Four-velocity and four-acceleration of a future timelike worldline

Let $c>0$ and let $x:I\to A$ be a $C^1$ future timelike worldline with proper-time function $\tau$ and inverse $\rho$ of Proper time along a future timelike worldline and The proper-time function is a strictly increasing C^1 parametrization with C^1 inverse.

(a) Four-velocity. The four-velocity of $x$ is the map $U:I\to V$ defined by

$U(s):=\frac{x^{\mathrm{\prime }}(s)}{{\tau }^{\mathrm{\prime }}(s)}=\frac{c{x}^{\mathrm{\prime }}(s)}{\sqrt{-\eta (x^{\mathrm{\prime }}(s),x^{\mathrm{\prime }}(s))}};$

equivalently, $U(s)$ is the tangent at parameter value $s$ of the proper-time parametrized worldline $x\circ \rho$, read at the proper-time parameter $\tau (s)$: by clause (iv) of The proper-time function is a strictly increasing C^1 parametrization with C^1 inverse, $(x\circ \rho {)}^{\mathrm{\prime }}(\tau (s))=x^{\mathrm{\prime }}(s)\mathrm{/}{\tau }^{\mathrm{\prime }}(s)=U(s)$. The map $U$ is continuous, $U(s)$ is future timelike for every $s\in I$, and $U$ does not depend on the base point chosen for $\tau$ (changing the base point adds a constant to $\tau$).

(b) Four-acceleration. If $x$ is $C^2$, that is, $x^{\mathrm{\prime }}$ is differentiable with continuous derivative (Curves, regular causal curves and future timelike worldlines in affine Minkowski space), the four-acceleration of $x$ is the map $A:I\to V$ defined by

$A(s):=\frac{U^{\mathrm{\prime }}(s)}{{\tau }^{\mathrm{\prime }}(s)}=\frac{c{U}^{\mathrm{\prime }}(s)}{\sqrt{-\eta (x^{\mathrm{\prime }}(s),x^{\mathrm{\prime }}(s))}};$

equivalently, $A(s)$ is the derivative with respect to the proper-time parameter of the four-velocity along the worldline: by the chain rule, $A(s)=(U\circ \rho {)}^{\mathrm{\prime }}(\tau (s))$. The map $A$ is continuous.

(c) Observer. The dimensionless four-velocity of $x$ is $u:=U\mathrm{/}c$. A point observer of the model at an event is a future unit timelike vector $n\in V$, $\eta (n,n)=-1$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones); the four-velocity of the observer's own worldline, the straight future timelike line through the event with tangent $cn$, is $cn$ at each of its events.

(d) Names and scope. The names four-velocity and four-acceleration are the model's names for $U$ and $A$; the physical interpretation, including the units in which $U$ and $A$ are measured, is supplied with the coordinate conventions of the framework and is not part of the affine geometry. The definition makes no claim that $U$ has norm $-c^2$ or that $A$ is orthogonal to $U$; those are proved in Norms of four-velocity and four-acceleration, and the observer rest-space decomposition.

• Well-definedness. (i) The division by ${\tau }^{\mathrm{\prime }}(s)$ is legitimate because ${\tau }^{\mathrm{\prime }}(s)>0$ for every $s$ by clause (i) of The proper-time function is a strictly increasing C^1 parametrization with C^1 inverse. (ii) The identification $U(s)=(x\circ \rho {)}^{\mathrm{\prime }}(\tau (s))$ is clause (iv) of that lemma. (iii) Continuity of $U$: in an inertial chart the coordinate components of $U$ are $U^j=y^{j^{\mathrm{\prime }}}\mathrm{/}{\tau }^{\mathrm{\prime }}$, quotients of the continuous functions $y^{j^{\mathrm{\prime }}}$ (the components of the continuous coordinate derivative) and the continuous positive ${\tau }^{\mathrm{\prime }}$, hence continuous (A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions, Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function); by the componentwise characterisation of continuity for ${\mathbb{R}}^4$-valued maps and the identification of $U$ with $E(U_{\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{r}\mathrm{d}})$ this gives continuity of the $V$-valued map $U$. (iv) $U(s)$ is future timelike because it is the positive multiple ${\tau }^{\mathrm{\prime }}(s{)}^{-1}{x}^{\mathrm{\prime }}(s)$ of the future timelike vector $x^{\mathrm{\prime }}(s)$: in a future $\eta$-orthonormal basis the future timelike set is $\{v:v^0>\mathrm{\mid }{v}_{\mathrm{s}\mathrm{p}}\mathrm{\mid }\}$, stable under multiplication by positive scalars (The timelike set has exactly two convex components, described in orthonormal coordinates, Time orientation, future and past timelike/null/causal vectors, and the closed cones). (v) $A$ is defined only when $x$ is $C^2$, in which case $U$ is $C^1$: the coordinate expression $g_x=(y^{0^{\mathrm{\prime }}}{)}^2-{\sum }_{j=1}^3(y^{j^{\mathrm{\prime }}}{)}^2$ is $C^1$, and the square-root chain rule on $g_x>0$ makes ${\tau }^{\mathrm{\prime }}=c^{-1}\sqrt{g_x}$ $C^1$ (Proper time along a future timelike worldline, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$). Its chart components $y^{j^{\mathrm{\prime }}}\mathrm{/}{\tau }^{\mathrm{\prime }}$ are therefore quotients of $C^1$ functions with nonvanishing denominator (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$), so $U^{\mathrm{\prime }}$ exists, is continuous (A function differentiable at $c$ is continuous at $c$, Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function), and $A=U^{\mathrm{\prime }}\mathrm{/}{\tau }^{\mathrm{\prime }}$ is continuous as another quotient. (vi) The observer clause is consistent with (a): the straight line with tangent $cn$ has tangent of squared $\eta$-length $\eta (cn,cn)=-c^2$, so its proper-time derivative is $c^{-1}\sqrt{c^2}=1$ and its four-velocity is $cn$.
• What is deliberately not asserted here. Neither the norm of $U$ nor the orthogonality of $A$ with $U$ is asserted above, so the definition cannot be used as if those identities were definitional; they are theorems of Norms of four-velocity and four-acceleration, and the observer rest-space decomposition. No choice principle is used: all objects are defined by explicit formulas from the given worldline.