JHEP09(2017)145 Springer August 20, 2017 : September 10, 2017 September 28, 2017 : : g known to provide article interacting with Received . It is then shown how andard Belinfante method appropriate superpotential the Einstein tensor; ii) the Accepted Published en. It is used to recast the vity as a theory of the self- ation of the graviton vertices, related definitions of energy- ts the graviton field equation d of converting to a covariant nstructed which connect to: i) the Published for SISSA by https://doi.org/10.1007/JHEP09(2017)145 . 3 1708.03977 The Authors. c Classical Theories of Gravity, Models of Quantum Gravity

The so-called ΓΓ-form of the gravitational Lagrangian, lon , [email protected] Los Angeles, CA 90095-1547, U.S.A. E-mail: Mani L. Bhaumik InstituteDepartment for of Theoretical Physics Physics, and Astronomy, UCLA, Open Access Article funded by SCOAP ArXiv ePrint: common definition consisting simplyLandau-Lifshitz of definition. the nonlinear part of Keywords: applicable to any fieldin carrying a nonzero spin. formrepresentations complying This based represen with on the other,momentum precepts at (pseudo)tensors of face are standard value allterms. field completely related Specifically, un theory by the the superpotentials addition are of explicitly co is taken as theinteracting graviton. starting point A forformulation straightforward but by discussing general the General metho introductionEinstein Relati of field a equation referencethe as metric canonical the energy-momentum is equation tensor giv symmetrized of by motion the of st a spin-2 p Abstract: its most compact expression as well as the most efficient gener E.T. Tomboulis On the ‘simple’ formthe of self-interacting the graviton gravitational action and JHEP09(2017)145 ) 1 4 7 12 κλ , or g (1.4) (1.3) (1.5) gives β µν ∂ µν ( gg g i − βλ √ s in a more 1), and we use g − = , ακ 1 g µν − , µν g 1 g − , ) in terms of ], the Einstein-Hilbert + 2 , or 1 1.3 µν µκ g g n ( . βν = diag(1  . φ g Einstein field equations. It is , or η ] (1.2) y (GR) [ satisfies homogeneity properties
f the Lagrangian, which contains β αβ α αλ µν ) (1.1) g Γ L α µν g g B 2 ( Γ α α µν ∂ − Γ µν gR g + − κλ − g − L , but, after the isolation of the boundary √ β αν L − µν x [ ν µν Γ g 4 Γ . When needed, functional dependence is, as usual, x d α µβ 4 αβ αµ Γ d ... of theory with fields – 1 – g Γ) + . Z g  ∂ ) − S 2 − Z 1 g κ µν − κλ 2 λ µλ 1 g νλ − κ Γ Γ gg β g = ν ∂ √ ∂ ∂ − ( = µκ EH g )( g = √ S g EH − = + α ( αβ = S √ g B h L λ µλν ) , the Minkowski metric is denoted by R ) = µν g αµνβκλ πG g = ( α K ] for action functional ∂ ) µν gR ( φ = 16 [ R g µν − ) 2 S g κ L − √ α , and their respective derivatives. We return to these point − ∂ √ g ( ( that gives this ΓΓ-form of the Lagrangian. x − 4 x x ], e.g., L · 1 4 4 d √ d d / − Z µν Z Z 2 g 2 2 1 κ 1 1 4 κ κ = Expressing the metric connections (Christoffel symbols) Γ i In the following 1 = ≡ = µν S only first derivatives ofnoteworthy the that metric. Its variation gives the Dropping the total divergence gives the so-called ΓΓ-form o curvature conventions can be written in the form indicated by [ general context below. and (EH) action with g term, it is 1 Introduction As it is well-known since the early days of General Relativit 3 The self-interacting graviton 4 Conclusion Contents 1 Introduction 2 Covariant formulation and may be viewed equally well as a function of either JHEP09(2017)145 . µν on. h (1.6) (1.8) g both i ) 3) are then λ , the starting ) is not a vector asically simple α ≥ g β ices it introduces an B n as field variables. ( ) in powers of κ g g ( µν g µν ) (1.7) K g ) as the starting point, , κλ ng + h 1.5 =0 mmon textbook definition nteracting graviton, which tensor of the gravity field. β ν s 2 alar density which depends | ) ∂ m, the auxiliary field being, l vertices ( ].
