Existence, uniqueness and linearity of Fermi-Walker transport

Work in the inertial chart of one inertial frame and identify $V$ with ${\mathbb{R}}^4$ by its ordered basis, so that fields along $x$ are ${\mathbb{R}}^4$-valued functions; by [F1] and [F2] the transport equation is the linear system $X^{\mathrm{\prime }}=C(s)X$ with coefficient matrix $C^a{}_b(s)={\tau }^{\mathrm{\prime }}(s)c^{-2}(U^a(s)A_b(s)-A^a(s)U_b(s))$. By [F3] the entries of $C$ are continuous when $x$ is $C^2$, so $C:I\to M_4(\mathbb{R})$ is a continuous matrix-valued function on the interval $I$; and if $x$ is $C^k$ with $k\ge 2$ then $U$ is $C^{k-1}$ and $A$ is $C^{k-2}$, so $C$ is $C^{k-2}$.

Fundamental solution on $I$: by [F7], every point $s\in I$ lies in a nondegenerate compact interval $K\subseteq I$ containing ${\tau }_0$; if $s$ is interior to $I$, choose $K$ with $s$ interior to $K$, and use the corresponding one-sided endpoint reading otherwise. Such intervals exist because $I$ is order-convex and has at least two points; enlarge the segment between $s$ and ${\tau }_0$ using another point when necessary. By [F4], ${\mathrm{\Phi }}^{\mathrm{\prime }}=C\mathrm{\Phi }$, $\mathrm{\Phi }({\tau }_0)=I_4$ has a unique solution on every such $K$. On the intersection of two intervals the solutions agree by [F4] if the intersection is nondegenerate, and by their common initial value if it is the singleton $\{{\tau }_0\}$. Thus $\mathrm{\Phi }(s)$ is their unique common value. Using a compact interval containing a neighbourhood of each interior $s$, or a relative one-sided neighbourhood at an included endpoint, shows that $\mathrm{\Phi }$ is $C^1$, ${\mathrm{\Phi }}^{\mathrm{\prime }}=C\mathrm{\Phi }$ and $\mathrm{\Phi }({\tau }_0)=I_4$ on $I$. Unique values define the map without a choice principle.

Uniqueness of the vector initial value problem: let $\tilde{X}:I\to V$ solve $X^{\mathrm{\prime }}=CX$ with $\tilde{X}({\tau }_0)=X_0$. Form the matrix $\mathrm{\Psi }$ whose first column is $\tilde{X}$ and whose other columns are zero, and let ${\mathrm{\Psi }}_0$ be the corresponding constant initial matrix with first column $X_0$. Then ${\mathrm{\Psi }}^{\mathrm{\prime }}=C\mathrm{\Psi }$ and $\mathrm{\Psi }({\tau }_0)={\mathrm{\Psi }}_0$. The matrix $\mathrm{\Phi }{\mathrm{\Psi }}_0$ satisfies the same problem by step 2.1 and the product rule with constant right factor. Uniqueness in [F4] on every nondegenerate compact subinterval containing the initial point gives $\mathrm{\Psi }=\mathrm{\Phi }{\mathrm{\Psi }}_0$ on $I$. Their first columns therefore give $\tilde{X}=\mathrm{\Phi }{X}_0$, proving uniqueness.

Existence and linearity: with $\mathrm{\Phi }$ as in step 2.1 and $X:=\mathrm{\Phi }{X}_0$, the field $X$ is $C^1$, satisfies $X^{\mathrm{\prime }}={\mathrm{\Phi }}^{\mathrm{\prime }}{X}_0=C\mathrm{\Phi }{X}_0=CX$ and $X({\tau }_0)=\mathrm{\Phi }({\tau }_0)X_0=X_0$; hence $X$ is a $C^1$ solution on $I$ of the transport equation with initial value $X_0$, and step 3.1 shows that it is the only one. Since matrix multiplication is linear in its second factor, the map $X_0\mapsto X=\mathrm{\Phi }{X}_0$ is linear from $V$ to the space of $C^1$ fields along $x$ satisfying the transport equation.

Regularity: assume $x$ is $C^k$ with $k\ge 2$, so that $C$ is $C^{k-2}$ by step 1.1, and let $X$ be the solution of step 4.1, which satisfies $X^{\mathrm{\prime }}=CX$. For $k=2$ the $C^1$ conclusion is already step 4.1. For $k\ge 3$, one proves by induction on $j=1,\dots ,k-2$ that $X$ is $C^j$: for $j=1$ this is step 4.1; if $X$ is $C^j$ with $j\le k-2$, then the product $CX$ is $C^j$ by [F6], because $C$ is $C^{k-2}$ and $j\le k-2$; hence $X^{\mathrm{\prime }}=CX$ is $C^j$, so $X$ is $C^{j+1}$. The induction ends at $j=k-2$, giving that $X$ is $C^{k-1}$.

The initial-tetrad clause: let $E_0({\tau }_0),\dots ,E_3({\tau }_0)$ be a tetrad at $x({\tau }_0)$ with $E_0({\tau }_0)=U({\tau }_0)\mathrm{/}c$ (Charts, tetrads, observer congruences and adapted charts in affine Minkowski space). By steps 4.1 and 3.1, for each $a$ there is exactly one $C^1$ Fermi–Walker transported field $E_a:I\to V$ along $x$ with $E_a({\tau }_0)$ equal to the prescribed value. For $a=0$ the field $U\mathrm{/}c$ is $C^1$, satisfies $D_{\mathrm{F}\mathrm{W}}(U\mathrm{/}c)\mathrm{/}d\tau =0$ by clause (b) of Fermi-Walker transport along a timelike worldline, and has the same initial value $U({\tau }_0)\mathrm{/}c$; by the uniqueness of step 3.1, $E_0=U\mathrm{/}c$. Hence there is a unique tuple of Fermi–Walker transported fields with the prescribed initial values, and its time component is the four-velocity up to the factor $1\mathrm{/}c$.

Conclusion: step 4.1 and step 3.1 establish existence, uniqueness and linearity in the initial value for the $C^1$ solution of the Fermi–Walker transport equation on $I$; step 5.1 establishes the $C^{k-1}$ regularity when the worldline is $C^k$ with $k\ge 2$ (in particular $C^1$ when $k=2$); and step 5.2 establishes the uniquely determined tuple with prescribed tetrad initial values and identifies its time component as $U\mathrm{/}c$. Whether such a tuple remains a tetrad at every time is not claimed here and is proved in the following item.