structure coefficients fCABare given byfaµν = −Faµν ,fbµa = −fbaµ = ∂aA bµ ,fCAB= 0,otherwise.

(2.12)The virtue of this basis is that it brings the metric (2.1) into a block diagonal formgAB = γµν00φab,(2.13)which drastically simplifies the computation of the scalar curvature. In this basis theLevi-Civita connections are given byΓCAB = 12gCD(ˆ∂AgBD + ˆ∂BgAD −ˆ∂DgAB) + 12gCD(fABD −fBDA −fADB),(2.14)where fABC = gCDfDAB .

For completeness, we present the connection coefficients incomponents,Γαµν = 12γαβˆ∂µγνβ + ˆ∂νγµβ −ˆ∂βγµν,Γaµν = −12φab∂bγµν −12Faµν ,4

J.H. Yoon :(1+1)-Dimensional MethodsΓνµa = Γνaµ = 12γνα∂aγµα + 12γναφabFbµα ,Γbµa = 12φbc ˆ∂µφac + 12∂aA bµ −12φbcφae∂cA eµ ,Γbaµ = 12φbc ˆ∂µφac −12∂aA bµ −12φbcφae∂cA eµ ,Γµab = −12γµν ˆ∂νφab + 12γµνφac∂bA cν + 12γµνφbc∂aA cν ,Γcab = 12φcd∂aφbd + ∂bφad −∂dφab.

(2.15)For later purposes it is useful to have the following identities,Γααµ = 12γαβ ˆ∂µγαβ,Γaaµ = 12φab ˆ∂µφab −∂aA aµ ,(2.16a)Γββa = 12γαβ∂aγαβ,Γbba = 12φbc∂aφbc. (2.16b)The curvature tensors are defined asRDABC= ˆ∂AΓDBC −ˆ∂BΓDAC+ ΓDAE ΓEBC −ΓDBE ΓEAC −fEAB ΓDEC ,RAC = RBABC ,R = gACRAC.

(2.17)Explicitly, the scalar curvature R is given byR = γµν(Rαµαν+ Raµaν ) + φab(Rcacb + Rµaµb ) ,(2.18)which becomes, after a lengthy computation,R=γµνRµν + φacRac + 14φabγµνγαβFaµα Fbνβ+14γµνφabφcdn(Dµφac)(Dνφbd) −(Dµφab)(Dνφcd)o+14φabγµνγαβn(∂aγµα)(∂bγνβ) −(∂aγµν)(∂bγαβ)o+ ∇AjA,(2.19)where Rµν and Rac are defined byRµν = ˆ∂µΓααν −ˆ∂αΓαµν + Γαµβ Γβαν −Γββα Γαµν ,(2.20a)Rac = ∂aΓbbc −∂bΓbac + Γbad Γdbc −Γddb Γbac . (2.20b)The last term in (2.19) is given by∇AjA = ∇µjµ + ∇aja,(2.21a)∇µjµ =ˆ∂µ + Γααµ + Γccµjµ,(2.21b)∇aja =∂a + Γcca + Γααaja,(2.21c)5

J.H. Yoon :(1+1)-Dimensional Methodswhere jµ and ja are given byjµ = γµνφab ˆ∂νφab −2∂aA aν,ja = φabγµν∂bγµν.

(2.22)That ∇AjA is a surface term in the action integral can be seen easily, using (2.16).For instance let us show that √−γ√φ∇µjµ is a surface term, where γ = detγµν andφ = detφab. From (2.21b) we have√−γqφ∇µjµ = √−γqφh∂µjµ −A aµ ∂ajµ +Γααµ + Γccµjµi.

(2.23)The first term in the r.h.s. of (2.23) can be written as√−γqφ∂µjµ = −12√−γqφγαβ∂µγαβ + φab∂µφabjµ + ∂µ√−γqφjµ,(2.24)and for the second term, we have√−γqφA aµ ∂ajµ = −√−γqφhnA aµ (Γααa + Γbba ) + ∂aA aµojµi+ ∂a√−γqφA aµ jµ.

(2.25)The last two terms in the r. h. s. of (2.23) becomes, using (2.16),√−γqφΓααµ + Γccµjµ=√−γqφ12γαβ∂µγαβ −A aµ Γααa + 12φab∂µφab−A aµ Γbba −∂aA aµjµ. (2.26)Putting (2.24), (2.25), and (2.26) into (2.23), we find that it is a total divergenceterm,√−γqφ∇µjµ = ∂µ√−γqφjµ−∂a√−γqφA aµ ja,(2.27)which we may ignore.

