Physlib.Relativity.Tensors.ComplexTensor.Metrics.Pre

The contravariant Minkowski metric $\eta^{ab}$ defined as an element of the tensor product of two complex contravariant Lorentz vector spaces.

Let $\{e^\mu\}_{\mu=0}^3$ be the basis vectors of the complex contravariant Lorentz vector space. The contravariant Minkowski metric $\eta^{ab}$ is given by the expansion: $$\eta^{ab} = e^0 \otimes e^0 - e^1 \otimes e^1 - e^2 \otimes e^2 - e^3 \otimes e^3$$ where $e^0$ corresponds to the basis vector indexed by `Sum.inl 0`, and $e^1, e^2, e^3$ correspond to the basis vectors indexed by `Sum.inr 0`, `Sum.inr 1`, and `Sum.inr 2` respectively.

The contravariant Minkowski metric $\eta^{ab}$ is defined as a morphism $\mathbb{1} \to V \otimes V$ in the category of representations of $SL(2, \mathbb{C})$, where $\mathbb{1}$ denotes the trivial representation and $V$ denotes the complex contravariant Lorentz vector representation (`complexContr`). Specifically, this morphism maps a complex scalar $a$ to the element $a \cdot \eta^{ab}$ in the tensor product space, where $\eta^{ab}$ is the contravariant metric tensor. The definition includes the proof that this map is an intertwining operator, formalizing the invariance of the Minkowski metric under the action of $SL(2, \mathbb{C})$.

The result of applying the underlying linear map of the contravariant Minkowski metric morphism $\eta^{ab}: \mathbb{1} \to V \otimes V$ to the complex scalar $1$ is equal to the contravariant Minkowski metric tensor $\eta^{ab}$ (denoted by `contrMetricVal`). Here, $\mathbb{1}$ represents the trivial representation and $V$ represents the complex contravariant Lorentz vector representation.

The covariant Minkowski metric tensor, typically denoted with lower indices as \( \eta_{\mu\nu} \), defined as an element of the underlying vector space of the tensor product of two complex covariant Lorentz vector representations.

In the basis $\{e^0, e^1, e^2, e^3\}$ of the complex covariant Lorentz representation, where $e^0$ represents the temporal basis vector (indexed by `Sum.inl 0`) and $e^1, e^2, e^3$ represent the spatial basis vectors (indexed by `Sum.inr 0`, `Sum.inr 1`, and `Sum.inr 2` respectively), the covariant Minkowski metric tensor $\eta$ is expanded as: $$\eta = e^0 \otimes e^0 - e^1 \otimes e^1 - e^2 \otimes e^2 - e^3 \otimes e^3$$ where $\otimes$ denotes the tensor product over the complex numbers $\mathbb{C}$.

The definition `Lorentz.coMetric` defines the covariant Minkowski metric $\eta_{\mu\nu}$ as a morphism in the category of representations of $SL(2, \mathbb{C})$ over the complex numbers. Specifically, it is a morphism from the monoidal unit object $\mathbb{1}$ (the trivial representation on $\mathbb{C}$) to the tensor product representation $V \otimes V$, where $V$ is the complex covariant Lorentz representation (`complexCo`). The morphism maps a scalar $a \in \mathbb{C}$ to $a \cdot \eta_{\mu\nu}$, where $\eta_{\mu\nu}$ is the metric tensor with components $(1, -1, -1, -1)$ represented by `coMetricVal`. This definition ensures that the metric is invariant under the action of $SL(2, \mathbb{C})$.

Applying the covariant Minkowski metric morphism $\eta_{\mu\nu} : \mathbb{1} \to V \otimes V$ to the identity element $1 \in \mathbb{C}$ yields the covariant Minkowski metric tensor value $\eta_{\mu\nu} \in V \otimes V$, where $V$ is the complex covariant Lorentz representation space and $\mathbb{1}$ is the monoidal unit (the trivial representation on $\mathbb{C}$).

In the context of the representation theory of $SL(2, \mathbb{C})$, let $\eta^{\mu\nu}$ be the contravariant Minkowski metric (the element of $V \otimes V$ corresponding to `contrMetric`) and $\eta_{\alpha\beta}$ be the covariant Minkowski metric (the element of $V^* \otimes V^*$ corresponding to `coMetric`). When these two tensors are composed and contracted—specifically by pairing the second index of $\eta^{\mu\nu}$ with the first index of $\eta_{\alpha\beta}$ using the natural contraction morphism—the resulting $(1,1)$-tensor is the identity tensor $\delta^\mu_\beta$ (represented by the unit of the adjunction `coContrUnit`). In standard index notation, this expresses the identity: $$\eta^{\mu\rho} \eta_{\rho\nu} = \delta^\mu_\nu$$

In the context of the representation theory of $SL(2, \mathbb{C})$, let $\eta_{\mu\nu}$ be the covariant Minkowski metric (the element of $V^* \otimes V^*$ corresponding to `coMetric`) and $\eta^{\mu\nu}$ be the contravariant Minkowski metric (the element of $V \otimes V$ corresponding to `contrMetric`). When these two tensors are composed and contracted—specifically by pairing the second index of $\eta_{\mu\nu}$ with the first index of $\eta^{\rho\sigma}$ using the natural contraction morphism—the resulting $(1,1)$-tensor is the identity tensor $\delta_\mu^\nu$ (represented by the unit of the adjunction `contrCoUnit`). In standard index notation, this expresses the identity: $$\eta_{\mu\rho} \eta^{\rho\nu} = \delta_\mu^\nu$$