Physlib.Relativity.Tensors.RealTensor.Metrics.Basic

The notation $\eta'$ denotes the contravariant Lorentz metric tensor (the cometric), which is defined in this context as a complex Lorentz tensor.

The notation $\eta$ represents the contravariant Minkowski metric tensor $\eta^{\mu\nu}$ (also known as the cometric) within the framework of Lorentz tensors.

For any dimension $d \in \mathbb{N}$, the covariant Minkowski metric tensor $\eta'_d$ is equal to the rank-2 tensor constructed from the representation-theoretic morphism $\text{Lorentz.preCoMetric } d$ using the `fromConstPair` construction.

For any spacetime dimension $d \in \mathbb{N}$, the contravariant metric tensor $\eta$ is equal to the rank-2 tensor constructed from the Lorentz-invariant Minkowski metric morphism `Lorentz.preContrMetric d` via the `fromConstPair` operation.

For any dimension $d \in \mathbb{N}$, the covariant Minkowski metric tensor $\eta'_d$ is equal to the rank-2 tensor obtained by applying the $k$-linear map `fromPairT` to the element $\text{Lorentz.preCoMetricVal } d$, which represents the metric value in the tensor product of the Lorentz representation spaces.

For any spacetime dimension $d \in \mathbb{N}$, the contravariant Minkowski metric tensor $\eta_d$ is equal to the rank-2 tensor obtained by applying the $k$-linear map `fromPairT` to the element `Lorentz.preContrMetricVal d`, which represents the contravariant metric value in the tensor product of the Lorentz representation spaces.

For any natural number $d$ and any element $g$ of the Lorentz group $\text{LorentzGroup}(d)$, the covariant metric tensor $\eta'$ is invariant under the group action of $g$. That is, $g \cdot \eta' = \eta'$.

For any natural number $d$ and any element $g$ of the Lorentz group $\text{LorentzGroup}(d)$, the contravariant metric tensor $\eta$ is invariant under the group action of $g$. That is, $g \cdot \eta = \eta$.

For any spatial dimension $d \in \mathbb{N}$, let $\eta'$ be the covariant Minkowski metric tensor (represented as a rank-2 tensor with indices of type `Color.down`). For any multi-index $b = (b_0, b_1)$ in the set of tensor component indices, the component of $\eta'$ with respect to the canonical basis is equal to the $(b_0, b_1)$-th entry of the Minkowski matrix $M = \mathrm{diag}(1, -1, \dots, -1)$. That is, $$[\eta']_{b_0, b_1} = M_{b_0, b_1}.$$

In $d+1$-dimensional spacetime, let $\eta$ be the contravariant Minkowski metric tensor. For any multi-index $b = (b_0, b_1)$ characterizing a component of a rank-2 contravariant tensor, the value of the component of $\eta$ at index $b$ with respect to the canonical basis is equal to the entry of the Minkowski matrix at the corresponding indices. That is, $$[\eta]_{b_0, b_1} = (\text{minkowskiMatrix})_{\mu, \nu}$$ where $\mu$ and $\nu$ are the spacetime indices corresponding to $b_0$ and $b_1$, and the Minkowski matrix is defined as $\text{diag}(1, -1, \dots, -1)$.

For a $(d+1)$-dimensional spacetime, let $g^{(c)}$ be the metric tensor associated with an index color $c$ (representing the type of indices, such as covariant or contravariant). For any multi-index $\phi = (\phi_0, \phi_1)$ used to index the components of a rank-2 tensor, the component of $g^{(c)}$ with respect to the canonical basis is equal to the $(\phi_0, \phi_1)$-th entry of the Minkowski matrix. That is, $$[g^{(c)}]_{\phi_0, \phi_1} = \eta_{\phi_0, \phi_1}$$ where $\eta = \mathrm{diag}(1, -1, \dots, -1)$ is the Minkowski matrix.

In $(d+1)$-dimensional spacetime, let $t$ be a real Lorentz tensor of rank $n+1$. Let $i \in \{0, \dots, n\}$ specify an index to be raised or lowered. Let $T'$ be the tensor obtained by applying the dualization map to the $i$-th index of $t$ (i.e., $T' = \text{toDualMapAtIndex}_i(t)$). For any multi-index $\phi$, the component of $T'$ is given by: $$[T']_{\phi} = \sum_{x} [t]_{(\phi_0, \dots, \phi_{i-1}, x, \phi_{i+1}, \dots, \phi_n)} \cdot \eta_{x, \phi_i}$$ where $[t]$ denotes the components of the tensor with respect to the standard basis, and $\eta$ is the Minkowski matrix $\text{diag}(1, -1, \dots, -1)$.

For a real Lorentz tensor $t$ of rank $n+1$ in $d$ spatial dimensions, let $t'$ be the tensor obtained by raising or lowering the $i$-th index of $t$. For any multi-index $\varphi$, the component of $t'$ at $\varphi$ is equal to the component of $t$ at $\varphi$ multiplied by the corresponding diagonal entry of the Minkowski metric $\eta$: $$(t')_{\varphi} = t_{\varphi} \cdot \eta_{\varphi_i \varphi_i}$$ where $\eta = \text{diag}(1, -1, \dots, -1)$.