The scalar-field matter model of the classical gravitational framework

Let $(M,g,N)$ be an oriented time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Oriented smooth manifolds and oriented charts), let $c>0$ be a fixed real constant, and let $\mathrm{\nabla }$ be the metric-selected connection of $g$ (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).

(a) Data. A scalar-field matter model on $M$ is a triple $(\phi ,V,K)$ consisting of

• a smooth function $\phi :M\to \mathbb{R}$ ($C^r$ and smooth maps between smooth manifolds, with target the standard one-dimensional smooth manifold $\mathbb{R}$);
• a smooth function $V:\mathbb{R}\to \mathbb{R}$, the potential; and
• a real constant $K>0$, the coupling.

The physical reading of $K$ and of the units of $\phi$ and $V$ is fixed by the framework's matter postulate, not by this definition.

(b) Scalar-action density and action differences. For every smooth function ${\phi }^{\mathrm{\prime }}$ and every smooth Lorentzian metric $g^{\mathrm{\prime }}$ of signature $(-,+,+,+)$ on $M$ define the scalar-action density

$\mathrm{\ell }({\phi }^{\mathrm{\prime }},g^{\mathrm{\prime }}):=-\frac{K}{c}[\frac{1}{2}{g}^{\mathrm{\prime }ab}{\mathrm{\partial }}_a{\phi }^{\mathrm{\prime }}{\mathrm{\partial }}_b{\phi }^{\mathrm{\prime }}+V({\phi }^{\mathrm{\prime }})]{\mathrm{v}\mathrm{o}\mathrm{l}}_{g^{\mathrm{\prime }}}\in {\mathrm{\Omega }}^4(M),$

where $g^{\mathrm{\prime }ab}$ are the inverse-metric components (The inverse metric, musical isomorphisms and index raising) and ${\mathrm{v}\mathrm{o}\mathrm{l}}_{g^{\mathrm{\prime }}}$ the metric volume form (The metric volume form, current flux through 3-faces and finite-chain charges). For two pairs $(\phi ,g)$ and $({\phi }^{\mathrm{\prime }},g^{\mathrm{\prime }})$ with ${\phi }^{\mathrm{\prime }}-\phi$ and all components of $g^{\mathrm{\prime }}-g$ supported in a compact set, the difference $\mathrm{\ell }({\phi }^{\mathrm{\prime }},g^{\mathrm{\prime }})-\mathrm{\ell }(\phi ,g)$ is a compactly supported smooth $4$-form and the action difference

$\mathrm{\Delta }S:={\int }_M[\mathrm{\ell }({\phi }^{\mathrm{\prime }},g^{\mathrm{\prime }})-\mathrm{\ell }(\phi ,g)]$

is a real number (Finite chart localization gives choice-free integration and compact Stokes), the integral being independent of how the pair $(\phi ,g)$ is extended outside the support. Only such differences are used; no integrality of the total action on a noncompact $M$ is asserted.

(c) Scalar equation and candidate stress-energy tensor. The model's scalar equation of motion is

${\mathrm{\square }}_g\phi -V^{\mathrm{\prime }}(\phi )=0,{\mathrm{\square }}_g\phi :={\mathrm{\nabla }}^a{\mathrm{\partial }}_a\phi ,$

and the model's candidate stress-energy tensor is the smooth symmetric covariant $2$-tensor field

$T_{ab}:=K[{\mathrm{\partial }}_a\phi {\mathrm{\partial }}_b\phi -g_{ab}(\frac{1}{2}{\mathrm{\partial }}^c\phi {\mathrm{\partial }}_c\phi +V(\phi )\left)\right],$

with ${\mathrm{\partial }}^c\phi {\mathrm{\partial }}_c\phi =g^{cd}{\mathrm{\partial }}_c\phi {\mathrm{\partial }}_d\phi$ (The inverse metric, musical isomorphisms and index raising, Smooth covariant 2-tensor fields on a smooth manifold). That this candidate is indeed the Hilbert stress-energy tensor of the model, and that the scalar equation implies its conservation, are proved in First variation, Hilbert tensor and conservation of the scalar-field model ↗; the definition itself asserts neither proved property, and this item does not assert that any solution of the scalar equation exists.

• Sign conventions. The overall sign of $\mathrm{\ell }$ is chosen so that, under the Hilbert convention of The Hilbert stress-energy tensor of a metric-dependent action, the metric variation of the action difference reproduces the tensor displayed in clause (c). The kinetic term has the sign convention $+\frac{1}{2}{g}^{ab}{\mathrm{\partial }}_a\phi {\mathrm{\partial }}_b\phi$; the sign of the potential term is $+V(\phi )$ inside the bracket of $\mathrm{\ell }$ and $-V(\phi )$ inside the metric bracket of $T_{ab}$.

• Scope. The item defines a model on the given smooth $(M,g,N)$; it makes no claim that the model is realized by any physical system, that solutions exist for given initial data, or that the scalar field is fundamental.