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General formulation of general-relativistic higher-order gauge-invariant perturbation theory

Gauge-invariant treatments of general-relativistic higher-order perturbations on generic background spacetime is proposed. After reviewing the general framework of the second-order gauge-invariant perturbation theory, we show the fact that the linear-order metric perturbation is decomposed into gauge-invariant and gauge-variant parts, which was the important premis of this general framework. This means that the development the higher-order gauge-invariant perturbation theory on generic background spacetime is possible. A remaining issue to be resolve is also disscussed.


Introduction
Perturbation theories are powerful techniques in many area of physics and lead phyisically fruiteful results. In particualr, in general relativity, the construction of exact soltuions is not so easy and known exact solutions are often too idealized, though there are many known exact solutions to the Einstein equation [1]. Of course, some exact solutions to the Einstein equation well describe our universe, gravitational field of stars, and black holes. However, in natural phenomena, there always exist "fluctuations". To describe these fluctuations, the linear perturbation theories around some background spacetime are developed [2], and are used to describe fluctuations of our universe, gravitational field of stars, and gravitational waves from strongly gravitating sources.
Besides the development of the general-relativistic linear-order perturbation theory, higher-order general-relativistic perturbations also have very wide applications, for example, cosmological perturbations [3,4,5,6,7], black hole perturbations [8], and perturbation of a neutron star [9]. In spite of these wide applications, there is a delicate issue in the treatment of general-relativistic perturbations, which is called gauge issue. General relativity is based on general covariance. Due to this general covariance, the gauge degree of freedom, which is an unphysical degree of freedom of perturbations, arises in general-relativistic perturbations. To obtain physical results, we have to fix this gauge degree of freedom or to treat some invariant quantities in perturbations. This situation becomes more complicated in higher-order perturbations. Therefore, it is worthwhile to investigate higher-order gauge-invariant perturbation theory from a general point of view.
According to this motivation, the general framework of higher-order generalrelativistic gauge-invariant perturbation theory has been discussed [10,11] and applied to cosmological perturbations [6,7]. However, this framework is based on a conjecture (Conjecture 1 below) which roughly states that we have already known the procedure to find gauge-invariant variables for a linear-order metric perturbations. The main purpose of this letter is to give the outline of a proof of this conjecture. Due to this proof, a formulation of the higher-order general-relativistic gauge-invariant perturbation theory is almost completed on generic background spacetime. The details of the ingredients of this Letter are explained in Ref. [14].

