Tensor Gauge Condition and Tensor Field Decomposition

8/18/2019 Tensor Gauge Condition and Tensor Field Decomposition

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a r X i v : 1 1 0 1 . 2 8 0 9 v 5

[ g r - q c ] 2 8 N o v 2 0 1 1

Tensor gauge condition and tensor field decomposition

Xiang-Song Chen1,2,3,∗ and Ben-Chao Zhu11Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, China

2Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing 210093, China 3Kavli Institute for Theoretical Physics China, Chinese Academy of Science, Beijing 100190, China

(Dated: November 30, 2011)

We discuss various proposals of separating a tensor field into pure-gauge and gauge-invariant

components. Such tensor field decomposition is intimately related to the effort of identifying thereal gravitational degrees of freedom out of the metric tensor in Einstein’s general relativity. Weshow that, as for a vector field, the tensor field decomposition has exact correspondence to, andcan be derived from, the gauge-fixing approach. The complication for the tensor field, however, isthat there are infinitely many complete gauge conditions, in contrast to the uniqueness of Coulombgauge for a vector field. The cause of such complication, as we reveal, is the emergence of apeculiar gauge-invariant pure-gauge construction for any gauge field of spin ≥ 2. We make anextensive exploration of the complete tensor gauge conditions and their corresponding tensor fielddecompositions, regarding mathematical structures, equations of motion for the fields, and nonlinearproperties. Apparently, no single choice is superior in all aspects, due to an awkward fact that nogauge-fixing can reduce a tensor field to be purely dynamical (i.e., transverse and traceless), as canthe Coulomb gauge in a vector case.

PACS numbers: 11.15.-q, 04.20.Cv

I. INTRODUCTION

It is a familiar practice to separate a vector field A

into transverse and longitudinal parts, A ≡ A⊥ + A, de-

fined by ∂ · A⊥ = 0 and ∂ × A = 0. One may naturallythink of an analogous separation for a tensor field. Suchfield decompositions are well motivated in physics. For

the vector case, e.g., it is the transverse field A⊥ thatdescribes a real photon. The need of decomposing thetensor field is closely related to gravity. The Einsteinequivalence principle dictates that gravitational effect is

described by the metric tensor, which also characterizesthe inertial effect associated with coordinate choice. Butsometimes, one does face the necessity of identifying thereal gravitational degrees of freedom out of the metric,e.g., in associating a meaningful energy to gravitationalradiation, in analyzing canonical structure of gravitationand thus quantizing it to define a physical graviton. Theattempt to separate gravity from inertial effect dates backto the bimetric theory of Rosen in 1940 [1]. This theoryis yet formal because Rosen does not give any actual pre-scription to separate a background metric from the exper-imentally measured total metric. After about 20 years,Arnowitt, Deser and Misner (ADM) proposed the famous

transverse-traceless (TT) decomposition of a symmetrictensor in their canonical formulation of general relativity[2]. Based upon the quantum action principle, Schwingerpresently a slightly different TT decomposition in hisattempt of quantizing the gravitational field [3]. Lateron, such tensor decomposition was further developed byDeser [4], and York [5], with the aim of going covariantly

∗Electronic address: [email protected]

beyond the linear approximation. These decompositionsare really operational by giving explicit expression of theseparated component in terms of the total tensor.

Separating the metric tensor is much more involvedthan separating A into A⊥ + A, due to one more in-dex and also the nonlinearity of gravity. Regarding non-linearity, it should be mentioned that in the Yang-Millstheory separating the physical degrees of freedom fromthe gauge freedom is not so trivial either. This problemhas revived recently in connection with the nucleon spinstructure [6]. A possible solution is presented by Chenet al. [7, 8], and is extended successfully to gravity by

Chen and Zhu (CZ) [9].

The decompositions of ADM, Schwinger, Deser, York,and CZ are motivated from different aspects, and thusnaturally contain notable differences. The CZ decom-position is specifically aimed at a clear separation of apure-gauge background, while the others are closely as-sociated with dynamical structure of gravity. In this pa-per, we explore and compare these decompositions fromseveral perspectives. After a brief review of these decom-positions, we first look at the relatively simple linearizedgravity, for which we show that the aforementioned de-compositions can all be conveniently rederived from agauge-fixing approach. In so doing, we reveal an inter-esting mathematical structure that the decompositionsof ADM, Schwinger, Deser and York contain doubly-nonlocal operation (namely, with the inverse Laplacianoperator 1 ∂ 2 used quadratically), while that of CZ is the

unique singly-nonlocal choice. We then go beyond thelinear order and demonstrate that the CZ decompositiondiffers critically from those of Deser and York in theirphysical implications: The physical and pure-gauge (orinertial) effects are cleanly kept apart to all orders in theCZ formulation, but start to mix in the Deser and York

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approaches beyond linear order. We also explain howthe Deser decomposition can be modified in the spirit of CZ formulation. Finally, we give some further discussionabout how to analyze the physical and dynamical con-tent of gravity, regarding the selection of constraints (orgauge-fixing) and the equations of motion. Apparently,there is no universally superior choice of gauge for a ten-sor field. The significant complication from the vector

case to the tensor case (and virtually all higher-rank ten-sors) is revealed to arise from two facts: (i) It is possibleto construct a peculiar gauge-invariant pure-gauge field,thus there are infinitely many ways to fix the gauge com-pletely; (ii) however, no gauge-fixing can pick out directlythe purely dynamical (i.e., transverse and traceless) com-ponent of a tensor, and extra extraction is needed.

