Physlib.Relativity.Tensors.LeviCivita.Basic

The Levi-Civita tensor $\epsilon^{\mu\nu\rho\sigma}$ is a rank-4 contravariant real Lorentz tensor in 4-dimensional spacetime ($d=3$ spatial dimensions). For a multi-index $f = (\mu, \nu, \rho, \sigma)$, the component of the tensor is defined by the generalized Kronecker delta $\delta^{\mu\nu\rho\sigma}_{0123}$. This value corresponds to the sign of the permutation $f$ if $f$ is a permutation of the indices $(0, 1, 2, 3)$, and is $0$ otherwise. Consequently, the tensor is totally antisymmetric and satisfies the normalization $\epsilon^{0123} = 1$.

The notation $\epsilon_4$ represents the rank-four Levi-Civita tensor $\epsilon^{\mu\nu\rho\sigma}$ within the framework of real Lorentz tensors. This tensor is totally antisymmetric and is normalized such that its component in the standard basis $\epsilon^{0123} = 1$.

The rank-four contravariant Levi-Civita tensor $\epsilon^{\mu\nu\rho\sigma}$ in $(1+3)$-dimensional spacetime is equal to the tensor whose components are given by the generalized Kronecker delta $\delta^{\mu\nu\rho\sigma}_{0123}$. Specifically, for any multi-index $f = (\mu, \nu, \rho, \sigma)$, the component of the tensor is determined by the sign of the permutation relative to the identity sequence $(0, 1, 2, 3)$. This theorem states that $\epsilon^{\mu\nu\rho\sigma}$ can be constructed by lifting these integer-valued generalized Kronecker delta components into the real Lorentz tensor space using the `TensorInt.toTensor` function.

Given a sequence of four indices represented by a function $g : \{0, 1, 2, 3\} \to \{0, 1, 2, 3\}$, this function returns the value of the 4-dimensional Euclidean Levi-Civita symbol $\epsilon_{g(0)g(1)g(2)g(3)}$. The value is defined as the generalized Kronecker delta $\delta^{g(0)g(1)g(2)g(3)}_{0123}$, which is equal to the sign of the permutation $g$ (i.e., $+1$ for even permutations and $-1$ for odd permutations) if $g$ is a permutation, and $0$ otherwise.

In 4-dimensional spacetime ($d=3$ spatial dimensions), the components of the rank-four contravariant Levi-Civita tensor $\epsilon^{\mu\nu\rho\sigma}$ in the standard basis, for a given multi-index $b = (\mu, \nu, \rho, \sigma)$ where each index is in $\{0, 1, 2, 3\}$, are given by the generalized Kronecker delta: $$\epsilon^{\mu\nu\rho\sigma} = \delta^{\mu\nu\rho\sigma}_{0123}$$ Here, the generalized Kronecker delta $\delta^{\mu\nu\rho\sigma}_{0123}$ is defined as the determinant of the $4 \times 4$ matrix whose entries are the Kronecker deltas $\delta_{\alpha\beta}$ for $\alpha \in \{\mu, \nu, \rho, \sigma\}$ and $\beta \in \{0, 1, 2, 3\}$.

Let $\epsilon^{\mu\nu\rho\sigma}$ be the rank-four contravariant Levi-Civita tensor in $3+1$ dimensional spacetime. For any multi-index $b = (b_0, b_1, b_2, b_3)$, if there exist distinct index positions $i, j \in \{0, 1, 2, 3\}$ such that the indices at these positions are equal (i.e., $b_i = b_j$), then the component of the tensor in the standard basis is zero: $$\epsilon^{b_0 b_1 b_2 b_3} = 0$$

In 4-dimensional spacetime (with $d=3$ spatial dimensions), the rank-4 contravariant Levi-Civita tensor $\epsilon^{\mu\nu\rho\sigma}$ is antisymmetric with respect to its first two indices. That is, for any indices $\mu, \nu, \rho, \sigma \in \{0, 1, 2, 3\}$, it holds that: $$\epsilon^{\mu\nu\rho\sigma} = -\epsilon^{\nu\mu\rho\sigma}$$

The rank-four Levi-Civita tensor $\epsilon^{\mu\nu\rho\sigma}$ in $3+1$ dimensional spacetime is antisymmetric with respect to its middle two indices, satisfying the identity $\epsilon^{\mu\nu\rho\sigma} = -\epsilon^{\mu\rho\nu\sigma}$.

The rank-four Levi-Civita tensor $\epsilon^{\mu\nu\rho\sigma}$ is antisymmetric in its last two indices. For any choice of indices $\mu, \nu, \rho, \sigma$, it satisfies the relation: $$\epsilon^{\mu\nu\rho\sigma} = -\epsilon^{\mu\nu\sigma\rho}$$

For the rank-four Levi-Civita tensor $\epsilon^{\mu\nu\rho\sigma}$ in $3+1$ dimensional spacetime, the contraction with its covariant counterpart over three indices satisfies the identity: $$\epsilon^{\mu\nu\rho\sigma} \epsilon_{\mu\nu\rho\lambda} = -6 \delta^\sigma_\lambda$$ where $\delta^\sigma_\lambda$ is the Kronecker delta (the unit tensor).

In 4-dimensional spacetime (with $d=3$ spatial dimensions), the full contraction of the rank-four Levi-Civita tensor $\epsilon^{\mu\nu\rho\sigma}$ with its covariant form $\epsilon_{\mu\nu\rho\sigma}$ (where indices are lowered using the Minkowski metric $\eta$) is equal to $-24$: $$\epsilon^{\mu\nu\rho\sigma} \epsilon_{\mu\nu\rho\sigma} = -24$$ where the indices $\mu, \nu, \rho, \sigma$ are summed over $\{0, 1, 2, 3\}$.