β propriate interaction current, ] long time. 4 ng the Einstein field equations g )( , all graviton vertices are then ing ( α η ] for a proof). As a result the ( ( in tensor to the r.h.s. of the field n [ , amounts to simple insertion of 5 µν µ n, the introduction of an auxiliary ic part of the action: sh, h g h αβ h κλ + + host Lagrangian, a disaster for any actual ) g dependence in η s αµνβκλ [ ( µν 2 ) η µν K S νβ g and Γ taken as independent variables, one ) ) + ( κλ g 2) = ) g g µν − νλ ), for many manipulations the symmetrized s n ( g h µν ( µν s β α g g ) 1.5 ∂ – 2 – µα ∂ µ g ( − g x − κ ν is not always apparent. Similar, and related, issues 2)! ( ) 4 α λ d 1 L = − g g κ ] conclusively demonstrating that the self-coupling of a Z n ( + µν ( 3 µ 2 ) 1 , g κ µκ ( 2 sh g ) may be somewhat further simplified by a gauge choice involvin ] = β αβ ) + g ] = 1.5 ν h η η, h g [ ), is not a scalar density and, correspondingly, ( λ g s n ( η, h [ α ∂ − S 1.3 2 g ) gives the most compact form known of the gravitational acti ] that ( S ( √ 4 4 1 − 1.5 . Though this can give some simplification of the graviton vert ) = s µν ( : g ) = ) contains only up to cubic interactions. This was, of course µν g ( g s given by ( in the kinetic term with no change of derivative terms. This b 1.3 ∂ h L αµνβκλ K αµνβκλ K Now, If one adopts a first order formalism with and it inverse It is pointed out in [ 2 µν infinite number ofamplitude interaction computations. vertices in the Fadeev-Popov g should be used. ( obtained by the expansion of the inverse metric The definition of theconsists latter of is simply shifting highly the non-unique. nonlinear part The of most the Einste co come up in thewill representation be of our main GR focus as inas the the the following. theory equation of This of motion amounts the to ofwhich writi self-i a in spin-2 this particle coupled case to is, an ap of course, the energy-momentum pseudo g covariance of many results involving field, in terms ofof which course, one essentially reverts Γ. to This the has first recently order been formalis revisited i point of Deser’s seminalmassless work [ spin-2 particle bootstrapsits to simple structure GR. allows, Conversely, by tak simple Gaussian integratio A very similar, somewhat even simpler form is obtained byThe usi free quadratic part form of Expanding about flat background metric, powers of structure of the gravitational vertices has been known for a density. This is assolely it on should the be since metric there and does its not first exist derivatives a (see, sc e.g., [ Though irrelevant for writing the action ( notes that ( which, via is the Fierz-Paulirecursively action generated for by differentiation a of massless this free spin-2 kinet particle. Al JHEP09(2017)145 r ]. 11 field , is not 10 L variant formulation stein field equations s in fact motivated the heories. by the introduction of a ions of the field equations ] in discussing some of the ears. Different definitions agrangian density pplies to any metric gravity efinitions in asymptotically finitions must be related by below it is shown how this es in specific computations lts and explicitly construct, definitions for the total field um vector in asymptotically ing graviton. This we do in 13 nte symmetrization) [ al energy-momentum tensor or. These and other ad hoc o different representations. A ntly unrelated definitions has 2 ic angular momentum and can nifest. This can only be done metrized canonical energy mo- erpotential is the “improved” energy- we give the non-obvious expres- of a reference metric has indeed in [ andard textbook tensor and the e background metric if the usual f this connection is taken to be a priate superpotential term to the andard procedures of field theory. t face value unrelated definitions. nergy-momentum tensor. Of par- . Doing so means giving a precise 2 ], and references therein, for the purpose of 9 – 3 – and pseudotensors are converted to actual tensors and tenso L The gravitational field equations are then transformed as the 3 ]. 8 ] ensuring vanishing trace in (classical) scale invariant t – 12 6 using the covariant formulation of section 3 Our second and main goal is to relate various ways that the Ein Our purpose in this note is to do two things. First, to give a co Another well-known example of modification by addition of a sup 3 can be written assection the equation of motion of the self-interact addition of appropriate superpotential termsconfusion was in stressed the literature concerningsions these for matters. the Here superpotentials connecting these different, a equation of a spin-2mentum particle tensor. coupled We to then the proceedare Belinfante-sym to related show to how the otherin canonical representat particular, one. the We superpotential present termsLL several that tensor new connect to resu the the st canonical one. The fact that such different de The canonical energy-momentum tensorsymmetric. associated