# Different versions of covariant Maxwell's equations

• ShayanJ
In summary, there are multiple ways of writing Maxwell's equations in covariant form, including using a vector potential, forms, or Geometric Algebra. However, all of these approaches ultimately represent the same physical laws.
ShayanJ
Gold Member
The standard way of writing Maxwell's equations is by assuming a vector potential ## A^\mu ## and then defining ## F_{\mu \nu}=\partial_\mu A_\nu-\partial_\nu A_\mu ##. Then by considering the action ## \displaystyle \mathcal S=-\int d^4 x \left[ \frac 1 {16 \pi} F_{\mu \nu}F^{\mu\nu}+\frac 1 c J_\mu A^\mu \right] ##, the Maxwell's equations will be given by the e.o.m. of the action, which is ## \partial_\mu F^{\mu \nu}=\frac {4\pi} c J^\nu ##, and the Bianchi identity, ## \partial_\lambda F_{\mu \nu}+\partial_\nu F_{\lambda\mu}+\partial_\mu F_{\nu \lambda}=0 ##.But Maxwell's equations can also be written in the language of forms ## \mathbf{d\star F=4\pi \star J} \ , \ \mathbf{d F}=0 ## Where ## \mathbf F ## is the E.M. two-form and ## \star \mathbf S ## is the Hodge dual of the p-form ## \mathbf S ##.Yet another way of writing Maxwell's equations, is by considering two actions ## \displaystyle \mathcal S=-\int d^4 x \left[ \frac 1 {16 \pi} F_{\mu \nu}F^{\mu\nu}+\frac 1 c J_\mu A^\mu \right] ## and ## \displaystyle \mathcal S=\int d^4 x F_{\mu \nu}\mathcal F^{\mu\nu} ## where ## \mathcal F^{\mu \nu}=\frac 1 2 \varepsilon^{\mu \nu \lambda \sigma}F_{\lambda \sigma} ## represents the components of ## \star \mathbf F ##. These two actions give the equations ## \partial_\mu F^{\mu \nu}=\frac {4\pi} c J^\nu ## and ## \partial_\mu \mathcal F^{\mu \nu}=0 ##.1) Is there any other way of writing Maxwell's equations in covariant form?
2) Is there any relationship between all the above ways or are they just different independent ways of doing it?

Thanks

In the language of Geometric Algebra, you can write $DF = \mu_0 J$, where F is the EM Field bi-vector, J is the current vector, and D is the geometric derivative. This combines the above two separate relations into a single relation, since $D\cdot F = \mu_0 J$ and $D \wedge F = 0$. Of course, these are all just different notational ways of packaging the same physics.

## 1. What are the covariant Maxwell's equations?

The covariant Maxwell's equations are a set of four equations that describe the fundamental laws of electromagnetism. They were first developed by James Clerk Maxwell in the 1860s and are written in a way that is consistent with the principles of special relativity.

## 2. How do the covariant Maxwell's equations differ from the traditional Maxwell's equations?

The traditional Maxwell's equations are written in terms of electric and magnetic fields, while the covariant Maxwell's equations are written in terms of four-vector fields that combine the electric and magnetic fields into a single entity known as the electromagnetic field tensor. This allows for a more elegant and concise formulation of the equations.

## 3. What are the advantages of using covariant Maxwell's equations?

One of the main advantages is that they are compatible with the principles of special relativity, making them more useful for describing electromagnetic phenomena in the context of high speeds and strong gravitational fields. They also allow for a more unified and elegant treatment of electric and magnetic fields.

## 4. Are there different versions of covariant Maxwell's equations?

Yes, there are different formulations of covariant Maxwell's equations, such as the classical version and the quantum version. These different versions are used in different contexts, such as classical electrodynamics and quantum field theory.

## 5. How are the covariant Maxwell's equations used in modern physics?

The covariant Maxwell's equations are an integral part of many modern theories and models in physics, including quantum electrodynamics, general relativity, and string theory. They are also used in practical applications such as designing electronic devices and understanding the behavior of electromagnetic radiation.

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