General framework of higher-order gauge-invariant perturbation theory
In this section, we review the framework of the gauge-invariant perturbation theory [10,11]. In any perturbation theory, we always treat two spacetime manifolds. One is the physical spacetime (M,ḡ ab ), which is our nature itself, and we want to describe (M,ḡ ab ) by perturbations. The other is the background spacetime (M 0 , g ab ), which is prepared as a reference by hand. We note that these two spacetimes are distinct.
Further, in any perturbation theory, we always write equations for the perturbation of the variable Q like Q("p") = Q 0 (p) + δQ(p). (1) Equation (1) gives a relation between variables on different manifolds. Actually, Q("p") in Eq. (1) is a variable on M, while Q 0 (p) and δQ(p) are variables on M 0 . Since we regard Eq. (1) as a field equation, Eq. (1) includes an implicit assumption of the exitence of a point identification map M 0 → M : p ∈ M 0 → "p" ∈ M. This idenification map is a gauge choice in perturbation theories [12].
To develop this understanding of the "gauge", we introduce an infinitesimal parameter λ for perturbations and (n + 1) + 1-dimensional manifold N = M × R (n + 1 = dim M) so that M 0 = N | λ=0 and M = M λ = N | R=λ . On N , the gauge choice is regarded as a diffeomorphism X λ : N → N such that X λ : M 0 → M λ . Further, we introduce a gauge choice X λ as an exponential map with a generator X η a which is chosen so that its integral curve in N is transverse to each M λ everywhere on N [4,5]. Points lying on the same integral curve are regarded as the "same" by the gauge choice X λ .
The first-and the second-order perturbations of the variable Q on M λ are defined by the pulled-back X * λ Q on M 0 , which is induced by X λ , and expanded as Since Eq. (2) is just the perturbative expansion of X * λ Q λ , the first-and the second-order perturbations of Q are given by (1) When we have two gauge choices X λ and Y λ with the generators X η a and Y η a , respectively, and when these generators have the different tangential componetns to each M λ , X λ and Y λ are regarded as different gauge choices. The gauge-transformation is regarded as the change of the gauge choice X λ → Y λ , which is given by the From general arguments of the Taylor expansion [5], the pull-back Φ * λ is expanded as where ξ a (1) and ξ a (2) are the genertors of Φ λ . From Eqs. (2) and (3), each order gaugetransformation is given as We also employ the order by order gauge invariance as a concept of gauge invariance [7]. We call the kth-order perturbation Based on the above set up, we proposed a procedure to construct gauge-invariant variables of higher-order perturbations [10]. First, we expand the metric on the physical spacetime M λ , which is pulled back to the background spacetime M 0 through a gauge choice X λ as Although the expression (6) depends entirely on the gauge choice X λ , henceforth, we do not explicitly express the index of the gauge choice X λ in the expression if there is no possibility of confusion. The important premise of our proposal was the following conjecture [10] for h ab : For a second-rank tensor h ab , whose gauge transformation is given by (4), there exist a tensor H ab and a vector X a such that h ab is decomposed as where H ab and X a are transformed as under the gauge transformation (4), respectively.
We call H ab and X a are the gauge-invariant part and the gauge-variant part of h ab , respectively.
Although Conjecture 1 is nontrivial on generic background spacetime, once we accept this conjecture, we can always find gauge-invariant variables for higher-order perturbations [10]. Using Conjecture 1, the second-order metric perturbation l ab is decomposed as where Y L ab − X L ab = 0 and Y Y a − X Y a = ξ a (2) + [ξ (1) , X] a . Furthermore, using the firstand second-order gauge-variant parts, X a and Y a , of the metric perturbations, gaugeinvariant variables for an arbitrary tensor field Q other than the metric can be defined by (1) (2) These definitions (10) and (11) also imply that any perturbation of first and second order is always decomposed into gauge-invariant and gauge-variant parts as (2) respectively. Actually, the perturbations of the Einstein tensor are given in the same form Eqs. (12) and (13) : (1) where (1) G b a [ * ] is the gauge-invariant linear terms and (2) G b a [ * , * ] are collections of quadratic terms of gauge-invaraint linear metric perturbations. On the other hand, the energy momentum tensor on M λ is also expanded as and its first-and the second-order perturbations (1) T b a and (2) T b a are decomposed as Eqs. (12) and (13) : These were confirmed in the case of a perfect fluid, an imperfect fluid, and a scalar field [7]. Imposing order by order Einstein equations the first-and the second-order perturbation of the Einstein equations are automatically given in gauge-invariant form : Further, the perturbative equations of motion for matter fields, which are derived from the divergence of the energy momentum tensor, are also automatically given in gaugeinvariant form [7]. Thus, based only on Conjecture 1, we have developed the general framework of second-order general relativistic perturbation theory. We also note that this general framework of the second-order gauge-invariant perturbation theory are independent of the explicit form of the background metric g ab , except for Conjecture 1.

