Worked Fermi-Walker transport along uniformly accelerated hyperbolic motion

Derivatives and causal character: by [L4], $x^{\mathrm{\prime }}(\tau )=\frac{c^2}{\alpha }\cdot \frac{\alpha }{c}(\cosh \theta ,\sinh \theta ,0,0)=c(\cosh \theta ,\sinh \theta ,0,0)=U$, so $U$ is $C^1$ and $x$ is $C^2$; further $U^{\mathrm{\prime }}=c\cdot \frac{\alpha }{c}(\sinh \theta ,\cosh \theta ,0,0)=\alpha (\sinh \theta ,\cosh \theta ,0,0)=A$. Evaluating the diagonal form, $\eta (U,U)=c^2(-{\cosh }^2\theta +{\sinh }^2\theta )=-c^2$, so $x$ is future timelike with $U$ future (time component $c\cosh \theta >0$) and $\tau$ is a proper-time parameter by [L3]; $\eta (A,A)={\alpha }^2({\cosh }^2\theta -{\sinh }^2\theta )={\alpha }^2$ and $\eta (U,A)=c\alpha (-\cosh \theta \sinh \theta +\sinh \theta \cosh \theta )=0$; in particular $A\mathrm{\ne }0$ because $\alpha >0$.

Orthonormal pair: $X_0=U\mathrm{/}c=(\cosh \theta ,\sinh \theta ,0,0)$ and $X_1=(\sinh \theta ,\cosh \theta ,0,0)$ satisfy $\eta (X_0,X_0)=-{\cosh }^2\theta +{\sinh }^2\theta =-1$, $\eta (X_0,X_1)=-\cosh \theta \sinh \theta +\sinh \theta \cosh \theta =0$ and $\eta (X_1,X_1)=-{\sinh }^2\theta +{\cosh }^2\theta =1$ by [L4]; and $X_0^{\mathrm{\prime }}=U^{\mathrm{\prime }}\mathrm{/}c=A\mathrm{/}c$.

$X_1$ is Fermi-Walker transported: $X_1^{\mathrm{\prime }}=\frac{\alpha }{c}(\cosh \theta ,\sinh \theta ,0,0)=\frac{\alpha }{c}{X}_0$, and $\frac{1}{c^2}(\eta (A,X_1)U-\eta (U,X_1)A)=\frac{1}{c^2}(\alpha U-0)=\frac{\alpha }{c}{X}_0$, because $\eta (A,X_1)=\alpha (-{\sinh }^2\theta +{\cosh }^2\theta )=\alpha$ and $\eta (U,X_1)=c(-\cosh \theta \sinh \theta +\sinh \theta \cosh \theta )=0$ by [L4]; the right-hand side equals $X_1^{\mathrm{\prime }}$, so $X_1$ is transported.

$X_0$ is Fermi-Walker transported: by [L1] and steps 1.1-2.1, $\frac{1}{c^2}(\eta (A,X_0)U-\eta (U,X_0)A)=\frac{1}{c^2}(0-(-c)A)=A\mathrm{/}c=X_0^{\mathrm{\prime }}$, because $\eta (A,X_0)=\eta (A,U)\mathrm{/}c=0$ and $\eta (U,X_0)=\eta (U,U)\mathrm{/}c=-c$.

Difference from parallel transport and norm preservation: $X_1^{\mathrm{\prime }}=\frac{\alpha }{c}{X}_0\mathrm{\ne }0$ since $\alpha >0$ and $X_0\mathrm{\ne }0$, so $X_1$ is not parallel transported by [L1]; Fermi-Walker transport therefore differs from parallel transport along this accelerated worldline. The norm is preserved: $\eta (X_1,X_1)=1$ is constant, in agreement with the preservation clause [L2].

Tetrad: the constant fields ${\mathrm{\partial }}_y=(0,0,1,0)$ and ${\mathrm{\partial }}_z=(0,0,0,1)$ have ${\mathrm{\partial }}_y^{\mathrm{\prime }}={\mathrm{\partial }}_z^{\mathrm{\prime }}=0$ and $\eta (A,{\mathrm{\partial }}_y)=\eta (U,{\mathrm{\partial }}_y)=0$, likewise for ${\mathrm{\partial }}_z$, so the right-hand side of the Fermi-Walker rule vanishes and both are transported by [L1]. Hence $(U\mathrm{/}c,X_1,{\mathrm{\partial }}_y,{\mathrm{\partial }}_z)=(X_0,X_1,{\mathrm{\partial }}_y,{\mathrm{\partial }}_z)$ is an orthonormal tetrad every member of which is Fermi-Walker transported, as asserted in the Example; this also realizes the observer-adapted tetrad of [L2].

Conclusion: steps 1.1-2.1 verify the kinematic data, steps 3.1 and 2.2 verify that both legs $X_0=U\mathrm{/}c$ and $X_1$ satisfy the Fermi-Walker rule, step 3.2 exhibits the nonzero derivative $X_1^{\mathrm{\prime }}=\frac{\alpha }{c}{X}_0$ showing that Fermi-Walker transport is not parallel transport for $\alpha >0$, and step 4.1 completes the transported orthonormal tetrad. All identities are exact evaluations in the hyperbolic functions and the diagonal form.