Physlib.Relativity.Tensors.RealTensor.Vector.MinkowskiProduct

For a given spatial dimension $d$, the Minkowski product of two Lorentz vectors $p, q \in \text{Vector}_d$ is the real-valued scalar defined by the contraction of the Minkowski metric tensor $\eta_{\mu\nu}$ with the vectors $p^\mu$ and $q^\nu$. Following the $+ - - - \dots$ metric signature convention, the product is given by: $$p \cdot q = p^0 q^0 - \sum_{i=1}^d p^i q^i$$ where $p^0, q^0$ are the temporal components and $p^i, q^i$ are the spatial components of the vectors. In the formal implementation, this is calculated by taking the tensor product of the metric $\eta$ and the vectors $p$ and $q$, contracting the corresponding indices, and converting the resulting rank-0 tensor into a real number.

For any spatial dimension $d \in \mathbb{N}$ and Lorentz vectors $p, q \in \text{Vector}_d$, the Minkowski product is given by the coordinate formula: $$p \cdot q = p^0 q^0 - \sum_{i=0}^{d-1} p^i q^i$$ where $p^0$ and $q^0$ are the temporal components (indexed by $\text{inl } 0$) and $p^i, q^i$ are the spatial components (indexed by $\text{inr } i$ for $i \in \{0, \dots, d-1\}$).

For any spatial dimension $d \in \mathbb{N}$ and any Lorentz vectors $p, q \in \text{Vector}_d$, the Minkowski product is symmetric, satisfying: $$p \cdot q = q \cdot p$$ where $\cdot$ denotes the Minkowski product `minkowskiProductMap`.

For any spatial dimension $d \in \mathbb{N}$ and Lorentz vectors $p, q, r \in \text{Vector}_d$, the Minkowski product satisfies additivity in its first argument: $$(p + q) \cdot r = p \cdot r + q \cdot r$$ where $\cdot$ denotes the Minkowski product $p \cdot q = p^0 q^0 - \sum_{i=1}^d p^i q^i$.

For any spatial dimension $d \in \mathbb{N}$ and Lorentz vectors $p, q, r \in \text{Vector}_d$, the Minkowski product satisfies additivity in its second argument: $$p \cdot (q + r) = p \cdot q + p \cdot r$$ where $\cdot$ denotes the Minkowski product, defined using the $(+1, -1, \dots, -1)$ metric signature as $p \cdot q = p^0 q^0 - \sum_{i=1}^d p^i q^i$.

For any spatial dimension $d \in \mathbb{N}$, any scalar $c \in \mathbb{R}$, and any two Lorentz vectors $p, q \in \text{Vector}_d$, the Minkowski product of $c \cdot p$ and $q$ satisfies: $$(c \cdot p) \cdot q = c (p \cdot q)$$ where $p \cdot q$ denotes the Minkowski product, defined by the metric signature $(+1, -1, \dots, -1)$ as $p^0 q^0 - \sum_{i=1}^d p^i q^i$.

For any spatial dimension $d \in \mathbb{N}$, any scalar $c \in \mathbb{R}$, and any two Lorentz vectors $p, q \in \text{Vector}_d$, the Minkowski product of $p$ and $c \cdot q$ satisfies: $$p \cdot (c \cdot q) = c (p \cdot q)$$ where the Minkowski product $p \cdot q$ is defined by the metric signature $(+1, -1, \dots, -1)$ as $p^0 q^0 - \sum_{i=1}^d p^i q^i$, with $p^0, q^0$ being the temporal components and $p^i, q^i$ the spatial components.

For a given spatial dimension $d \in \mathbb{N}$, the Minkowski product is the continuous bilinear map from the space of Lorentz vectors $\text{Vector}_d$ to its dual space (the space of continuous linear forms on $\text{Vector}_d$). Given two Lorentz vectors $p, q \in \text{Vector}_d$, the product is defined by the metric signature $(+1, -1, \dots, -1)$ as: $$\langle p, q \rangle_m = p^0 q^0 - \sum_{i=1}^d p^i q^i$$ where $p^0, q^0$ represent the temporal components and $p^i, q^i$ represent the spatial components. This definition provides the continuous linear map version of the underlying Minkowski scalar product.

The notation $\langle p, q \rangle_m$ denotes the Minkowski product of two Lorentz vectors $p$ and $q$ of dimension $d$, which evaluates to a real number.

For any spatial dimension $d \in \mathbb{N}$ and any two Lorentz vectors $p, q \in \text{Vector}_d$, the value of the continuous bilinear Minkowski product $\langle p, q \rangle_m$ evaluated at $p$ and $q$ is equal to the value defined by the Minkowski product map $p \cdot q$. Here, $\text{Vector}_d$ is the space of Lorentz vectors of dimension $d$, and the product represents the contraction of the vectors with the Minkowski metric.

