Fermi-Walker transport along a timelike worldline

Let $x:I\to A$ be a $C^2$ future timelike worldline with four-velocity $U=dx\mathrm{/}d\tau$ and four-acceleration $A=dU\mathrm{/}d\tau$ (Four-velocity and four-acceleration of a future timelike worldline, Curves, regular causal curves and future timelike worldlines in affine Minkowski space), so that $\eta (U,U)=-c^2$ and $\eta (U,A)=0$ (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition), write $s$ for its given parameter and $\tau$ for its proper-time function, with ${\tau }^{\mathrm{\prime }}(s)>0$, and let $X:I\to V$ be a $C^1$ field along $x$ (Vector field and section along a smooth curve).

(a) Fermi–Walker derivative. The Fermi–Walker derivative of $X$ is

$\frac{D_{\mathrm{F}\mathrm{W}}X}{d\tau }:=\frac{X^{\mathrm{\prime }}}{{\tau }^{\mathrm{\prime }}}-\frac{1}{c^2}(\eta (A,X)U-\eta (U,X)A),$

where $X^{\mathrm{\prime }}$ denotes differentiation with respect to $s$ and $dX\mathrm{/}d\tau :=X^{\mathrm{\prime }}\mathrm{/}{\tau }^{\mathrm{\prime }}$. Here ${\tau }^{\mathrm{\prime }}=c^{-1}\sqrt{-\eta (x^{\mathrm{\prime }},x^{\mathrm{\prime }})}>0$ by Four-velocity and four-acceleration of a future timelike worldline. The derivative $X^{\mathrm{\prime }}$ is the ordinary $V$-valued derivative of $X$, which is the covariant derivative of $X$ along $x$ in the flat model (The flat covariant derivative on affine Minkowski space and fields along curves, clause (d)). The field $X$ is Fermi–Walker transported along $x$ when $D_{\mathrm{F}\mathrm{W}}X\mathrm{/}d\tau =0$, that is, when

$X^{\mathrm{\prime }}=\frac{{\tau }^{\mathrm{\prime }}}{c^2}(\eta (A,X)U-\eta (U,X)A)\mathrm{.}$

(b) The four-velocity is Fermi–Walker transported. Substituting $X=U$ and using $\eta (A,U)=0$ and $\eta (U,U)=-c^2$ gives $D_{\mathrm{F}\mathrm{W}}U\mathrm{/}d\tau =U^{\mathrm{\prime }}\mathrm{/}{\tau }^{\mathrm{\prime }}-c^{-2}(\eta (A,U)U-\eta (U,U)A)=A-A=0$.

(c) Comparison with parallel transport. If $A=0$ on an interval $J\subseteq I$, then on $J$ the Fermi–Walker equation reduces to $X^{\mathrm{\prime }}=0$ and coincides with ordinary parallel transport; if $A$ is not zero, the definition includes exactly the correction that also transports the time leg $U\mathrm{/}c$ along the worldline.

(d) Fermi–Walker transported tetrad. A Fermi–Walker transported tetrad along $x$ is a tuple $(E_0,E_1,E_2,E_3)$ of $C^1$ fields along $x$ with $\eta (E_a(s),E_b(s))=\mathrm{diag}(-1,1,1,1{)}_{ab}$ at every $s\in I$, $E_0=U\mathrm{/}c$ future, and every $E_a$ Fermi–Walker transported along $x$. These pairings make the four vectors a basis: pairing a linear relation with each $E_b$ forces every coefficient to vanish. Existence and uniqueness from initial data are proved in the following lemma; metric preservation is proved in the subsequent theorem. The physical interpretation as a nonrotating spatial frame carried by an accelerated observer is not asserted here; only the transport rule is defined.

• Well-definedness. For a $C^2$ worldline the fields $U$ and $A$ are continuous and $U$ is nowhere zero, so the coefficients ${\tau }^{\mathrm{\prime }}\eta (A,\cdot )\mathrm{/}{c}^2$ and ${\tau }^{\mathrm{\prime }}\eta (U,\cdot )\mathrm{/}{c}^2$ of the transport equation are continuous functions along $x$; the equation $X^{\mathrm{\prime }}={\tau }^{\mathrm{\prime }}{c}^{-2}(\eta (A,X)U-\eta (U,X)A)$ is therefore a first-order linear differential equation for the $V$-valued field $X$ (Vector field and section along a smooth curve, The flat covariant derivative on affine Minkowski space and fields along curves, clause (d)). This definition fixes the rule and does not assert that solutions exist; existence and uniqueness are proved in the following lemma, and preservation of the tetrad conditions in the subsequent theorem.

• Clause (b) is a computation, not a definition. The two identities used, $\eta (U,U)=-c^2$ and $\eta (U,A)=0$, are clauses (a) and (b) of Norms of four-velocity and four-acceleration, and the observer rest-space decomposition; the cancellation displayed in (b) is then immediate from the bilinearity of $\eta$. The definition deliberately introduces no norm or orthogonality claim of its own.

• Fermi–Walker transport of tensors along a worldline. Write $D$ for the vector Fermi–Walker derivative in (a). For a covector field $\alpha$ along $x$, define $(D\alpha )(X):=d[\alpha (X)]\mathrm{/}d\tau -\alpha (DX)$; for a type-$(r,s)$ tensor $T$, define $(DT)({\alpha }_1,\dots ;X_1,\dots )$ as $d[T({\alpha }_1,\dots ;X_1,\dots )]\mathrm{/}d\tau$ minus the evaluation terms in which each argument in turn is replaced by its $D$ derivative (C^k tensor fields on affine Minkowski space and their inertial components). In a constant basis the ordinary product rule cancels argument derivatives, leaving a well-defined tensor; in this framework only the vector case with the fields $E_a$ and the four-velocity is used, and no additional structure (connection coefficients, Christoffel symbols, curvature) is needed because the model's covariant derivative along a curve is the ordinary $V$-valued derivative.