Similarly, √−γ√φ∇aja is also a surface term. This altogethershows that √−γ√φ∇AjA is indeed a total divergence term.At this point it is important to notice the followings.

First, Dµφab, written asDµφab=∂µφab −£Aµφab=∂µφab −nA cµ (∂cφab) + (∂aA cµ )φcb + (∂bA cµ )φaco(2.28)indeed transforms covariantly under the infinitesimal diffeomorphism (2.4),δ(Dµφab) = −£ξ(Dµφab) = −[ξ, Dµφ]ab. (2.29)Second, the derivative ˆ∂µ, when applied to γµν, becomes the covariant derivativeˆ∂µγαβ = ∂µγαβ −£Aµγαβ = Dµγαβ,(2.30)6

J.H. Yoon :(1+1)-Dimensional Methodsso that ˆ∂µγαβ transforms covariantlyδ(ˆ∂µγαβ) = −£ξ(Dµγαβ) = −[ξ, Dµγαβ].

(2.31)These observations play an important role when we discuss the gauge invariance ofthe theory under diffN2. It is worth mentioning here that, from (2.20a) and (2.30),Rµν becomes the ‘covariantized’ Ricci tensorRµν = DµΓααν −DαΓαµν + Γαµβ Γβαν −Γββα Γαµν ,(2.32)as Γαµν ’s do not involve the ‘internal’ indices a, b, etc.

Thus we might call γµνRµν asthe ‘gauged’ gravity action in (1+1)-dimensions [12].With the scalar curvature at hand, one can easily write down the lagrangian for theEinstein-Hilbert action on P4. From (2.19) we haveL2=−√−γqφhγµνRµν + φabRab + 14φabFaµν F µνb+14γµνφabφcdn(Dµφac)(Dνφbd) −(Dµφab)(Dνφcd)o+14φabγµνγαβn(∂aγµα)(∂bγνβ) −(∂aγµν)(∂bγαβ)oi,(2.33)neglecting the total divergence term (2.27).

Clearly the action principle describes a(1+1)-dimensional field theory which is invariant under the gauge transformation ofdiffN2, as the gauge field A aµ couples minimally to both γµν and φab. Therefore eachterm in (2.33) is invariant under diffN2.

To understand the physical contents of thetheory we notice the followings. First, unlike the ordinary gravity, the metric γµνof M1+1 here is ‘charged’, because it couples to A aµ (with the coupling constant 1).Second, the metric φab of N2 can be identified as a non-linear sigma field, whose self-interaction potential is determined by the scalar curvature φabRab of N2.

The theorytherefore describes a gauge theory of diffN2 interacting with the ‘gauged’ gravity andthe non-linear sigma field on M1+1.3.Algebraically special class of space-timesIn contrast to the cases of the self-dual spaces and black-hole space-times, the (1+1)-dimensional action principle for general space-times, as we derived in the previoussection, appears to be rather formal and consequently, of little practical use. In thissection we therefore draw attention following the Petrov classification to a specificclass of space-times, namely, the algebraically special class, and interpret the entireclass from the (1+1)-dimensional point of view.

It turns out that space-times of thisclass can be formulated as (1+1)-dimensional field theory in a remarkably simpleform.7

J.H. Yoon :(1+1)-Dimensional MethodsLet us consider a class of space-times that contain a twist-free null vector field kA.These space-times belong to the algebraically special class of space-times, accordingto the Petrov classification.

This class of space-times is rather broad, since most ofthe known exact solutions of the Einstein’s equations are algebraically special. Beingtwist-free, the null vector field may be chosen to be a gradient field, so that kA = ∂Aufor some function u.

The null hypersurface N2 defined by u = constant spans the2-dimensional subspace for which we introduce two space-like coordinates ya. Thegeneral line element for this class has the form [5, 7]ds2 = φabdyadyb −2du(dv + madya + Hdu),(3.1)where v is the affine parameter, and φab, ma and H are functions of all of the fourcoordinates (u, v, ya), as we assume no Killing vector fields.For the class of space-times (3.1), we shall find the (1+1)-dimensional action principledefined on the (u, v)-surface.

