Physlib.Relativity.LorentzGroup.Boosts.Apply

In a spacetime with $d$ spatial dimensions, consider a Lorentz vector $p \in \text{Vector}_d$ and a Lorentz boost in the $i$-th spatial direction with velocity $\beta \in \mathbb{R}$, where $|\beta| < 1$. The temporal component (indexed by $0$) of the vector resulting from the boost action, $(\text{boost}(i, \beta) \cdot p)_0$, is given by: $$(\text{boost}(i, \beta) \cdot p)_0 = \gamma(\beta) (p_0 - \beta p_i)$$ where $p_0$ is the temporal component of $p$, $p_i$ is its $i$-th spatial component, and $\gamma(\beta)$ is the Lorentz factor.

For a Lorentz vector $p$ in a spacetime with $d$ spatial dimensions, let $B_i(\beta)$ be a Lorentz boost in the $i$-th spatial direction with velocity $\beta$, where $|\beta| < 1$. The $i$-th spatial component of the boosted vector $p' = B_i(\beta) \cdot p$ is given by: $$(p')^i = \gamma(\beta) (p^i - \beta p^0)$$ where $p^0$ is the temporal component (indexed by `Sum.inl 0`), $p^i$ is the $i$-th spatial component (indexed by `Sum.inr i`), and $\gamma(\beta)$ is the Lorentz factor.

In a spacetime with $d$ spatial dimensions, let $p$ be a Lorentz vector. Suppose we apply a Lorentz boost in the direction of the $i$-th spatial coordinate with velocity $\beta$, where $|\beta| < 1$. For any spatial index $j$ such that $j \neq i$, the $j$-th spatial component of the boosted vector remains unchanged. That is, $(\Lambda_{i}(\beta) \cdot p)_j = p_j$, where $\Lambda_{i}(\beta)$ is the boost transformation and $p_j$ denotes the $j$-th spatial component of the vector.

In a spacetime with $d$ spatial dimensions, let $p$ be a Lorentz vector. A Lorentz boost in the $i$-th spatial direction with velocity $\beta$ (where $|\beta| < 1$) transforms $p$ into a boosted vector $p' = \text{boost}(i, \beta) \cdot p$. The components of the boosted vector $(p')^j$ are given by the following piecewise definition: - For the temporal component ($j = 0$): $(p')^0 = \gamma(\beta)(p^0 - \beta p^i)$ - For the $i$-th spatial component: $(p')^i = \gamma(\beta)(p^i - \beta p^0)$ - For any other spatial component $j \neq i$: $(p')^j = p^j$ where $p^0$ and $p^k$ represent the temporal and spatial components of the original vector $p$, and $\gamma(\beta)$ is the Lorentz factor.