Adopted Einstein field equation in the direct-equation presentation

Adopt as the field law of the direct-equation presentation of the classical gravitational framework, for models satisfying Adopted spacetime model and classical domain of the gravitational framework and Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions, the following statement.

(i) The equation. Let $(M,g,N)$ be the model's smooth time-oriented Lorentzian four-manifold with metric-selected connection $\mathrm{\nabla }$ (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields) and let $G_{ab}={\mathrm{Ric}}_{ab}-\frac{1}{2}S{g}_{ab}$ be the Einstein tensor of that connection (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors). Let $T$ be the model's stress-energy tensor field on its matter domain $U\subseteq M$ (Stress-energy tensor field of a model) and let $\mathrm{\Lambda }\in \mathbb{R}$ be a constant parameter with dimension ${\mathrm{m}}^{-2}$ and $G_N>0$, $c>0$ the constants of SI unit and coordinate conventions of the classical gravitational framework. Adopt

$G_{ab}+\mathrm{\Lambda }{g}_{ab}=\frac{8\pi G_N}{c^4}{T}_{ab}\text{on }U$

as the field law of the model. Writing $\kappa :=8\pi G_N\mathrm{/}{c}^4$ and $T_{ab}=g_{ac}{g}_{bd}{T}^{cd}$ for the lowered tensor, the equation reads $G_{ab}+\mathrm{\Lambda }{g}_{ab}=\kappa T_{ab}$

(ii) Matter content. The equation constrains the metric and the matter of the model jointly. A model specifies its matter sector as in Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions: a scalar-field model with its action and stress-energy tensor (The scalar-field matter model of the classical gravitational framework), or a dust or perfect-fluid model with prescribed stress-energy tensor and matter equations. The consistency condition that every solution must satisfy, ${\mathrm{\nabla }}^a{T}_{ab}=0$, is not adopted separately; it is a consequence of the equation and of the contracted second Bianchi identity and is derived in Consistency conditions of the adopted field law: conservation, trace, parameter variation and the energy conditions of a perfect fluid.

(iii) Alternative presentation. The action formulation of the same field law is adopted separately in Adopted classical gravitational action and variational field law. A model adopts exactly one of the two presentations. They are alternative presentations of one classical field law: the action-to-equation theorem of this page identifies the equation obtained from stationarity of the total action with the equation displayed in clause (i), and the direct presentation is not an additional premise of an action-formulated model.

(iv) Status and scope. The equation is an adopted law of the model. It is not derived from mathematical symmetry, from the equivalence principle, or from observation; covariance and correspondence arguments motivate but do not prove it. No numerical value of $c$, $G_N$ or $\mathrm{\Lambda }$ is asserted, and no claim is made that any particular metric or matter configuration solves the equation or describes nature. The smooth-branch exclusions of Adopted spacetime model and classical domain of the gravitational framework and Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions apply: distributional or lower-dimensional sources, boundaries with junction conditions, and non-smooth fields are outside this formulation. Comparisons of a model's predictions with data test the combined model — geometry, matter content and this law — and agreement is evidence about that combination, not a proof of the postulate.

• Restricted branches. A model may fix $\mathrm{\Lambda }=0$ or choose a particular matter sector; each such restriction is part of the model, not of the framework.

• No well-posedness claim. The postulate asserts the equation on the matter domain; it asserts neither existence nor uniqueness of solutions for any data, which are separate mathematical problems with their own hypotheses.