II. VARIOUS TENSOR DECOMPOSITIONS

The original TT decomposition of ADM is linear andfor a symmetric spatial tensor:

hij ≡ hT T ij + hT ij + h

Lij, (1)

where hT T ij is TT: hT T ij,i = h

T T kk = 0, h

T ij ≡

12(δ ij h

T −1

∂ 2hT ,ij ) is constructed to be transverse: h

T ij,i = 0, and

hLij ≡ f i,j + f j,i is longitudinal (pure-gauge). (Notations:Greek indices run from 0 to 3, Latin indices run from 1to 3, and repeated indices are summed over, even whenthey both appear raised or lowered. ∂ or comma denotesordinary derivative, and ∇ denotes covariant derivative.)The four unknowns, hT and f i, are solved as follows:hLij,j = hij,j gives f i, then h

T kk + h

Lkk = hkk gives h

T . Theresults are:

f i = 1 ∂ 2(hik,k − 1

21

∂ 2hkl,kli ), (2)

hT = hkk − 1

∂ 2hkl,kl. (3)

Inserting them into Eq. (1) gives the explicit form of the TT component:

hT T ij = hij − 1

2δ ij (hkk −

1

∂ 2hkl,kl)

− 1

∂ 2(hik,kj + hjk,ki −

1

2hkk,ij −

1

2

1

∂ 2hkl,klij ). (4)

It can be easily checked that both hT T ij and h

T ij are in-

variant under the linear gauge-transformation δhij =ξ i,j + ξ j,i [10].

Schwinger also formulated a TT decomposition, by aslightly different parametrization [3]:

hij = hT T ij +

1

2(q i,j + q j,i) − δ ijq k,k + q ,ij . (5)

Here the terms containing q i are constructed to be“doubly transverse”: [12(q i,j + q j,i)− δ ij q k,k ],ij = 0. As a

result, q is given by ( ∂ 2)2q = hij,ij . Then by examiningthe trace and divergence of Eq. (5) one obtains

q i = 1

∂ 2(2hij,j −

1

2hjj,i −

3

2

1

∂ 2hjk,jki). (6)

The TT component hT T ij is the same as that in Eq. (4).Later, Deser presented a covariant decomposition [4]:

hij = ψ T ij + ∇iV j + ∇j V i. (7)

Here ψT ij is covariantly transverse: ∇iψT ij = 0. Deser does

not extract further a TT part since the covariant exten-sion of the linear hT ij is rather involving. The unknowncovariant vector V i is to be solved iteratively, and theleading term is just f i in (2).

By a clever parametrization, York obtained a covariantTT decomposition [5]:

hij = hT T ij +(∇iW j +∇jW i−

2

3gij∇kW

k)−1

3gijh

kk. (8)

Here the middle term is constructed to be traceless.

Again, the unknown covariant vector W i is to be solvediteratively. The leading term is

W i = 1

∂ 2(hij,j −

1

4hjj,i −

1

4

1

∂ 2hjk,jki), (9)

and the leading TT component is the same as that in Eq.(4).

The CZ decomposition has a very different formulation[9]. It applies to space-time instead of just space, and isdesigned to be a clean separation of gravitational andpure-gauge degrees of freedom up to moderately strongfield. The defining equations are

gµν ≡ ĝµν + ḡµν ; (10a)R̄ρσµν (ḡαβ ) = 0, (10b)

gijΓ̂ρij = 0. (10c)

Here ḡµν is intended as a pure-gauge background met-ric. Namely, it has an inverse ḡµν and can be used todefine a background connection Γ̄ρµν ≡

12

ḡρσ(∂ µḡσν +∂ ν ḡσµ −∂ σ ḡµν ) with a vanishing Riemann curvature ten-sor R̄ρσµν ≡ ∂ µΓ̄

ρσν − ∂ ν Γ̄

ρσµ + Γ̄

ραµΓ̄

ασν − Γ̄

ραν Γ̄

ασµ = 0.

The intended physical term ĝµν satisfies a delicate con-

straint g ij Γ̂ρij = 0. Here Γ̂ρµν = Γ

ρµν − Γ̄

ρµν is not an affine

connection, and ĝµν ≡ gµν − ḡµν is not the inverse of ĝµν .The CZ decomposition is also to be solved iteratively.