This with is the a feature L be of symmetrized any for field any with spin nonzeroThe by intrins explicit a construction standard involves procedure the addition (Belinfa canonical of an definition. appro relation between various definitions ofticular the interest gravitational here e is a formulation that adheres to the st being used before inverifying special agreement between schemes the as LL4-momentum. in and The the [ simple global procedure geometric belowtheory. is very The general reference and metric a iscase naturally of identified propagation with th over a background is considered. can be accomplishedsymmetric in but a otherwise arbitrary straightforward reference butmetric connection. one, general I a manner reference metric is introduced. Introduction so that quantities such as momentum tensor [ flat spacetimes, butis independence not of obvious theformulation due choice of to of more coordinat the abstractflat lack global spacetimes geometric of [ 4-momentum manifest d covariance. This ha frequently used one isdefinitions the make, Landau-Lifshitz in (LL)definition. particular, pseudotens no The existence referencebeen of to all the the these source canonic disparate, ofshould and some give appare the confusion same in result the for literature the total over field the energy-moment y equations. Other manipulations of the field equations lead t densities, and coordinate independencewith of the all introduction results of is some ma extra structure. In section JHEP09(2017)145 (2.5) (2.9) (2.4) (2.8) (2.2) (2.6) (2.1) (2.7) (2.3) . Note l µν α g ··· γ i ¯ Γ connection ··· µν , valid for any 1 ) to g k α α ρ . β B ··· ¯ ) that the relation 2.1 ∇ α 1 αν ρ β ∂ ρ µν of another arbitrary 2.1 ˆ Γ T ˆ Γ of the metric connec- j = ¯ ) is obtained from the µσ ∇ ms of the + α . α γ µα ∇ ] ˆ Γ ˆ Γ 2.2 B α µν
α − g , j B . ¯ ν µν ∇ X , α . ator g L ¯ ασ  ¯ ∂ ∇ Γ. ( δ σ 2 ρ  − ) may be expressed in terms of υ ˆ Γ ¯ ρ µρ β αβ l − + ∇ α µν α ˆ ˆ Γ Γ 2.5 µν L ˆ Γ − = ··· ) to obtain the useful identity α γαβ − ) and note that 1 α µν δ ˆ Γ h one has i k − µν α ˆ Γ νσ R β 2.1 l g 1.1 g µν + α ··· Z µν − µ g = γ − ··· ¯ ρ R ¯ + 1 µσ ∇ ··· γ ρ β αν ¯ 1 k α ∇ ν µν υ α µν ) yields β β ˆ ˆ Γ Γ + ), in ( ˆ ) Γ ν ρ µν ··· T B α gg 1 α µβ ˆ ¯ 2.3 Γ i α ∇ µσ αµ β – 4 – 2.4 ˆ β µγ − Γ ∇ g g ¯ ∇ T + ˆ β Γ  ν −  √ [ ¯ ∇ g Z i µν µν µν x ρ g X − 4 − ∇ = ν ˆ gg d Γ ) then ( √ ασ β + σ ¯ i ∇ − g l ∇ Z ¯ and briefly remark on the possibility of generalizing the = ∇ ˆ 2.1 Γ is a tensor. Indeed, it follows from ( 2 1 α ρ µν = 0 together with ( √ µρ ρ α 2 ··· ˆ Γ α ˆ 4 1 Γ + ), κ 1 ∇ = = ρ ˆ Γ is the difference of two connections and hence a tensor. µν ( k α − B g ¯ , as given by ( β ∇ h L = 2.2 α µν µνσ ··· ∇ ρ ˆ 1 − Γ = 0, in terms of the derivative operator µρν β ¯ EH ρ R S g T ρ µρ ¯ Γ. For any tensor R µ = ˆ Γ ∇ ¯ Γ). ν ¯ = ( ∇ ) then, and use of the divergence identity ¯ ∇ µνσ = h µν ρ µνσ 2.7 = 0 which is then easily verified to hold by applying ( l ρ R R α µν R ··· ), ( µν 1 gg g ˆ ≡ Γ must hold, i.e., k α − β 2.5 ∇ ··· √ factors in the square bracket on the r.h.s. in ( is any covector. Using ( ), for which 1 µνσ denoting the covariant derivative with connection ¯ Γ + ), ( β g ρ g υ Z ¯ T R ¯ ¯ ∇ ∇ µ 2.4 Insert now It is now straigtforward to express the Riemann tensor in ter We summarize in section ) = ∇ g ˆ with Here By ( Γ’s by using the fact that and condition Γ( theory of the self-interacting graviton beyond GR. 2 Covariant formulation The basic step is to expressed the covariant derivative oper vector density, one obtains The symmetric connection tion Γ( with starting from the defining relation that, as it is manifest from ( where now we define where JHEP09(2017)145 , of µν µν g g tials µν (2.10) (2.15) (2.11) (2.14) (2.12) (2.13) (2.16) g ), with α ¯ ∇ 1 and 1.1 , or of − : g , µν , respectively. and g , i µν α i g ) are well-known µν ρσ ne may specialize ¯ ) ∇ α g g , β ) is then the covari- ¯ κλ ∇ g ¯ 2.13 µν of degree ∇ et. In fact, the same β g ) reduces to ( ¯ µν 2.15 ∇ )–( and µν g , g . ). α )( 2.9 µν  here to generalize in terms ). ( ) g ¯ ∇ g 2.10 ( form und which then serves as the ρ νρ 1.2 . These properties are easily µν
ρσ 1.6 ˆ ¯ Γ g ction as introduced above. g µν ρσ α σ µ  g g g ∂ . βν α β w.r.t. β ρβ g L αµνβκλ becomes + − L ¯ ¯ ∇ ∇ ˆ Γ L − K αµ L , ρ µρ νσ µν g ρ µν ) = 2 = ) ¯ g ˆ g Γ ˆ Γ = 0, and ( ˆ µν Γ µ α µν α ν g ) ∂ g + 2 g 2.2 ¯ − µν L ∇ α µν  g ∂ + g ¯ νσ ∂ − ∇ ρ αν L 2 1 ] α g ˆ ∂ Γ L ), are manifestly a scalar density and a vector µσ β , gives ¯ µν ∇ νσ + ¯ g g – 5 – + ( g ( ¯ α ρα g ∇ ν − 2.6 is a homogeneous function of ∂ β ˆ Γ ∂ α µν = µν ) µρ ), is again given by ( ( ¯  ˆ µν ∇ Γ g ¯ g L g R ¯ ( µν ρσ α R − − µρ g ¯ g µν ¯ g ∇ α µν 1 2 = = α ) and ( ¯ gg βρ ∇ ¯ ) gg ∇ ( g − µν L αµνβκλ 2.8 − possesses several interesting properties. Taking differen µν ) = g ∂ ασ √ g g ∂ K √ ρσ L h g (¯ L [ α of degree +1, and, correspondingly, g ∂ ¯ x