Decomposition of the linear-order metric perturbation
Now, we give the outline of a proof of Conjecture 1. To do this, we only consider the background spacetimes which admit ADM decomposition [15]. Therefore, the background spacetime M 0 considered here is n + 1-dimensional spacetime which is desribed by the direct product R × Σ. Here, R is a time direction and Σ is the spacelike hypersurface (dim Σ = n). The background metric g ab is given as In this letter, we only consider the case where α = 1 and β i = 0, for simlicity. The most general case where α = 1 and β i = 0 is discussed in Ref. [14].
To consider the decomposition (7) of h ab , first, we consider the components of the metric h ab as Under the gauge-transformation (4), in the case where α = 1 and β i = 0, these components {h tt , h ti , h ij } are transformed as where K ij is the extrinsic curvature of Σ and D i is the covariant derivative associate with the metric q ij (D i q jk = 0). In our case, K ij = − 1 2 ∂ t q ij . Inspecting gauge-transformation rules (25) and (26), we introduce a new symmetric tensorĤ ab whose components are given bŷ Here, we assume the existence of the variable X t whose gauge-transformation rule is given by Y X t − X X t = ξ t . This assumption is confirmed later soon. Since the componentŝ H ti andĤ ij are regarded as a vector and a symmetric tensor on Σ, respectively,Ĥ ti and H ij are decomposed as [13] To confirm the one-to-one correspondence between {Ĥ ti ,Ĥ ij } and we have to discuss the boundary conditions for the variables However, in this paper, we do not discuss these boundary conditions in detail. Instead, we assume the existence of the Green functions of two elliptic derivative operators ∆ := D i D i and D ij := q ij ∆ + 1 − 2 n D i D j + (n) R ij , where (n) R ij is the Ricci curvature on Σ. The boundary conditions for the variables are implicitly included in the Green functions of these derivative operator and the one-to-one correspondence of the sets are guaranteed by these two Green functions. Further, we also decompose the component ξ i of the generator of gauge transformation as ξ i =: We first find the variable X t in Eq. (27). From the above gauge-transformation rules, we see that the combination satisfy Y X t − X X t = ξ t . We also find the variable X i Inspecting gauge-transformation rules (31)-(37) and using the variables X t and X i defined by Eqs. (38)-(39), we find gauge-invariant variables as follows: Actually, it is straightforward to confirm the gauge-invariance of these varaibles. In terms of the variables Φ, Ψ, ν i , χ ij , X t , and X i , original components of h ab is given by Comparing Eq. (7), a natural choice of H ab and X a are These show that the linear-order metric perturbation h ab is decomposed into the form Eq. (7).

Summary and discussions
In summary, we showed the outline of a proof of Conjecture 1 which is the important premise of our general framework of gauge-invariant perturbation theory. Although we only consider the background spacetime with α = 1 and β i = 0, the above proof is extended to general case where α = 1 and β i = 0 [14]. Further, in our proposal, we applied the ADM decomposition to identify the gauge-ivariant variable H ab and the gauge-variant variable X a , and our choice of the components of H ab depends on the choice of the spacelike hypersurface Σ in M 0 . However, this Σ dependence of our choice does not break the covariance in the statement of Conjecture 1. This can be easily see from the extension of our proof to the most general case where α = 1 and β i = 0 in Ref. [14]. We also note that the choice the decomosition (7) is not unique as pointed out in Ref. [7]. What we showed is a procedure to carry out the decomposition (7) to emphasize the existence of H ab and X a in Eq. (7). In our proof, we assumed the existence of the Green functions for the derivative operators ∆ and D ij . In this sense, we have specified the boundary conditions for the perturbative variables at the boundary ∂Σ of Σ, because explicit expression of Green functions are depends on the boundary conditions. This also implies that we have ingored the "zero-mode" which belong to the kernel of these derivative operators. These zero-mode corresponds to the degree of freefom of the boundary condition at the boundary ∂Σ. To includes these modes into our consideration, different treatments perturbations and the careful arguments of the boundary conditions for the perturbations will be necessary. We call this problem as zero-mode problem. Even in the cosmological perturbations, zero-mode problem exists. We leave the resolution of this zero-mode problem as a future work.
Although this zero-mode problem should be resolved, we confirmed the important premise of our general framework of second-order gauge-invaraint perturbation theory on generic background spacetime. Due to this, we have the possibility of applications of our framework for the second-order gauge-invariant perturbation theory to perturbations on generic background spacetime. Actually, in the cosmological perturbation case, we have developed the second-order cosmological perturbations along this general framework [6,7]. The similar development will be also possible for the any order perturbation in two-parameter case [10]. Therefore, we may say that the wide appicaltions of our gauge-invariant perturbation theory are opened. We also leave these developments as future works.