A small diffeomorphism changes metric components but preserves proper time

Lift the circle map to $F_{ϵ}(x)=x+ϵ\sin x$ on $\mathbb{R}$. By [F8], repeated differentiation makes $\sin$ smooth, and its first derivative gives $F_{ϵ}^{\mathrm{\prime }}(x)=1+ϵ\cos x\ge 1-\mathrm{\mid }ϵ\mathrm{\mid }>0$. The mean-value theorem [F11] makes $F_{ϵ}$ strictly increasing. Periodicity [F9] gives $F_{ϵ}(x+2\pi )=F_{ϵ}(x)+2\pi$, and boundedness [F10] gives $F_{ϵ}(x)-x$ bounded; hence its limits at the two ends of $\mathbb{R}$ are $\pm \mathrm{\infty }$. For any real target, choose endpoints with the target between their images and apply [F12]; strict increase gives uniqueness of the preimage. Thus $F_{ϵ}$ is bijective. Its derivative never vanishes, so [F4] makes the inverse smooth locally at every point, hence globally smooth. The translation identity makes the map and inverse descend to smooth inverse maps on the circle in [F14]; extending by the identity in the other coordinates gives a bijective smooth map with smooth inverse, hence the diffeomorphism ${\mathrm{\Phi }}_{ϵ}$ by [F13].

In these coordinates $d{\mathrm{\Phi }}_{ϵ}$ is diagonal with entries $1$, $1+ϵ\cos x^1$, $1$, $1$. Applying [F3] to the diagonal bilinear form gives $({\mathrm{\Phi }}_{ϵ}^{\ast }g)=-(d{x}^0{)}^2+L^2((1+ϵ\cos x^1{)}^2(d{x}^1{)}^2+(d{x}^2{)}^2+(d{x}^3{)}^2)\mathrm{.}$ At $x^1=0$, $(1+ϵ{)}^2\mathrm{\ne }1$ because $0<\mathrm{\mid }ϵ\mathrm{\mid }<1$, so the component arrays differ. Since $1+ϵ\cos x^1\ge 1-\mathrm{\mid }ϵ\mathrm{\mid }>0$, the displayed diagonal matrix has one negative and three positive entries and is Lorentzian by [F5].

The curve $\gamma (\lambda )=(\lambda ,x_{\ast })$ has tangent ${\mathrm{\partial }}_0$ and is future timelike for $g$ with $g(\dot{\gamma },\dot{\gamma })=-1$. Since ${\mathrm{\Phi }}_{ϵ}$ fixes $x^0$ and acts only on spatial coordinates, ${\mathrm{\Phi }}_{ϵ}\circ \gamma$ is also a unit-coordinate-time axis and $g(({\mathrm{\Phi }}_{ϵ}\circ \gamma {)}^{\mathrm{\prime }},({\mathrm{\Phi }}_{ϵ}\circ \gamma {)}^{\mathrm{\prime }})=-1$. By [F3], $({\mathrm{\Phi }}_{ϵ}^{\ast }g)(\dot{\gamma },\dot{\gamma })=g(({\mathrm{\Phi }}_{ϵ}\circ \gamma {)}^{\mathrm{\prime }},({\mathrm{\Phi }}_{ϵ}\circ \gamma {)}^{\mathrm{\prime }})=-1$. Formula [F6] therefore gives proper time $c^{-1}{\int }_0^{\mathrm{\ell }}1d\lambda =\mathrm{\ell }\mathrm{/}c$ in seconds on both curves.

The proper-time value is one of the finite-domain geometric observables treated in [F7]; here its equality is exact. The metric component change in step 2.1 is compatible with that equality because [F1] and [F2] define the analytic estimate in one fixed gauge, whereas [F3] compares the pulled-back geometric field. Thus the coordinate component difference alone is not a difference of the corresponding proper-time observable.