For any spatial dimension $d \in \mathbb{N}$ and any two Lorentz vectors $p, q \in \text{Vector}_d$, the continuous bilinear Minkowski product $\langle \cdot, \cdot \rangle_m$ is symmetric, satisfying: $$\langle p, q \rangle_m = \langle q, p \rangle_m$$

For any spatial dimension $d \in \mathbb{N}$ and Lorentz vectors $p, q \in \text{Vector}_d$, the Minkowski product $\langle p, q \rangle_m$ is given by the coordinate formula: $$\langle p, q \rangle_m = p^0 q^0 - \sum_{i=0}^{d-1} p^i q^i$$ where $p^0, q^0$ represent the temporal components (indexed by $\text{inl } 0$) and $p^i, q^i$ represent the spatial components (indexed by $\text{inr } i$ for $i \in \{0, \dots, d-1\}$).

For any spatial dimension $d \in \mathbb{N}$ and Lorentz vectors $p, q \in \text{Vector}_d$, the Minkowski product $\langle p, q \rangle_m$ is equal to the sum over all spacetime indices $\mu$ of the product of the diagonal entries of the Minkowski matrix $\eta$ and the components of the vectors $p$ and $q$ at those indices: $$\langle p, q \rangle_m = \sum_{\mu} \eta_{\mu\mu} p^\mu q^\mu$$ where $\mu \in \text{Fin } 1 \oplus \text{Fin } d$ indices the time and space components, and $\eta$ is the Minkowski matrix $\mathrm{diag}(1, -1, \dots, -1)$.

For any spatial dimension $d \in \mathbb{N}$, any two Lorentz vectors $p, q \in \text{Vector}_d$, and any element $\Lambda$ of the Lorentz group $\text{LorentzGroup}(d)$, the Minkowski product $\langle \cdot, \cdot \rangle_m$ is invariant under the action of $\Lambda$. That is, $$\langle \Lambda \cdot p, \Lambda \cdot q \rangle_m = \langle p, q \rangle_m$$ where $\Lambda \cdot p$ and $\Lambda \cdot q$ represent the transformed vectors under the Lorentz group action.

For any spatial dimension $d \in \mathbb{N}$ and any two Lorentz vectors $p, q \in \text{Vector}_d$, the Minkowski product $\langle p, q \rangle_m$ is equal to the product of their temporal components minus the Euclidean inner product of their spatial parts: $$\langle p, q \rangle_m = p^0 q^0 - \langle \mathbf{p}, \mathbf{q} \rangle_{\mathbb{R}}$$ where $p^0, q^0$ are the time components and $\mathbf{p}, \mathbf{q} \in \mathbb{R}^d$ are the spatial parts of $p$ and $q$, respectively.

For any spatial dimension $d \in \mathbb{N}$ and any Lorentz vector $p \in \text{Vector}_d$, the Minkowski product of $p$ with itself is equal to the square of the norm of its temporal component minus the square of the norm of its spatial part: $$\langle p, p \rangle_m = \|p^0\|^2 - \|\mathbf{p}\|^2$$ where $p^0 \in \mathbb{R}$ is the time component and $\mathbf{p} \in \mathbb{R}^d$ is the spatial part of the vector $p$.

For any spatial dimension $d \in \mathbb{N}$ and any Lorentz vector $p \in \text{Vector}_d$, the Minkowski product of $p$ with itself is less than or equal to the square of its temporal component: $$\langle p, p \rangle_m \leq (p^0)^2$$ where $\langle \cdot, \cdot \rangle_m$ is the Minkowski product and $p^0$ is the time component of the vector $p$.

For any spatial dimension $d \in \mathbb{N}$, Lorentz vector $p \in \text{Vector}_d$, and index $\mu \in \text{Fin } 1 \oplus \text{Fin } d$, the Minkowski product of the $\mu$-th standard basis vector and $p$ is given by: $$\langle \text{basis } \mu, p \rangle_m = \eta_{\mu\mu} p^\mu$$ where $\text{basis } \mu$ is the $\mu$-th standard coordinate basis vector, $\eta_{\mu\mu}$ is the $\mu$-th diagonal entry of the Minkowski matrix $\mathrm{diag}(1, -1, \dots, -1)$, and $p^\mu$ is the $\mu$-th component of the vector $p$.

For any spatial dimension $d \in \mathbb{N}$, Lorentz vector $p \in \text{Vector}_d$, and index $\mu \in \text{Fin } 1 \oplus \text{Fin } d$, the Minkowski product of $p$ and the $\mu$-th standard basis vector is given by: $$\langle p, \text{basis } \mu \rangle_m = \eta_{\mu\mu} p^\mu$$ where $\text{basis } \mu$ is the $\mu$-th standard coordinate basis vector, $\eta_{\mu\mu}$ is the $\mu$-th diagonal entry of the Minkowski matrix $\mathrm{diag}(1, -1, \dots, -1)$, and $p^\mu$ is the $\mu$-th component of the vector $p$.

For any spatial dimension $d \in \mathbb{N}$ and any Lorentz vector $p \in \text{Vector}_d$, the Minkowski product $\langle p, q \rangle_m$ is equal to zero for all vectors $q \in \text{Vector}_d$ if and only if $p = 0$. Here, $\text{Vector}_d$ is the space of $(d+1)$-dimensional vectors (one temporal and $d$ spatial components), and $\langle \cdot, \cdot \rangle_m$ denotes the Minkowski scalar product defined by the metric signature $(1, -1, \dots, -1)$.

