Scope of normal coordinates and of the local-inertial-frame reading

This remark separates the mathematics of Point-normal and exponential coordinate charts of the metric-selected connection and The geodesic initial-value problem and the exponential map of the metric-selected connection from the physical reading usually attached to them, and is bookkeeping: it establishes no mathematical result and no physical law.

• What the construction gives. For every event $p$ there is a chart with $g_{ab}(p)={\eta }_{ab}$, ${\mathrm{\Gamma }}^a{}_{bc}(p)=0$ and ${\mathrm{\partial }}_a{g}_{bc}(p)=0$. In the exponential chart from an orthonormal basis, each geodesic through $p$ is the straight radial coordinate line while its initial vector remains in the chosen star-shaped exponential domain (Point-normal and exponential coordinate charts of the metric-selected connection). The identities are first order at the single point $p$.

• What it does not give. The construction asserts nothing about the second and higher derivatives of the metric, and therefore nothing about the curvature of the model; a chart of this kind is not an isometry and does not make any neighborhood of $p$ identical with a region of Minkowski space. The remark makes no curvature claim here; the second-order behavior is the subject of the curvature material of this track, which is a later page of the same category.

• No global inertial frame, no congruence. A chart with the normal-form identities at one event is not a global label of events by inertial coordinates, and the exponential charts at different points are not compatible frames. An "observer" in this library is a single future unit timelike vector at a point, and a "congruence" is a field of such vectors on an open set (Observers, rest spaces and the spatial projection at an event, Observer congruences, their observer curves and adapted charts); a freely falling observer is one timelike geodesic, not a global reference frame. Charts adapted to a congruence are a different construction from the normal coordinates of a geodesic through a point.

• The physical reading is adopted, not derived. The statement that an ideal freely falling test body or observer, using normal coordinates around an event of its own worldline, finds the local form of the metric to be that of Minkowski space at first order — and the associated local inertial-frame reading of the equivalence principle — depends on the adopted clock and free-fall assumptions of Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework. It is a statement about the model in the test-body idealization, not a derivation from the geometry alone and not an empirical claim that real experiments cannot detect a gravitational field locally. The limitations recorded in the postulate — finite-size and tidal couplings of extended bodies, self-force, and nonlocal measurements — are precisely what the first-order identities cannot exclude.