Adopted spacetime model and classical domain of the gravitational framework

Adopt as physical assumptions of the classical gravitational framework the following statements.

(i) Spacetime model. The spacetime of a model of the framework is a time-oriented smooth Lorentzian four-manifold $(M,g,N)$ as in Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model: a smooth four-manifold $M$ without boundary, a Lorentzian metric $g$ of signature $(-,+,+,+)$ on $M$ (Smooth manifolds and their smooth charts), and a time orientation $N$. Its points represent idealized physical events; the vector $N$ fixes which causal directions are future-directed. At each event $p\in M$, every nonzero tangent vector $v\in T_{p}M$ is classified by the sign of $g_p(v,v)$ as in Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, and each nonzero causal vector is future- or past-directed according to the sign of $g_p(v,N_p)$. The zero vector is neither timelike, null, spacelike nor causal. This is a pointwise rule and does not require choosing a curve through $p$. The metric $g$ is the gravitational field of the model: it is the object that the field law of the framework constrains, and the curvature $R^g$ of the uniquely constructed torsion-free $g$-compatible connection ${\mathrm{\nabla }}^g$ of Metric compatibility, torsion freedom and the metric-selected connection class is the model's representation of gravity. That supplier defines $R^g(X,Y)Z={\mathrm{\nabla }}_X^g{\mathrm{\nabla }}_Y^{g}Z-{\mathrm{\nabla }}_Y^g{\mathrm{\nabla }}_X^{g}Z-{\mathrm{\nabla }}_{[X,Y]}^{g}Z$ and proves the connection exists; this clause adds no independent connection field. The rules by which ideal clocks, test bodies and light rays couple to $g$ are adopted separately as physical assumptions of the framework.

(ii) Classical domain. Geometry and matter are treated as classical continuum fields: the model describes macroscopic configurations of matter and a classical metric, with no quantum description of either. The framework makes no claim that a real spacetime is exactly such a manifold, that matter is exactly a classical continuum, or that the smooth description applies to arbitrarily small scales or arbitrarily strong fields; those are domain limitations of the model family, not consequences of it.

(iii) Unity of the model. A model of the framework is specified by its geometry $(M,g,N)$ together with its matter content and the field law adopted for it; the geometry alone selects no matter configuration and no dynamics (the matter assumptions are adopted separately, in the companion postulate on matter content). Physical parameters enter through the unit conventions of SI unit and coordinate conventions of the classical gravitational framework; no numerical value of a constant is asserted here.

(iv) Smooth-branch restriction. Each field of a model is defined and $C^{\mathrm{\infty }}$ at every point of its specified open domain in $M$; the metric is $C^{\mathrm{\infty }}$ on all of $M$ and $M$ has no boundary. A field is smooth here when its component functions in the model's smooth coordinate and bundle charts are $C^{\mathrm{\infty }}$ on that domain. Configurations with a field undefined or nonsmooth at any point of its stated domain are outside this smooth branch, even when the exceptional set has empty interior. Matter concentrated on a lower-dimensional set in a way that produces a distributional stress-energy tensor, and manifolds with boundary, are also outside this branch and require additional mathematics. A model may instead specify an open domain excluding an exceptional locus; its smooth-branch claims then apply only on that domain and assert no extension across the excluded locus.

(v) Status. These are adopted assumptions. They are not derived from the mathematics of manifolds, from any symmetry principle, or from experiment; the equivalence principle and covariance arguments motivate but do not prove them. Testing the framework means comparing predictions computed from a fully specified model — geometry, matter content and field law — with observations, and agreement is evidence about the model, not a proof of these postulates.

• Adopted, not deduced. Clause (i) adopts the geometric arena; clauses (ii)-(iv) delimit its intended regime. Nothing in this item selects the metric of any model, asserts that a candidate metric satisfies any field equation, or asserts that any observed spacetime is of this form.

• Empirical content is downstream. Empirical significance lies in the predictions of specified models and in their comparison with reported observations; those comparisons carry their own apparatus, calibration, sampling and uncertainty qualifications and are recorded where the framework treats them. This postulate itself makes no quantitative empirical claim.

• No scope creep. The absence of a boundary, the dimension four, and the global time orientation are part of the adopted model, not conclusions. Where later items treat initial value problems, asymptotics or maximal extensions, they state their additional hypotheses explicitly.