x µν ρ µν ∇ 4 4 µν ( g ) is the covariant generalization of ( µν ¯ ). Γ ) shows that d d g 2 ∂ replaced by the ordinary derivative ( g − 2.9 αβ 1.5 Z Z ¯ ∇ g 2.13 2 2 h 1 1 κ κ ), after a bit of algebra one obtains the relations g ¯ Γ is itself the Christoffel of some symmetric rank-2 tensor ¯ . Note that setting ¯ g − = = 2.7 ) display the homogeneity properties of , now given by ( √ may be equally well considered either a function of S 4 1 ) and ( B L 2.13 ¯ Γ has been a symmetric but otherwise arbitrary connection. O = 2.12 L and − , as required for consistency. , and satisfies homogeneity properties with respect to each s L ) and ( R µν So far Expressed in terms of the metric by use of ( The Lagrangian density g = α 2.12 ¯ ¯ of degree +1. With homogeneity relations hold also w.r.t. to the set deduced by considering ∇ to the case where of the identity ( degree +1. ant generalization of ( of the covariant derivative of an arbitrary symmetric conne relations useful in deriving many GR results. They are shown density, respectively. ( where, in symmetrized form, from which one may further obtain so that dropping the total divergence the action assumes the R Here, Adding ( ( Most often one is interestedreference in metric expanding about ¯ some backgro JHEP09(2017)145 . ], 3) 15 , ≥ (2.21) (2.22) (2.18) (2.17) (2.19) in this ). The n 14 ( n 1.6 S ), and the g , ). The free ) )–( ed flat space on of motion g (¯ ms, one has ) in the special ˆ Γ is now easily 1.5 2.15 3 (2.23) n manifestly co- ) = 0. In general ) in powers of the γ ˆ Γ = Γ( 2.15 ≥ ( of ( , ¯ Γ + 1.1 . αµνβκλ i ] ) K . 2 sh ) = κλ Riem − g h ) can be generated [ n + , n =0 β rentiations w.r.t. the back- s ]  | g ¯ ∇ ) [¯
. ] (2.20) ] y rmulation ( 2.20 ¯ Γ = 0, thus )( ( EH on Γ( g, h it: [¯ ert action ( =0 g δ S s 2 g, h (¯ ρσ | [¯ ) ¯ g S
sh, h n e this better consider the straight- ] x δ satisfying ( 2 S ) + δ sh − γ , then y αβ g n ( =2 η ∞ [¯ ¯ g αµνβκλ +  X n = 2 δ ρσ ) g S ) K g [¯ x ) x ( 2) ] + y h ( EH δ g 4 − µν [¯ αβ n S d αβ ( ¯ h ) give its most compact form as the theory of g s n s EH δ α ∂ – 6 – Z ∂ ) S x h ¯  ∇ x ! ) 4 ( 2.19 ( 1 d n h 2)! taken to transform as tensors, each term κλ ] = αβ )–( 1 x h h h − Z 4 β ] = d n + x h ¯ ∇ ( ) as the free kinetic term. 4 2.17 ] = g about a background satisfying the classical field equation and [¯ γ d Z )( ) then, the entire expansion ( g, h [¯ 2 sh g EH µν ] = n µν 1 = Z κ h S h S  + ] quadratic action, i.e., one has: g 2.22 α ) reduces to the apparently non-covariant form ( ! g g, h + 2 [¯ ¯ [¯ n ∇ ] = ¯ Γ, i.e., that for a metric ¯ g, h n ( [¯ EH µν x S 2.15 2 g 4 S g, h ] = S s [¯ d 2 ∂ = ¯ is absent when the background satisfies the classical equati Z S g, h 1 [¯ 6= 0 and manifest covariance is regained with the covariantiz µν S n g ) ) (i.e., with ¯ : 2)! S γ is taken to be the Minkowski metric h 1 = 0. We now note that, to within possible total derivative ter − g ¯ Γ( n 2.17 ( µν ] from the ¯ g ) gives the most compact form of the GR gravitational action i : 16 g /δ ] = ] g 2.15 [¯ = 0, the action is simply given by the second term on the r.h.s. Now, if ¯ Expanding ( g, h [¯ EH µν n ¯ S covariant formulation ( explicitly verified. expansion is of manifestly covariant form. Also, the relati δS The linear term with cf. also [ graviton field a self-interacting massless spin-2forward particle. expansion To of appreciat the original form of the Einstein-Hilb latter can then always becase properly of viewed as a the flat covariant connection fo part is now By repeated application of ( This can be moreground neatly ¯ expressed in terms of functional diffe R and all graviton vertices are obtained by differentiation of It should be noted that, with ¯ coordinates FP action ( variant form. Correspondingly, ( JHEP09(2017)145 , .) λ 2 µν and ¯ g (3.2) (3.1) g ) = 0, (2.24) (2.25) α − above, µν γ ¯ ¯ ∇ √ ( R L , R f µν g Riem . i as the “equivalent” ) L ), where, after the κλ g β ), the differentiations erentiations, however, 4 . ¯ 2.19 ∇ used in them to obtain, ). ] g )( 2.15 [¯ g to it. Thus, with a cosmo- lation obeyed by the canon- have been carried out. This ( 2.19 EH n S s set replaced by S l action contribution 2 ns of covariant derivatives, etc, αβ  for the graviton now propagating g ) αµνβκλ now gives a surviving contribution ν . x ( ¯ K = 0. ∇ δ to in some literature as ‘miraculous’, ‘hidden µν , ) , terms containing factors of ) αβ µ ν ¯ g h t ¯ g ν to be a flat background: λ µν λµ αβ µ δ ¯ ] is also of manifestly covariant form with g ∇ r y year old identification of g ) ¯ γ L . This is a tremendous simplification com- α ∇ = λ µ 14 ). It is only after laborious rearrangements x g ∂ ¯ ¯ ( ∇ ¯ ∇ = ∇ )[ µν ( and − g αβ ¯ ∂ R 2.23 h – 7 – = ) + ( , 2.24 x h − λ g µ ν 4 t 2 L d )–( 2 µ ν 2 − κ δ Z  = 2.23 µν satisfying ) in the above discussion, but it is, of course, straightfor- ¯ 1 2! µ ν . The canonical energy-momentum tensor density for the R t g 2 