At linear order, gµν ≡ ηµν + hµν with ηµν the Minkowskimetric and |hµν | ≪ 1, an elegant, gauge-invariant expres-sion is obtained after a fairly lengthy calculation [9]:

ĥCZµν = hµν − 1

∂ 2(hµi,iν + hνi,iµ − hii,µν ). (11)

Its trace is of special use and worth recording:

ĥCZii = 2(hii − 1

∂ 2hij,ij ) = 2

1

∂ 2∂ j (hii,j − hji,i). (12)


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Moreover, an elegant relation can be derived for ĥCZij and

the (fairly complicated) hT T ij [9]:

hT T ij = ĥCZij −

1

4(δ ij ĥ

CZkk +

1

∂ 2ĥCZkk,ij ). (13)

III. DERIVATION OF THE LINEAR

DECOMPOSITION BY GAUGE-FIXING

Before discussing the highly tricky difference betweenDeser, York, and CZ at higher orders, we first take acloser look at their linear-order results, which we red-erive from a convenient gauge-fixing approach. The lineof our derivation will show clearly the one-to-one corre-spondence between complete gauge conditions and gauge-

covariant decompositions.

Consider the linear gauge-transformation: h′µν = hµν −∂ µξ ν − ∂ ν ξ µ. We show that the gauge parameter ξ µ thatbrings h′µν (x) into the “generalized” transverse gauge,

∂ ih′

i0 +a∂

0h′

ii = 0,

(14a)∂ ih′ij + b∂ j h

′ii = 0, (14b)

is essentially unique. Here a, b can take any value exceptb = −1, as Eq. (12) indicates that (hji,i − hii,j ) has agauge-invariant divergence, thus cannot be used to definea gauge. Various combinations of a, b have been studiedin the literature. The gauge a = −14 , b = 0 was employedby ADM [2], who also suggested an important choice of a = −12 , b = −

13 , which agrees with the Dirac gauge at

linear approximation [11]. The gauge a = −23 , b = −13

was encountered by Weinberg and termed “too ugly todeserve a name” [12], while the special properties of a =b = −1

2 was recently revealed by CZ [13].

We cast Eqs. (14) into the equations for ξ µ:

∂ 2ξ 0 + (1 + 2a)∂ 0∂ iξ i = ∂ ihi0 + a∂ 0hii, (15a)

∂ 2ξ j + (1 + 2b)∂ j∂ iξ i = ∂ ihij + b∂ j hii. (15b)

To solve, act on both sides of (15b) with ∂ j and sum over j, we get

ξ i,i = 1

2(1 + b)(bhii +

1

∂ 2hij,ij ). (16)

We remind that inversion of the Laplacian operator ∂ 2

implies a vanishing boundary value at infinity. Substi-

tuting ξ i,i back into Eqs. (15), we get

ξ 0 = 1

∂ 2[h0i,i +

2a − b

2(1 + b)hii,0 −

1 + 2a

2(1 + b)

1

∂ 2hik,ik0],(17a)

ξ j = 1

∂ 2[hij,i +

b

2(1 + b)hii,j −

1 + 2b

2(1 + b)

1

∂ 2hik,ikj ].(17b)

These are the unique ξ µ that bring h′µν into the gauge

(14). The above derivation also show clearly that Eqs.(14) fix the gauge completely: If hµν is already in

this gauge, then we get ξ µ ≡ 0. Namely, no gauge-transformation can preserve Eqs. (14).

The uniqueness of ξ µ suggests the definition of a gauge-

invariant tensor, ĥ(ab)µν = hµν − ξ µ,ν − ξ ν,µ, which satisfies

ĥ(ab)0i,i + aĥ

(b)ii,0 = 0 and

ĥ(b)ji,i + bĥ

(b)ii,j = 0. (We put a super-

script (ab) to remind the dependence on the parametersa, b. The purely spatial components actually involve onlyb, and (b) is used for clarity.) The explicit expression of

ĥ(ab)µν is

ĥ(ab)00 = h00 −

1

∂ 2(2hk0,k0 − hkk,00)

−1 + 2a

1 + b

1

∂ 2(hkk −

1

∂ 2hkl,kl),00, (18a)

ĥ(ab)0j = h0j −

1

∂ 2(hk0,kj + hkj,k0 − hkk,0j )

−1 + a + b

1 + b

1

∂ 2(hkk −

1

∂ 2hkl,kl),0j , (18b)

ĥ(b)ij = hij −

1

∂ 2(hki,kj + hkj,ki − hkk,ij)

−1 + 2b

1 + b1

∂ 2(hkk −

1 ∂ 2

hkl,kl),ij . (18c)

We have organized ĥ(ab)µν in a form which displays ev-

ident gauge-invariance. By Eqs. (18), the tensor hµν

is separated into a gauge-invariant part, ĥ(ab)µν , plus a

pure-gauge part, h̄µν = hµν − ĥ(ab)µν = ξ µ,ν + ξ ν,µ. No-

tice that this is an operational separation since ĥ(ab)µν and

h̄µν are both expressed explicitly in terms of the givenhµν . We thus see that each complete gauge conditionhi0,i +ahii,0 = 0, hij,i +bhii,j = 0 (specified by a, b) corre-sponds to one ways of decomposing hµν into an invariant

part plus a pure-gauge part.The expressions for ĥ

(ab)µν look rather complicated, but

the spatial trace ĥ(b)ii is remarkably simple for all choice

of b:

