All transverse and TT tensors in flat spaces of any dimension

We present general formulas for transverse and transverse-traceless (TT) symmetric tensors in flat spaces. TT tensors in conformally flat spaces can be obtained by means of a conformal transformation.


Introduction
A symmetric tensor T ij is called transverse if it satisfies where ∇ i denotes the covariant derivative with respect to metric g ij of a Ddimensional space. Condition (1) occurs in general relativity as an analog of the conservation law for energy and momentum, as a gauge condition for the metric tensor or as the momentum constraint in the initial data problem for the vacuum Einstein equations. In the last case one often assumes that, additionally, If (1) and (2) are satisfied then the tensor T is called transverse-traceless (TT). TT tensors were introduced by Arnowitt, Deser and Misner in 1959 [1]. Equation (1) is particularly simple if ∇ i is the covariant derivative corresponding to a flat connection. Even then its solutions are interesting, especially if (2) is satisfied, since they can be used to construct initial data with a conformally flat metric by means of the Lichnerowicz-York method [2,3]. This method is based on the invariance of equations (1)-(2) under the transformation with a nonvanishing function ψ. Examples of data obtained in this way can be found in classical papers by Misner [4], Brill and Lindquist [5], Bowen and York [6] and Brandt and Brügmann [7]. In this article we present general solution of equations with or without condition (2) with constant metric g ij . They are expressed in terms of potentials which undergo gauge transformations. From the traceless solutions we can obtain all TT tensors in conformally flat spaces by means of transformation (3).
For obvious reasons we assume D ≥ 2. All results are based on the Poincare lemma about local exactness of closed forms. For our purposes it is convenient to use the following form of this lemma where tensors α and β are completely antisymmetric in indices i 1 , i 2 , ... and I denotes a set of additional indices. Integer p satisfies 1 ≤ p ≤ D − 1.
2 Solutions of T ij ,j = 0 Proposition 1. All solutions of equation (4) have the form where R ikjp is a tensor possessing all algebraic symmetries of the Riemann tensor.
Proof. Let us apply the Poincare lemma (5) with p = 1 and I = i to equation (4). Hence, there exists tensor S ijk such that Condition Taking into account (5) with I = [ij] we obtain potentials V ijkp such that and A combination of equations (9) for different permutations of indices ijk leads to Substituting (11) to (7) yields Let R ikjp be defined by where the square bracket denotes antisymmetrization over all indices inside the bracket. Then (12) takes the form (6). Tensor R ijkp has algebraic symmetries of the Riemann tensor. It is antisymmetric in the first and second pair of indices, symmetric with respect to interchange of these pairs and R [ijkp] = 0 (hence also R i[jkp] = 0).
For D = 2 one has where R is arbitrary function and ǫ ij is the standard completely antisymmetric tensor. It follows from (6) that For D ≥ 3 tensor R ijkp contains more components than number of degrees of freedom admitted by equation (4). Arbitrariness in choice of R ijkp describes the following proposition.
For D ≥ 3 tensor R ikjp is given up to the translation by Proof. For D = 2 an addition to R does not change T ij if all its second derivatives vanish. Thus, it has to be linear in all coordinates.
In order to show that for D ≥ 3 all solutions of the equation with symmetries of the Riemann tensor, have the form (16) let us write (17) as From the Poincare lemma (5) with I = i one obtains where tensor V ijpk is antisymmetric in the last 3 indices, Thanks to R ′[ijk]p = 0 an antisymmetrization of (19) over the indices ij leads to Using again (5), now with I = [ij], we obtain where ξ ipjkr = ξ [ip] [jkr] . Antisymmetrization of (21) over ijp yields Taking a suitable combination of equations (22) with different permutations of indices allows to express tensor V in the form Substituting (23) to (21) leads to (16).
In dimension D = 2 gauge transformations (15) can be used to remove a constant term and terms linear in coordinates in an expansion of R around a fixed point. In higher dimensions tensor R ikjp can be decomposed into a traceless part C ikjp , corresponding to the Weyl tensor in general relativity, and the rest which is defined by an analog of the Ricci tensor Here R = R i i and .
Tensor C ikjp vanishes identically for D = 3. For D ≥ 4 we can try to achieve this condition by means of the gauge transformation (16).

Proposition 3. In dimension D = 3 every solution and in D ≥ 4 every analytic solution of (4) has the form
where R ij is an arbitrary symmetric tensor undergoing the gauge transformation with arbitrary functions ξ i .
Proof. Equation (26) follows if (24) with C ikjp = 0 is substituted to (6). In order to prove that for D ≥ 4 condition C ikjp = 0 is available one could look for an appropriate gauge transformation of the form given by Proposition 2.
In our opinion it is easier to prove solvability of (26) with respect to R ij if a transverse tensor T ij is given. Let us distinguish coordinate x 1 , which together with x a , a = 2, ...D, composes a Cartesian system of coordinates. Concerning evolution of R ij with respect to x 1 equations (26) with indices 11 and 1a are constraints since they do not contain second derivatives of R ij over x 1 . It is easy to show that they are preserved in x 1 if they are satisfied at x 1 = 0 and remaining equations (26) are fulfilled. If functions T ij are analytic then the constraints at x 1 admit solutions and from the Cauchy-Kowalewska theorem we obtain analytic solutions R ij of all equations (26). This situation is similar to that in general relativity. Equations (26) are identical with the linearized Einstein equations if R ij − 1 D−1 Rg ij is identified with the first corrections to the constant metric g ij . This analogy suggests the gauge transformations (27). It is easy to show that they preserve the rhs of (26). Counting number of components of R ij and ξ i we can be sure that transformations (27) are general up to functions of D − 1 variables. In order to exclude such additional functions one should find all transformations (16) preserving C ikjp = 0 and investigate their efect on R ij . It hasn't been done.
Remark: If g ij has the Lorentzian signature then equations (26) for R ij are hyperbolic and the assumption about analyticity in Proposition 3 is not necessary.

