Adopted inertial covariance of the model equations

Adopt as physical assumptions of the special-relativity framework the following two statements.

(i) Spacetime. The oriented time-oriented affine Minkowski space $(A,V,\eta ,C^{+})$ of Oriented time-oriented affine Minkowski space (the spacetime primitives) is the spacetime of the framework. Its points represent idealized physical events; the displacement $q-p$ of two events represents an idealized spacetime separation of those events; and the sign of $\eta (q-p,q-p)$ plus the chosen component $C^{+}$ is the adopted causal structure: the timelike, null, spacelike, future-causal and past-causal classifications of displacements are those of Timelike, null, spacelike and causal vectors of a Lorentz vector space and Time orientation, future and past timelike/null/causal vectors, and the closed cones.

(ii) Inertial changes of description. Write $F:A\to A$ for a map for which there is a linear map $L:V\to V$ satisfying

$F(q)-F(p)=L(q-p)\text{for all }p,q\in A\mathrm{.}$

Such an $L$ is unique when it exists: for each $v\in V$ and any fixed $p\in A$, applying the display to the pair $(p,p+v)$ gives $Lv=F(p+v)-F(p)$, so the value $Lv$ is determined by $F$. An inertial change of description is such a map $F$ whose linear part $L$ satisfies

$\eta (Lv,Lw)=\eta (v,w)\text{for all }v,w\in V,\det L=+1\text{ relative to the fixed orientation of }V,L(C^{+})=C^{+}\mathrm{.}$

The condition $\det L=1\mathrm{\ne }0$ makes $L$ invertible by A finite-dimensional linear operator over a field is invertible if and only if its determinant is nonzero. Fix $p_0\in A$; then $F(p)=F(p_0)+L(p-p_0)$ by the affine condition, and the map $r\mapsto p_0+L^{-1}(r-F(p_0))$ is its inverse, so $F$ is a bijection.

For clarity, the induced transformation of tensors has the following precise meaning. The algebraic dual $V^{\ast }$ is the vector space of linear maps $\alpha :V\to \mathbb{R}$ (Linear functionals and the algebraic dual $V^{\ast }=\mathcal{L}(V,F)$). For integers $r,s\ge 0$, a type $(r,s)$ tensor is a map $T:(V^{\ast }{)}^r\times V^s\to \mathbb{R}$ linear in each argument; write ${\mathcal{T}}_s^r(V)$ for their vector space with pointwise operations (A type $(r,s)$ tensor on a finite-dimensional vector space). A tensor field here is an assignment $T:A\to {\mathcal{T}}_s^r(V)$; each particular model specifies its tensor types, domains and required regularity. Its transformed field is defined by

$(F_{\ast }T{)}_{F(p)}({\alpha }_1,\dots ,{\alpha }_r,w_1,\dots ,w_s):=T_p({\alpha }_1\circ L,\dots ,{\alpha }_r\circ L,L^{-1}{w}_1,\dots ,L^{-1}{w}_s)\mathrm{.}$

This is well defined since $F$ is bijective and all slot maps are linear; for a restricted domain $D\subseteq A$ the new domain is $F(D)$. In particular scalar fields transform by composition with $F^{-1}$ and vector fields by $v_p\mapsto L{v}_p$ at $F(p)$, since $\alpha (L{v}_p)=(\alpha \circ L)(v_p)$.

Adopt the assumption that every solution configuration of each specified mechanical or electromagnetic model is carried to a solution of that same model by relabelling events with $F$ and applying the displayed transformations to its fields and displacements. This is the meaning of covariance here; which equations and admissible configurations constitute a model is specified when that model is introduced.

• Adopted, not deduced. Both clauses are adopted assumptions of the framework. They are not deduced from experiment in this item, and they are not derived from the verbal postulates of Einstein; together with the ideal-clock and ideal-light assumptions of the companion postulate they fix the framework's physical interpretation.
• What covariance is claimed for. Clause (ii) asserts covariance only for the equations of the mechanical and electromagnetic models developed in this framework from the present primitives. It does not select a force law, a matter model or an equation of motion, and it does not assert that every conceivable equation written in the coordinates of the model is covariant; a previously adopted background structure would not be excluded by a purely syntactic covariance test. The formulation is a constraint on the form of the models, not a derivation of dynamics.
• Scope. The scope of the assumption is flat classical spacetime with idealized events and inertial descriptions only: gravitational effects (no curvature of the affine model is introduced), globally accelerated or rotating frame structure, and quantum effects are outside it. The companion item Adopted ideal-clock and ideal-vacuum-light interpretation states the corresponding ideal-clock and ideal-light assumptions.
• No claim about nature. Covariance of the model equations under the maps above is a property of the specified model. It carries no claim that the physical world is exactly described by the model, and experimental agreement with a model prediction would be evidence about the specified model rather than a proof of this postulate.