ĥ(b)ii =

1

1 + b(hii −

1

∂ 2hij,ij ) =

1

2(1 + b)ĥCZii . (19)

The gauge-invariant ĥ(ab)µν relates to the gauge-

invariant ĥCZµν by

ĥ(ab)00 =

ĥCZ00 − 1 + 2a

2(1 + b)

1

∂ 2(ĥCZkk ),00, (20a)

ĥ(ab)0j = ĥCZ0j −

1 + a + b

2(1 + b)

1

∂ 2 (ĥCZkk ),0j , (20b)

ĥ(b)ij =

ĥCZij − 1 + 2b

2(1 + b)

1

∂ 2(ĥCZkk ),ij . (20c)

Another important relation is between hT T ij and ĥ(b)ij :

hT T ij = ĥ(b)ij −

1 + b

2 δ ijh

(b)kk +

1 + 3b

2

1

∂ 2h(b)kk,ij. (21)

This agrees with Eq. (4) in the gauge hij,i + bhii,j = 0.


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For comparison, we recall the parallel construction fora vector field Aµ with the gauge-transformation A′µ(x) =Aµ(x) − ∂ µΛ(x). The parameter Λ that brings A′µ into

the Coulomb gauge, ∂ iA′i = 0, is also unique: Λ =

1 ∂ 2

Ai,i.

This says that Coulomb gauge is a complete gauge for avector field. We can again define a gauge-invariant quan-tity, Âµ = Aµ − ∂ µ

1 ∂ 2

Ai,i; and make the decomposition

Aµ ≡ Âµ + Āµ, with Āµ = ∂ µ1

∂ 2

Ai,i being a pure-gauge.

The spatial components, ̂A and ̄A, are nothing but the

transverse field A⊥ and the longitudinal field A.Eqs. (17) and (18) are far more complicated than their

counterparts for vector field, not only by more indices,but by a doubly-nonlocal operation 1 ∂ 2

1 ∂ 2

which also ap-

pears in (2), (4), (6) and (9).Some special choices of a, b are particularly interesting:(i) a = b = −12 kills the doubly-nonlocal terms in Eqs.

(17) and (18), and gives the CZ decomposition in Eq.(11), with the pure-gauge part:

h̄CZµν = 1

∂ 2(hµi,iν + hνi,iµ − hii,µν )

≡ ξ CZµ,ν + ξ CZν,µ, (22a)

ξ CZµ = 1

∂ 2(hµi,i −

1

2hii,µ). (22b)

(ii) b = 0 gives a gauge-transformation parameter ξ iequal to f i in Eq. (2), and thus the pure-gauge part of the ADM decomposition of a spatial tensor:

h̄ADMij = 1


1

∂ 2hkl,klij )

≡ ξ ADMi,j + ξ ADMj,i , (23a)

ξ ADMi = f i = 1

∂ 2(hik,k −

1

2

1

∂ 2hkl,kli ). (23b)

This is also the linear-order result of the covariant Deserdecomposition in Eq. (7). The choice b = 0 is not asconvenient as one may expect, as the doubly-nonlocalstructure remains. This is similar to the familiar fact ingravity that the gauge ∂ µhµν = 0 is not convenient, and acleverer choice is the harmonic gauge hµν,µ −

12h

µµ,ν = 0.

The two pure-gauge terms h̄ADMij and h̄CZij differ by

another pure-gauge term:

h̄ADMij = h̄CZij +

1

2

1

∂ 2(ĥCZkk ),ij . (24)

(iii) b = −13 gives a gauge-transformation parameter ξ iequal to W i in Eq. (9), and thus the pure-gauge part of York decomposition at linear order:

h̄Yorkij = 1


1

2hkk,ij −

1

2

1

∂ 2hkl,klij )

≡ ξ Yorki,j + ξ Yorkj,i , (25a)

ξ Yorki = W i = 1

∂ 2(hik,k −

1

4hkk,i −

1

4

1

∂ 2hkl,kli ). (25b)

h̄Yorkij differs from h̄CZij by yet another pure-gauge term:

h̄Yorkij = h̄CZij +

1

4

1

∂ 2(ĥCZkk ),ij . (26)

When applied to general relativity, the pure-gaugeterm h̄µν = ξ µ,ν + ξ ν,µ relates to the inertial effect asso-ciated with coordinate choice. They differ in the decom-

positions of ADM (Deser), York, and CZ. Accordingly,the gauge-invariant parts in these decompositions, whichare essentially the tensors defined in the gauges hji,i = 0,hji,i −

13

hii,j = 0, and hji,i −12

hii,j = 0, respectively, alsodiffer. We will discuss further the properties of thesedifferent gauge-invariant quantities in later sections.

