Proper acceleration of a massive worldline and its agreement with the acceleration of an observer congruence

Let $c>0$ and let $x:I\to A$ be a $C^2$ future timelike worldline in the affine Minkowski model (Oriented time-oriented affine Minkowski space (the spacetime primitives), Curves, regular causal curves and future timelike worldlines in affine Minkowski space), with generic parameter $s\in I$. Write $\tau :I\to \tau [I]$ for its proper-time function and $U(s):=x^{\mathrm{\prime }}(s)\mathrm{/}{\tau }^{\mathrm{\prime }}(s)$, $A(s):=U^{\mathrm{\prime }}(s)\mathrm{/}{\tau }^{\mathrm{\prime }}(s)$ for its four-velocity and four-acceleration (Four-velocity and four-acceleration of a future timelike worldline). These are proper-time derivatives evaluated at the event $x(s)$; $s$ need not itself be proper time. Pointwise $\eta (U,U)=-c^2$, $\eta (U,A)=0$, and $A(s)$ is spacelike whenever it is nonzero (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition (a),(b)).

(a) Definition. The proper acceleration of $x$ at the parameter value $s$ is the real number

$a(s):=\sqrt{\eta (A(s),A(s))},$

the nonnegative square root of Rational powers $a^r$ of a positive base. This is well defined for every $s\in I$. Indeed, fix an inertial chart of $A$ (Inertial frames and inertial charts of the affine Minkowski model); since $x$ is $C^2$, the component functions of $A$ are continuous, and $\eta (A,A)$ is a sum of products of continuous component functions, hence continuous, by 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. If $A(s)=0$ then $\eta (A,A)=0$ at $s$, and if $A(s)\mathrm{\ne }0$ then $A(s)$ is spacelike and $\eta (A,A)>0$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space); in both cases the argument of the square root is nonnegative, so $a(s)$ is a well-defined nonnegative real number for every $s\in I$. With the SI coordinate and unit conventions of SI units and coordinate conventions of the special-relativity framework, in which $x^0=ct$ with $t$ in seconds and the spatial coordinates are in metres while $\eta$ has the dimensionless matrix $\mathrm{diag}(-1,1,1,1)$ in an inertial chart, the components of $A$ carry the unit metre per second squared and $\eta (A,A)$ carries the unit $(\text{metre per second squared}{)}^2$, so $a$ carries the unit metre per second squared. The worldline is uniformly accelerated when $a$ is constant on $I$.

(b) Agreement with the acceleration of a congruence. Let $\mathrm{\Omega }$ be a nonempty open subset of $A$ and let $n:\mathrm{\Omega }\to V$ be an observer congruence with acceleration field $a_{\mathrm{c}\mathrm{g}}:={\mathrm{\nabla }}_{n}n$, where $\mathrm{\nabla }$ is the flat covariant derivative of The flat covariant derivative on affine Minkowski space and fields along curves (Kinematics of an observer congruence: acceleration, expansion, shear, vorticity and Born rigidity (b)). If $x$ is a flow line of $n$ and $U=cn$ along $x$, then $A=c^2{a}_{\mathrm{c}\mathrm{g}}$ along $x$: the derivative along the worldline of the field $n$ is the covariant derivative ${\mathrm{\nabla }}_{dx\mathrm{/}d\tau }n={\mathrm{\nabla }}_{cn}n=c{\mathrm{\nabla }}_{n}n=c{a}_{\mathrm{c}\mathrm{g}}$, by the defining properties of the flat covariant derivative along curves (The flat covariant derivative on affine Minkowski space and fields along curves (d)), so $A=d(cn)\mathrm{/}d\tau =c{\mathrm{\nabla }}_{dx\mathrm{/}d\tau }n=c^2{a}_{\mathrm{c}\mathrm{g}}$. Consequently $a=c^2\sqrt{\eta (a_{\mathrm{c}\mathrm{g}},a_{\mathrm{c}\mathrm{g}})}$ along $x$: the proper acceleration is $c^2$ times the magnitude of the congruence acceleration. In particular the proper acceleration of a congruence worldline is determined by the congruence field alone, and such a worldline is uniformly accelerated exactly when $\eta (a_{\mathrm{c}\mathrm{g}},a_{\mathrm{c}\mathrm{g}})$ is constant along it.

(c) Reading and scope. This item defines a quantity of the model and fixes its name, its unit and its physical reading. The model reads $a(s)$ as the magnitude that an ideal accelerometer carried along the worldline would register at $s$; this ideal-instrument identification is adopted with the model in the same sense as the framework's ideal-clock reading of proper time and is not derived here. No accelerometer response law, calibration, frequency dependence, force law or interaction is specified or asserted; the definition asserts nothing about which bodies follow which worldlines; no measurement outcome and no statistical, uncertainty or empirical claim is made. No choice principle is used.

• Nonnegativity is the content of the well-definedness clause. The projection of the four-acceleration on the orthogonal complement of the four-velocity is spacelike or zero (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition (b)), so $\eta (A,A)\ge 0$ always, and $a$ vanishes exactly at the parameter values where $A=0$, i.e. at the events where the worldline has vanishing four-acceleration. Nothing in the definition presumes that such events exist, or fail to exist.
• Changing the proper-time origin. In this model the proper-time origin is arbitrary up to an additive constant (Proper time along a future timelike worldline); $a$ is a function of the generic parameter $s$; changing the proper-time origin does not change $U(s)$, $A(s)$ or $a(s)$. When expressed as a function of proper time instead, its argument shifts by that additive constant.