= Solution
There are two distinct explanatory questions. One asks why a proposed system of equations has a particular <dynamical symmetry>. The other asks why the common spacetime geometry is the right geometry for the behavior of physical rods, clocks and fields. A geometric account and a dynamical account give different directions of explanation, and their merits cannot be decided by a change of notation alone.
For the geometric account, <Minkowski spacetime> has a fixed <Lorentzian metric>; its <spacetime isometries> form the <Poincaré group>. If the matter equations use that <metric tensor> and no additional symmetry-breaking <background structure>, with couplings and boundary assumptions appropriate to the transformation, these <spacetime isometries> take solutions into solutions. For example, the <Klein-Gordon equation>
$$
\left(\eta^{\mu\nu}\partial_\mu\partial_\nu-\frac{m^2c^2}{\hbar^2}\right)\phi=0
$$
in <metric signature> $(-,+,+,+)$ is invariant under the <Poincaré group>. This provides one common reason for the same transformation rules appearing in many laws. An individual solution, such as a particular <plane wave>, need not possess every symmetry of the equation.
Geometry alone is insufficient unless the assumed coupling to geometry is included. In the <covariant equation with a preferred timelike vector>,
$$
(\eta^{\mu\nu}-\alpha n^\mu n^\nu)\partial_\mu\partial_\nu\phi=0,
$$
the fixed timelike $n$ selects a preferred rest frame. All the quantities can be transformed consistently under a <coordinate transformation>, but boosts that change $n$ are not symmetries with the physical background fixed. Thus <general covariance>, tensor notation and the mere presence of the <Minkowski metric> do not by themselves establish <Einstein's principle of relativity>.
A dynamical account instead starts with the common invariance of the actual interactions and asks why their composite rods and clocks measure a common interval. If their equations and suitable equilibrium solutions transform consistently under <Lorentz transformations>, their moving configurations can exhibit <length contraction> and their periodic processes <time dilation>. This is the explanatory strategy defended by <Harvey R. Brown> and <Oliver Pooley>, who deny that merely redescribing these effects geometrically supplies their microscopic explanation.<Minkowski space-time: a glorious non-entity by Harvey R. Brown and Oliver Pooley>[sections 3–5: constructive explanation, rods and clocks, and the direction of explanation]{ref} Such an account must still explain why independent interactions have the same relevant symmetry and why a particular clock is a good clock; it cannot simply infer those facts from the word “dynamics”.
The positions therefore need not disagree about experimental predictions. The geometric presentation makes a shared constraint explicit and permits economical deductions. The constructive presentation investigates why material systems realize that constraint. Defining geometry from universal dynamical behavior is a substantive proposal; deriving behavior from geometry also requires substantive matter-coupling assumptions. Neither direction is made compulsory by the formal equivalence of the descriptions.
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