As far as gauge transformation is concerned, the CZgauge with a = b = −12 is the most convenient one: It re-lates to all gauge configurations by just a singly-nonlocalmanipulation, while doubly-nonlocal manipulation maybe needed when transiting between two other gauges.The choice b = −13 , however, possesses a most impor-tant feature that it kills the non-local term in Eq. (21),thus simplifies the extraction of hT T

ij to be just algebraic.

This feature leads to various important applications:(iv) Weinberg combines b = − 13 with a = −

23 , which

simplifies hT T ij and kills the doubly-nonlocal term in Eq.(18b). By this choice Weinberg was able to demonstratequantum Lorentz invariance for the gravitational cou-pling [12].

(v) The linearized Dirac gauge is to combine b = −13with a = −12 . It simplifies h

T T ij and kills the doubly-

nonlocal term of the time-time component in Eq. (18a),which closely relates to the Newtonian potential. Thisgauge is thus especially advantageous in post-Newtoniandynamics [14, 15], and is also referred to as the ADM-TTgauge. It should be clarified that the term “TT gauge”here just means that in this gauge the TT componenthT T ij relates to hij locally : h

T T ij = hij −

13δ ij hkk , not that

hij is purely TT.

IV. BEYOND THE LINEAR ORDER

The difference between Deser, York, and CZ beyondthe linear order, on the other hand, is much more dras-tic. The delicate formulation of CZ via the Riemann cur-vature guarantees that ḡµν is a pure-gauge backgroundto all orders. However, in the Deser or York formula-tion, ∇iV j + ∇j V i or ∇iW j + ∇j W i is not a pure-gaugebeyond linear order, namely, it does not give a vanish-ing Riemann curvature. A correct parametrization of apure-gauge background metric can be obtained throughcoordination transformation x′µ = xµ + ξ µ from theMinkowski metric ηµν :

ḡµν = ∂x′ρ

∂xµ∂x′σ

∂xν ηρσ

= ηµν + ηρν ∂ξ ρ

∂xµ + ησµ

∂ξ σ

∂xν + ηρσ

∂ξ ρ

∂xµ∂ξ σ

∂xν .(27)


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The Riemann curvature of this metric is identicallyzero: R̄ρσµν (ḡαβ) ≡ 0. In principle, one may use sucha parametrization to formulate a clean separation of apure-gauge background, equivalent to what CZ achieve:

gµν ≡ ĝµν + ηµν + ηρν ξ ρ,µ + ησµ ξ

σ,ν + ηρσ ξ

ρ,µξ

σ,ν ; (28a)

gij{Γρij − Γ̄ρij[ḡαβ(ξ

λ)]} = 0. (28b)

It can be seen, however, that this formulation is no lessdemanding than that of CZ, since (except at linear order)the equations for ξ µ are even more involving than thosefor ḡµν .

Analogously, when decomposing a Yang-Mills field:Aµ = Āµ + Âµ, it is not really easier to parameterizethe pure-gauge field Āµ explicitly as U ∂ µU −1 and try tosolve U = eiω

aT a , compared to defining Āµ implicitly by a vanishing field strength F̄ µν ( Āρ) = 0 and solvingĀµ as Chen et al. do [7, 8]. Though being tedious, thefield decomposition in Refs. [7–9] can indeed be solvedstraightforwardly to any desired order.

We point out that the Deser decomposition can be

modified to pick out a clean pure-gauge background toall orders, following the construction line of CZ:

gµν = ψT µν + ḡµν ; (29a)

R̄ρσµν (ḡαβ) = 0, (29b)

∇iψT iµ = 0. (29c)

This would be equivalent to, but easier to solve thanparameterizing the background explicitly as in Eq. (27):

gµν ≡ ψT µν + ηµν + ηρν ξ

ρ,µ + ησµ ξ

σ,ν + ηρσ ξ

ρ,µξ

σ,ν ; (30a)

∇

iψT

iµ = 0. (30b)

At linear order, Eqs. (29) or (30) give the same ψT ijas by Eq. (7). But beyond the linear order, Eqs. (29)or (30) can guarantee that (gµν − ψ

T µν ) is a pure-gauge,

while Eq. (7) cannot.

V. EQUATIONS OF MOTION FOR THE

GAUGE-INVARIANT FIELDS

To look into the physical implications of the tensordecompositions, we derive here the equations of motionfor the various gauge-invariant quantities constructed byADM, Deser, York, and CZ. As we will see, this pro-vides a very illuminating perspective on the dynamics of general relativity, and sheds important light on the ques-tion of what are the most appropriate physical variables(or equivalently, the most appropriate gauge) for gravi-tational field.

The unconstrained (gauge-dependent) hµν satisfies thelinearized Einstein equation

hµν − ∂ µ∂ ρhρ

ν − ∂ ν ∂ ρhρ

µ + ∂ µ∂ ν hρ

ρ = −S µν . (31)

Here ≡ ∂ 2 − ∂ 2t , S µν ≡ T µν − 12ηµν T

ρρ, and we put

16πG = 1. Of the gauge-invariant quantities built out

of hµν , h(ab)µν stands a primary role. Other quantities like

hCZ µν or ψT µν are just special cases of h

(ab)µν for certain a, b.

