The passive Lorentz boost with parameter beta

Let $(o;e_0,e_1,e_2,e_3)$ be an inertial frame of the model (Inertial frames and inertial charts of the affine Minkowski model), and let $x=(x^0,x^1,x^2,x^3)$ be its coordinates, so that a vector is written $x=x^0{e}_0+x^1{e}_1+x^2{e}_2+x^3{e}_3$ and the spatial part is the coordinate column $x_{\mathrm{v}\mathrm{e}\mathrm{c}}=(x^1,x^2,x^3)$ of ${\mathbb{R}}^3$ with its standard Euclidean structure (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

For $\mathit{\beta }\in {\mathbb{R}}^3$ with $\mathrm{\mid }\mathit{\beta }\mathrm{\mid }<1$, where $\mathrm{\mid }\mathit{\beta }\mathrm{\mid }$ is the Euclidean norm, set

$\gamma (\mathit{\beta }):=(1-\mathrm{\mid }\mathit{\beta }{\mathrm{\mid }}^2{)}^{-1\mathrm{/}2}\ge 1,$

with equality exactly for $\mathit{\beta }=\mathbf{0}$ and strict inequality for $\mathit{\beta }\mathrm{\ne }\mathbf{0}$; here the power is the rational power (Rational powers $a^r$ of a positive base, Laws of rational exponents). The passive Lorentz boost with parameter $\mathit{\beta }$ is the linear map $B(\mathit{\beta }):V\to V$ (Linear map between vector spaces over the same field) defined by its action on coordinates: for $\mathit{\beta }\mathrm{\ne }\mathbf{0}$,

$x^{\mathrm{\prime }0}=\gamma (x^0-\mathit{\beta }\cdot x_{\mathrm{v}\mathrm{e}\mathrm{c}}),x_{\mathrm{v}\mathrm{e}\mathrm{c}}^{\mathrm{\prime }}=x_{\mathrm{v}\mathrm{e}\mathrm{c}}+\frac{\gamma -1}{\mathrm{\mid }\mathit{\beta }{\mathrm{\mid }}^2}(\mathit{\beta }\cdot x_{\mathrm{v}\mathrm{e}\mathrm{c}})\mathit{\beta }-\gamma \mathit{\beta }{x}^0,$

and for $\mathit{\beta }=\mathbf{0}$, $B(\mathbf{0}):={\mathrm{id}}_V$. Here $x^{\mathrm{\prime }}$ denotes the coordinates of $B(\mathit{\beta })$ applied to the vector with coordinates $x$, and $\mathit{\beta }\cdot x_{\mathrm{v}\mathrm{e}\mathrm{c}}$ is the Euclidean scalar product. Equivalently, in the ordered basis $(e_0,e_1,e_2,e_3)$ the matrix of $B(\mathit{\beta })$ is

$\left(\begin{array}{cc}\gamma & -\gamma {\mathit{\beta }}^T\\ -\gamma \mathit{\beta }& I_3+{\displaystyle \frac{\gamma -1}{\mathrm{\mid }\mathit{\beta }{\mathrm{\mid }}^2}}\mathit{\beta }{\mathit{\beta }}^T\end{array}\right),$

that is, $B^0{}_0=\gamma$, $B^0{}_i=B^i{}_0=-\gamma {\beta }_i$ for $i=1,2,3$, and $B^i{}_j={\delta }^i{}_j+\frac{\gamma -1}{\mathrm{\mid }\mathit{\beta }{\mathrm{\mid }}^2}{\beta }_i{\beta }_j$ for $i,j=1,2,3$, with the spatial block replaced by $I_3$ when $\mathit{\beta }=\mathbf{0}$. In the $1+1$ case $\mathit{\beta }=\beta e_1$ this reads $x^{\mathrm{\prime }0}=\gamma (x^0-\beta x^1)$ and $x^{\mathrm{\prime }1}=\gamma (x^1-\beta x^0)$. The vector $\mathit{\beta }$ is called the boost parameter; it is dimensionless and mathematically independent of any unit convention.

