Accelerated Observers

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.

Let $\mathrm{\Omega }$ be a nonempty open subset of affine Minkowski space $A$, $c>0$, and let $n:\mathrm{\Omega }\to V$ be a $C^1$ future unit timelike field, $\eta (n,n)=-1$. Define the following first-order quantities in any inertial chart:

$h^a{}_b:={\delta }^a{}_b+n^a{n}_b,h_{ab}:={\eta }_{ab}+n_a{n}_b,a^b:=n^c{\mathrm{\partial }}_c{n}^b,{\theta }_{ab}:=h_a{}^c{h}_b{}^d{\mathrm{\partial }}_{(c}{n}_{d)},$

where symmetrization means half the sum, $n_a:={\eta }_{ab}{n}^b$, and $h(v,w):=h_{ab}{v}^a{w}^b$. Here $h$ is $C^1$ and $a,\theta$ are continuous. The rest projection $h(v):=v+\eta (n,v)n$ is as in Norms of four-velocity and four-acceleration, and the observer rest-space decomposition (c). For $C^1$ fields and tensors define

$(L_{n}h{)}_{ab}:=n^c{\mathrm{\partial }}_c{h}_{ab}+h_{cb}{\mathrm{\partial }}_a{n}^c+h_{ac}{\mathrm{\partial }}_b{n}^c\mathrm{.}$

For smooth data, define the ordinary Lie derivative by $({\mathcal{L}}_{n}h)(X,Y):=n(h(X,Y))-h([n,X],Y)-h(X,[n,Y])$ for smooth vector fields $X,Y$; the brackets have the coordinate formula of Coordinate formula for the Lie bracket. These first-order coordinate formulas transform tensorially under affine inertial chart changes and agree, on smooth fields, with Kinematics of an observer congruence: acceleration, expansion, shear, vorticity and Born rigidity, The flat covariant derivative on affine Minkowski space and fields along curves and the ordinary smooth Lie derivative. Say the $C^1$ field is Born rigid when $L_{n}h=0$; the identity $L_{n}h=2\theta$ makes this equivalent to $\theta =0$.

(a) Let $J$ be a nonempty open interval, $B\subseteq {\mathbb{R}}^3$ a nonempty open set, and $\mathrm{\Phi }:J\times B\to \mathrm{\Omega }$ an injective $C^2$ immersion with ${\mathrm{\partial }}_s\mathrm{\Phi }=n(\mathrm{\Phi })$. Put $e_I:={\mathrm{\partial }}_{{\xi }^I}\mathrm{\Phi }$, $I=1,2,3$, and $G_{IJ}(s,\xi ):=h(e_I,e_J)$ at $\mathrm{\Phi }(s,\xi )$. Then $n,e_1,e_2,e_3$ are linearly independent, $G$ is symmetric positive definite at every point, and its entries are $C^1$. Thus for each $s$ it is a $C^1$ positive-definite material metric on $B$. If $n$ and $\mathrm{\Phi }$ are smooth, $G$ is a smooth Riemannian metric, as in the original smooth branch.

(b) For all $I,J$,

${\mathrm{\partial }}_s{G}_{IJ}=(L_{n}h)(e_I,e_J)=2\theta (e_I,e_J),{\mathrm{\partial }}_{\tau }{G}_{IJ}=2c\theta (e_I,e_J),$

where $\tau =s\mathrm{/}c$ is the proper-time parameter along each flow line, whose four-velocity is $cn$. For a nonzero label direction $e_I$, put ${\mathrm{\ell }}_I:=\sqrt{h(e_I,e_I)}>0$ and $u_I:=h(e_I)\mathrm{/}{\mathrm{\ell }}_I$. This $u_I$ is a horizontal unit rest vector, and

${\mathrm{\partial }}_{\tau }{\mathrm{\ell }}_I=c\theta (u_I,u_I){\mathrm{\ell }}_I\mathrm{.}$

The alternative normalized material vector $e_I\mathrm{/}{\mathrm{\ell }}_I$ need not be horizontal, but has the same $\theta$ contraction because $\theta$ is horizontal. Positive/negative expansion gives positive/negative fractional change of this material rest length per proper time.

(c) If $\theta =0$, then ${\mathrm{\partial }}_s{G}_{IJ}=0$ and $G_{IJ}(s,\xi )$ is constant in $s$ for each fixed label, with no restriction on material-label separation: the entire internal metric is preserved exactly. Conversely, a nonzero contraction $\theta (e_I,e_J)$ gives a nonzero derivative of that metric component at the point and hence nonconstancy on a neighbourhood. The smooth Born-rigid branch and its original conclusions are included.

(d) This is a local statement about the given family and its material metric. Integrating the material metric at a common flow parameter gives an internal label-space length; it does not identify that length with a finite endpoint distance on a single observer's orthogonal simultaneity hyperplane without an additional argument. No matter model, elasticity, field equation, global rigidity theorem, boundary condition or choice principle is used.

