Wave propagation interpretation

[microsansfil]

Hello,

We read very often that wave propagation électro-magnetism is the result of an interaction between electric field and magnetic field.

However, Maxwell's equations are not causal.


These equations have no delay between electric field and magnetic field, they simply pose relations between the two fields, which can be read both in one direction than the other, or as a single equalities.

Nothing in Maxwell's equations imply causality, an "induction" of a field by the other..

Why this interpretation ?

Patrick

 

[Orodruin]

Maxwell's equations are causal, they imply that disturbances in the fields cannot propagate faster than the speed of light in vacuum (and do so in vacuum). They are local in the sense that the rate of change of the fields at a given point in space-time will only depend on the field and charge configuration at that particular space-time point. Any influence from one point in space to another must occur through propagating a field disturbance (at c) or changing the charge distribution (at less than c).

 

[microsansfil]

Maxwell's equations are causal, they imply that disturbances in the fields cannot propagate faster than the speed of light in vacuum (and do so in vacuum).

Isn't it two different things ? What can we derive from Maxwell's equation to assert that each field are causally dependent from each other ? what is the propagation by "local" effect ?

Patrick

 

[Orodruin]

Locality means that the equations of how the field at a point evolves depend only on quantities specified at that point in that instant. From Maxwell's equations follow that disturbances in the electromagnetic field travel at speed c and special relativity tells us that charges travel at lower speeds. Thus, the value of a field at a given space-time point can only depend on the charge and field configuration at earlier times within the past light-cone. This is the essence of causality, what happens only depends on what unambiguously is in the past.

Note that Maxwell's equations are differential equations, they relate the change of the fields with time to the change of the fields in space and so relates the field values at different points - and the speed with which disturbances travel between the points is predicted to be c. This means that the field at a given point will be influenced by what the field was at other places, but at earlier times.

 

[WannabeNewton]

These equations have no delay between electric field and magnetic field, they simply pose relations between the two fields, which can be read both in one direction than the other, or as a single equalities.

Yes, exactly!

Nothing in Maxwell's equations imply causality, an "induction" of a field by the other..
Why this interpretation ?

Yes, you're correct. Strictly speaking that interpretation is flawed. It only holds in a reduction of order perturbation theory. See: https://www.physicsforums.com/showpost.php?p=4783107&postcount=4 and https://www.physicsforums.com/showpost.php?p=4783132&postcount=6

 

[Orodruin]

Yes, you're correct.

So you are claiming Maxwell's equations are not causal in structure? Do you wish to build QED on a non-causal field theory? I certainly do not.

 

[WannabeNewton]

So you are claiming Maxwell's equations are not causal in structure? Do you wish to build QED on a non-causal field theory? I certainly do not.

Who claimed that? That wasn't the OP's claim at all. He/she is talking about something completely different. He/she is saying that there is no notion of electromagnetic induction ("change in magnetic field induces electric field and change in electric field induces change in magnetic field") when we consider Maxwell's equations as a system of 2 simultaneous dynamical equations and 2 simultaneous constraint equations for the electric and magnetic fields, which is definitely true.

 

[Orodruin]

The OP makes this claim twice:

However, Maxwell's equations are not causal.

...

Nothing in Maxwell's equations imply causality, an "induction" of a field by the other..

 

[WannabeNewton]

It's a poor choice of words on the OP's part. He/she is using "causality" to mean "induction" as opposed to causality in the strict sense one presumes in physics.

 

[Orodruin]

In that case I would just like to mention this to the OP: "Causality" in physics is typically reserved for a particular property in physical theories, essentially that an event can only be influenced by other events if the other events are in the past (or, equivalently, an event may only influence other events if they are in the future).

I have also never come across the induced-by-inductuon problem, but seen several examples of people who think causality is broken somewhere, which is why my backbone reaction was to go in that direction.

 

[microsansfil]

He is saying that there is no notion of electromagnetic induction ("change in magnetic field induces electric field and change in electric field induces change in magnetic field") when we consider Maxwell's equations as a system of 2 simultaneous dynamical equations and 2 simultaneous constraint equations for the electric and magnetic fields, which is definitely true.

yes this is the meaning of my question with my english.

Patrick

 

[microsansfil]

In that case I would just like to mention this to the OP.

OP ?

Patrick

 

[microsansfil]

"Causality" in physics is typically reserved for a particular property in physical theories,

In Maxwell equation there is only symbol =. What is this meaning in physics reserved word ?

Patrick

 

[WannabeNewton]

yes this is the meaning of my question with my english.

Look at the threads I linked.

 

[microsansfil]

Look at the threads I linked.

OK Thank


Relativity shows that the division between electric field and magnetic field is artificial. in the 4D formalism appears only an electromagnetic field without division. it will appear only as the choice of a reference.

Patrick

 

[WannabeNewton]

Relativity shows that the division between electric field and magnetic field is artificial. in the 4D formalism appears only an electromagnetic field without division. it will appear only as the choice of a reference.

Yes indeed so this is another reason why the usual notion of induction, while true in the reduction of order approximation scheme, is not conceptually correct in the exact formalism. In reality we only have the dynamical equation $\partial^{\mu}F_{\mu\nu} = -4\pi j_{\nu}$ for the electromagnetic field $F_{\mu\nu}$, along with the constraint equation $\partial_{[\mu}F_{\nu\gamma]} = 0$. If we solve these simultaneously in some coordinate system and then reexpress the solution in a Lorentz covariant form we have found the electromagnetic field associated with the given charge distribution.