Consistency of the definition. The displayed formulas assign to every coordinate pair $(x^0,x_{\mathrm{v}\mathrm{e}\mathrm{c}})$ a pair $(x^{\mathrm{\prime }0},x_{\mathrm{v}\mathrm{e}\mathrm{c}}^{\mathrm{\prime }})$; writing the result as the matrix product above and using that ${\mathbb{R}}^4$ is identified with coordinates through A finite list $v:n\to V$ is an ordered basis if and only if every $x\in V$ equals ${\sum }_{i<n}{\lambda }_i{v}_i$ for exactly one $\lambda :n\to F$; those scalars are the coordinates of $x$ in that ordered basis (see Coordinate columns $[v{]}_{\mathcal{B}}$ and matrices $[T{]}_{\mathcal{B}}^{\mathcal{C}}$ of linear maps relative to ordered bases), the map $v\mapsto E(M(E^{-1}(v)))$ is the unique linear map with that matrix. The case $\mathit{\beta }=\mathbf{0}$ is declared separately because the expression $(\gamma -1)\mathrm{/}\mathrm{\mid }\mathit{\beta }{\mathrm{\mid }}^2$ is undefined there. That the two cases fit together is the limit computation recorded next.

The limit as $\mathit{\beta }\to \mathbf{0}$. For $\mathit{\beta }\mathrm{\ne }\mathbf{0}$ the identity ${\gamma }^2-1=\mathrm{\mid }\mathit{\beta }{\mathrm{\mid }}^2{\gamma }^2$ gives

$\frac{\gamma -1}{\mathrm{\mid }\mathit{\beta }{\mathrm{\mid }}^2}=\frac{{\gamma }^2}{\gamma +1}\mathrm{.}$

Since $\mathrm{\mid }\mathit{\beta }{\mathrm{\mid }}^2\le 3\mathrm{/}4$ forces $1\le \gamma \le 2$, the bounds $\mathrm{\mid }{\beta }_i\mathrm{\mid }\le \mathrm{\mid }\mathit{\beta }\mathrm{\mid }$ and $\mathrm{\mid }{\beta }_i{\beta }_j\mathrm{\mid }\le \mathrm{\mid }\mathit{\beta }{\mathrm{\mid }}^2$ (from the Euclidean norm convention, Absolute value in an ordered field, Basic properties of the absolute value) show that $\gamma \to 1$ as $\mathit{\beta }\to \mathbf{0}$; here continuity of the positive-base power at fixed exponent $-1\mathrm{/}2$ is Continuity and derivatives of positive-base real powers, with real and rational powers identified by The exponential definition of real powers agrees with the existing rational powers. Hence $(\gamma -1)\mathrm{/}\mathrm{\mid }\mathit{\beta }{\mathrm{\mid }}^2\to 1\mathrm{/}2$, so $B^i{}_j={\delta }^i{}_j+\frac{1}{2}{\beta }_i{\beta }_j+o(\mathrm{\mid }\mathit{\beta }{\mathrm{\mid }}^2)$ and $B(\mathit{\beta })\to I$ entrywise, without a hidden division by zero. Here $o(\mathrm{\mid }\mathit{\beta }{\mathrm{\mid }}^2)$ denotes a remainder whose ratio to $\mathrm{\mid }\mathit{\beta }{\mathrm{\mid }}^2$ tends to $0$ through nonzero parameters: the ratio of each spatial remainder is bounded by $\mathrm{\mid }{\gamma }^2\mathrm{/}(\gamma +1)-1\mathrm{/}2\mathrm{\mid }\to 0$. The time-space entries obey $\mathrm{\mid }\gamma {\beta }_i\mathrm{\mid }\le 2\mathrm{\mid }\mathit{\beta }\mathrm{\mid }\to 0$, and the time-time entry tends to $1$ by $\gamma \to 1$. The definition asserts no Lorentz property: that $B(\mathit{\beta })$ is a proper orthochronous Lorentz map with inverse $B(-\mathit{\beta })$ is proved in The passive boost is a proper orthochronous Lorentz map with inverse B(-beta), and its collinear composition law in Collinear boost composition and the relativistic velocity-addition law. The physical reading of $c\mathit{\beta }$ as a relative velocity is not part of this definition (Linear and affine Lorentz maps, and the Lorentz group).