1.1 µν µν κ g ] = ( γ γ g ) can be shown to be equivalent to ( − − g, h [¯ √ 2 √ h S 2.23 − the differentiations needed to generate x 4 µν 2 generated by ( ) is modified to d g chosen as the variable set. (By the homogeneity properties o ≥ Z = after 2.15 n µν 2 1 g κ µν ], ) is given by α h ¯ until = ∇ g, h transforming as tensors. In carrying out the successive diff , 1 [¯ 2.15 S n − h µν n S g S We have only considered ( The field equations are in fact equivalent to the following re This is to be contrasted with the much simpler expressions ( ) is, by the equations of motion, conserved: Instances of this equivalence have on occasion been referred 4 are generated. The background equations of motion cannot be background equation of motion is first immediately used in ( and 3.1 ¯ ¯ ¯ GR Lagrangian. In this section we take the reference metric ¯ gives the vertices due to theon cosmological top term interaction of this background. 3 The self-interacting graviton and define action ( the same expression holds, as can be easily checked, with thi pared to the result of the direct expansion ( Each logical term, ( of terms, integrations by parts,that use the of expressions commutation ( relatio ward to include a cosmological term and/or matter couplings involved are those of simple algebraic powers of ¯ simplicity’ and the like, but, in fact, originate in the ninet with g g ( with Expansion about a background ¯ ical energy-momentum tensor: from the first term on the r.h.s. This, apart from the classica progressively results into extremely lengthy expressions R say, in addition to terms formed only out of ¯ JHEP09(2017)145 ν λµ r (3.8) (3.7) (3.4) (3.9) (3.6) (3.3) (3.5) (3.10) ] by ex- 17 . i . contains only . λσ # ) g βλ α β µνλ g ρν ¯ ∇ β r S ) m the canonical one ¯ ∇ − νβ rinsic angular momen- σ ρ γ α δ . . 1 2 νρ ρσ r = 0, by the antisymmetry  γ ( n, however, that it can be ], known as Belinfante sym- αβ − ) under general linear trans- iven by . ν ained by Tolman [ µ g ymmetric energy-momentum µβ − µα ons. By explicit computation 11 ) ensity is then  T σλ γ γ , ν ial” which is indeed constructed g . νβ 2.15 αβ ρ ) gives να ¯ g ∇ λνµ g = 0. 10 ) shows that g L ¯ µβ ) + s ∇ λ ρσ µ ν ∂ γ α 3.6 2 1 , t ¯ − g ∇ 3.7 + β . µν ( µ ( r λν + νλ ∂ ¯ νβ µ µν ∇ r − .( γ µ h ν +  λµν S δ s − α λ νσ µα κσ 1 2 µλν g νµ g ¯ g + ∇ νβ β r α  S − ( γ ¯ + ∇ ¯ ) β ∇ − ) µνλ ρα – 8 – ) can be expressed in terms of the quantity ν µ µα λµ s t γ αβ r λα g λβ =  , it contains only quadratic and higher powers in g g ) L 3.5 γ − λ ν µ = = 2 1 ∂ t ) να − ¯ ρσ ν ∇ µ − µνλ g α ( γ L T λ κλ ∂ µρ µνλ ∂ = S r s − ¯ g ∇ 2 ( α = 2 + κ ∂ λβ ¯ ∇ µνλ α g ) immediately implies ≡ ρµ µνλ σα S ρσ r s g ν g ( 2 term: (  3.2 2 λµ κ h να r . It is automatically conserved, ) one finds κρ γ S µρ g " ¯ γ ∇ 2 1 σνλ 3.3 ) in ( 1 2 2 1 S − = − 3.4 . Like the canonical ): ρ µσ = = λσ γ νλ µ r 3.2 ) in the definition of the superpotential ( S = µνρ µνλ 3.9 S νλ S µ 2 2 2 κ 2 S κ Define now a new energy-momentum tensor density obtained fro As it is well known, however, for any field system carrying int . We now note that the spin density ( h property of where by the addition of a occurring in ( quadratic and higher powers of the graviton field Note the antisymmetry property tum (nonzero spin) the canonicaltensor. definition The does standard not procedure yield rendering a it s symmetric [ Note that using ( Inserting ( metrization, is to add to itout the of divergence the of a field “superpotent spin density. The spin tensor density is here g from the definition ( plicit computation using themore simply equations derived of by using motion. theformations invariance of of It the the is action coordinates ( know and enforcing the field equati where By explicit computation one finds that The non-covariant form of this notable relation was first obt The appropriate superpotential constructed from the spin d JHEP09(2017)145 , µ ν h t κ ngly, (3.14) (3.11) (3.12) (3.13) → h ing matter. , quantum- . y presented in # ) . tic prescription, λ quation achieved µν derivatives and an µν r the spin-2 graviton, T ¯ ∇ + . λ πG νµ  oint to be made here is r t is needed is to replace ( udo-tensor. This has the , two νβ = 8 ver, to bear no relation to h g e interaction current, which αλ  µ λ t, the suggestion seems to be he addition of superpotentials. γ nsion, the r.h.s. has an expan- δ ary Belinfante-symmetrization ), however, it should be that αβ 1 2 ) on the l.h.s. and shuffling the − ty in any local definition of field g ) R − 3.13 µν λ νλ µ µν γ ], simply rewrites the Einstein field να µβ g r S ; whereas the r.h.s., as already noted, factors of g 2 1 + ). 