Physlib.Relativity.Tensors.RealTensor.Velocity.Basic

For a given spatial dimension $d$, the set of Lorentz velocities $\text{Velocity}_d$ is equipped with a topological space structure. This structure is the subspace topology inherited from the natural topology of the space of Lorentz vectors $\text{Vector}_d$.

For any two Lorentz velocities $v$ and $w$ of dimension $d$, if their underlying Lorentz vectors are equal ($v.1 = w.1$), then the velocities themselves are equal ($v = w$). A Lorentz velocity is defined as a Lorentz vector with a Minkowski norm $\langle v, v \rangle_m = 1$ and a positive time component.

For a Lorentz vector $v$ in a Minkowski space with $d$ spatial dimensions, $v$ is a Lorentz velocity if and only if its Minkowski inner product with itself $\langle v, v \rangle_m$ is equal to $1$ and its time component $v^0$ is strictly positive ($v^0 > 0$).

For any Lorentz velocity $v$ in a space with $d$ spatial dimensions, the Minkowski inner product of its underlying vector with itself is equal to $1$, denoted as $\langle v, v \rangle_m = 1$.

For any Lorentz velocity $v$ in a Minkowski space with $d$ spatial dimensions, the temporal component $v^0$ of the vector is strictly positive, i.e., $v^0 > 0$.

For any Lorentz velocity $v$ in a Minkowski space with $d$ spatial dimensions, the temporal component $v^0$ of its underlying vector is non-negative, i.e., $v^0 \ge 0$.

For any Lorentz velocity $v$ in a Minkowski space with $d$ spatial dimensions, the absolute value of its time component $v^0$ is equal to the time component itself, i.e., $|v^0| = v^0$.

For any Lorentz velocity $v$ in a Minkowski spacetime with $d$ spatial dimensions, let $\mathbf{v} \in \mathbb{R}^d$ be its spatial part and $v^0 \in \mathbb{R}$ be its temporal component. The Euclidean norm of the spatial part is less than or equal to the absolute value of the temporal component, i.e., $\|\mathbf{v}\| \leq |v^0|$.

For any Lorentz velocity $v$ in $(1+d)$-dimensional Minkowski spacetime, let $\mathbf{v}$ denote its spatial part and $v^0$ denote its temporal component. The square of the Euclidean norm of the spatial part satisfies the relation $\|\mathbf{v}\|^2 = (v^0)^2 - 1$.

For any two Lorentz velocities $u$ and $v$ in a Minkowski spacetime with $d$ spatial dimensions, their Minkowski inner product is non-negative, i.e., $0 \le \langle u, v \rangle_m$. Here, a Lorentz velocity is defined as a future-directed Lorentz vector whose Minkowski norm is equal to 1.

For any two Lorentz velocities $u$ and $v$ in $(1+d)$-dimensional Minkowski spacetime, the sum of one and their Minkowski inner product is non-zero, i.e., $1 + \langle u, v \rangle_m \neq 0$. Here, a Lorentz velocity is defined as a future-directed Lorentz vector with Minkowski norm equal to 1.

For a given Lorentz vector $u \in \text{Vector}_d$, the function $v \mapsto \langle u, v \rangle_{\eta}$ mapping a Lorentz velocity $v \in \text{Velocity}_d$ to its Minkowski inner product with $u$ is continuous. Here, $\langle \cdot, \cdot \rangle_{\eta}$ denotes the Minkowski inner product, and $\text{Velocity}_d$ is equipped with the subspace topology inherited from the space of Lorentz vectors $\text{Vector}_d$.

For a fixed Lorentz vector $u \in \text{Vector}_d$, the function $v \mapsto \langle v, u \rangle_{\eta}$ mapping a Lorentz velocity $v \in \text{Velocity}_d$ to its Minkowski inner product with $u$ is continuous. Here, $\langle \cdot, \cdot \rangle_{\eta}$ denotes the Minkowski inner product, and $\text{Velocity}_d$ is equipped with the subspace topology inherited from the space of Lorentz vectors $\text{Vector}_d$.

In $(1+d)$-dimensional Minkowski space, the rest velocity $0 \in \text{Velocity}_d$ is the Lorentz vector defined by having its temporal component equal to $1$ and all its $d$ spatial components equal to $0$. Mathematically, it is represented by the basis vector $e_0$ corresponding to the temporal index in the standard basis of $\mathbb{R}^{1,d}$. This vector is future-directed and satisfies the unit norm condition $\langle e_0, e_0 \rangle_{\eta} = 1$ required for a Lorentz velocity.

This definition designates the rest velocity $u = (1, \mathbf{0})$ as the canonical zero element $0$ for the type of Lorentz velocities $\text{Velocity}_d$ in $(1+d)$-dimensional Minkowski space. The Lorentz velocity $0$ corresponds to a vector with a temporal component of $1$ and spatial components equal to $0$, satisfying the unit norm and future-directed conditions.

For the zero element $0$ (the rest velocity) in the space of Lorentz velocities $\text{Velocity}_d$ in $(1+d)$-dimensional Minkowski space, its temporal component is equal to $1$.

For a given Lorentz velocity $u$ in $(1+d)$-dimensional Minkowski spacetime, this definition constructs a continuous path $\gamma: [0, 1] \to \text{Velocity}_d$ connecting the rest velocity $\mathbf{0} = (1, \mathbf{0})$ to $u$. For any $t \in [0, 1]$, the path is defined as: $$\gamma(t) = \left( \sqrt{1 + t^2 \|\mathbf{u}\|^2} - t u^0 \right) e_0 + t u$$ where $u^0$ is the temporal component of $u$, $\mathbf{u} \in \mathbb{R}^d$ is the spatial part of $u$, and $e_0$ is the unit temporal basis vector. The definition ensures that $\gamma(0) = \mathbf{0}$, $\gamma(1) = u$, and that for every $t$, $\gamma(t)$ is a valid Lorentz velocity (i.e., it is future-directed and has Minkowski norm equal to 1).

The space of Lorentz velocities $\text{Velocity}_d$ in $(1+d)$-dimensional Minkowski spacetime is path-connected. This means that for any two Lorentz velocities $u, v \in \text{Velocity}_d$ (future-directed Lorentz vectors with Minkowski norm equal to 1), there exists a continuous path $\gamma: [0, 1] \to \text{Velocity}_d$ such that $\gamma(0) = u$ and $\gamma(1) = v$. The topology on $\text{Velocity}_d$ is the subspace topology inherited from the standard topology on the space of Lorentz vectors.