# All transverse and TT tensors in flat spaces of any dimension

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## Abstract

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.

## Keywords

Transverse tensors TT tensors Conformal method## 1 Introduction

*D*-dimensional space. Condition (1) occurs in general relativity as an analog of the conservation law for energy and momentum, as the harmonic coordinate condition or as the momentum constraint in the initial data problem for the vacuum Einstein equations. In the last case one often assumes that, additionally,

*I*denotes a set of additional indices. Integer

*p*satisfies \(1\le p\le D-1\).

## 2 Solutions of \(T^{ij}_{\ ,j}=0\)

## Proposition 1

## Proof

*ijk*leads to

\(\square \)

*R*is arbitrary function and \(\epsilon ^{ij}\) is the standard completely antisymmetric tensor. It follows from (6) that

## Proposition 2

For \(D=2\) function *R* is given up to the translation

## 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.

*ij*leads to

*ijp*yields

*V*in the form

*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 \(R_{kp}=R^i_{\ kip}\)

## Proposition 3

## Proof

Equation (26) follows if (24) with \(C^{ikjp}=0\) is substituted to (6). In order to prove that for \(D\ge 4\) 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 \(x^1\) Eq. (26) with indices 11 and 1*a* 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 Eq. (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 Eq. (26). This situation is similar to that in general relativity. Equations (26) are identical with the linearized Einstein equations if \(R_{ij}-\frac{1}{D-1}Rg_{ij}\) 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 \(\xi _{i}\) 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. \(\square \)

## 3 TT tensors

## Proposition 4

*R*satisfies

## Proof

For \(D=2\) there is \(R^{ij}=\frac{1}{2}Rg^{ij}\), so (28) reduces to Eq. (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).

*pk*yields (30). \(\square \)

Potentials \(S^{ijk}\) are not uniquely defined. Their arbitrariness can be easily defined in the case of gauge \(C^{ijkp}=0\).

## Proposition 5

## Proof

A description of TT tensors in dimension \(D=3\) is much simpler than in \(D\ge 4\). In this case Propositions 4 and 5 lead to the following result.

## Proposition 6

## Proof

*A*can be arbitrary, but only its symmetric part appears in \(G_{ls}\). Without loss of generality we can assume

*A*can be further simplified by means of (34). Since for \(D=3\)

## Remark

If \(D\ge 4\) one can consider gauge conditions for a TT tensor other than \(C^{ijkr}= 0\). A natural candidate is condition \(R^{ij}=0\).

## Proposition 7

## Proof

If \(R^{ij}=0\) then gauge transformations (16) are restricted by

## 4 Discussion

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 follows from the paper of Conboye and Ó 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).

For a general metric *g* we are not able to write down solutions of (4), with or without \(H=0\), in terms of potentials. In the case of \(D=3\) and axially symmetric fields Eq. (4) can be completely solved in generic case (see Propositions 2.1 and 2.2 in [10]), but then condition \(H=0\) becomes a differential equation.

In this paper we focused on mathematical description of transverse and transverse-traceless tensors. Concerning physical applications we are mainly interested in using results of Sect. 3 in order to construct initial data within the conformal approach to the vacuum Einstein equations. Following results of Maxwell [11] one can include in these data marginally trapped surfaces which are expected to develop into black hole horizons. Such configurations would generalize those of Bowen and York [6]. They are often considered as nonphysical since the Kerr metric does not admit any conformally flat section. However, the conformal flatness property will be spoiled during the time evolution. It would be interesting to see if the Kerr like black holes will be created in agreement with the cosmic censor hypothesis. We are going to consider a related but simpler problem: using our data and Maxwell’s approach we will test the Penrose inequality involving the surface area of an initial trapped surface and the total mass of the initial surface.

Another possible application of our approach would be a construction of cosmological initial data describing perturbations of the standard cosmological models (note that surfaces of constant time in the Friedmann-Robertson-Walker spacetimes are always conformally flat).

*A*satisfying the unconstrained wave equation (65). Note that functions \(A_{ij}\) can be found e.g. by means of separation of variables what is not possible for \(h_{ij}\) because of conditions (64).

We hope that other physical applications appear with time.

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