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 $\text{diag}(1, -1, \dots, -1)$.