For this purpose let us first introduce the ‘light-cone’coordinates (u, v) such thatu = 1√2(x0 + x1),v = 1√2(x0 −x1),(3.2)and define A au and A avA au = 1√2(A a0 + A a1 ),A av = 1√2(A a0 −A a1 ). (3.3)For γµν, we assume the Polyakov ansatz [13]γµν = −2h−1−10,γµν = 0−1−12h,(detγµν = −1),(3.4)in the (u, v)-coordinates.

Then the line element (2.1) becomesds2=φabdyadyb −2dudv −2h(du)2 + φab(A au du + A av dv)(A bu du + A bv dv)+2φab(A au du + A av dv)dyb. (3.5)If we choose the ‘light-cone’ gauge 4 A av = 0, then this becomesds2 = φabdyadyb −2 duhdv −φabA bu dya +h −12φabA au A budui.

(3.6)A comparison of (3.1) and (3.6) tells us that if the following identificationsma = −φabA bu ,H = h −12φabA au A bu(3.7)4 Here we are referring to the disposable gauge degrees of freedom in the action. There could betopological obstruction against globalizing this choice, as the general coordinate transformation ofN2 corresponds to the gauge transformation.8

J.H. Yoon :(1+1)-Dimensional Methodsare made, then the two line elements are the same.

This shows that the Polyakovansatz (3.4) amounts to the restriction (modulo the gauge choice A av= 0) to thealgebraically special class of space-times that contain a twist-free null vector field.Let us now examine the transformation properties of h, φab, A au , and A av under thediffeomorphism of N2,y′a = y′a(yb, u, v),u′ = u,v′ = v.(3.8)Under these transformations, we find thath′(y′, u, v) = h(y, u, v),(3.9a)φ′ab(y′, u, v) = ∂yc∂y′a∂yd∂y′bφcd(y, u, v),(3.9b)A′au (y′, u, v) = ∂y′a∂yc A cu (y, u, v) −∂uy′a,(3.9c)A′av (y′, u, v) = −∂vy′a,(3.9d)which become, under the infinitesimal variations, δya = ξa(y, u, v) and δxµ = 0,δh = −[ξ, h] = −ξa∂ah,(3.10a)δφab = −[ξ, φ]ab = −ξc∂cφab −(∂aξc)φcb −(∂bξc)φac,(3.10b)δA au = −Duξa = −∂uξa + [Au, ξ]a,(3.10c)δA av = −∂vξa. (3.10d)This shows that h and φab are a scalar and tensor field, respectively, and A au andA avare the gauge fields valued in the infinite dimensional Lie algebra associatedwith the group of diffeomorphisms of N2.

That A avis a pure gauge is clear, as itdepends on the gauge function ξa only. Therefore it can be always set to zero, atleast locally, by a suitable coordinate transformation (3.8).

To maintain the explicitgauge invariance, however, we shall work with the line element (3.5) in the following,with the understanding that A av is a pure gauge.Let us now proceed to write down the action principle for (3.5) in terms of the fields h,φab, A au , and A av . For this purpose, it is convenient to decompose the 2-dimensionalmetric φab into the conformal classesφab = Ωρab,(Ω> 0 and detρab = 1).

(3.11)The kinetic term K of φab in (2.33) then becomesK≡14√−γqφγµνφabφcdn(Dµφac)(Dνφbd) −(Dµφab)(Dνφcd)o=−(DµΩ)22Ω+ 14Ωγµνρabρcd(Dµρac)(Dνρbd)=−12eσ(Dµσ)2 + 14eσγµνρabρcd(Dµρac)(Dνρbd),(3.12)9

J.H. Yoon :(1+1)-Dimensional Methodswhere we defined σ by σ = lnΩ, and the covariant derivatives DµΩ, Dµρab, and DµσareDµΩ= ∂µΩ−A aµ ∂aΩ−(∂aA aµ )Ω,(3.13a)Dµρab = ∂µρab −[Aµ, ρ]ab + (∂cA cµ )ρab,(3.13b)Dµσ = ∂µσ −A aµ ∂aσ −∂aA aµ ,(3.13c)respectively, where [Aµ, ρ]ab is given by[Aµ, ρ]ab = A cµ ∂cρab + (∂aA cµ )ρcb + (∂bA cµ )ρac.