TT tensors
For T ij given by expression (6) the traceless condition (2) leads to the following equation for the "Ricci tensor" Equation (28) can be easily solved in terms of potentials.
Proposition 4. For D = 2 TT tensors are given by (14), where R satisfies For D ≥ 3 TT tensors are given by (6) and where Proof. For D = 2 there is R ij = 1 2 Rg ij , so (28) reduces to equation (29), which can be further integrated by means of holomorphic functions (for a definite signature of g ij ) or functions of null coordinates (for mixed signature).
Potentials S ijk are not uniquely defined. Their arbitrariness can be easily defined in the case of gauge C ijkp = 0.

Proposition 5. If (26) and (30) are satisfied then potentials S ijk undergo gauge transformations
where Proof. In order to prove (34) one should solve condition for an addition to S ijk which does not influence R ij given by (30). From (36) and (5) it follows that where V ijkr = V (ij) [kr] . If we define χ ijkr = 3 8 V i[jkr] then V ijkr = 4χ (ij)kr . Taking into account the identity Denoting two last expressions in (39) by η ijk leads to transformation (34).
An application of transformation (34) with ξ k = −S ik i /(D−1) and vanishing all other terms allows to obtain S ik i = 0 and R = 0. Then substituting (30) into (26) yields An appropriate choice of η allows to get S [ijk] = 0. Further gauge conditions can be assumed thanks to free functions χ ijkr . A description of TT tensors in dimension D = 3 is much simpler than in D ≥ 4. In this case Propositions 4 and 5 lead to the following result.

Proposition 6. In dimension D = 3 every TT tensor is given by
where A ij is a symmetric tensor undergoing the gauge transformations with arbitrary functions χ i and η.
Proof. For D = 3 one has C ikjp = 0 and where G ls = R ls − 1 2 Rg ls corresponds to the Einstein tensor. Substituting (43) to (6) leads to the following simplified version of formula (26) Gauge transformation (27) is now the most general since should be substituted into (16). If T i i = 0 then R pk is given by (30), where Tensor A can be arbitrary, but only its symmetric part appears in G ls . Without loss of generality we can assume hence the gauge condition S ik i = 0 is satisfied. Substituting (46) to (30) and using (44) leads to (41).
Remark: If T ′ij is a transverse tensor then is a TT tensor. Case (41) corresponds to If D ≥ 4 one can consider gauge conditions for a TT tensor other than C ijkr = 0. A natural candidate is condition R ij = 0.

Proposition 7. In dimension D ≥ 4 every analytic TT tensor has the form
where C ikjp is a tensor with all algebraic properties of the Weyl tensor.
Proof. Formula (52) is true if transformations (16) allow to kill R ij . Let us look for solutions of equations where Let x 1 and x a be coordinates introduced in the proof of Proposition 3. Equation (53) with k = p = 1 is a constraint regarding evolution of ξ kpr with respect to x 1 . The derivative of this constraint over x 1 leads to the new constraint Condition (55) is preserved if T i i = 0 since then R kp ,kp = 0. Thus, if we assume (53) with k = p = 1 and (55) on the surface x 1 = 0 then, in analytic case, we obtain solution of all equations (53). It remains to prove that for this solution there is a tensor ξ ikjpr which satisfies (54). It is the case since every such tensor admits decomposition whereξ jpr ik is traceless and It follows from (56) that tensorsξ ikjpr and ξ kpr can be defined independently.

Discussion
All calculations were performed in Cartesian coordinates x i . If we replace partial derivatives with respect to x i by covariant derivatives ∇ k then we can pass to other systems of coordinates. In particular formula (6) then takes the form For a nonflat metric expression (59) fails to satisfy (1) because of nonvanishing commutators of covariant derivatives. Given solutions of equations (1) and (2) in a flat space we can obtain all TT tensors in conformally flat spaces by means of transformation (3). The author is not able to express in general these solutions in terms of covariant derivatives of potentials with no use of the conformal factor ψ. The only exception is the case of spaces of constant curvature (space forms corresponding to flat space, sphere or hyperbolic space). Then R ij kp = λ(δ i k δ j p − δ j k δ i p ) , λ = const (60) and equation (1) is still satisfied if formula (59) is replaced by In order to satisfy T i i = 0 it is sufficient, but may be not necessary, to assume the covariant version of (30) R pk = ∇ r S (pk)r (62) We expect that for D ≥ 4 the gauge condition C ijkl = 0 is available for transverse tensors and condition R ij = 0 for TT tensors. Let us assume that space is flat but a TT tensor is invariant under a symmetry of metric. Such solutions can be obtained by means of invariant tensor potentials appearing in Propositions 1-7. Such description is not necessarily optimal as it is shown in the paper of Conboye andÓ Murchadha [8] and Conboye [9] in dimension D = 3. Their expressions for TT tensors contain only two arbitrary functions and at most their second derivatives, not third as in (41). It is difficult to say if a similar simplified description exists in higher dimensions.