First Order Symmetry Operators for the Linearized Field Equation of Metric Perturbations

We determine the general form of the first order linear symmetry operators for the linearized field equation of metric perturbations in the spacetimes of dimension D>=4. Apart from the part derived easily from the invariance under general coordinate transformations, we find a part consisting of a Killing-Yano 3-form.


Introduction
It is often necessary to solve the linearized equations of motion of various fields in given back- solutions to easier equations, have also been considered. The following is a partial list of recent discussion: For spin 0 fields in generalized Kerr-NUT-(A)dS spacetimes, see [1]. For spin 1/2 fields see e.g. [2,3,4,5]. For spin 1 fields under some ansatz in Kerr-NUT-(A)dS spacetimes see [6,7], and in 4 dimensions see [8]. For spin 3/2 fields see e.g. [9]. For spin 2 and lower spin fields in Petrov type D spacetimes and higher dimensional extensions, see e.g. [10,11,12].
In this paper, we make no assumption about the form and the signature of the background geometry except that the dimension D is greater than or equal to 4, and determine the

Preliminaries
We consider the metric g µν in D-dimensional spaces, which we decompose into the background metric G µν and the perturbation h µν : g µν = G µν + h µν . No assumption about the signature of g µν is made.
Einstein-Hlibert Lagrangian for g µν (up to the overall constant factor) is given by The part linear in h µν is given by G µν R(G) + ΛG µν h µν + (total derivative term), (2.2) which shows that if there is no matter field the background equation of motion is the following vacuum Einstein equation: or equivalently, 4) and the following also holds: Unless otherwise stated, we do not impose these equations on the background. The part quadratic in h µν is given by where M λρµν is the following hermitian operator: and where L ǫ is the Lie derivative operator along ǫ µ : Under this transformation, δL = ∂ µ (ǫ µ L) for arbitrary h µν , and therefore If ǫ µ is a Killing vector of the background geometry, δh µν contains only terms linear in h µν , and from the terms quadratic in h µν in the above, Therefore we expect that L ǫ commutes with M λρ µν : Indeed this can be directly confirmed by the fact that L ǫ commutes with ∇ µ and L ǫ R λ ρµν = 0. Note that (2.17) is true even when the background geometry does not satisfy the equation of motion. If the background geometry satisfies the vacuum equation of motion, L has no linear term in h µν , and from the terms in δL linear in h µν , Since this is true for arbitrary h µν and ǫ µ , From the part proportional to h µν , In the following sections we solve these conditions. However in 2 dimensions, R λρ µν = Rδ [λ µ δ ρ] ν , and any manifold satisfies vacuum Einstein equation with no cosmological constant. Indeed it can be shown that M λρ µν h µν = 0 holds for any h µν if Λ = 0. In 3 dimensions, independent components of Riemann tensor are given in terms of Ricci tensor: and this means that any background satisfying vacuum Einstein equation is a space of constant curvature, which has been well studied. Therefore in the following sections we only consider cases of D ≥ 4.

Results and conclusions
In this section we summarize our results and give conclusions so that readers who are not interested in the details of the procedure for solving the equations (2.23), (2.24), (2.25), and (2.26) can skip them. The details will be explained in the next section.
First we show the solution to equation (2.23) which is purely algebraic: where K µ is an arbitrary vector, and Y µνλ is an arbitrary antisymmetric tensor. F λρ φ and H λρ φ are arbitrary tensors satisfying Using the above we obtain the following solution to (2.24): where c is a scalar function, and it turns out that K µ must be a conformal Killing vector, and Y µνλ must be a Killing-Yano tensor.
Then we find that (2.25) is equivalent to the following three relations: is satisfied, and (3.6) and (3.7) mean that c and ∇ κ K κ are constants.
Finally we find that (2.26) is equivalent to the following three relations: where c = c− 4 D ∇ κ K κ is a constant, Y µνλ is a Killing-Yano 3-form, H µν λ = H νµ λ is an arbitrary tensor, F λρ µ = F ρλ µ is an arbitrary tensor, and K µ is a conformal Killing vector satisfying the condition ∇ κ K κ = const. if Λ = 0, or K µ is a Killing vector if Λ = 0.
The above result is obtained for D ≥ 4, but it can be confirmed that it gives a solution even for D = 3, although it may not be general solution.
The terms proportional to c in (3.13) and (3.14) correspond to (2.11). The term consisting of H µν λ in (3.14) corresponds to (2.19), and the term consisting of F λρ µ in (3.13) corresponds to (2.20). The first terms in the right hand side of (3.13) and (3.14) correspond to (2.17).  [14]. In the case of Λ = 0, nonzero constant ∇ κ K κ gives a noncommuting symmetry operator, and it is also interpreted as a result of general coordinate transformation, because if h µν is a solution, ∇ µ K ν + ∇ ν K µ + L K h µν is also a solution, which is in the form of (2.13). In fact, the conformal transformation with ∇ κ K κ = const. is a constant rescaling, and the difference between S λρ µν and Q λρ µν comes from the rescaling of the background metric in M λρ µν .
By analyses similar to ours given in the next section we can construct general form of higher order symmetry operators in principle. However such calculations become increasingly difficult as the order of the operators becomes higher and higher. One of immediate methods to give higher order operators is to take products of first order operators. Although it gives no more information on the solutions than the first order operators, it may give a hint about the general forms of higher operators.

A procedure for solving the conditions for the symmetry operators
In this section we show the details of the procedure for solving the conditions (2.23), (2.24), Our basic strategy is to express tensors in terms of tensors with fewer free indices.
Analysis of (2.23) for S λρ µν φ First let us solve (2.23), which is purely algebraic and does not contain derivatives. By contracting σ and τ in (2.23), we obtain = (terms proportional to Q γδ αβǫ ). The difference of (4.5) and (4.6) gives a relation containing only Q γδ αβǫ . Using it we can simplify (4.5): By using this (4.4) can be simplified further: Each term in the right hand side of (2.23) can be rewritten by this, and we obtain +(terms proportional to Q γδ αβǫ with some pairs of indices contracted). (4.9) Note that this no longer contains S γδ αβ ǫ . Therefore (4.8) exhaustively contains information on S γδ αβ ǫ .
Analysis of (2.25) Next let us analyze (2.25  (4.59) The first two lines of the above can be rewritten by using  we obtain 0 = 1 4 ∇ (λ (∇ ρ) ∇ κ − ∇ |κ| ∇ ρ) ) + (∇ (λ ∇ |κ| − ∇ κ ∇ (λ )∇ ρ) H µνκ +(terms proportional to ∇ α ∇ β c, Λ∇ ǫ H αβγ , R γδζη ∇ ǫ H αβθ , ∇ ǫ R γδζη H αβθ , (4.69) The first, second and third lines of the above can be simplified by replacing the commutators of covariant derivatives by Riemann tensors, and the fourth line is rewritten as follows: (4.70) The first line of the above can be simplified by (A.7), and the second line can be simplified by using (A.4): The last term of the above cancels the fifth line of (4.69). The sixth and seventh line are simplified by the following relations derived from (A.3): where we used Then terms proportional to ∇ [µ K ν] from the seventh line cancel the eighth and ninth lines of (4.69).