Adopted classical gravitational action and variational field law

Adopt as physical assumptions of the classical gravitational framework, for models whose geometric data satisfy Adopted spacetime model and classical domain of the gravitational framework and whose matter sector satisfies Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions, the following statements. All symbols $c,G_N,\mathrm{\Lambda },\kappa$ are those of SI unit and coordinate conventions of the classical gravitational framework.

(i) Oriented models. The spacetime of a model in this formulation carries, in addition to $(M,g,N)$, an orientation (Oriented smooth manifolds and oriented charts); the metric volume form ${\mathrm{v}\mathrm{o}\mathrm{l}}_g$ of The metric volume form, current flux through 3-faces and finite-chain charges and its integrals are understood with respect to it. A model without a chosen orientation, or with a nonorientable $M$, requires the density formulation of the action, which is not developed here and is outside the scope of this postulate.

(ii) Einstein-Hilbert action density. The Einstein-Hilbert action density of a smooth Lorentzian metric $g^{\mathrm{\prime }}$ of signature $(-,+,+,+)$ on $M$ is the smooth $4$-form

${\mathrm{\ell }}_{\mathrm{E}\mathrm{H}}(g^{\mathrm{\prime }}):=\frac{c^3}{16\pi G_N}(S_{g^{\mathrm{\prime }}}-2\mathrm{\Lambda }){\mathrm{v}\mathrm{o}\mathrm{l}}_{g^{\mathrm{\prime }}},$

where $S_{g^{\mathrm{\prime }}}$ is the scalar curvature of $g^{\mathrm{\prime }}$ (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors) and $\mathrm{\Lambda }\in \mathbb{R}$ is a constant parameter of dimension ${\mathrm{m}}^{-2}$. For two metrics whose difference is supported in a compact set, the action difference $\mathrm{\Delta }{S}_{\mathrm{E}\mathrm{H}}:={\int }_M[{\mathrm{\ell }}_{\mathrm{E}\mathrm{H}}(g^{\mathrm{\prime }})-{\mathrm{\ell }}_{\mathrm{E}\mathrm{H}}(g)]$ is a real number. Only such differences are used; no value of the total action of a noncompact spacetime is asserted. The coupling

$\kappa :=\frac{8\pi G_N}{c^4}$

has dimension ${\mathrm{m}}^{-1}{\mathrm{k}\mathrm{g}}^{-1}{\mathrm{s}}^2$.

(iii) Matter sectors and available presentations. A model adopting this action presentation uses the supplied scalar-field action in sector (1). Sectors (2) and (3) are described here only to distinguish the alternative direct-equation presentation; they do not adopt this variational postulate. The three sector descriptions use the units of SI unit and coordinate conventions of the classical gravitational framework:

1. a scalar-field sector, specified by the data $(\phi ,V,K)$ of The scalar-field matter model of the classical gravitational framework with the physical readings: $\phi$ dimensionless, $V$ a geometric potential with values in ${\mathrm{m}}^{-2}$, and $K>0$ in $\mathrm{J}{\mathrm{m}}^{-1}$. Length coordinates give ${\mathrm{\partial }}_a\phi$ dimension ${\mathrm{m}}^{-1}$, so the kinetic term and $V$ have matching dimensions; the physical potential energy density is $KV$ in $\mathrm{J}{\mathrm{m}}^{-3}$. The sector's action differences are those defined there, and its stress-energy tensor is the candidate tensor of that definition;
2. a dust sector, specified by $(\rho ,u)$ of The dust matter model: rest density, four-velocity, stress-energy and dust system, whose unit-tangent coefficient is the rest energy density $\epsilon :=\rho$ in $\mathrm{J}{\mathrm{m}}^{-3}$; the rest-mass density is ${\rho }_m=\epsilon \mathrm{/}{c}^2$ in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$ and the physical four-velocity is $U=cu$ with $g(U,U)=-c^2$. Its stress-energy tensor $T^{ab}=\rho u^a{u}^b={\rho }_m{U}^a{U}^b$ is prescribed by the model and its matter equations are the dust system of that definition;
3. a perfect-fluid sector, specified by $(\epsilon ,p,n)$ (with optional equation of state) of The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy with $\epsilon ,p$ in $\mathrm{J}{\mathrm{m}}^{-3}$, whose stress-energy tensor is prescribed by the model and whose matter equations are the fluid equations of that definition.

In sectors (2) and (3) no matter action is supplied: the model prescribes $T$ and the matter equations directly and adopts the direct-equation presentation, with its consistency conditions. Clause (iv) does not apply to these sectors. In sector (1) the matter action is the scalar action of The scalar-field matter model of the classical gravitational framework; its metric variation is required to be represented by its stress-energy tensor in the Hilbert convention

$\delta S_m=-\frac{1}{2c}{\int }_M{T}_{ab}\delta g^{ab}{\mathrm{v}\mathrm{o}\mathrm{l}}_g\text{off shell in matter variables,}$

which is the content of The Hilbert stress-energy tensor of a metric-dependent action clause (b) for that action; for the scalar action this representation is a proved property of the model, recorded in First variation, Hilbert tensor and conservation of the scalar-field model clause (b), and is adopted here as the reading of $T$ in the framework's units.

(iv) Variational field law. For the supplied scalar-field action, the classical field law adopted for the model is: the metric $g$ is stationary for the total action difference $\mathrm{\Delta }{S}_{\mathrm{t}\mathrm{o}\mathrm{t}}:=\mathrm{\Delta }{S}_{\mathrm{E}\mathrm{H}}+\mathrm{\Delta }{S}_m$ under every jointly smooth metric family with compactly supported variation as defined in The Hilbert stress-energy tensor of a metric-dependent action clause (a), and the model's matter configuration satisfies its matter equations of motion. This variational principle is the adopted field law of the action presentation; the displayed field equation it is equivalent to is derived, not adopted a second time, in Stationarity of the adopted total action is equivalent to the Einstein equation with the framework's coupling.

(v) Alternative presentation. The direct-equation presentation of the same field law is adopted separately in Adopted Einstein field equation in the direct-equation presentation. A model adopts exactly one of the two presentations; they are alternative presentations of one classical field law and are never combined as two independent premises.

(vi) Status and scope. Clauses (i)-(v) are adopted assumptions. No geometry, symmetry principle or observation derives them; the action formulation is not claimed to be selected by nature, and no numerical value of $c$, $G_N$ or $\mathrm{\Lambda }$ is asserted. 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 verbatim: lower-dimensional or distributional sources, boundaries with junction conditions, and non-smooth fields require mathematics and physical assumptions not supplied here, and the comparison of any model prediction with data tests the combined model, including this postulate, rather than proving it.

• Deductions are conditional. Every quantity derived from this postulate — the field equations, conservation identities and consistency conditions — is a logical consequence of these adopted assumptions and the mathematics of the framework, not a statement about nature.

• Choices inside the formulation. The cosmological parameter may be fixed to zero for a restricted branch of a model; the matter sector is a model input; an equation of state in the fluid sector is a further assumption of that model. Different choices give different models of the same framework.

• No boundary terms are adopted. Nonzero variations of the metric on a boundary are not covered by clause (iv); their treatment requires additional boundary terms that this framework does not supply.