  We analyze the weak-field limit of General Relativity with matter and its
possible quantisations. This analysis aims towards a predictive quantum theory
to provide a first-principles description of gravitational effects in
macroscopic quantum systems. This includes recently proposed experiments on the
generation of (Newtonian) gravitational forces from quantum distributions of
matter, and phenomena like gravity-induced entanglement, gravitational cat
states, gravity-induced Rabi oscillations, and quantum causal orderings of
events. Our main results include: (i) The demonstration that these phenomena do
not involve true gravitational degrees of freedom. (ii) We show that, unlike
full general relativity, weak gravity with matter is a parameterised field
theory, i.e., a theory obtained by promoting spacetime coordinates to
`dynamical' variables. (iii) Quantisation via gauge-fixing leads to an
effective field theory that account for some phenomena, but at the price of
gauge dependence that manifests more strongly on spacetime observables. This
ambiguity is a manifestation of the problem of time that persists even in weak
gravity. (iv) A consistent quantisation of parameterised field theories is
essential for a predictive and spacetime covariant theory of weak gravity that
describes gravitational effects in macroscopic quantum systems. We also discuss
the implication of our results to gravitational decoherence theories, the
notion of locality in gravity vis-a-vis quantum information theory, and the
intriguing possibility that proposed solutions to the problem of time can be
tested in weak-gravity quantum experiments.
