Physlib.Relativity.Tensors.RealTensor.Metrics.Pre

For any natural number $d$, the contravariant Minkowski metric tensor value $\text{preContrMetricVal}(d)$ can be expanded in terms of the contravariant basis vectors $e^\mu$ as: $$\text{preContrMetricVal}(d) = e^0 \otimes e^0 - \sum_{i=0}^{d-1} e^i \otimes e^i$$ where $e^0$ is the time-like basis vector (defined as `contrBasis d (Sum.inl 0)`) and $e^i$ are the spatial basis vectors (defined as `contrBasis d (Sum.inr i)` for $i \in \{0, \dots, d-1\}$). The operator $\otimes$ denotes the tensor product over $\mathbb{R}$.

For any natural number $d$, the contravariant Minkowski metric tensor value $\text{preContrMetricVal}(d)$ can be expanded as the sum over all indices $i$ of the diagonal entries of the Minkowski matrix $\eta$ multiplied by the tensor product of the corresponding basis vectors $e^i$ with themselves: $$\text{preContrMetricVal}(d) = \sum_i \eta_{ii} (e^i \otimes e^i)$$ where $\eta_{ii}$ are the diagonal entries of the $(d+1)$-dimensional Minkowski matrix (represented by `minkowskiMatrix i i`), $e^i$ are the contravariant basis vectors (represented by `contrBasis d i`), and $\otimes$ denotes the tensor product over $\mathbb{R}$.

For any natural number $d$, the evaluation of the linear map representing the contravariant Lorentz metric $\text{preContrMetric}(d)$ at the scalar $1 \in \mathbb{R}$ is equal to the contravariant metric tensor value $\text{preContrMetricVal}(d)$.

For any natural number $d$, the covariant Minkowski metric tensor value $\eta$ in $d+1$ dimensions (denoted by `preCoMetricVal d`) can be expressed in terms of the dual basis vectors $e^0, e^1, \dots, e^d$ (denoted by `coBasis d`). Specifically, the metric tensor is the difference between the tensor product of the temporal basis vector with itself and the sum of the tensor products of the spatial basis vectors with themselves: $$\eta = e^0 \otimes e^0 - \sum_{i=1}^d e^i \otimes e^i$$ where $e^0$ corresponds to the index `Sum.inl 0` and $e^i$ corresponds to the index `Sum.inr i`.

For any natural number $d$, the covariant Minkowski metric tensor $\eta$ in $d+1$ dimensions (denoted by `preCoMetricVal d`) is equal to the sum over all indices $i$ of the diagonal elements of the Minkowski matrix $\eta_{ii}$ (denoted by `minkowskiMatrix i i`) multiplied by the tensor product of the $i$-th dual basis vector $e^i$ (denoted by `coBasis d i`) with itself: $$\eta = \sum_i \eta_{ii} (e^i \otimes e^i)$$

For any natural number $d$, applying the covariant metric map $\text{preCoMetric}(d)$ to the real number $1$ yields the covariant metric tensor value $\text{preCoMetricVal}(d)$.

In $(1+d)$-dimensional Minkowski spacetime, let $\eta^{\mu\nu}$ denote the contravariant metric tensor and $\eta_{\rho\sigma}$ denote the covariant metric tensor. When the tensor product of these two metrics is formed and the second index of the contravariant metric is contracted with the first index of the covariant metric, the resulting tensor (after swapping the remaining indices) is the Kronecker delta $\delta^\mu_\sigma$, which represents the identity tensor in the representation $\text{Co } d \otimes \text{Contr } d$. In component notation, this expresses the identity: \[ \sum_{\rho} \eta^{\mu\rho} \eta_{\rho\sigma} = \delta^\mu_\sigma \]

For any dimension $d$, let $\eta_{\mu\nu}$ be the covariant Minkowski metric tensor (represented by `preCoMetric d`) and $\eta^{\rho\sigma}$ be the contravariant Minkowski metric tensor (represented by `preContrMetric d`). The theorem states that if one takes the tensor product of these two metrics and contracts the second index of the covariant metric with the first index of the contravariant metric (using the morphism `coContrContract`), the resulting tensor—after reordering the remaining indices via the braiding map $\beta$—is the identity tensor $\delta_\mu^\rho$ (represented by `preContrCoUnit d`). In standard index notation, this corresponds to the identity: $$\sum_{\nu} \eta_{\mu\nu} \eta^{\nu\rho} = \delta_\mu^\rho$$ where $\delta_\mu^\rho$ is the Kronecker delta.