19 λ ν λ + µν n δ ¯ − ∇ να λ h 2 1 g 3.13 αν + r µν − µβ ( µ ν R γ T µλ νµ – 9 – . Thus, after the customary rescaling γ − g + ) was first obtained, in non-covariant form, by β λ ) and ( µν : δ ν µ µα h t ν ) + σ g  λ ) gives then the Tolman relation in terms of the new 3.13 T 3.11 = β νβ 2) containing µα ¯ r ν µσ γ ∇ µ 3.8 is a symmetric tensor. ≥ γ T + − = n µν λ = λ µν αµ T µν r g ) finally yields µν r ) ( ) with ( T αβ + νλ γ 3.11 γ λ 3.11  " is the matter energy-momentum tensor density. Correspondi 3.10 νµ β α r ν ¯ ¯ ∇ ∇ ) one gets the explicit simple result: µ α M 2 1 ) above, which then gives the general Tolman relation includ ¯ ∇ ) with ( 3.4 − gT ] long ago, from a different perspective than the present-day ) and ( 1 2 -th order term ( 3.2 as the basic variables. This is not, however, how it is usuall factor. − ) that is now used in ( n = 18 ) this tensor is given precisely by the standard field-theore 3.2 3.12 √ 1) in ( µν µν − g 3.14 µ ν = n ) appears to be the most concise version of the graviton field e ) recasts the Einstein field equations as the field equation of T 3.13 ( 2 T ) the l.h.s. is linear in the graviton field κ 2 µ ν κ T + 3.13 3.13 ( It is straightforward to include matter in the above. All tha ( Now, from ( µ ν 3.13 t overall sion with the giving the graviton field conventional boson field mass-dime and it is ( one defines by advantage of being automaticallythe symmetric. canonical field It energy-momentumthat (pseudo)tensor; seems, no in howe fac such connection exists. Having established ( rest onto the r.h.s., which then defines the graviton field pse such different representations reflect theenergy-momentum density, inherent and ambigui thus be equivalent to within t equation by keeping the linearized part of ( Here that in ( i.e., the canonical energy-momentum tensorappropriate with for the necess aPapapetrou spin-2 [ particle.motivated ( view of the propagating graviton on a background. the l.h.s. being thefor free gravity kinetic is, part of and course, the the r.h.s. energy-momentum being tensor. th The p energy-momentum tensor contains second and higher powers of Combining ( As it is manifest from ( Combining ( In ( by use of the literature. The customary presentation, e.g., [ JHEP09(2017)145 hat (3.20) (3.17) (3.16) (3.22) (3.18) (3.19) (3.15) (3.21) ) ) νρ ασ ) γ γ νβ µα e appropriate g νβ γ γ µα − γ − αρ ) by the addition of ) in the alternative − νσ γ g γ , αβ µν 3.14 g . αβ γ
, ( γ
µν (
σα nsity µνλ . I βσ γ g om ( µρ γ ergy-momentum tensor den- cal energy-momentum tensor µ ρ S γ + . µν δ µσνα λ µσνα rm. − ˜ . T − U ¯ . − ∇ ) µα U ) α ) g α as our basic variables. In terms of νρ σµρα ρσ + ¯ πG σν σµ ∇ ¯ g ασ ∇ g νβ 8 g U g g β νµ γ νλ α α ρ νλ − µα g µνλ µα g ¯ δ ∇ g νβ ¯ γ ∇ − g ) gives the Tolman equation (including ) can be re-expressed as follows. Let # = γ µ S to µ ) α ) − − λ ¯ µν ∇ 3.2 ¯ − ¯ ∇ ρσ ∇ g ρλ µα ¯ 2 ∇ 3.13 µν g αρ 1 g 1 2 σα γ ρλ κ g νσ g β αβ µ g g g νβ γ µνλ + ¯ ¯ µν ∇ ) + I ∇ = ( µν ) in ( – 10 – γ ( µ λ αβ α γ g ) may be rewritten in the form S σ λ ( 2 ρσ γ ) = T ¯ 1 − λ ∇ ( κ T ) can then be written in the form g σ λ ) βσ µσνα 3.18 ¯ 3.4 = ( + ∇ T γ µρ ρλ g µν  − ( U ν λ g g γ g γ + t α + 3.13  σ λ ( t ) gives precisely what one gets by putting the linearized  − − + ¯ g γ σ λ αβ ∇ µσνα µν t g γ µλ ρλ γ √ √ ( − − νρ ( ρσ g U γ T − − LL g αµνβρσ 3.17  √ √ ρλ T ρσ √ . Using ( 2 1 √ g = = = ] 1 2 K given by (  γ 2 2  in ( κ ρ = να ρσ µν − + + LL " ][ µσ ρ η ˜ T − T γ r µσ µσ ) on the l.h.s. and all the rest onto the r.h.s. It is thus seen t [ r − = R U √ γ 1 2 µν ) is now the covariantized Fierz-Pauli free kinetic part, th = g 1 2 = 3.17 − µσνα µνα I U µν S ). Setting R 2 2 κ 3.15 ) gives is symmetric and conserved. ( To make this connection we revert from The expression for µν 3.20 ˜ Thus, if we now define form the latter one finds that, after some algebra, ( The l.h.s. of ( matter and after raising the covariant index by the metric de where part of ( the superpotential Note that be a redefined energy-momentum tensor density that differs fr ( T this procedure amounts to modifyingdensity the by symmetrized the canoni addition of an appropriate superpotential te self-interactions for which aresity ( given on the r.h.s. by the en JHEP09(2017)145 5 . γ , it ) may citly − ρσ M κρ (3.24) (3.23) (3.25) (3.26) √ T ικ ]. h ) / , i.e the 9 γ T g σ λ ) ¯ ρσ t , but it is ∇ , LL − γρ T )( ρλ ρσ LL h t g ικα ιλ ) σ = ( LL ) = 0. As seen h g ¯ σ λ σ λ ∇ S σ − . This raising by T α T ), which can then . ¯ )( ( ], may be directly and ∇ g 0 ¯ ( # ∇ ρλ ρ ρλ M λσ 20 ≡ g 2 g ρσ T h σγ 2 LL αβ , to form κ γ g T being a tensor density + h y flat spacetimes, one + γ ρσ ¯ LL β + -symmetrized canonical ∇ λρ 0 ( T i ρσ g ρ LL ¯ ρσ ∇ LL ) LL t ) is the LL form of the field rived at it via the Tolman . ιβ λρ α T ) in the presence of matter, T + )( ( γδ g the LL tensor and revert to γ ¯ ∇ . ρ ch as g h ric density ρσ   M ικ ρ ) 3.22 κα ¯ ) one finds ικ i ∇ T − g etric, as, e.g., was done in [ at since we have the relation ld not have the expected vector g ¯ h ∇ ικα 1 2 ργ LL κβ + − h )( ), between − h 3.25 S λ ρσ LL σǫ κβ ια γ ¯ t ) + ικ ∇ h γ h h 1 − 3.21 )( λ )( αβ β ια g − ¯ h √ ∇ g ισ ¯ ∇ β − h  )( α − αβ γ ¯ ∇ α ¯ h