Frame change of a declared protocol: how the prediction map and an additive error transform between two inertial frames

Fix one particle label ($N=1$), the force branch and a model as in Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error, and fix two inertial frames $F$, $F^{\mathrm{\prime }}$ of the model with adapted coordinates $(t,x)$ and $(t^{\mathrm{\prime }},x^{\mathrm{\prime }})$. By Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes there are $b\in \mathbb{R}$, $a_0,v\in {\mathbb{R}}^3$ and $R\in SO(3)$ with $t^{\mathrm{\prime }}(p)=t(p)+b,x^{\mathrm{\prime }}(p)=Rx(p)+vt(p)+a_0\text{for every event }p\mathrm{.}$

The protocol in $F$. Take a protocol of Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error with a single observation time $t_1\in I_0$ and the position readout $\mathrm{\Phi }(\theta ,z):=x_{\theta ,z}(t_1)\in {\mathbb{R}}^3$ of dimension $\mathrm{\ell }$, read in metres (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework); the observation is $Y=\mathrm{\Phi }(\theta ,z)+\epsilon$ for an error random element $\epsilon :\mathrm{\Omega }\to {\mathbb{R}}^3$ on a probability space $(\mathrm{\Omega },\mathfrak{F},\mathbb{P})$.

Domains under the coordinate change. The transformed configuration set at time $t^{\mathrm{\prime }}$ is $Q_{t^{\mathrm{\prime }}}^{\mathrm{\prime }}:=RQ+v(t^{\mathrm{\prime }}-b)+a_0$, so the transformed time-configuration domain $\{(t^{\mathrm{\prime }},x^{\mathrm{\prime }}):t^{\mathrm{\prime }}\in I_0+b,x^{\mathrm{\prime }}\in Q_{t^{\mathrm{\prime }}}^{\mathrm{\prime }}\}$ is open (it is the image of $I_0\times Q$ under the invertible affine frame change). It need not be a product with one fixed configuration set, for example when a boost translates bounded $Q$. The following formulas always describe the coordinates of the same worldline and recorded events. They define a force-branch protocol in $F^{\mathrm{\prime }}$ only when the transformed model has an admissible fixed-domain presentation, in particular when $Q={\mathbb{R}}^3$; no fixed-domain model is asserted in the general moving-domain case.

The coordinate readout in $F^{\mathrm{\prime }}$. The same event at which the readout is taken has $F^{\mathrm{\prime }}$-time $t_1+b$; reading the same label position in $F^{\mathrm{\prime }}$ at that time defines the transformed prediction map on the same parameter/preparation labels, ${\mathrm{\Phi }}^{\mathrm{\prime }}(\theta ,z):=x_{\theta ,z}^{\mathrm{\prime }}(t_1+b)=R\mathrm{\Phi }(\theta ,z)+v{t}_1+a_0,$ and the $F^{\mathrm{\prime }}$-coordinates of the recorded point are $Y^{\mathrm{\prime }}:=RY+v{t}_1+a_0={\mathrm{\Phi }}^{\mathrm{\prime }}(\theta ,z)+R\epsilon ,$ so the additive-error form is preserved with the rotated error ${\epsilon }^{\mathrm{\prime }}:=R\epsilon$ and the transformed open error domain $V^{\mathrm{\prime }}:=RV$, which contains $0$ and contains every value of ${\epsilon }^{\mathrm{\prime }}$.

Transform of the error law. Suppose $\epsilon$ is square-integrable with mean $E[\epsilon ]=0$ and covariance matrix $\mathrm{\Sigma }$ in the sense of Mean vector and covariance matrix of a square-integrable random vector. Then ${\epsilon }^{\mathrm{\prime }}=R\epsilon$ has mean $0$ and covariance $\mathrm{Cov}({\epsilon }^{\mathrm{\prime }})=R\mathrm{\Sigma }{R}^{\mathsf{T}},$ and $\mid {\epsilon }^{\mathrm{\prime }}\mid =\mid \epsilon \mid$ in the Euclidean norm because $R$ is orthogonal. In particular an additive error declared in one frame need not have the same law in another: its law is the pushforward under $R$, which equals the original law precisely when that law is invariant under this rotation, and the prediction map acquires the known affine term $v{t}_1+a_0$. A protocol must declare the frame of its readout and of its error model.