Particle spacetime, observers and worldlines — Examples

The quotient coordinate $[s{]}_L$ has local lifts to real intervals; on every such chart the metric components are constant Minkowski components. The curve $\gamma (s)=([s{]}_L,0,0,0)$ is locally a coordinate line with tangent ${\mathrm{\partial }}_{\theta }$, norm $-1$, and future sign. By [F2] it is a future timelike affine geodesic. Since $\gamma (s+L)=\gamma (s)$, it is closed.

In the slab, $\beta (s)=(s,0,0,0)$ for $s\in (0,\mathrm{\ell })$ is a vertical coordinate line with unit future timelike tangent; [F2] makes it an affine geodesic on this finite interval. If it extended continuously to $s=0$ as a curve in $S_{\mathrm{\ell }}$, its first coordinate would have to equal $0$, which is excluded from the slab; the same obstruction holds at $s=\mathrm{\ell }$. Therefore it cannot be extended within this manifold despite the locally constant metric.

For the lab, $d{x}^0\mathrm{/}dt=c$ and $d\mathbf{x}\mathrm{/}dt=0$, so $-g(dx\mathrm{/}dt,dx\mathrm{/}dt)=c^2$ and the clock density is $1$. Its elapsed reading is therefore $\mathrm{\Delta }{\tau }_L=2T$.

On either traveler leg, $d{x}^0\mathrm{/}dt=c$ and $\mathrm{\mid }d\mathbf{x}\mathrm{/}dt\mathrm{\mid }=v$, so $-g(dx\mathrm{/}dt,dx\mathrm{/}dt)=c^2-v^2>0$. The density is the constant $\sqrt{1-v^2\mathrm{/}{c}^2}$ on both legs. Additivity across the single corner follows from the piecewise definition [F2], giving $\mathrm{\Delta }{\tau }_R=2T\sqrt{1-v^2\mathrm{/}{c}^2}<2T=\mathrm{\Delta }{\tau }_L$. This is a deduction from the ideal-clock model and specified paths, not a measured result.

Define $x_a({\lambda }_a)=x(a{\lambda }_a)$. Then $d{x}_a\mathrm{/}d{\lambda }_a=aP=:P_a$, which remains nonzero future null. Since $D_{\lambda }P=0$, the constant rescaling also gives $D_{{\lambda }_a}{P}_a=0$; the image of $x_a$ equals the image of $x$. Thus both parameterizations satisfy the free null equations of [F1].

For any ${\lambda }_0\in I$, the rescaled curve satisfies $x_a({\lambda }_0\mathrm{/}a)=x({\lambda }_0)$ and $P_a({\lambda }_0\mathrm{/}a)=aP({\lambda }_0)$. For an observer $n$ at this common event, [F2] gives $E_n(P_a)=a{E}_n(P)$. Hence the same unparameterized ray supports distinct positive momentum and energy scales; the path image alone does not select one.

Differentiating $x(\tau )=p+\tau U_0$ gives $U=U_0$, which is constant in the flat affine connection. It is future timelike by assumption, and $D_{\tau }U=0$, so it solves the free massive equation of [F1]. The mechanical momentum $P=m{U}_0$ is also constant.

In the inertial chart of [F3], write $U_0=(U^0,\mathbf{U})$. Since $\chi (x(\tau ))=\chi (p)+\tau (U^0,\mathbf{U})$ and $t=x^0\mathrm{/}c$, the coordinate velocity is $\mathbf{v}=c\mathbf{U}\mathrm{/}{U}^0$. The norm gives $(U^0{)}^2=c^2+\mathrm{\mid }\mathbf{U}{\mathrm{\mid }}^2$, while future direction gives $U^0>0$, so $\mathrm{\mid }\mathbf{v}\mathrm{\mid }<c$. The observer $n=e_0$ is fixed under the affine tangent identification; [F2] and [F4] therefore make $E_n(P)=-c\eta (P,e_0)$ constant because $P$ is constant.