The flat model: Minkowski space R^4, its standard Lorentz form, inertial charts and cone sets

(a) The underlying manifold. Let $M:={\mathbb{R}}^4$ be Euclidean four-space as a set and real vector space (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$), equipped with its standard smooth structure, in which the identity map $x=(x^0,x^1,x^2,x^3):M\to {\mathbb{R}}^4$ is a global smooth chart (Smooth manifolds and their smooth charts, Open subsets of Euclidean space have the standard smooth structure). The coordinate vector fields ${\mathrm{\partial }}_a:=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^a$, $a=0,1,2,3$, are smooth vector fields on $M$ (A smooth vector field is a smooth section of the tangent bundle), and at every $p\in M$ the tuple $({\mathrm{\partial }}_0{\mathrm{\mid }}_p,{\mathrm{\partial }}_1{\mathrm{\mid }}_p,{\mathrm{\partial }}_2{\mathrm{\mid }}_p,{\mathrm{\partial }}_3{\mathrm{\mid }}_p)$ is a basis of $T_{p}M$ (Derivations at a point and the tangent space).

(b) The metric. The flat metric $\eta$ is the smooth covariant $2$-tensor field on $M$ whose components in the chart $x$ are the constant functions ${\eta }_{ab}=\mathrm{diag}(-1,1,1,1)$ (Smooth covariant 2-tensor fields on a smooth manifold), that is $\eta =-(d{x}^0{)}^2+(d{x}^1{)}^2+(d{x}^2{)}^2+(d{x}^3{)}^2\mathrm{.}$ It is symmetric, and at every $p$ its matrix in the basis of (a) is $\mathrm{diag}(-1,1,1,1)$: diagonal, with one negative and three positive entries and no zero entry. Hence ${\eta }_p$ is nondegenerate with inertia $(3,1,0)$, that is, a Lorentz form on $T_{p}M$ (Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space), and $(M,\eta )$ is a smooth Lorentzian four-manifold of signature $(-,+,+,+)$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model). The restriction of ${\eta }_p$ to $\mathrm{span}({\mathrm{\partial }}_1{\mathrm{\mid }}_p,{\mathrm{\partial }}_2{\mathrm{\mid }}_p,{\mathrm{\partial }}_3{\mathrm{\mid }}_p)$ is positive definite.

(c) Time orientation. The field $N:={\mathrm{\partial }}_0$ satisfies $\eta (N,N)=-1<0$ pointwise, so $N$ is a time orientation of $(M,\eta )$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model clause (c)). The triple $(M,\eta ,N)$ is the flat model. At every $p$ the pointwise vocabulary of Timelike, null, spacelike and causal vectors of a Lorentz vector space and Time orientation, future and past timelike/null/causal vectors, and the closed cones applies, and a nonzero causal $v\in T_{p}M$ is future-directed exactly when ${\eta }_p(v,N_p)<0$ (Classification of future and past causal vectors by a timelike field).

(d) Events, displacements and straight paths. The points of $M$ are the events of the model. Since $M={\mathbb{R}}^4$ is a vector space, for events $p,q\in M$ the displacement from $p$ to $q$ is the vector $q-p\in M$, and the straight path from $p$ to $q$ is the curve $s\mapsto p+s(q-p)$ on $[0,1]$, whose chart representative has the constant velocity $q-p$. For $v\in {\mathbb{R}}^4$ the translation is the map ${\tau }_v(p):=p+v$ of $M$.

(e) Inertial charts and frames. A chart of $M$ with domain $M$ is called a global chart. A global chart $y=(y^0,y^1,y^2,y^3):M\to {\mathbb{R}}^4$ is inertial when its components of $\eta$ are the constant functions $\mathrm{diag}(-1,1,1,1)$ on $M$ and its first coordinate vector field $\mathrm{\partial }\mathrm{/}\mathrm{\partial }{y}^0$ is future-directed at every point. The chart $x$ of (a) is inertial: its components of $\eta$ are $\mathrm{diag}(-1,1,1,1)$ by (b), and its first coordinate vector field is $N$, future-directed by (c). If $y$ is inertial and $o\in M$, the tuple $(o;\mathrm{\partial }\mathrm{/}\mathrm{\partial }{y}^0{\mathrm{\mid }}_o,\mathrm{\partial }\mathrm{/}\mathrm{\partial }{y}^1{\mathrm{\mid }}_o,\mathrm{\partial }\mathrm{/}\mathrm{\partial }{y}^2{\mathrm{\mid }}_o,\mathrm{\partial }\mathrm{/}\mathrm{\partial }{y}^3{\mathrm{\mid }}_o)$ is an inertial frame at $o$; its vectors satisfy $\eta (e_a,e_b)=\mathrm{diag}(-1,1,1,1{)}_{ab}$ and $e_0$ is future-directed.

(f) Cone sets. Let a vector $v\in T_{p}M$ have coordinates $(v^0,v^1,v^2,v^3)$ in an inertial chart, write $v_s:=(v^1,v^2,v^3)$ and $\mathrm{\mid }{v}_s\mathrm{\mid }:=\sqrt{(v^1{)}^2+(v^2{)}^2+(v^3{)}^2}$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$). Write $C^{+}:=\{v\in {\mathbb{R}}^4:v^0>\mathrm{\mid }{v}_s\mathrm{\mid }\}$ and $\overline{C^{+}}:=\{v\in {\mathbb{R}}^4:v^0\ge \mathrm{\mid }{v}_s\mathrm{\mid }\}$ for the coordinate cone sets. In every inertial chart and at every event $p$, the future cone $C_p^{+}$ of (c) consists of exactly those $v$ with $v^0>\mathrm{\mid }{v}_s\mathrm{\mid }$, its closure of exactly those with $v^0\ge \mathrm{\mid }{v}_s\mathrm{\mid }$, and the future-directed causal vectors of $T_{p}M$ are exactly the nonzero vectors with $v^0\ge \mathrm{\mid }{v}_s\mathrm{\mid }$; moreover the future cone is closed under sums. All of this is proved in The causal cones of the flat model in inertial coordinates, and closure of the future cone under sums ↗, where the independence of the inertial chart is also established.

(g) Scope. This is a mathematical definition; it asserts nothing about nature. The physical reading of the flat model is recorded with the framework postulates of this library and is not part of this definition, and the flat examples and counterexamples of this page are conditional statements inside the model.

• Why the flat model is stated here. The causal-example items of this page need one fixed flat model with its cone calculus, so that page states the model, its inertial charts and its cone sets locally instead of depending on an external supplier for them; the criterion is then a local lemma.
• What is not asserted. Nothing here claims that a general Lorentzian manifold is locally isometric to the flat model, and no curvature statement is made; the components of $\eta$ are constant in the chart $x$, and that is all the description uses.