The frame-time parameter condition of a worldline: failure under rescaling and behaviour under time shifts

The standard chart is adapted and its coordinates are the standard ones: $o=0$, $u=(1,0,0,0)$ has $\tau (u)=1$, the standard basis is orthonormal and positively oriented, and for $p=(x^0,x^1,x^2,x^3)$ the expansion $p=o+x^{0}u+{\sum }_j{x}^j{e}_j$ exhibits the adapted coordinates as $t=x^0$, $x=(x^1,x^2,x^3)$ by [L1]. The assigned curve $\gamma (t)=tu$ is $C^{\mathrm{\infty }}$ on $\mathbb{R}$, has $F$-coordinates $(t,0,0,0)$ and satisfies $\tau (\gamma (s)-o)=s$. Thus its assignment to label $1$ with the specified fixed positive mass satisfies [L2]; worldline status uses that assignment, not the parameter condition alone.

The curve $\kappa (s)=2su$ has $\tau (\kappa (s)-o)=2s$, which differs from $s$ for every $s\mathrm{\ne }0$, for instance at $s=1$ the value is $2$; therefore $\kappa$ fails the parameter condition and is not a worldline of the model, although it is of class $C^{\mathrm{\infty }}$ and its image is the same time axis as that of $\gamma$; it is the reparametrization $\kappa (s)=\gamma (2s)$. The parameter condition is thus a genuine restriction, and the image of a curve does not determine whether it is a worldline.

Under a change of frame the parameter condition is consistent with the reparametrization: by [L3] the same worldline $\gamma$ satisfies $t^{\mathrm{\prime }}(\gamma (s))=s+b$, so the curve ${\gamma }_b(t^{\mathrm{\prime }}):=\gamma (t^{\mathrm{\prime }}-b)$, defined on the shifted interval, satisfies $t^{\mathrm{\prime }}({\gamma }_b(t^{\mathrm{\prime }}))=t^{\mathrm{\prime }}$ and is the description of the motion in $F^{\mathrm{\prime }}$ in the sense of [L2]; the time shift $b$ is absorbed into the reparametrization. For the explicit worldline of step 1.1 this reads ${\gamma }_b(t^{\mathrm{\prime }})=(t^{\mathrm{\prime }}-b)u$.

For the affine reparametrizations in one fixed frame, $\tau ({\kappa }_{\lambda ,\mu }(s)-o)=t(\gamma (\lambda s+\mu ))=\lambda s+\mu$ by [L1] and step 1.1, and the equation $\lambda s+\mu =s$ holds for all real $s$ exactly when $\lambda =1$ and $\mu =0$. Hence no nontrivial affine reparametrization preserves the parameter condition in a fixed frame: the parameter of a worldline is fixed to be the frame time, and a reparametrized curve with $\lambda \mathrm{\ne }1$ becomes a worldline only after the inverse reparametrization. The example confirms that both the parameter and the frame enter the definition of a worldline, as recorded in [L2].