(3.14)The inclusion of the divergence term ∂aA aµ in (3.13) is necessary to ensure (3.13)transform covariantly (as the tensor fields) under diffN2, since Ωand ρab are thetensor densities of weight −1 and +1, respectively. Using the ansatz (3.4), the kineticterm (3.12) becomesK=eσ(D+σ)(D−σ) −12eσρabρcd(D+ρac)(D−ρbd)−heσn(D−σ)2 −12ρabρcd(D−ρac)(D−ρbd)o,(3.15)where +(−) stands for u(v).

The Polyakov ansatz (3.4) simplifies enormously theremaining terms in the action (2.33), as we now show. Let us first notice that detγµν =−1.

Therefore the term√−γqφφacRac =qφφacRac(3.16)can be removed from the action being a surface term. Moreover, since we haveγµν∂aγµν =2√−γ ∂a√−γ = 0,(3.17)the last term in the action (2.33) vanishes.

Furthermore, one can easily verify thatφabγµνγαβ(∂aγµα)(∂bγνβ)=φab(∂aγ++)γ+−(∂bγ−α)γα+=0,(3.18)since ∂bγ−α = 0. The only remaining terms that contribute to the action (2.33) arethus the (1+1)-dimensional Yang-Mills action and the ‘gauged’ gravity action.

TheYang-Mills action becomes14φabFaµν F µνb = −12eσρabFa+−Fb+−. (3.19)To express the ‘gauged’ Ricci scalar γµνRµν in terms of h and A av , etc., we have tocompute the Levi-Civita connections first.

They are given byΓ+++ = −D−h,Γ−++ = D+h + 2hD−h,Γ−+−= Γ−−+ = D−h,(3.20)10

J.H. Yoon :(1+1)-Dimensional Methodsand vanishing otherwise.

Thus the ‘gauged’ Ricci tensor becomesR+−= R−+ = −D2−h,R−−= 0. (3.21)From (3.4) and (3.21), the ‘gauged’ Ricci scalar γµνRµν is given byγµνRµν = 2γ+−R+−= 2D2−h,(3.22)since γ++ = R−−= 0.

Putting together (3.15), (3.19), and (3.22) into (2.33), theaction becomesL2=−12e2σρabFa+−Fb+−+ eσ(D+σ)(D−σ) −12eσρabρcd(D+ρac)(D−ρbd)+heσn12ρabρcd(D−ρac)(D−ρbd) −(D−σ)2o+ 2eσD2−h. (3.23)The last term in (3.23) can be expressed aseσD2−h=eσ∂−−A b−∂b∂−h −A a−∂ah=eσn∂2−h −∂−A a−∂ah−A a−∂a(D−h)o=−(∂−eσ)(∂−h) + (∂−eσ)A a−∂ah+ ∂aeσA a−(D−h)+∂−eσ∂−h−∂−eσA a−∂ah−∂aeσA a−D−h≃−eσ(∂−σ)(D−h) + eσA a−(∂aσ)(D−h) + eσ(∂aA a−)(D−h)=−eσ(D−σ)(D−h),(3.24)where we dropped the surface term and used (3.13c).

This can be written aseσ(D−σ)(D−h)=eσ(D−σ)∂−h −A a−∂ah=−h∂−eσD−σ+ h∂aeσA a−D−σ+ ∂−heσD−σ−∂aheσA a−D−σ≃−heσ(∂−σ)(D−σ) −heσ∂−(D−σ) + heσA a−(∂aσ)(D−σ)+heσ(∂aA a−)(D−σ) + heσA a−∂a(D−σ)=−heσnD2−σ + (D−σ)2o. (3.25)We therefore haveeσD2−h ≃heσnD2−σ + (D−σ)2o,(3.26)neglecting the surface terms.

The resulting (1+1)-dimensional action principle there-fore becomesL2=−12e2σρabFa+−Fb+−+ eσ(D+σ)(D−σ) −12eσρabρcd(D+ρac)(D−ρbd)+heσn2D2−σ + (D−σ)2 + 12ρabρcd(D−ρac)(D−ρbd)o,(3.27)11

J.H. Yoon :(1+1)-Dimensional Methodsup to the surface terms.

