Freely falling frame and the use of special relativity? Let observer $\mathscr{O}_1$ be in a freely falling frame and following the trajectory $\gamma_1(\tau)$. Consider another observer $\mathscr{O}_2$ who is not in a freely falling frame and has trajectory $\gamma_2(\tau)$. Let also the trajectories $\gamma_1(\tau)$ and $\gamma_2(\tau)$ meet at the space time event $A$. We also assume that we have two particles particle $P_1$ has a world line that also passes through event $A$ and particle $P_2$ has a world line that does not. 
My question is:
To what extent can the observers $\mathscr{O}_1$ and $\mathscr{O}_2$  describe the properties of the particles $P_1$ and $P_2$ using the formula of Special relativity?
My Guess
(The following is provided as an idea of what I am looking for, this probably isn't correct)
I have been reading around and although am still fairly confused about this topic, think I have some idea.This answer seems to indicate the following: Observer $\mathscr{O}_2$   can explain the properties of the particle $P_2$ using special relativity, as long as no derivatives are used. They can do this using a special set of coordinates called a 'local observers frame' (see Moore 2013, p. 145) in which the metric is Minkowski. This means that this measures for particle $P_1$ an energy:
$$E(P_1)=-p^\mu(P_1) u_{\mu}(\mathscr{O}_2) $$
but for particle $P_2$ the same relation does not hold:
$$E(P_2)\ne-p^\mu(P_2) u_{\mu}(\mathscr{O}_2) $$
Observer $\mathscr{O}_1$ can also do one step better then $\mathscr{O}_2$ in that they can explain the properties of $P_1$ using special relativity with formulas that have up to one derivative but it to can't use special relativistic formulas to explain the properties of $P_2$.
 A: Here is my answer after some further research. I believe it is correct but would like confirmation. If it is indeed so I will award the bounty to an expansion on this answer - maybe including a explanation of the case without diagonal metrics.
The Hypothesis of Locality
The idea set out in this question is based on the 'Hypothesis of locality' which as stated in [1] (which is the source unless otherwise stated) is given by:

Hypothesis of locality: Locally (i.e., over infinitesimal regions of space and time), neither gravity nor acceleration changes the length of a standard rod or the rate of a standard clock relative to a nearby freely falling (i.e., inertial) standard rod, or standard clock instantaneously co-moving with it.

This hypothesis is, however, not generally true - but is true in the case where the metric in the non-inertial frame is diagonal or can be written in a diagonal form. 
This is, however, nearly always the case.
The momentum-energy example
The argument that:
$$E(P_1)=-p^\mu(P_1) u_{\mu}(\mathscr{O}_2)$$
then proceeds as follows. Let $\mathscr{O}_3$ be a freely-falling observer co-moving with $\mathscr{O}_2$ at event $A$. Then for $\mathscr{O}_3$ the space-time is locally Minkowskian and he/she therefore measures an energy given by the special relativistic formula:
$$\tilde E(P_1)=-p^\mu(P_1) u_{\mu}(\mathscr{O}_3)$$
On the assumption of the validity of the Hypothesis of locality we must therefore have 
$$E(P_1)=\tilde E(P_1)$$
where $E(P_1)$ is the energy measured by observer $\mathscr{O}_2$ and $\tilde E(P_1)$ is that measured by observer $\mathscr{O}_3$. Since $u_\mu(\mathscr{O}_3)=u_\mu(\mathscr{O}_2)$ by definition we therefore have:
$$E(P_1)=-p^\mu(P_1) u_{\mu}(\mathscr{O}_2)$$
as stated in the question.
References
R.D.Klauber, ”Toward a Consistent Theory of Relativistic Rotation”, in ”Relativity in Rotating Frames”, ed.
G.Rizzo and M.L.Ruggiero, ed. Kluwer, 2004.
