The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one recasts the Einstein equations as a first-order evolutionary problem with constraints. The main idea is the following: Let $\Sigma$ be a 3-dimensional (smooth, connected) manifold and fix one-parameter families of functions $\{N_{t}\}_{t}\subset C^{\infty}(\Sigma)$ and vector fields $\{X_{t}\}_{t}\subset\Gamma^{\infty}(T^{\ast}\Sigma)$ on $\Sigma$, smoothly depending on $t$ in a suitable sense. Then, we consider the equations
\begin{align*}
\text{Evolution}:
\begin{cases}
\partial_{t}h=&-2Nk+\mathcal{L}_{X}h\\
\partial_{t}k=&N\bigg[\mathrm{Ric}(h)+\mathrm{tr}_{h}(k)k-2(k\times k)\bigg]-\mathrm{Hess}_{h}(N)+\mathcal{L}_{X}k
\end{cases}
\end{align*}
\begin{align*}
\text{Constraints}:
\begin{cases}
\Vert k\Vert^{2}_{h}-\mathrm{tr}_{h}(k)^{2}-\mathrm{Scal}(h)&=0\\
\nabla\mathrm{tr}(k)-\mathrm{div}(k)&=0.
\end{cases}
\end{align*}
where $\Vert k\Vert_{h}:=k^{ij}k_{ij}$, $(k\times k)_{ij}:={k_{i}}^{k}k_{kj}$ and $\mathrm{Hess}_{h}(N):=\nabla_{i}^{h}\nabla_{j}^{h}N$. These are equations for a $t$-dependent Riemannian metric $h_{t}$ on $\Sigma$ and a symmetric 2-tensor field $k_{t}$. Then, if $(-\varepsilon,\varepsilon)\ni t\mapsto (h_{t},k_{t})$ is a (local) solution to this system, the Lorentzian metric $g$ on $M_{\varepsilon}:=(-\varepsilon,\varepsilon)\times\Sigma$ given by
$$g=-(N^{2}+X^{i}X_{i})\mathrm{d} t^{2}+2X_{i}d t\otimes \mathrm{d}x^{i}+h_{ij}\mathrm{d}x^{i}\otimes\mathrm{d}x^{j}$$
is a (local in time) solution to the Einstein equations $\mathrm{Ric}(g)-\frac{1}{2}\mathrm{Scal}(g)g=0$, where $k_{t}$ is the second fundamental form of $\Sigma$. There has been a vast literature on the mathematical side, especially by Fischer, Marsden and Moncrief in the 1970s and 1980s on this approach (see e.g. [1] and [2] below for reviews). The constraint equations can be seen as a hyperbolic system and hence provide constraints on the choice of initial data. The solvability of these equations for a given $(N,X)$ is reviewed extensively in [3].
Now, lets say I am interested in the problem of linearized gravity in this setting. As a background, I want to consider a globally-hyperbolic manifold, i.e. a manifold globally of the form $M=\mathbb{R}\times\Sigma$ with $g=-\beta^{2}d t^{2}+h_{t}$ where $\Sigma$ is a Cauchy-hypersurface, $\beta$ a positive function and $h_{t}$ a one-parameter family of Riemannian metrics on $\Sigma$. Then, the Einstein equations for $g$ are equivalent to the ADM equations with $N:=\beta$ and $X:=0$. Now, lets say I linearize these equations around $(h,k)$ where $k:=\frac{1}{2\beta}\partial_{t}h$. Then, I get linear equations on the perturbations $(\overline{h},\overline{k})$.
Question: What should I do with the lapse and shift in the linearized setting? Now, my background manifold by definition has $N:=1$ and $X:=0$ globally. But if I just linearize these equations around $(h,k)$, then I only obtain perturbations $\overline{g}$ with the property $\overline{g}_{00}=\overline{g}_{0i}=0$ (which somehow seems to be related to a gauge choice for the perturbation), which seems somehow restricted compared to the usual approach of linearized gravity (i.e. taking the 4D Einstein equations). On the other hand, if I take the full ADM equations above with non-trivial $N,X$ and linearize $(h,k,N,X)$ with background $(N,X)=(1,0)$, then I don't really know how I should handle the perturbations $(\overline{N},\overline{X})$, since they are in some sense just external parameters, which are freely specifiable.
Let me mention that the linearized constraint equations are discussed frequently in the literature, especially in relation to stability problems in mathematical relativity. The linearized evolution equations, however, are discussed much less, at least as far as I am aware of. An exception is a short discussion by Fischer-Marsden contained in Section 4 of the review [2] below.
Literature:
- [1]: Fischer, Marsden: The initial value problem and the dynamical formulation
of general relativity. In S. W. Hawking and W. Israel, editors, General relativity:
an Einstein centenary survey, pages 138–211. Cambridge University Press 1979.
- [2]: Fischer, Marsden: Topics in the Dynamics of General Relativity. In
J. Ehlers, editors, Isolated gravitating systems in general
relativity, pages 322–395. North Holland Publishing Company 1979.
- [3]: Choquet-Bruhat. General Relativity and the Einstein Equations. Oxford University Press 2009.