Let $c>0$ and $\alpha >0$, set $R:=c^2\mathrm{/}\alpha$, and let $(A,V,\eta ,C^{+})$ be the affine Minkowski space with an inertial chart $\chi =(x^0,x^1,x^2,x^3)$, $x^0=ct$ (Oriented time-oriented affine Minkowski space (the spacetime primitives), Inertial frames and inertial charts of the affine Minkowski model). Write $x^a(\tau ):={\chi }^a(x(\tau ))$ for the coordinate components of the curve $x:\mathbb{R}\to A$ defined by $x(\tau ):={\chi }^{-1}(R\sinh \theta ,R\cosh \theta ,0,0)$, where $\theta :=\alpha \tau \mathrm{/}c$.

(a) Regularity, causal character and parametrization. $x$ is $C^{\mathrm{\infty }}$, and $x^{\mathrm{\prime }}(\tau )=c(\cosh \theta ,\sinh \theta ,0,0)$ is future timelike for every $\tau$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones), so $x$ is a future timelike worldline (Curves, regular causal curves and future timelike worldlines in affine Minkowski space). Since $\eta (x^{\mathrm{\prime }},x^{\mathrm{\prime }})=-c^2$ is constant, the parameter $\tau$ is a proper-time parameter of the curve: the proper-time function $T$ of Proper time along a future timelike worldline satisfies $T(b)-T(a)=b-a$ for all $a<b$ in $\mathbb{R}$ (Proper time along a future timelike worldline, The proper-time function is a strictly increasing C^1 parametrization with C^1 inverse), and the four-velocity is $U:=dx\mathrm{/}d\tau =c(\cosh \theta ,\sinh \theta ,0,0)$ with $\eta (U,U)=-c^2$ (Four-velocity and four-acceleration of a future timelike worldline, Norms of four-velocity and four-acceleration, and the observer rest-space decomposition (a)).

(b) Constant four-acceleration magnitude. The four-acceleration $A:=dU\mathrm{/}d\tau =\alpha (\sinh \theta ,\cosh \theta ,0,0)$ satisfies $\eta (A,A)={\alpha }^2$ and $\eta (U,A)=0$. The chart coordinate velocity is $d{x}^1\mathrm{/}dt=c\tanh \theta$, so $\mathrm{\mid }d{x}^1\mathrm{/}dt\mathrm{\mid }<c$ for every $\tau$, and $x^1=R\cosh \theta >R\mathrm{\mid }\sinh \theta \mathrm{\mid }=\mathrm{\mid }{x}^0\mathrm{\mid }$ for every $\tau$: the worldline lies in the right wedge $W:=\{x^1>\mathrm{\mid }{x}^0\mathrm{\mid }\}$ and its speed in the chart tends to $c$ only as $\tau \to \pm \mathrm{\infty }$.

(c) Rindler flow line. Let $\mathrm{\Phi }(\theta ,\rho ,y,z):={\chi }^{-1}(\rho \sinh \theta ,\rho \cosh \theta ,y,z)$ on $\mathbb{R}\times (0,\mathrm{\infty })\times {\mathbb{R}}^2$ and let $n:={\rho }^{-1}{\mathrm{\partial }}_{\theta }$ be the Rindler congruence of Fermi-Walker transport preserves the metric, and explicit rigid and non-rigid observer congruences (b), whose pullback metric is $-{\rho }^{2}d{\theta }^2+d{\rho }^2+d{y}^2+d{z}^2$ and whose congruence acceleration $a_{\mathrm{c}\mathrm{g}}:={\mathrm{\nabla }}_{n}n$ is ${\rho }^{-1}{\mathrm{\partial }}_{\rho }$, with $\eta (a_{\mathrm{c}\mathrm{g}},a_{\mathrm{c}\mathrm{g}})={\rho }^{-2}$ (The flat covariant derivative on affine Minkowski space and fields along curves, Kinematics of an observer congruence: acceleration, expansion, shear, vorticity and Born rigidity). Then $x$ is exactly the flow line $\mathrm{\Phi }(\cdot ,R,0,0)$ reparametrized by $\theta =\alpha \tau \mathrm{/}c$, and along $x$ one has $A=c^2{a}_{\mathrm{c}\mathrm{g}}$: the four-acceleration of the worldline is $c^2$ times the congruence acceleration, and $\eta (a_{\mathrm{c}\mathrm{g}},a_{\mathrm{c}\mathrm{g}})=1\mathrm{/}{R}^2={\alpha }^2\mathrm{/}{c}^4$ on the flow line $\rho =R$.

(d) Asymptotic behaviour. Along $x$ one has $x^1-x^0=R{e}^{-\theta }\to 0$ with $x^0\to +\mathrm{\infty }$ as $\tau \to +\mathrm{\infty }$, and $x^1+x^0=R{e}^{\theta }\to 0$ with $x^0\to -\mathrm{\infty }$ as $\tau \to -\mathrm{\infty }$: the worldline approaches each of the null planes $\{x^1=x^0\}$ and $\{x^1=-x^0\}$ asymptotically without meeting either, and the chart coordinate speed tends to $c$ at both ends.

(e) Scope. This is an exact kinematic computation in the affine model. It uses no force law, no mass, no clock hypothesis and no empirical input; beyond identifying the displayed parameter with the proper time of the curve, only the flow parametrization of the congruence is used. The curve is not asserted to be a geodesic and no statement is made about the forces that would maintain it. No choice principle is used.