Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors

Let an inertial frame $(o,u,(e_1,e_2,e_3))$ of the model with adapted coordinates $(t,x)$ be fixed, and let $\gamma :I\to A$ be the worldline of a particle label described in this frame as in Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses; thus $\gamma$ is of class $C^2$, parametrized so that $t(\gamma (s))=s$ for all $s\in I$, and

$\gamma (s)=o+su+{\sum }_{j=1}^3{\xi }_j(s)e_j$

for the unique coordinate functions ${\xi }_j:I\to \mathbb{R}$ of the point $\gamma (s)$ in the adapted chart (Affine time space with Euclidean simultaneous slices and adapted charts, Existence and uniqueness of adapted coordinates, and the affine structure of simultaneous slices).

(a) The coordinate curve. The coordinate curve of $\gamma$ in the frame is

$x_{\gamma }:=({\xi }_1,{\xi }_2,{\xi }_3):I⟶{\mathbb{R}}^3\mathrm{.}$

It is of class $C^2$ by the regularity clause of Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses, and it determines $\gamma$ by the display above, since the parameter condition fixes the time coordinate. It has dimension $[{\xi }_j]=\mathrm{\ell }$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework).

(b) Coordinate velocity and acceleration. The derivative of a vector-valued map is taken componentwise: for the map $x_{\gamma }$ above, its derivative at $s$ is

$x_{\gamma }^{\mathrm{\prime }}(s):=({\xi }_1^{\mathrm{\prime }}(s),{\xi }_2^{\mathrm{\prime }}(s),{\xi }_3^{\mathrm{\prime }}(s)),$

where each ${\xi }_j^{\mathrm{\prime }}(s)={\lim }_{h\to 0}({\xi }_j(s+h)-{\xi }_j(s))\mathrm{/}h$ is the ordinary derivative of a real function. The coordinate velocity of the label in the frame is

$w(s):=x_{\gamma }^{\mathrm{\prime }}(s)\in {\mathbb{R}}^3,$

with components $w_j={\xi }_j^{\mathrm{\prime }}$, and the coordinate acceleration is the coordinate velocity of the velocity,

$a(s):=w^{\mathrm{\prime }}(s)=x_{\gamma }^{\mathrm{\prime }\mathrm{\prime }}(s)\in {\mathbb{R}}^3\mathrm{.}$

Their dimensions are $[w_j]=\mathrm{\ell }{\tau }^{-1}$ and $[a_j]=\mathrm{\ell }{\tau }^{-2}$, read in metres per second and metres per second squared. The triple $w(s)$ is the triple of numbers read in the frame's axes: it is a coordinate description, not an element of spacetime.

(c) Spatial velocity and acceleration vectors. For the affine curve $\gamma :I\to A$, subtraction of events uses the affine displacement in $V$. Define its derivative by the limit in the finite-dimensional vector space $V$, ${\gamma }^{\mathrm{\prime }}(s):={\lim }_{h\to 0}\frac{\gamma (s+h)-\gamma (s)}{h}=u+{\sum }_j{\xi }_j^{\mathrm{\prime }}(s)e_j,$ and define ${\gamma }^{\mathrm{\prime }\mathrm{\prime }}(s)$ as the derivative of the $V$-valued map ${\gamma }^{\mathrm{\prime }}$. This is the canonical identification of each affine tangent space with $V$; the $C^2$ coordinate regularity guarantees both limits. The frame's time direction $u$ and spatial basis identify two elements of the spatial displacement space $K$ attached to the motion:

$\underset{‾}{v}(s):={\gamma }^{\mathrm{\prime }}(s)-u={\sum }_{j=1}^3{w}_j(s)e_j\in K,\underset{‾}{a}(s):={\gamma }^{\mathrm{\prime }\mathrm{\prime }}(s)={\sum }_{j=1}^3{a}_j(s)e_j\in K\mathrm{.}$

Here ${\gamma }^{\mathrm{\prime }}(s)-u$ indeed lies in $K=\ker \tau$, because $\tau ({\gamma }^{\mathrm{\prime }}(s))=\frac{d}{ds}\tau (\gamma (s)-o)=1$ by linearity of $\tau$ and the parameter condition. The vectors $\underset{‾}{v}(s)$ and $\underset{‾}{a}(s)$ are the spatial velocity vector and spatial acceleration vector of the label in the frame; the velocity depends on the frame time direction $u$, whereas the acceleration is an intrinsic element of $K$ unchanged by an inertial frame change (time shifts have derivative $1$). Their coordinates in the frame's basis $(e_1,e_2,e_3)$ are the triples $w(s)$ and $a(s)$.

(d) Speed. The speed of the label in the frame at time $s$ is the Euclidean norm $\mathrm{\parallel }w(s){\mathrm{\parallel }}_2=\sqrt{{\sum }_j{w}_j(s{)}^2}$, of dimension $\mathrm{\ell }{\tau }^{-1}$; equivalently $\mathrm{\parallel }\underset{‾}{v}(s){\mathrm{\parallel }}_2=\sqrt{g(\underset{‾}{v}(s),\underset{‾}{v}(s))}$, the norm induced by the spatial metric, since the basis is $g$-orthonormal.

(e) Transformation under a change of inertial frame (recorded). Let a second inertial frame be given, with coordinates $(t^{\mathrm{\prime }},x^{\mathrm{\prime }})$ and transformation parameters $(b,a,v,R)$ as in Inertial frames, stationary observers and the adopted Galilean transformations between frames, and let $\eta (t^{\mathrm{\prime }}):=x^{\mathrm{\prime }}(\gamma (t^{\mathrm{\prime }}-b))\in {\mathbb{R}}^3$ be the description of the motion in the second frame. Then

${\eta }^{\mathrm{\prime }}(t^{\mathrm{\prime }})=Rw(t^{\mathrm{\prime }}-b)+v,{\eta }^{\mathrm{\prime }\mathrm{\prime }}(t^{\mathrm{\prime }})=Ra(t^{\mathrm{\prime }}-b),$

that is, in the differentiated variables and abbreviated notation $w^{\mathrm{\prime }}=Rw+v$ and $a^{\mathrm{\prime }}=Ra$. This law is clause 2 of the mathematical transformation law of Transformation between adapted charts and its kinematic consequences, recorded here in the model's notation; the definition itself asserts neither law. The model-level statement of the law, including the frame-independence of the property $a\equiv 0$, is proved as a theorem later on this page.

• Velocity is frame-dependent; acceleration rotates. By clause (e) the coordinate velocity changes under a boost, $w^{\mathrm{\prime }}=Rw+v$, while the coordinate acceleration transforms by the rotation alone, $a^{\mathrm{\prime }}=Ra$; the speed is invariant under the rotation part of a frame change and changes under boosts. These are kinematic properties of the model's frames and hold before any dynamics is introduced.

• Spatial vectors and their coordinates. $\underset{‾}{v}={\gamma }^{\mathrm{\prime }}-u$ changes when the frame time direction changes. By contrast $\underset{‾}{a}={\gamma }^{\mathrm{\prime }\mathrm{\prime }}$ has the same value in $K$ in every inertial frame at the corresponding event; its coordinate triple rotates by clause (e). A change of basis changes coordinates without changing this affine vector.

• No dynamical content. Nothing in this item says how $a(s)$ is determined: the force law and the equation of motion are adopted on the later pages of the framework. In particular the identification of $ma$ with a force is not made here.