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It is always said that Maxwell equations from classical electrodynamics are itself relativistic but I often encounter the term "relativistic electrodynamics". What is the difference between "electrodynamics" and "relativistic electrodynamics" if electrodynamics is itself "relativistic"? What is the new physics in the relativistic version of electrodynamics? I mean, what phenomena could relativistic electrodynamics describe but classical electrodynamics can not?

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    $\begingroup$ Hint: Electrodynamics involves more than just Maxwell's equations. What equation(s) govern the behavior of the charged matter? $\endgroup$ – Chiral Anomaly Apr 7 at 12:47
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You do not give links, but I suspect that "electrodynamics" could describe the electricity and magnetism relations found that established the laws of Gauss Faraday and Ampere, which Maxwell's equations established as connected in one formulation.

Maxwell's electrodynamics is inherently relativistic, that is where Lorenz transformations were established, so this must be the relativistic case. I think that "relativistic" is such an obvious attribute that it is generally omitted when discussing electrodynamics.Maybe it is in older books that a distinction is made?

Of course there is also quantum electrodynamics , which is another story.

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Electrodynamics could be considered the study of time-variable electric and magnetic fields via Maxwell's equations.

The distinction I would draw with Relativistic Electrodynamics is when one starts to consider how those fields would appear/behave in other frames of reference, although some would offer the valid argument that the distinction is when you start treating the electric and magnetic fields as components of a single electromagnetic field, rather than as separate, though connected, entities.

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In addition to using the relativistic expressions and the formalism involving the electromagnetic fields to write down Maxwell's Equations,
I think the adjective "relativistic" means that one needs to be more careful to treat the whole system of fields and sources [and media ] and measuring apparatuses relativistically (with relative speeds comparable to the speed of light).

In addition, one needs to pay attention to the fact that disturbances are not propagated instantaneously, but travel on the light-cones. This can affect measurements.

E.g. Lienard-Weichert potentials, Radiation from accelerated charges, ....

So, "effects" that were [possibly implicitly] negligible in the slow and quasi-static cases may no longer be negligible.

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In relativistic electrodynamics, the mathematical apparatus of special relativity is used to cast Maxwell's equations in a manifestly covariant form. This apparatus wasn't present at the time Maxwell invented his equations.

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At the moment I'm writing this, there are already five answers, so what could I add more? Let us see:

"Classical electrodynamics (CED)" is to be used as an opposite to "Quantum electrodynamics (QED)" and means "the theory of electromagnetic interaction between matter (sources for fields) and electromagnetic fields (i.e. time-variable electric and magnetic fields) using the laws of classical physics". The fundamental equations of CED are Maxwell's equations (written using $\vec{E},\vec{B}$ in vacuum, or $\vec{E},\vec{B}, \vec{H}, \vec{D}$ in material media --> non-manifestly covariant form, or using $F_{\mu\nu}$ in vacuum, or $F_{\mu\nu},G_{\mu\nu}$ in material media --> manifestly covariant form).

On the other hand, "relativistic electrodynamics" (as shorthand for "specially relativistic electrodynamics") is a term used to describe the above-defined CED using either the mathematical apparatus of special relativity (Minkowski spacetime, tensors, and forms in Minkowski spacetime, Lorentz transformations written in Minkowski spacetime) --> manifestly covariant relativistic electrodynamics (born in a paper of Hermann Minkowski), or CED using the regular 3D/Euclidean vectors and tensors, and Lorentz transformations --> non-manifestly covariant relativistic electrodynamics (born in papers of Einstein, but also Poincare and Lorentz around 1900-1905). The latter is now considered obsolete for teaching in universities because using Minkowski spacetime is: a) indispensable for studying special relativity (even in absence of electromagnetic fields and interactions), b) allows one to move smoothly to CED in curved spacetime, c) allows one to move smoothly to QED.

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I partially agree with @DescheleSchilder, but I also think that by relativistic electrodynamics one usually means quantum electrodynamics (QED). The reason is that in QED all particles are treated in relativistic way, on equal footing - this is how the term relativistics, which referred to the teratment of electrons, protons, etc. has become attached to electrodynamics. Moreover, the QED (including all the particles) forms a unified treatment of electromagnetic interactions, which Maxwell equations alone do not (since they do not include the material equatiosn describing teh charged partciles).

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