We first display the relevant equations for h(ab)µν , then

explain their derivations and physical meanings:

∂ 2ĥ(ab)00 = −S 00 +

1 + 2a

1 + b

1 ∂ 2

T 00,00

= ∂ 2h00(hi0,i+ahii,0=0, hij,i+bhii,j=0) , (32a)

∂ 2ĥ(ab)0i = −S 0i +

1 + a + b

1 + b

1

∂ 2T 00,0i

= ∂ 2h0i(hi0,i+ahii,0=0, hij,i+bhii,j=0) , (32b)

∂ 2ĥ(ab)ii = −

1

1 + bT 00 = ∂

2hii(hij,i+bhii,j=0) , (32c)

∂ 2ĥ(b)ji,i =

b

1 + bT 00,j = ∂

2hji,i(hij,i+bhii,j=0) ,(32d)

ĥ(b)ij = −Ŝ

(b)ij = hij

(hij,i+bhii,j=0) . (32e)

The source term Ŝ (b)ij is built with S ij in the same way

as ĥ(b)ij is built with hij in Eq. (18c):

Ŝ (b)ij = S ij −

1

∂ 2(S ki,kj + S kj,ki − S kk,ij )

−1 + 2b

1 + b

1

∂ 2(S kk −

1

∂ 2S kl,kl),ij . (33)

The reason is that the equation for the gauge-invariant

quantity h(b)ij must be gauge-invariant, therefore can be

derived in any convenient gauge. Eqs. (32e) and (33)are direct consequence of the equation in the harmonicgauge, hµν = −S µν .

We have organized Eqs. (32) to show an important anddelicate dual relation between gauge-invariant equationsand gauge-fixed equations: If there exists an X gaugein which a gauge-invariant quantity Y µν = hµν , thenthe gauge-invariant equations for the gauge-invariant Y µν take the same form as the gauge-dependent equations forthe gauge-dependent hµν in the specific X gauge. This

dual relation can be very handy in deriving equations.E.g., to seek the equations of motion for hµν in the gaugehi0,i + ahii,0 = 0, hij,i + bhii,j = 0, we can instead lookat the equations of motion for the gauge-invariant quan-

tity ĥ(ab)µν in Eqs. (18), which then can be derived in the

convenient harmonic gauge. Even more conveniently, we

see in Eqs. (20) that the expression of ĥ(ab)µν via ĥCZµν is

fairly simple, so we can derive the equations of motion

for ĥ(ab)µν via the equations of motion for ĥCZµν , which in

turn are just the equations of motion for hµν in the gauge


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hiµ,i − 12hii,µ = 0, as we recently derive in Ref. [13]:

∂ 2ĥCZ0µ = −S 0µ = ∂ 2h0µ

(hiµ,i−

1

2hii,µ=0)

, (34a)

∂ 2ĥCZii = −2T 00 = ∂ 2hii

(hiµ,i−

1

2hii,µ=0) , (34b)

∂ 2ĥCZji,i = −T 00,j = ∂ 2hji,i

(hiµ,i−

1

2hii,µ=0)

, (34c)

ĥCZ

ij

= −Ŝ CZij

= hij

(hiµ,i−

1

2 hii,µ=0) . (34d)

Like Ŝ (b)ij ,

Ŝ CZij is built with S ij in the same way as ĥCZij

is built with hij; and Ŝ (b)ij relates to

Ŝ CZij in the same way

as h(b)ij relates to h

CZij :

Ŝ CZij = S ij − 1 ∂ 2

(S ik,kj + S jk,ki − S kk,ij ), (35a)

Ŝ (b)ij =

Ŝ CZij − 1 + 2b

2(1 + b)

1

∂ 2(Ŝ CZkk ),ij . (35b)

For a consistency check: as a = b = −12 , Eqs. (32)reduce to Eqs. (34), which are just the special case of Eqs. (32) with the simplest sources terms.

The instantaneous Laplacian operator ∂ 2 in Eqs. (32a-

32d) means that the gauge-invariant quantities ĥ(ab)0µ ,

ĥ(b)ii , and h

(b)ij,j are non-dynamical or non-propagating .

Equivalently, in the gauge hi0,i+ahii,0 = 0, hij,i+bhii,j =0, the component h0µ, the spatial trace hii and the spa-tial divergence hij,j , are non-dynamical. We give a noteon the equations for the spatial trace and divergence.They are certainly consistent with, and can be carefullyreorganized from, the apparently “propagating-looking”equations in Eqs. (32e). But a more elucidating wayto reveal the non-dynamical character of hii and hij,j isthrough Eq. (32b) and the constraint hi0,i + ahii,0 =

0, hij,i + bhii,j = 0: First, h0i,i + ahii,0 = 0 says thatthe trace hii has the same property as h0i, which is non-dynamical by Eq. (32b). Then, hji,i +bhii,j = 0 says thatthe spatial divergence hji,i is non-dynamical as well.

