4 2. Second kind gauge The second kind gauge appears in perturbation theories in a theory with general covariance. This notion of the second kind “gauge” is the main issue of this paper. To explain this, we have to remind what we are doing in perturbation theories. First, in any perturbation theories, we always treat two spacetime manifolds. One is the physical spacetime M . This physical spacetime M is our nature itself and we want to describe the properties of this physical spacetime M through perturbations. The other is the background spacetime M$_{0}$ . This background spacetime have noth- ing to do with our nature and this is a fictitious manifold which is prepared by us. This background spacetime is just a reference to carry out perturbative analyses. We emphasize that these two spacetime manifolds M and M$_{0}$ are distinct. Let us denote the physical spacetime by ( M , ¯ g$_{ab}$ ) and the background spacetime by ( M$_{0}$ , g$_{ab}$ ), where ¯ g$_{ab}$ is the metric on the physical spacetime mani- fold, M , and g$_{ab}$ is the metric on the background space- time manifold, M$_{0}$ . Further, we formally denote the spacetime metric and the other physical tensor fields on M by Q and its background value on M$_{0}$ by Q$_{0}$ . Second, in any perturbation theories, we always write equations for the perturbation of the physical variable Q in the form Usually, this equation is simply regarded as a relation be- tween the physical variable Q and its background value Q$_{0}$ , or as the definition of the deviation δQ of the phys- ical variable Q from its background value Q$_{0}$ . However, Eq. (2.5) has deeper implications. Keeping in our mind that we always treat two different spacetimes, M and M$_{0}$ , in perturbation theory, Eq. (2.5) is a rather curi- ous equation in the following sense: The variable on the left-hand side of Eq. (2.5) is a variable on M , while the variables on the right-hand side of Eq. (2.5) are variables on M$_{0}$ . Hence, Eq. (2.5) gives a relation between vari- ables on two different manifolds. Further, through Eq. (2.5), we have implicitly identi- fied points in these two different manifolds. More specifi- cally, Q (“ p ”) on the left-hand side of Eq. (2.5) is a field on M , and “ p ” ∈ M . Similarly, we should regard the back- ground value Q$_{0}$ ( p ) of Q (“ p ”) and its deviation δQ ( p ) of Q (“ p ”) from Q$_{0}$ ( p ), which are on the right-hand side of Eq. (2.5), as fields on M$_{0}$ , and p ∈ M$_{0}$ . Because Eq. (2.5) is regarded as an equation for field variables, it implicitly states that the points “ p ” ∈ M and p ∈ M$_{0}$ are same. This represents the implicit assumption of the ex- istence of a map M$_{0}$ → M : p ∈ M$_{0}$ ↦→ “ p ” ∈ M , which is usually called a gauge choice (of the second kind) in perturbation theory[17]. It is important to note that the second kind gauge choice between points on M$_{0}$ and M , which is estab- lished by such a relation as Eq. (2.5), is not unique to the theory with general covariance. Rather, Eq. (2.5) in- volves the degree of freedom corresponding to the choice of the map X : M$_{0}$ ↦→ M . This is called the gauge de- gree of freedom (of the second kind). Such a degree of freedom always exists in perturbations of a theory with general covariance. General covariance intuitively means that there is no preferred coordinate system in the theory as mentioned above. If general covariance is not imposed on the theory, there is a preferred coordinate system in the theory, and we naturally introduce this preferred co- ordinate system onto both M$_{0}$ and M . Then, we can choose the identification map X using this preferred co- ordinate system. However, there is no such coordinate system in general relativity due to the general covari- ance, and we have no guiding principle to choose the identification map X . Indeed, we may identify “ p ” ∈ M with q ∈ M$_{0}$ ( q = p ) instead of p ∈ M$_{0}$ . In the above understanding of the concept of “gauge” (of the second kind) in general relativistic perturbation theory, a gauge transformation is simply a change of the map X . These are the basic ideas of gauge degree of freedom (of the second kind) in the general relativistic perturbation theory which are pointed out by Sacks[16] and mathe- matically clarified by Stewart and Walker[17]. Based on these ideas, higher-order perturbation theory has been developed in Refs. [8–13, 15, 20–22]. To formulate the above understanding in more detail, we introduce an infinitesimal parameter λ for the per- turbation. Further, we consider the 4 + 1-dimensional C. Formulation of perturbation theory FIG. 2: The second kind gauge is a point-identification be- tween the physical spacetime M$_{λ}$ and the background space- time M 0 on the extended manifold N . Through Eq. (2.5), we implicitly assume the existence of a point-identification map between M$_{λ}$ and M 0 . However, this point-identification is not unique by virtue of the general covariance in the theory. We may chose the gauge of the second kind so that p ∈ M 0 and “ p ” ∈ M$_{λ}$ is same ( X$_{λ}$ ). We may also choose the gauge so that q ∈ M 0 and “ p ” ∈ M$_{λ}$ is same ( Y$_{λ}$ ). These are different gauge choices. The gauge transformation X$_{λ}$ → Y$_{λ}$ is given by the diffeomorphism Φ = X $_{−}$1 λ ◦ Y$_{λ}$ . Q (“ p ”) = Q$_{0}$ ( p ) + δQ ( p ) . (2.5)