σǫ ∇ ικ ¯ ∇ )( g ¯ ικ ∇ γ ( αβ ικ h γδ h ρσ ). Similarly, an explicit expression for h + αβ h g – 11 – g g α β − ικ − ) in the pure gravity case one obtains ¯ 3.1 ¯ κλ + ∇ ∇ T ( g δǫ κβ αβ g = h ] in a rather different manner that makes no reference − = 3.22 ]. With symmetric matter tensor γ β ) ) + σγ 20 ικ ¯ ικ 20 ∇ g ικα LL , known from the original derivation [ [ g LL ) and ( ργ κβ α ). Alternatively, using ( T h h h ρσ S ¯ )(2 β ∇ γ λ LL α ρσ 2 LL is indeed symmetric and that ¯ T 3.7 ) and ( ¯ ∇ ¯ 3.21 1 λρ − ια κ ∇ ∇ T for pure gravity and ( α g h ], it is important to note, leads to √ )( )( contains only first derivatives of the metric and can be expli ρσ ¯ LL ικ ∇ + ρσ 20 3.13 ια T g κσ LL γ ρσ h h LL T ια β − γ − ) and ( T h ¯ ¯ ∇ β ) ∇ √ κρ ( ( 3.1 ¯ 1 2 g ∇ " ) that 3.1 ρσ α ) with ( ιλ γ g = ) and are conserved. ¯ g ∇ 1 − ) and obtained an explicit relation, ( ιλ 3.22 (2 g κβ 3.23 √ 1 8 3.22 h  ιακβ 2 3.20 1 ) and ( κ U + − + ) was originally derived [ β = ¯ ∇ 3.21 3.22 α ) explicitly relates the (reweighted) LL and the Belinfante ικ is the LL gravitational energy-momentum density, and ( The explicit formula for ( This is actually something of a problem if, in asymptoticall ¯ This can always be done in formulations employing a reference m ∇ T 5 in the LL definitions [ ρσ 1 2 LL 3.25 Comparing ( is seen from ( from ( tensor densities of weight 1 one needs a reference metric, su be evaluated over a 2-spheretransformation at spatial property. infinity, since To it correct wou this when working with defines the total field momentum as the spatial integral of ( where ( the weight 2 densities, energy-momentum tensors by a superpotential term. obtained from our relation ( to the canonical energy-momentumequation tensor ( density. Here we ar equations given in terms of canonical density with the covariant index raised by the met computed from these relations. that enter ( In the formulation presented here this is naturally arrived T be obtained from ( of weight 2. One may define a corresponding tensor by letting g JHEP09(2017)145 ]. ) 11 νρ , h (3.28) (3.27) γ 10 ¯ ∇ )( µλ h en how it can σ ]. The present µν ¯ 9 ∇ h ( β ¯ σγ ∇ the entire dynamical g ], was carried out in α s the field equation of . ¯ ∇ λρ ent manner. In explicit 18 # g ) αβ ing the graviton vertices. eraction current being its tric for any field carrying t derivations and compact act analog of the Lorentz + h re. e.g., [ γδ , and here a natural choice he latter generally leads to rmula for the generation of h  − uced to those obtained from ρ ) ic version can be constructed at applies to any spin [ ntroducing a reference metric. relevant in perturbative quan- ained by expansion of the EH ) ] and [ riant can then be given. This ¯ ion. This arbitrary connection ∇ ργ energy-momentum tensor. The γρ h 17 y following standard field theory )( e usual setup of propagation over h λ σǫ σ ¯ ∇ h ¯ ∇ λ )( )( ¯ ∇ µσ , λσ )( h h γ σǫ γ ¯ g ∇ = 0 ¯ ( ∇ γδ ( µν g ρσ (the “ΓΓ” form), rather than the original EH λρ g g – 12 – g ν − νλ L ¯ ∇ µν . g δǫ g µ g ) is straightforward and can be applied to any metric 1 2 J σγ ) + g 2.4 = ργ ) + µ h )(2 λ νβ A λρ ¯ h ∇  g α )( ) and ( ¯ µν ∇ νσ g )( 2.2 h − γ µα ¯ )–( ∇ h νρ ( β g 2.1 ¯ ρσ ∇ µλ ( g g " µλ , however, is not an invariant density. In this note we have se ) assumes the noteworthy form γ (2 g L 8 1 1 −  √ 3.13 − + = ) gives perhaps the most compact expression in closed form of µν We next showed how the Einstein field equations may be recast a All our discussion above has been carried in a gauge independ h ¯  3.28 Reference (flat) metrics for this purpose have been used befo be covariantized by the introductionmay, in of particular, a be reference taken to connect be a metric connection, thus i expressions for many GR results.tum This gravity becomes where particularly itThe provides density the most concise way of generat form of the Lagrangian, has long been known to provide efficien 4 Conclusion Working with the Lagrangian density canonical definition of energy-momentumintrinsic angular tensor momentum, is i.e., not non-zeroby symme the spin. addition A of a symmetr superpotential term in a construction th the spin-2 graviton coupled to the gravitational canonical This allows one to exhibitprocedures. the graviton The self-interactions construction, b based on older work in [ was contrasted with theaction formula for in generating a vertices mannermuch obt preserving lengthier background expressions field that invariance.the can former. T be only laboriously red scheme based on ( gravity theory. It naturally includes,a as background. a special For case, perturbativegraviton th vertices gravity, a which simple is compact manifestly fo background field inva content of GR assymmetrized the field canonical equation energy-momentum of tensor.gauge the photon graviton field It with equation is the int the ex calculations one frequently hasis to the specify de a Donder