For any spatial dimension $d \in \mathbb{N}$ and any linear map $f : \text{Vector}_d \to \text{Vector}_d$ on the space of Lorentz vectors, the Minkowski product satisfies $\langle f(p), q \rangle_m = \langle p, q \rangle_m$ for all vectors $p, q \in \text{Vector}_d$ if and only if $f$ is the identity map $\text{id}$. Here, $\langle \cdot, \cdot \rangle_m$ denotes the Minkowski scalar product defined by the metric signature $(+1, -1, \dots, -1)$.

For a given spatial dimension $d \in \mathbb{N}$, the adjoint of a linear map $f: \text{Vector}_d \to \text{Vector}_d$ with respect to the Minkowski product is defined as the linear map whose matrix representation relative to the standard basis is the Minkowski dual of the matrix of $f$. Specifically, if $\Lambda$ is the matrix of $f$ with respect to the standard basis, the adjoint is the linear map corresponding to the matrix $\eta \Lambda^T \eta$, where $\eta = \text{diag}(1, -1, \dots, -1)$ is the Minkowski metric matrix.

For any spatial dimension $d \in \mathbb{N}$, let $f: \text{Vector}_d \to \text{Vector}_d$ be a linear map and $p, q \in \text{Vector}_d$ be Lorentz vectors. The Minkowski product $\langle \cdot, \cdot \rangle_m$ satisfies the adjoint property: $$\langle f(p), q \rangle_m = \langle p, f^\dagger(q) \rangle_m$$ where $f^\dagger$ is the adjoint of $f$ with respect to the Minkowski product.

For any spatial dimension $d \in \mathbb{N}$, let $f: \text{Vector}_d \to \text{Vector}_d$ be a linear map and $p, q \in \text{Vector}_d$ be Lorentz vectors. The Minkowski product $\langle \cdot, \cdot \rangle_m$ satisfies the adjoint property: $$\langle p, f(q) \rangle_m = \langle f^\dagger(p), q \rangle_m$$ where $f^\dagger$ is the adjoint of $f$ with respect to the Minkowski product.

A linear map $f: \text{Vector}_d \to \text{Vector}_d$ satisfies the property `IsLorentz` if it preserves the Minkowski product $\langle \cdot, \cdot \rangle_m$. Specifically, for all Lorentz vectors $p, q \in \text{Vector}_d$, the following equality must hold: $$\langle f(p), f(q) \rangle_m = \langle p, q \rangle_m$$ where $\langle \cdot, \cdot \rangle_m$ is the Minkowski product defined with the metric signature $(+1, -1, \dots, -1)$.

Let $f: \text{Vector}_d \to \text{Vector}_d$ be a linear map on the space of Lorentz vectors. The property `IsLorentz f` holds if and only if $f$ preserves the Minkowski product for all vectors $p, q \in \text{Vector}_d$. That is, $$\text{IsLorentz } f \iff \forall p, q \in \text{Vector}_d, \langle f(p), f(q) \rangle_m = \langle p, q \rangle_m$$ where $\langle \cdot, \cdot \rangle_m$ denotes the Minkowski product with metric signature $(+1, -1, \dots, -1)$.

Let $f: \text{Vector}_d \to \text{Vector}_d$ be a linear map on the space of Lorentz vectors. The property `IsLorentz f` holds if and only if $f$ preserves the Minkowski product for all pairs of standard basis vectors $\text{basis } \mu$ and $\text{basis } \nu$ indexed by $\text{Fin } 1 \oplus \text{Fin } d$. That is, $$\text{IsLorentz } f \iff \forall \mu, \nu \in \text{Fin } 1 \oplus \text{Fin } d, \langle f(\text{basis } \mu), f(\text{basis } \nu) \rangle_m = \langle \text{basis } \mu, \text{basis } \nu \rangle_m$$ where $\langle \cdot, \cdot \rangle_m$ denotes the Minkowski product with metric signature $(+1, -1, \dots, -1)$.

Let $f: \text{Vector}_d \to \text{Vector}_d$ be a linear map on the space of Lorentz vectors. The property `IsLorentz f` holds if and only if the composition of the adjoint of $f$ and $f$ is the identity map. That is, $$\text{IsLorentz } f \iff f^\dagger \circ f = \text{id}$$ where $f^\dagger$ denotes the adjoint of $f$ with respect to the Minkowski product $\langle \cdot, \cdot \rangle_m$.

Let $f: \text{Vector}_d \to \text{Vector}_d$ be a linear map on the space of Lorentz vectors. The property `IsLorentz f` (which signifies that $f$ preserves the Minkowski product $\langle \cdot, \cdot \rangle_m$) holds if and only if the matrix representation of $f$ with respect to the standard basis, $\text{toMatrix}(\text{basis, basis}, f)$, is an element of the Lorentz group $O(1, d)$.