Maxwell's eqs. & unification of electric & magnetic fields

[Frank Castle]

Maxwell's equations reveal an interdependency between electric and magnetic fields, inasmuch as a time varying magnetic field generates a rotating electric field and vice versa. Furthermore, the equations predict that even in the absence of any sources one can have self propagating electric and magnetic fields, so called electromagnetic waves.

However, is it correct to say that although Maxwell's equations show that electric and magnetic fields are interdependent, they do not imply that the two are different aspects of the same physical entity - the electromagnetic field?!

Given this, is it then correct to say that it is not until one takes into account special relativity that it becomes clear that electricity and magnetism are different aspects of the same physical entity - the electromagnetic field?

Indeed, if one considers a frame of reference in which only an electric (or magnetic) field is observed, then, upon a Lorentz transformation to another frame of reference, it is found that one will observe a combination of electric and magnetic fields. This implies that the two are not independent of one another, since there is no observer independent manner in which one can separate electric and magnetic fields, hence implying that they are different aspects of a single entity - the electromagnetic field?!

 

[Lucas SV]

Maxwell's equations have the Lorentz symmetry, that is, they are left unchanged under Lorentz transformations. Since SR tell us that Lorentz transformations are the transformations we should use to make a change of (inertial) coordinates, it is correct to say that one needs SR, in order to unify electricity and magnetism. In special relativity the combination becomes the tensorial object called electromagnetic field strength tensor, which is independent of coordinate system chosen. The Lorentz symmetry of Maxwell's equations was the great discovery made by Einstein, the other symmetry being gauge invariance.

 

[Frank Castle]

Maxwell's equations have the Lorentz symmetry, that is, they are left unchanged under Lorentz transformations. Since SR tell us that Lorentz transformations are the transformations we should use to make a change of (inertial) coordinates, it is correct to say that one needs SR, in order to unify electricity and magnetism. In special relativity the combination becomes the tensorial object called electromagnetic field strength tensor, which is independent of coordinate system chosen. The Lorentz symmetry of Maxwell's equations was the great discovery made by Einstein, the other symmetry being gauge invariance.

So is what I wrote in my original post a correct understanding then?

 

[vanhees71]

The OP is mostly correct. It's only a bit misleading to think that some components of the electromagnetic field are sources of the others. Thinking through Maxwell's equations reveals that writing the corresponding solutions in that way is overcomplicating things to apparently non-local looking equations. The more natural way is to think about the electromangnetic field as one entity, consisting of six real field components, which can be arranged either as two vectors under spatial rotaions, $\vec{E}$ and $\vec{B}$, an antisymmetric tensor $F_{\mu \nu}$ in Minkowski space, or as the Riemann-Silberstein complex vector $\vec{F}=\vec{E}+\mathrm{i} \vec{B}$, which latter idea leads to the not so well known representation of the proper orthocrhonous Lorentz group as $\mathrm{SO}(3,\mathbb{C})$, which is in fact very convenient for the Lorentz-transformation properties of the field components.

No matter how you think about it, the most natural interpretation is that the sources of the electromagnetic field are charge-current distributions (which build a Minkowski four-vector field $(j^{\mu})=(c \rho,\vec{j})$. The usually needed solutions of the Maxwell equations are the retarded solutions for the electromagnetic field, known as Jefimenko's equations, also they where already found much earlier by Ludvig Lorenz (the Danish physicist, often mixed up with the Dutch physicist Lorentz).

 

[Frank Castle]

The OP is mostly correct. It's only a bit misleading to think that some components of the electromagnetic field are sources of the others. Thinking through Maxwell's equations reveals that writing the corresponding solutions in that way is overcomplicating things to apparently non-local looking equations. The more natural way is to think about the electromangnetic field as one entity, consisting of six real field components, which can be arranged either as two vectors under spatial rotaions, $\vec{E}$ and $\vec{B}$, an antisymmetric tensor $F_{\mu \nu}$ in Minkowski space, or as the Riemann-Silberstein complex vector $\vec{F}=\vec{E}+\mathrm{i} \vec{B}$, which latter idea leads to the not so well known representation of the proper orthocrhonous Lorentz group as $\mathrm{SO}(3,\mathbb{C})$, which is in fact very convenient for the Lorentz-transformation properties of the field components.

No matter how you think about it, the most natural interpretation is that the sources of the electromagnetic field are charge-current distributions (which build a Minkowski four-vector field $(j^{\mu})=(c \rho,\vec{j})$. The usually needed solutions of the Maxwell equations are the retarded solutions for the electromagnetic field, known as Jefimenko's equations, also they where already found much earlier by Ludvig Lorenz (the Danish physicist, often mixed up with the Dutch physicist Lorentz).

Thanks for your detailed answer.

So is it correct to say that one cannot conclude that the electric and magnetic fields are components of a single entity, the electromagnetic field, until one takes into account special relativity and observes that they transform into one another as one moves between inertial frames?! (There is no consistent, i.e. observer independent, manner in which one can separate the electric field from the magnetic field, which implies that they are not distinct fields, but instead components of a unified field)

 

[Lucas SV]

(There is no consistent, i.e. observer independent, manner in which one can separate the electric field from the magnetic field, which implies that they are not distinct fields, but instead components of a unified field)

If you think of the 6 components of the EM field (evaluated at a point in spacetime) in a coordinate system you may separate into electric and magnetic field, and define two 3-vectors whose components in some basis are the components in that coordinate system. Rotations then may be thought as a change of basis, i.e. a passive transformation., and components of E will change independently of B.

