= Solution
The numerical coordinates assigned to an event can be changed without changing the event. It does not follow that every relation between coordinate systems is devoid of physical content. A standard <inertial frame> includes a family of resting measuring standards and synchronized <clocks>. Its coordinates are normalized so that the <Minkowski metric> has the same standard form. The relation between two such descriptions therefore records properties of the standards and of <spacetime>, not just an arbitrary choice of names.
Write $x^\mu=(ct,x,y,z)$ and use <metric signature> $(-,+,+,+)$. A <Lorentz transformation> satisfies
$$
x'^\mu=\Lambda^\mu{}_{\nu}x^\nu,\qquad
\Lambda^T\eta\Lambda=\eta,\qquad
\eta=\operatorname{diag}(-1,1,1,1).
$$
An origin translation can also be included without changing the <Minkowski interval>. The displayed condition preserves the interval between events, including the <null cone> and <proper time> along a <timelike curve>. Its physical significance lies in the claim that comparisons made with actual idealized rods and <clocks> obey this same metric structure.
A more general passive relabeling is allowed, but then the metric components must change with the labels. For example, let $t'=2t$ and leave $x,y,z$ fixed. In the new coordinates,
$$
ds^2=-\frac{c^2}{4}\,dt'^2+dx^2+dy^2+dz^2.
$$
Nothing physical has changed, but $t'$ no longer has the original clock-normalized meaning of $t$. Retaining the old metric components as well as the new labels would change the mathematical model. This distinction is part of the <physical content of a passive Lorentz transformation>: the transformation between standard inertial coordinate systems preserves a specified metric form, whereas arbitrary relabelings need not preserve that form.
For an operational example, a <clock> moving at <velocity> $v$ between two events on its <world line> accumulates
$$
\Delta\tau=\Delta t\sqrt{1-v^2/c^2}
$$
in the first frame. In its own rest frame the same result is simply the interval between its local readings. The equality is a physical agreement about one <clock>, derived using the preserved <Minkowski interval>. It is not a claim that the two coordinate-time differences are identical. Similarly, an observer measures a <light> angular <frequency> through $\omega=-k_\mu u^\mu$, where $k^\mu$ is the <wave four-vector> and $u^\mu$ the observer's <four-velocity>. Transforming both sets of components leaves this scalar unchanged. Comparing two different observers gives the physical <relativistic Doppler effect> rather than a disagreement over one observer's measurement.
There is a further distinction between passive transformations of the same situation and active changes to a situation. Re-expressing a particular experiment in different coordinates is always possible if all geometric fields are transformed consistently. Preparing a second, moving copy of the apparatus is a different physical arrangement. Its relation to the first is governed by a <dynamical symmetry> only if the relevant laws and background structure have the appropriate invariance. An <active Lorentz transformation of a vector field> makes the changed field configuration explicit.
This is why mere <Lorentz covariance> of notation must not be confused with the <Lorentz invariance> of the dynamics. A physically preferred <timelike vector field> can be transformed as a vector under passive relabeling while distinguishing particular states of motion. Holding that preferred field fixed can destroy the active boost symmetry of the matter dynamics. The <covariant equation with a preferred timelike vector> is a concrete example.
Coordinates are therefore labels, but the <Lorentz transformation> relates labels tied to shared physical standards and a definite metric structure. Its measurable consequences follow from that structure and the dynamical assumptions, not from the bare act of changing labels.<On the Electrodynamics of Moving Bodies by Albert Einstein>[sections 1–4: operational definitions of time and length, and the transformation of moving standards]{ref}
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