  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
spacetime observables becoming ambiguous. 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.
