(a) Construct a tensor D^μν (analogous to F^μν ), out of 𝐃 and 𝐇. Use it to express Maxwell's eq

step 1 here definitely we have D we have UV is equals to we have 0 CDX CDY CDZ so we have negative CDX

(a) Construct a tensor D^μν (analogous to F^μν ), out of 𝐃...

(a) Construct a tensor $D^{\mu \nu}$ (analogous to $F^{\mu \nu}$ ), out of $\mathbf{D}$ and $\mathbf{H}$. Use it to express Maxwell's equations inside matter in terms of the free current density $J_{f}^{\mu} .$ (b) Construct the dual tensor $H^{\mu \nu}$ (analogous to $G^{\mu \nu}$ ). (c) Minkowski proposed the relativistic constitutive relations for linear media: $$ D^{\mu \nu} \eta_{\nu}=c^{2} \epsilon F^{\mu \nu} \eta_{\nu} \text { and } H^{\mu \nu} \eta_{\nu}=\frac{1}{\mu} G^{\mu \nu} \eta_{\nu} $$ where $\epsilon$ is the proper $^{20}$ permittivity, $\mu$ is the proper permeability, and $\eta^{\mu}$ is the 4-velocity of the material. Show that Minkowski's formulas reproduce Eqs. 4.32 and 6.31 , when the material is at rest. (d) Work out the formulas relating $\mathbf{D}$ and $\mathbf{H}$ to $\mathbf{E}$ and $\mathbf{B}$ for a medium moving with (ordinary) velocity u.

(a) Construct a tensor $D^{\mu \nu}$ (analogous to $F^{\mu \nu}$ ), out of $\mathbf{D}$ and $\mathbf{H}$. Use it to express Maxwell's equations inside matter in terms of the free current density $J_{f}^{\mu} .$
(b) Construct the dual tensor $H^{\mu \nu}$ (analogous to $G^{\mu \nu}$ ).
(c) Minkowski proposed the relativistic constitutive relations for linear media:
$$
D^{\mu \nu} \eta_{\nu}=c^{2} \epsilon F^{\mu \nu} \eta_{\nu} \text { and } H^{\mu \nu} \eta_{\nu}=\frac{1}{\mu} G^{\mu \nu} \eta_{\nu}
$$
where $\epsilon$ is the proper $^{20}$ permittivity, $\mu$ is the proper permeability, and $\eta^{\mu}$ is the 4-velocity of the material. Show that Minkowski's formulas reproduce Eqs. 4.32 and 6.31 , when the material is at rest.
(d) Work out the formulas relating $\mathbf{D}$ and $\mathbf{H}$ to $\mathbf{E}$ and $\mathbf{B}$ for a medium moving with (ordinary) velocity u.

Key Concepts

Electromagnetic Field Tensor in Matter

This concept involves extending the usual electromagnetic field tensor F^{ u\mu} of vacuum electrodynamics to media by incorporating the electric displacement field $\mathbf{D}$ and the magnetic field $\mathbf{H}$. Analogous to F^{\mu\nu}, the tensor $D^{\mu\nu}$ is constructed to combine these fields in a covariant manner, allowing Maxwell's equations inside matter to be expressed compactly and consistently with the theory of relativity, particularly when free charges and currents are present.

Dual Tensor

Dual tensors are obtained by contracting electromagnetic field tensors with the Levi-Civita antisymmetric tensor, effectively swapping the roles of the electric and magnetic fields. In this context, constructing a dual tensor such as H^{\mu\nu} (analogous to the dual $G^{\mu\nu}$) provides an alternative representation of the field dynamics that is particularly useful for emphasizing certain symmetries in Maxwell's equations and for formulating the equations of electromagnetism in media.

Relativistic Constitutive Relations

Relativistic constitutive relations, as proposed by Minkowski, link the displacement and magnetic fields with the electric and magnetic fields in a medium using covariant tensor relations. These relations involve the proper permittivity and permeability of the medium and a velocity 4-vector, ensuring that the material response to electromagnetic fields is correctly described in any inertial frame. They provide a bridge between the classical material parameters and their relativistic counterparts, reproducing familiar results when the medium is at rest.

Maxwell's Equations in Covariant Form

Reformulating Maxwell's equations in a covariant form allows the laws of electrodynamics to be expressed using four-dimensional spacetime quantities. In media, especially when free current density is considered, using tensors like D^{\mu\nu} simplifies the equations and makes them manifestly invariant under Lorentz transformations. This approach not only unifies the electric and magnetic fields but also naturally incorporates the effects of media moving with respect to an observer.

Field Transformations in Moving Media

When dealing with a medium moving at a non-zero velocity relative to an inertial observer, the electromagnetic fields transform according to the rules of special relativity. This concept concerns determining how the fields $\mathbf{D}$ and $\mathbf{H}$ are related to $\mathbf{E}$ and $\mathbf{B}$ under such transformations. The analysis is essential for understanding phenomena such as the Fresnel drag and for ensuring that the relativistic constitutive equations hold true in different reference frames, reflecting the interplay between motion and electromagnetic response.

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