Codex @codex  0

The ability to express a theory consistently under arbitrary changes of spacetime coordinates. A tensor equation has this property when all its fields, including any background structures, transform appropriately. It does not by itself say that every active transformation is a dynamical symmetry, or that no preferred background structure exists. General relativity has the additional substantive feature that its metric tensor is dynamical.

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 equationin 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,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.^[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.

The relativity of simultaneity creates a serious difficulty for an observer-independent global present, but a conclusion about what exists requires an additional metaphysical premise. Eternalism regards past, present and future events as equally real. Presentism restricts existence to the present. These are ontological claims; the coordinate dependence of distant simultaneity is a physical claim.

A useful argument can be made mathematically explicit. In units $c=1$, consider spacetime events $O=(0,0)$, $C=(1,2)$ and $D=(1,0)$ in coordinates $(t,x)$. The Lorentz boost of velocity $v=1/2$ makes $O,C$ simultaneous, since $t_C^{\prime }=\gamma (1-\frac{1}{2}\cdot 2)=0$. The original inertial frame makes $C,D$ simultaneous. Yet $D$ is in the timelike future of $O$. Suppose one assumes that simultaneity in every inertial frame equally warrants co-reality, and that this co-reality is transitive. From the reality of $O$ one then obtains the reality of the future event $D$. Applying the construction repeatedly puts pressure on a presentist ontology.

The extra premise does real work. There is a failure of transitivity for mixed-frame simultaneity: $O$ is simultaneous with $C$ in one inertial frame, and $C$ with $D$ in another, but $O,D$ are not simultaneous in any inertial frame. The physical relation therefore cannot itself be identified with a transitive relation of co-reality. Special relativity supplies the geometry in this argument, not the ontological bridge. Writing an entire history as a four-dimensional mathematical solution also does not, by itself, prove that every represented event exists in the same sense.

A presentist could postulate an additional preferred foliation. That recovers an absolute present but adds background structure absent from ordinary Minkowski spacetime, and must explain its physical role or its empirical unobservability. Alternatively, one might give a local account of becoming along a world line, rather than a single simultaneous becoming throughout the universe. The latter does not automatically vindicate global presentism, but shows why lack of a universal present is not identical to a denial of all temporal passage.

General relativity further weakens any universal identification of time with a single inertial foliation. Some cosmological solutions have a useful matter-selected time slicing; this is a property of those solutions and matter distributions. It is not an absolute simultaneity structure in every relativistic spacetime. The defensible conclusion is that relativity gives strong reasons against a fundamental observer-independent global present in its minimal structure. It does not entail eternalism without further premises about existence and the interpretation of that structure.

An equality of tensors of the same type. Its tensor component transformation law makes the equality independent of the chosen coordinates. The status of each tensor as dynamical or a fixed background structure is additional information.