Notice that h is a Lagrange multiplier, whose variation yieldsthe constraintH0 = D2−σ + 12(D−σ)2 + 14ρabρcd(D−ρac)(D−ρbd) ≈0. (3.28)From this (1+1)-dimensional point of view, h is the lapse function (or a pure gauge)that prescribes how to ‘move forward in the u-time’, carrying the surface N2 at eachpoint of the section u = constant.

The constraint, H0 ≈0, is polynomial in σ andA a−, and contains a non-polynomial term of the non-linear sigma model type but in(1+1)-dimensions, where such models often admit exact solutions. This allows us toview the problem of the constraints of general relativity [14] from a new perspective.We now have the (1+1)-dimensional action principle for the algebraically special classof space-times that contain a twist-free null vector field.

It is described by the Yang-Mills action, interacting with the fields σ and ρab on the ‘flat’ (1+1)-dimensionalsurface, which however must satisfy the constraint H0 ≈0. (The flatness of the(1+1)-dimensional surface can be seen from the fact that the lapse function, h, canbe chosen as zero, provided that H0 ≈0 holds.) The infinite dimensional group ofthe diffeomorphisms of N2 is built-in as the local gauge symmetry, via the minimalcouplings to the gauge fields.Having formulated the algebraically special class of space-times as a gauge theory on(1+1)-dimensions, we may wish to apply varieties of field theoretic methods developedin (1+1)-dimensions.

For instance, the action (3.27) can be viewed as the bosonizedform [15] of some version of the (1+1)-dimensional QCD in the infinite dimensionallimit of the gauge group [16]. For small fluctuations of σ, the action (3.27) becomesL2 = −12ρabFa+−Fb+−+ (D+σ)(D−σ) −12ρabρcd(D+ρac)(D−ρbd),(3.29)modulo the constraint H0 ≈0.

It is beyond the scope of this article to investigatethese theories as (1+1)-dimensional quantum field theories. However, this formula-tion raises many intriguing questions such as: would there be any phase transitionin quantum gravity as viewed as the (1+1)-dimensional quantum field theories?

If itdoes, then what does that mean in quantum geometrical terms? Thus, general rela-tivity, as viewed from the (1+1)-dimensional perspective, renders itself to be studiedas a gauge theory in full sense [17], at least for the class of space-times discussed here.4.w∞-gravity as special casesIn the previous section we derived the action principle on (1+1)-dimensions as thevantage point of studying general relativity for this algebraically special class ofspace-times.

We now ask different but related questions: what kinds of other (1+1)-dimensional field theories related to this problem can we study? For these, let us12

J.H. Yoon :(1+1)-Dimensional Methodsconsider the case where the local gauge symmetry is replaced by the area-preservingdiffeomorphisms of N2.

(For these varieties of field theories, we shall drop the con-straint (3.28) for the moment. It is at this point that we are departing from generalrelativity.) This class of field theories naturally realizes the so-called w∞-gravity [8, 9]in a linear and geometric way, as we now describe.The area-preserving diffeomorphisms are generated by the vector fields ξa, tangentto the surface N2 and divergence-free,∂aξa = 0.

(4.1)Let us find the gauge fields A a± compatible with the divergence-free condition (4.1).Taking the divergence of both sides of (3.10c) and (3.10d), we have∂aδA a± = −∂±(∂aξa) + ∂a[A±, ξ]a. (4.2)This shows that the condition ∂aA a± = 0 is invariant under the area-preserving diffeo-morphisms, and characterizes a special subclass of the gauge fields, compatible withthe condition (4.1).

Moreover, when ∂aA a± = 0, the fields ρab and σ behave under thearea-preserving diffeomorphisms as a tensor and a scalar field, respectively, as (3.13b)and (3.13c) suggest. Indeed, the Jacobian for the area-preserving diffeomorphisms isjust 1, disregarding the distinction between the tensor fields and the tensor densities.The (1+1)-dimensional action principle now becomesL′2 = −12e2σρabFa+−Fb+−+ eσ(D+σ)(D−σ) −12eσρabρcd(D+ρac)(D−ρbd),(4.3)where Dµσ, Dµρab, and Fa+−areD±σ = ∂±σ −A a± ∂aσ,(4.4a)D±ρab = ∂±ρab −[A±, ρ]ab,(4.4b)Fa+−= ∂+A a−−∂−A a+ −[A+, A−]a.