Since ĥ(b)ii and h

(b)ij,j are non-dynamical, the truly dy-

namical component of ĥ(b)ij is its TT part. The TT part

of ĥ(b)ij actually equals h

T T ij in Eq. (4), because

ĥ(b)ij and

hij differ by a pure-gauge term which does not contributeTT component. The equation of motion for hT T ij is thusof special importance:

hT T ij = −S T T ij . (36)

The source term S T T

ij is the TT part of S ij. It relates toS ij and S

(b)ij in the same way as h

T T ij relates to hij and

h(b)ij :

S T T ij = S ij − 1

2δ ij (S kk −

1

∂ 2S kl,kl)

− 1

∂ 2(S ik,kj + S jk,ki −

1

2S kk,ij −

1

2

1

∂ 2S kl,klij )

= Ŝ (b)ij −

1 + b

2 δ ij S

(b)kk +

1 + 3b

2

1

∂ 2S (b)kk,ij. (37)

It should be noted that the equation for the TT compo-nent hT T ij does not have a gauge-fixing dual. The reasonis that hij cannot in general be reduced to contain onlyTT component, except for a pure wave without source[13]. In this regard, the tensor gauge field behaves ratherpeculiar that there are infinitely many gauges which canremove all nonphysical (gauge) degrees of freedom, butthere is no gauge which can directly pick out the dynam-

ical (TT) component. (As we explained in Section III,one can at best simply the extraction of hT T ij to be alge-braic and local.) Here one should notice the differencebetween “nonphysical” and “non-dynamical”. E.g., inelectrodynamics, the instantaneous Coulomb potential isnon-dynamical but physical, and must be included intothe total Hamiltonian of the system. Similarly, for grav-ity the instantaneous Newtonian interaction (contributedby h0µ and hii) is non-dynamical but physical, and mustbe included into the total Hamiltonian, especially in aquantum theory [12].

In electrodynamics, the Coulomb gauge is the uniqueconstraint to pick out the two physical (and at the same

time dynamical) components A⊥. The above peculiarfeature of tensor gauge field leads to an embarrassing

fact that among the infinitely many complete gauge con-ditions, hi0,i + ahii,0 = 0, hij,i + bhii,j = 0, no singlechoice is superior in all aspects. Equations (32) indicatethat by a = b = −12 we obtain the simplest form of equa-tions for all components of hµν . However, we still haveto extract the dynamical component hT T ij , and Eq. (21)

indicates that b = −13 leads to the simplest expression

for hT T ij . In Ref. [12], Weinberg chooses b = −13 to-

gether with a = −23 , which simplifies the equation forh0j , but leaves a complicated equation for h00. Sinceh00 is closely related to the Newtonian potential, post-

Newtonian dynamics favors a = −12 which leads to thesimplest equation for h00; but then one must choose be-

tween the convenience in dealing with h0j (which favorsb = −12) or the dynamical component h

T T ij (which favors

b = −13).

VI. ANOTHER PERSPECTIVE ON GAUGE

CONDITION AND GAUGE-FIELD

DECOMPOSITION

To understand better the trickiness in a tensor gaugetheory, in this section we approach the problem of gauge

condition and gauge-field decomposition from anotherperspective. In the decomposition of gauge fields intoan invariant part plus a pure-gauge part,

Ai ≡ Âi + Āi, hij ≡ ĥij + h̄ij , (38)

we had essentially followed in Sec. III a line of construct-ing the pure-gauge fields Āi and h̄ij. We now proceedby asking: What are the possible means of construct-ing the invariant quantities Âi and ĥij ? For the vector


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8

In Minkowski space-time, one needs to pay attentionthat only the Laplacian operator has a well-defined in-

verse, while the d’Alembert operator = ∂ 2 − ∂ 2t doesnot. (More exactly, inverting the Laplacian operatorrequires just boundary conditions, while inverting thed’Alembert operator would also require initial conditionswhich are unavailable.) Therefore, Âµ is to be con-structed with 1 ∂ 2

F jµ,j , instead of 1

F νµ,ν . Analogously,

ĥµν is to be constructed with 1 ∂ 2 Rµkνk and 1 ∂ 2

1 ∂ 2

Rµkνl,kl,

instead of 1

Rµρνρ and 1

1

Rµρνσ,ρσ . Another fact worth

noting is that ĥµν can posses two free parameters, not

just one. The reason is that ĥµν can be added by a gauge-invariant pure-gauge h̄µν ≡ ξ µ,ν + ξ µ,ν , where ξ 0 = αΞ,0and ξ j = β Ξ,j , with Ξ in Eq. (42) and α, β being twofree parameters. Accordingly, the complete tensor gaugeconditions in Minkowski space-time also contain two freeparameters, as formulated equivalently by a and b in Eqs.(14).

