The flat model spacetime: the Minkowski metric on R^4 and its inertial chart

Let $M:={\mathbb{R}}^4$ carry its standard smooth structure (Smooth manifolds and their smooth charts), and let $x=(x^0,x^1,x^2,x^3):M\to {\mathbb{R}}^4$ be the identity chart, $x(p)=p$; it is a global chart of the maximal smooth atlas, and the coordinate derivations ${\mathrm{\partial }}_0,{\mathrm{\partial }}_1,{\mathrm{\partial }}_2,{\mathrm{\partial }}_3$ form a basis of $T_{p}M$ at every $p\in M$ (Derivations at a point and the tangent space, Coordinate derivations form a basis of the tangent space). Write

$\eta :=\mathrm{diag}(-1,1,1,1)$

for the standard Lorentz form of inertia $(3,1,0)$ in this basis (Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space).

(a) The metric. Define the covariant $2$-tensor field $g$ on $M$ by the requirement that its components in the identity chart are the constants

$g_{ab}:={\eta }_{ab}(a,b=0,1,2,3),$

that is, $g_p({\mathrm{\partial }}_a{\mathrm{\mid }}_p,{\mathrm{\partial }}_b{\mathrm{\mid }}_p)={\eta }_{ab}$ for every $p\in M$ (Smooth covariant 2-tensor fields on a smooth manifold clause (a)). The component functions are constant, hence $C^{\mathrm{\infty }}$, so $g$ is a smooth symmetric covariant $2$-tensor field; at every $p$ the form $g_p$ is nondegenerate with inertia $(3,1,0)$. By Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model clause (a) the pair $(M,g)$ is therefore a smooth Lorentzian four-manifold of signature $(-,+,+,+)$.

(b) The inertial chart and frame. The chart $x$ is called the inertial chart of the flat model. In this chart the coordinate frame $({\mathrm{\partial }}_0,{\mathrm{\partial }}_1,{\mathrm{\partial }}_2,{\mathrm{\partial }}_3)$ is a $g$-orthonormal frame at every point, $g({\mathrm{\partial }}_a,{\mathrm{\partial }}_b)={\eta }_{ab}$; it is simultaneously a coordinate frame and a tetrad field in the sense of Local frame fields, coordinate frames and tetrad fields, defined on all of $M$.

(c) Time orientation and causal classification. Define the smooth vector field $N:={\mathrm{\partial }}_0$ on $M$ (A smooth vector field is a smooth section of the tangent bundle). Since $g_p(N_p,N_p)={\eta }_{00}=-1<0$ for every $p$, the field $N$ is everywhere timelike, so $(M,g,N)$ is a smooth time-oriented Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model clause (c)). For a nonzero tangent vector $v=v^a{\mathrm{\partial }}_a{\mathrm{\mid }}_p$ the definitions of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model clause (c) and Timelike, null, spacelike and causal vectors of a Lorentz vector space give

$g_p(v,v)=-(v^0{)}^2+{\sum }_{i=1}^3(v^i{)}^2,g_p(v,N_p)=-v^0\mathrm{.}$

so, for a nonzero causal $v$, $v$ is future-directed exactly when $v^0>0$, and

• $v$ is future timelike $⟺$ $v^0>0$ and $(v^0{)}^2>{\sum }_i(v^i{)}^2$;
• $v$ is future null $⟺$ $v^0>0$ and $(v^0{)}^2={\sum }_i(v^i{)}^2$;
• $v$ is spacelike $⟺$ $(v^0{)}^2<{\sum }_i(v^i{)}^2$;

consequently the future causal cone at every point is $\{v\mathrm{\ne }0:v^0\ge \sqrt{{\sum }_i(v^i{)}^2}\}$. These are the standard readings of the Minkowski form in the inertial chart; the chain of equivalences is the specialization of the causal classification to this pair $(g_p,N_p)$ and is recorded here for later use.

(d) Scope. This item defines geometric data only: a manifold, a metric field, a time orientation and a distinguished chart and frame. It makes no claim about curvature, about any field equation, about which geometry a physical model uses, or about nature; the physical reading of the data, the dimensions of the coordinates and the numerical values or signs of physical constants belong to the framework's unit conventions and postulates, which are separate items. In particular no relation between the coordinate functions $x^a$ and physical time or length measurements is asserted here.