gauge. gauge A common ( in which ( JHEP09(2017)145 ] ommons , Dover d form, such gr-qc/0411023 [ , then ask what modifi- ead to a theory of the , as well as the LL and (1970) 9 the exact relation between 1 total field 4-momentum for , itational energy-momentum ion of appropriate superpo- s the nonlinear part of the redited. ully clarified by the explicit tric. Other definitions of the ed. These points, the source action current being precisely y suggests the following ques- e literature before. The use of ility would seem to be to give ntial terms. This was done in hen shown to be related to the act several physical reasons for cases, the explicit form of the given here makes it also appar- This will be considered elsewhere. , 2nd edition, Cambridge University ] consistent self-coupling of a massless ]. 2 Gen. Rel. Grav. , SPIRE IN ]. ) above, in particular, expresses this in a form ][ – 13 – 3.13 SPIRE IN [ Hidden Simplicity of the Gravity Action ), which permits any use, distribution and reproduction in Tensors, differential forms and variational principles (1987) L99 arXiv:1705.00626 [ 4 variable in obtaining relatively concise formulas in close CC-BY 4.0 The mathematical theory of relativity g ), should be noted in this connection. This article is distributed under the terms of the Creative C 3.17 (2017) 002 Gravity From Selfinteraction in a Curved Background Selfinteraction and gauge invariance ]. 09 ) and ( SPIRE IN Class. Quant. Grav. JHEP [ Press (1924). (1989). Showing explicitly how various definitions of the local grav Our exploration of the self-interacting graviton naturall 3.26 [1] A.S. Eddington, [3] S. Deser, [4] C. Cheung and G.N. Remmen, [5] D. Lovelock and H. Rund, [2] S. Deser, any medium, provided the original author(s) and source are c References cations may be introducedself-interacting graviton in truly the beyondup above GR. the framework The assumption that only of possib locality wouldconsidering of the l interactions. delocalization There of are graviton in interactions. f Open Access. spin-2 particle leads tocomplying GR. to Equation the ( preceptsthe of standard symmetrized field canonical theory, the energy-momentum inter tensor. One may as ( Attribution License ( the Belinfante-symmetrized canonical tensors,appropriate superpotentials. and, in These have all the not metric appeared density in th gravitational field energy-momentum (pseudo)tensor werecanonical t version by theparticular addition for of appropriate the superpote Einstein standard tensor, textbook and the pseudotensor LLthe energy-momentum defined tensor. commonly a We employed gave LL tensor and the canonical definition the present covariantized framework with flat background me tion. As demonstrated by the general argument in [ tentials makes it apparent thatasymptotically they flat all spacetimes. must The leadent covariant to formulation that the this same agreementof is some independent of confusion the inresults coordinates the presented us here. literature over the years, are hopef are equivalent, just as in any other field theory, by the addit JHEP09(2017)145 , , A ] , , ivity ry . (1940) 1. 18 , (1939) 870 arXiv:0906.0926 (2010) 641 Proc. Roy. Irish Acad. [ 6 , 42 , 2nd edition, Pergamon Press s and conserved quantities at spatial Physica , , Gordon and Breach (1965). (2009) 084014 Mem. Acad. Roy. Belg. ]. , A new improved energy-momentum tensor ]. Gen. Rel. Grav. , D 80 Bootstrapping gravity: A consistent approach to SPIRE – 14 – , J. Wiley (1972). IN [ SPIRE IN ]. [ Gravitational Fields in Finite and Conformal Bondi ]. ]. A covariant formulation of the Landau-Lifshitz complex Phys. Rev. The classical theory of fields A unified treatment of null and spatial infinity in general , SPIRE . SPIRE IN ]. SPIRE [ . IN (1978) 1542 [ IN [ (1966) 1039 19 SPIRE IN (1979) 233 150 ][ (1968) 861 (1970) 42 11 The undor equation of the meson field (1967) 1195 Einstein’s Theory of Gravitation and Flat Space 9 (1930) 875 On the Use of the Energy-Momentum Principle in General Relat 59 Sur le tenseur d’impulsion-´energie Dynamical Theory of Groups and Fields Quantum Theory of Gravity. 2. The Manifestly Covariant Theo Gravitation and Cosmology Some Total Invariants of Asymptotically Flat Space-Times 162 35 Gravity from self-interaction redux ]. Phys. Rev. J. Math. Phys. , , (1948) 11. SPIRE IN arXiv:0910.2975 Phys. Rev. Phys. Rev. energy-momentum self-coupling [ Annals Phys. [ Gen. Rel. Grav. Frames J. Math. Phys. relativity. I — Universalinfinity structure, asymptotic symmetrie (1962). 52 [9] S. Persides and D. Papadopoulos, [7] J. Winicour, [8] A. Ashtekar and R.O. Hansen, [6] L.A. Tamburino and J.H. Winicour, [12] C.G. Callan Jr., S.R. Coleman and R. Jackiw, [17] R.C. Tolman, [18] A. Papapetrou, [16] L.M. Butcher, M. Hobson and A. Lasenby, [13] S. Deser, [14] B.S. DeWitt, [10] F.J. Belinfante, [11] L. Rosenfeld, [15] B.S. DeWitt, [20] L.D. Landau and E.M. Lifshitz, [19] S. Weinberg,