VII. SUMMARY AND DISCUSSION

In this paper we carefully examined and demonstratedwhy decomposing a tensor is much more tricky than de-composing a vector, even for the linear case. Concern-ing mathematical structures, it is the CZ decompositionthat has exact correspondence to the simple (and unique)vector decomposition. Namely, by a (singly) nonlocal

construction, a (uniquely) gauge-invariant field ĥCZµν canbe built out of hµν . Like the transverse vector current j⊥ = j − ∂

1 ∂ 2

∂ · j, the source for ĥCZµν also contains at

most a singly nonlocal structure.

If doubly nonlocal constructions are allowed, however,more nontrivial possibilities emerge for a tensor (but notfor a vector). Especially, a peculiar quantity which isgauge-invariant but at the same time a pure-gauge can beconstructed. It is essentially this peculiar quantity thatleads to infinitely many ways of fixing the gauge com-pletely, or equivalently, of decomposing the gauge fieldinto invariant and pure-gauge parts. (From our demon-

stration, it is clear that such phenomena would arise forall gauge fields of spin ≥ 2.)

A further and vital complication for the tensor gaugefield is that its truly dynamical component (the TT part)does not show up automatically after all gauge degreesof freedom have been removed; and extraction of the TTpart favors a different gauge than that of CZ. Our ob-servation reveals vividly that tensor coupling is indeedextraordinarily nontrivial, and requires further carefulstudies, even at linear approximation.

Beyond the linear order, separation of a pure-gaugebackground requires very careful formulation. Moreover,the nonlinear Einstein equations are so hard to solve that,

for the purpose of achieving certain sort of convenience,one may have to invent some particularly delicate fieldvariables and gauge conditions [16–18]; some gauges areeven too complicated to put into explicit forms [18]. Itshould be remarked, however, that the issue of gaugechoice may not be just a matter of convenience. As Wein-berg elaborated in Ref. [12], quantum Lorentz invariancefor gravitation might not be guaranteed with an arbitrarygauge-fixing.

This work is supported by the China NSF Grants10875082 and 11035003. X.S.C. is also supported by theNCET Program of the China Ministry of Education.

[1] N. Rosen, Phys. Rev. 57, 147 (1940).[2] R. Arnowitt, S. Deser, and C.W. Misner, in Gravita-

tion , L. Witten ed. (Wiley, New York, 1962), Chapter 7(posted as arXiv:gr-qc/0405109); and references therein.

[3] J. Schwinger, Phys. Rev. 130, 1253 (1963).[4] S. Deser, Ann. Inst. Henri Poincaré 7, 149 (1967).[5] J.W. York, J. Math. Phys. 14, 456 (1973); see also a

recent discussion by H.P. Pfeiffer and J.W. York, Phys.Rev. D 67, 044022 (2003).

[6] For a recent review of the nucleon spin problem, see, e.g.,F. Myhrer and A.W. Thomas, J. Phys. G 37, 023101(2010).

[7] X.S. Chen, X.F. Lü, W.M. Sun, F. Wang, and T. Gold-man, Phys. Rev. Lett. 100, 232002 (2008).

[8] X.S. Chen, W.M. Sun, X.F. Lü, F. Wang, and T. Gold-man, Phys. Rev. Lett. 103, 062001 (2009).

[9] X.S. Chen and B.C. Zhu, Phys. Rev. D 83, 084006(2011).

[10] C. Misner, K. Thorne, and J. Wheeler, Gravitation (Free-

man, San Francisco, 1973), pp. 948-949 (Box 35.1).[11] P.A.M. Dirac, Phys. Rev. 114, 924 (1959). See also the

Lecture notes by É. Gourgoulhon, arXiv:gr-qc/0703035.[12] S. Weinberg, Phys. Rev. 138, B988 (1965).[13] X.S. Chen and B.C. Zhu, Phys. Rev. D 83, 061501

(2011).[14] P. Jaranowski and G. Schaefer, Phys. Rev. D 57, 7274

(1998) [arXiv:gr-qc/9712075].[15] T. Damour, P. Jaranowski, G. Schaefer, Phys. Lett. B

513, 147 (2001) [arXiv:gr-qc/0105038].[16] B. Kol, arXiv:1009.1876.[17] T. Damour and G. Schaefer, Gen. Rel. Grav. 17, 879

(1985).[18] T. Ohta and T. Kimura, Prog. Theor. Phys. 79, 819

(1988); ibid. 76, 329 (1986); T. Ohta, H. Okamura, T.Kimura, and K. Hiida, ibid. 51, 1598 (1974); ibid. 50,492 (1973).
http://arxiv.org/abs/gr-qc/0405109http://arxiv.org/abs/gr-qc/0703035http://arxiv.org/abs/gr-qc/9712075http://arxiv.org/abs/gr-qc/0105038http://arxiv.org/abs/1009.1876http://arxiv.org/abs/1009.1876http://arxiv.org/abs/gr-qc/0105038http://arxiv.org/abs/gr-qc/9712075http://arxiv.org/abs/gr-qc/0703035http://arxiv.org/abs/gr-qc/0405109

Tensor Gauge Condition and Tensor Field Decomposition

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