(4.4c)Under the infinitesimal variationsδya = ξa(y, u, v),δxµ = 0,(∂aξa = 0),(4.5)the fields transform asδσ = −[ξ, σ] = −ξa∂aσ,(4.6a)δρab = −[ξ, ρ]ab = −ξc∂cρab −(∂aξc)ρcb −(∂bξc)ρac,(4.6b)δA a+ = −D+ξa = −∂+ξa + [A+, ξ]a,(4.6c)δA a−= −∂−ξa,(4.6d)which shows that it is a linear realization of the area-preserving diffeomorphisms.The geometric picture of the action principle (4.3) is now clear: it is equipped with13

J.H. Yoon :(1+1)-Dimensional Methodsthe natural bundle structure, where the gauge fields are the connections valued inthe Lie algebra associated with the area-preserving diffeomorphisms of N2.

Thus theaction principle (4.3) provides a field theoretical realization of w∞-gravity [8, 9] in alinear and geometric way, with the built-in area-preserving diffeomorphisms as thelocal gauge symmetry.With this picture of w∞-geometry at hands, we may construct as many differentrealizations of w∞-gravity as one wishes. The simplest example would be a singlereal scalar field representation, which we may writeL′′2 = −12Fa+−Fa+−+ (D+σ)(D−σ),(4.7)where we used δab in the summation, and D±σ and Fa+−are as given in (4.4a) and(4.4c).

By choosing the gauge A a−= 0 and eliminating the auxiliary field A a+ interms of σ using the equations of motion of A a+ , we recognize (4.7) a single realscalar field realization of w∞-gravity. In presence of the auxiliary field A a+ , (4.7)provides an example of the linearized realization of w∞-gravity for a single real scalarfield.

It would be interesting to see if the representation (4.7) is related to the onesconstructed in the literatures [8, 9].5.DiscussionIn this review, we examined space-times of 4-dimensions from a (1+1)-dimensionalpoint of view. That general relativity admits such a description is rather surpris-ing, even though the action principle in general appears rather formal.

For the alge-braically special class of space-times, however, the (1+1)-dimensional action principle,as we have shown here, is formulated as the Yang-Mills type gauge theories interact-ing with matter fields, where the infinite dimensional group of diffeomorphisms of the2-surface becomes the ‘internal’ gauge symmetry. The constraint conjugate to thelapse function appears partly as polynomial.

The non-polynomial part is a typicalnon-linear sigma model type in (1+1)-dimensions, where such models often admitexact solutions. We also discussed the so-called w∞-gravity as special cases of thealgebraically special class of space-times.

The detailed study of the w∞-gravity andits geometry in terms of the fibre bundle will be presented somewhere else.We wish to conclude with a few remarks. First, one might be interested in findingexact solutions of the Einstein’s equations in this formulation.

Various two (or more)Killing reductions of the Einstein’s equations have been known for sometime whichled to the discovery of many exact solutions to the Einstein’s equations, by makingthe system essentially two (or lower) dimensional. In our formulation, the Einstein’sequations are already put into a two dimensional form without such assumptions.This might be useful in finding new solutions of the Einstein’s equations, whichpossess no Killing symmetries5.5Interestingly, there are exact solutions of the Einstein’s equations which possess no space-time14

J.H. Yoon :(1+1)-Dimensional MethodsSecond, we need to find the constraint algebras for the algebraically special classof space-times explicitly in terms of the variables we used here.

As we have shownhere, the splitting of the metric variables into the gauge fields and the ‘matter fields’is indeed suitable for the description of general relativity as Yang-Mills type gaugetheories in (1+1)-dimensions. It remains to study the constraint algebras in detail tosee if the ordering problem in the constraints of general relativity becomes manageablein terms of these variables.Lastly, that the Lie algebra of SU(N) for large N can be used as an approximationof the infinite dimensional Lie algebra of the area-preserving diffeomorphisms of the2-surface has been suggested as a way of ‘regulating’ the area-preserving diffeomor-phisms.

In connection with the problem regarding the regularization of quantumgravity in this formulation, one might wonder as to whether it is also possible to ap-proximate the diffeomorphism algebras of the 2-surface in terms of finite dimensionalLie algebras in a certain limit. There seem to be many interesting questions to beasked about general relativity in this formulation.References[1] Q.H.

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