But when one considers boosts, the change in components of E will depend on B. If we admit boosts as a valid physical change of observer, and assume E and B are independent fields, then we have a contradiction because they look independent for one observer and dependent for another. I think what you are asking is the converse of this, i.e. dependent fields imply SR. I don't know if this is true, but i don't see why it should be.

the Riemann-Silberstein complex vector ⃗F=⃗E+i⃗BF→=E→+iB→\vec{F}=\vec{E}+\mathrm{i} \vec{B}

So I assume there is an equivalent to a Lorentz boost which acts in $\mathbb{C}^3$?

 

[Frank Castle]

I think what you are asking is the converse of this, i.e. dependent fields imply SR. I don't know if this is true, but i don't see why it should be.

No, I actually assuming SR in my question. What I meant was that the fact that boosts of electric field components are dependent on magnetic field components means that the two fields cannot be consistently be treated independently, since, like you said, in one frame they may appear independent, but if we boost to another frame this will not be the case. The only consistent approach is to consider them as components of a single entity, the electromagnetic field.

My question is, really, can one imply from Maxwell's equations alone that the electric and magnetic fields are different aspects of the same entity (the electromagnetic field), or does this only become apparent when SR is taken into account?

To me, it seems that it only truly becomes evident that the electric and magnetic fields are actually just components of the electromagnetic field (analogously to how it only becomes evident that space and time form a single entity, i.e. space-time, when one takes SR into account). Maxwell's equations seem to simply imply that the two fields are interdependent, but not necessarily parts of a single entity.

 

[Lucas SV]

My question is, really, can one imply from Maxwell's equations alone that the electric and magnetic fields are different aspects of the same entity (the electromagnetic field), or does this only become apparent when SR is taken into account?

If we do not assume the principle of relativity, not much can be implied. Then Maxwell's equations are just partial differential equations whose solutions are 6 functions of space and time. Suppose we have a weaker version of the principle of relativity saying that the field equations must be invariant under rotations, This is true for Maxwell's equations (since they are invariant under Lorentz transformations, which is a purely mathematical statement, they are in particular invariant under rotations), which means the equations are a good enough candidate for the laws of physics. In such world it makes sense to think of E and B as physical fields (without the weaker principle, this doesn't even make sense). Now if we generalize the set of admissible transformation the full electromagnetic field can now be thought as an independent field.

The key is the principle of relativity. This is the principle which tell us that we should be able to write the field equations in a form which does not depend on a coordinate system, and whose solutions are mathematical objects which are the real physical fields. So yes, I think you are right, we only discover the electromagnetic field is the field of the theory when SR is taken into account. It doesn't make sense to think of the electromagnetic field as an entity independent of coordinate system unless the principle of relativity is taken into account, and the set of admissible transformations is enlarged to include Lorentz transformations.

 

[Frank Castle]

If we do not assume the principle of relativity, not much can be implied. Then Maxwell's equations are just partial differential equations whose solutions are 6 functions of space and time. Suppose we have a weaker version of the principle of relativity saying that the field equations must be invariant under rotations, This is true for Maxwell's equations (since they are invariant under Lorentz transformations, which is a purely mathematical statement, they are in particular invariant under rotations), which means the equations are a good enough candidate for the laws of physics. In such world it makes sense to think of E and B as physical fields (without the weaker principle, this doesn't even make sense). Now if we generalize the set of admissible transformation the full electromagnetic field can now be thought as an independent field.

The key is the principle of relativity. This is the principle which tell us that we should be able to write the field equations in a form which does not depend on a coordinate system, and whose solutions are mathematical objects which are the real physical fields. So yes, I think you are right, we only discover the electromagnetic field is the field of the theory when SR is taken into account. It doesn't make sense to think of the electromagnetic field as an entity independent of coordinate system unless the principle of relativity is taken into account, and the set of admissible transformations is enlarged to include Lorentz transformations.

Ok great, thanks for your help. Things seem a bit clearer now!

By the way, is it possible to show how the electric and magnetic fields, $\mathbf{E}$ and $\mathbf{B}$, transform individually under Lorentz transformations without first introducing the field strength tensor, $F_{\mu\nu}$? Or, do we first need to note, that in a particular reference frame, and particular gauge, the electric and magnetic fields can be expressed as $$\mathbf{E}=-\nabla\phi -\frac{\partial\mathbf{A}}{\partial t}\\ \mathbf{B}=\nabla\times\mathbf{A}$$ where $\phi$ is a scalar potential and $\mathbf{A}$ is a vector potential. This suggests that we define an electromagnetic four potential $A^{\mu}$, as $$A^{\mu}=\left(\phi,\mathbf{A}\right)$$ which transforms under Lorentz transformations as $$A'^{\mu}=\Lambda^{\mu}_{\; \nu}A^{\nu}$$

 

[vanhees71]

To derive the transformation the latter way is the most direct way, but you can easily figure out that it's way easier to introduce the Riemann-Silberstein vector (in Heaviside-Lorentz or Gaussian units)
$$\vec{F}=\vec{E}+\mathrm{i} \vec{B}$$
This complex vector transforms under Lorentz transformations as an $\mathrm{SO}(3,\mathbb{C})$ transformation. The rotations are the usual ones with real rotation angles, while boosts are those with imaginary rotation angles, $\mathrm{i} \eta$, with $\eta$ the rapidity of the boost, i.e., $\beta=\tanh \eta$. Such a imaginary rotation around $\vec{n}$ corresponds to a boost in this direction.