The flat model: vanishing connection and curvature, translation Killing fields and straight geodesics

(The connection vanishes.) In the inertial chart all metric components are the constants ${\eta }_{ab}$ by The flat model spacetime: the Minkowski metric on R^4 and its inertial chart clause (a), so each partial derivative ${\mathrm{\partial }}_a{g}_{bc}$ vanishes identically. The formula of [L1] gives ${\mathrm{\Gamma }}^a{}_{bc}=\frac{1}{2}{\sum }_d{\eta }^{ad}(0+0-0)=0$ at every point of $M$, proving (a).

(The coordinate fields are Killing.) Fix $a\in \{0,1,2,3\}$ and consider the map ${\mathrm{\Psi }}_s(p):=p+s{e}_a$ in the identity chart, where $e_a$ is the $a$-th standard basis vector. The curve $s\mapsto {\mathrm{\Psi }}_s(p)$ has coordinate derivative $e_a$ at every parameter, so it is an integral curve of ${\mathrm{\partial }}_a$; it is defined for all $s\in \mathbb{R}$. By uniqueness of maximal integral curves in [L4], the maximal local flow of ${\mathrm{\partial }}_a$ is this global flow $\mathrm{\Psi }$. In the inertial chart the differential of ${\mathrm{\Psi }}_s$ is the identity of the chart basis and the metric components are the constants ${\eta }_{ab}$, so the components of the pulled-back field are $({\mathrm{\Psi }}_s^{\ast }g{)}_{ab}(p)=g_{ab}({\mathrm{\Psi }}_s(p))={\eta }_{ab}=g_{ab}(p)$ for all $p$; hence ${\mathrm{\Psi }}_s^{\ast }g=g$ for every $s$. By [L5], ${\mathcal{L}}_{{\mathrm{\partial }}_a}g=\frac{d}{ds}{\mid }_0{\mathrm{\Psi }}_s^{\ast }g=0$, so ${\mathrm{\partial }}_a$ is a Killing field. In particular $N={\mathrm{\partial }}_0$ is Killing, and $g(N,N)=-1$ by clause (c) of The flat model spacetime: the Minkowski metric on R^4 and its inertial chart, so it is unit timelike. This proves (c).

(The curvature vanishes.) Substituting $\mathrm{\Gamma }\equiv 0$ into the coordinate formula of [L2] gives $R^a{}_{bcd}\equiv 0$; since $\mathrm{Ric}$, $S$ and $G$ are obtained from $R^a{}_{bcd}$ and $g^{ab}$ by the contractions and products displayed in [L2], and $g^{ab}$ is the constant matrix $\mathrm{diag}(-1,1,1,1)$, all of them vanish identically. The curvature operator vanishes because its components in the chart vanish, and a smooth tensor field all of whose components in one chart vanish is the zero field. This proves (b).

(Geodesics are straight lines.) Let $z:=x\circ \gamma :I\to {\mathbb{R}}^4$ be the coordinate expression of a smooth curve. By [L3] and step 1.1, $\gamma$ is a geodesic if and only if ${\ddot{z}}^k=0$ for $k=0,1,2,3$ on every parameter subinterval, that is, if and only if $z(s)=x_0+sv$ for some $x_0,v\in {\mathbb{R}}^4$; then the maximal interval is all of $\mathbb{R}$, because the affine expression extends to $\mathbb{R}$ and is again a solution. The speed is constant for every geodesic by clause 3 of Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed, and inserting the affine expression gives the value $g({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})=\eta (v,v)$. This proves the first two claims of (d).

(Future timelike geodesics and proper time.) Suppose $\eta (v,v)=-{\lambda }^2<0$ with $v^0>0$. By clause (c) of The flat model spacetime: the Minkowski metric on R^4 and its inertial chart, $\gamma$ is a future timelike curve, and by [L6] the parameter $\tau (s)=\lambda (s-s_0)\mathrm{/}c$ is proper time, with $g(U,U)=-c^2$ for $U=d\gamma \mathrm{/}d\tau$. This proves the third claim of (d); for (e), write $t=x^0(\gamma (s))\mathrm{/}c=(x_0^0+s{v}^0)\mathrm{/}c$, so that $d\mathbf{q}\mathrm{/}dt=(c\mathrm{/}{v}^0)(v^1,v^2,v^3)$ is constant; from $\eta (v,v)<0$ and $v^0>0$ we have ${\sum }_i(v^i{)}^2<(v^0{)}^2$, hence [L7] gives $\mathrm{\mid }\mathbf{w}{\mathrm{\mid }}^2=c^2{\sum }_i(v^i{)}^2\mathrm{/}(v^0{)}^2<c^2$.