Consider the Lagrangian of a real, scalar field coupled to gravity via the metric $g_{\mu\nu}$ and covariant derivative $\nabla_\mu$
$$\mathcal{L} = \sqrt{-g} (-\frac{1}{2} g^{\mu\nu} \nabla_\mu \phi \nabla_\nu \phi -\frac{1}{2}m^2\phi^2). $$
After canonical quantization, we can expand the field operator in terms of positive-negative frequency modes as (putting the theory in a box) $$ \hat{\phi}(x) = \sum_k [\hat{a}_k u_k(x) + \hat{a}_k^\dagger u^*_k(x)]. $$
However, generally different observers wouldn’t agree on this split between the modes and identify particle states differently. So, this expansion is not unique and for each such expansion we need to find a complete set of orthonormal modes $\{v_k(x)\}$, where orthonormality is w.r.t. the Klein-Gordon inner product: $$ (\phi_1, \phi_2) := -i \int \sqrt{g_\Sigma} [\phi_1 \nabla_\mu \phi_2^* - \phi_2^* \nabla_\mu \phi_1] d\Sigma^\mu.$$ Here $\Sigma$ is a space-like hypersurface and $g_\Sigma$ the induced metric.
My question is about the uniqueness of this hypersurface. In particular, how arbitrary is this choice? For example, would choosing a coordinate chart or defining a worldline fix this surface? Also, if coordinates fix it, I’d expect isometries to result in equivalent definitions of vacuum. Is this correct?