When looking at $\nabla \times E = -\partial_t B$ and $\nabla \times B = \partial_t E + j$ then, it doesn't make sense to say $\partial_t B$ induces the circulation $\nabla \times E$ and similarly for the Ampere law, because there is only $F_{\mu\nu}$ which simultaneously contains the electric and magnetic components and whose equations of motion in a given choice of inertial frame split into the two dynamical equations above for $E$ and $B$. There is no causation involved wherein $B$ generates $E$ and the resulting $E$ generates $B$ and so on.

But as mentioned this is not how the problem of solving Maxwell's equations is usually presented in introductory books on EM. Therein one solves the equations perturbatively, using a reduction of order. When using a reduction of order method for a set of coupled dynamical equations it is exactly true that the nth order solutions generate the n+1th order solutions and this is usually what one means by $B$ generates $E$ etc. Indeed, it should really be $B^{(n)}(t,x)$, through $\nabla \times E^{(n+1)} = -\partial_t B^{(n)}$, induces $E^{(n+1)}$ and so on.

 

[Orodruin]

Yes indeed so this is another reason why the usual notion of induction, while true in the reduction of order approximation scheme, is not conceptually correct in the exact formalism. In reality we only have the dynamical equation $\partial^{\mu}F_{\mu\nu} = -4\pi j_{\nu}$ for the electromagnetic field $F_{\mu\nu}$, along with the constraint equation $\partial_{[\mu}F_{\nu\gamma]} = 0$. If we solve these simultaneously in some coordinate system and then reexpress the solution in a Lorentz covariant form we have found the electromagnetic field associated with the given charge distribution.

The heart of the matter is that Einstein arrived at special relativity by thinking mainly about the propagation of light and the Maxwell equations and their invariance is discussed in his original paper. The elegance and simplicity in writing the Maxwell equations in tensor notation should be quite astounding to anyone who first was taught the regular form. Even more astounding the fact that it is essentially the simplest thing you can do to obtain a 4-force field without a heat-like component. Then you go back to the original paper and you realize that Einstein actually wrote down Maxwell's equations component by component ...

 

[WannabeNewton]

Then you go back to the original paper and you realize that Einstein actually wrote down Maxwell's equations component by component ...

Haha. I guess it's a bit like when you learn Maxwell's equations in the usual form and conclude from their aesthetic elegance that Maxwell must have been an admirer of true beauty in physical equations but when you see his original magnum opus you see he wrote out everything in components in a most ugly fashion

 

[Orodruin]

Haha. I guess it's a bit like when you learn Maxwell's equations in the usual form and conclude from their aesthetic elegance that Maxwell must have been an admirer of true beauty in physical equations but when you see his original magnum opus you see he wrote out everything in components in a most ugly fashion

When I write down awful expressions in my papers, I would like to think that someone will write an elegant generalization of it a hundred years from now ... Probably too much to ask for ...

 

[WannabeNewton]

When I write down awful expressions in my papers, I would like to think that someone will write an elegant generalization of it a hundred years from now ... Probably too much to ask for ...

Hey, you never know!

 

[Orodruin]

Hey, you never know!

Hehe ... Want to give it a try?
$$
{\rm BR}(\tau \to \eta \ell) =
\left|
\mathscr{F}_{+}
\mathcal{C}_{LQ}^{1}
-
\left[
\mathcal{C}_{LQ'}^{1}
+
\mathcal{C}_{LQ'}^{3}
\right] \right.
\left.+
\frac{3 m_{\eta}^{2}}{4 m_{s} m_{\tau}}
\mathscr{F}_{-}
\left\{
\frac{1}{2}
\mathscr{F}'
\left[
\mathcal{C}_{EU}
+
\mathcal{C}_{ED}
\right]
-
\mathcal{C}_{ES}
\right\}
\right
|^{2}
+
\left(
\frac{3 m_{\eta}^{2}}{4 m_{s} m_{\tau}}
\right)^{2}
\mathscr{F}_{-}^{2}
\Bigl|
\frac{1}{2}
\mathscr{F}'
\left[
\mathcal{C}_{EU}^{\dagger}
+
\mathcal{C}_{ED}^{\dagger}
\right]
-
\mathcal{C}_{ES}^{\dagger}
\Bigr|^{2}
$$
Ugh!

At least some are nicer
$$
\frac XB \rightarrow -\frac{11}{14}
$$

But I am digressing, I should probably stop ... I think OP has had his questions answered apart from this:

OP ?

Patrick

OP = Original Post or Original Poster.

 

[WannabeNewton]

If you redefine everything in the RHS as a single variable different from the one on the LHS then you got yourself an elegant equation. Time to publish xP!

 

[microsansfil]

Thank you to everyone, this more clear for me now.Patrick

 

[USeptim]

Just a comment: Jefimenko's equations shows how electric and magnetic fields can be calculated from the existing four-currents and its time derivative. This way we get a casual expression for EM fields. In this sense, Maxwell equations would give an expression about how the wave propagates, both with homogeneous and inhomogeneous terms, but would not explain the "cause" of the field, that would be the charges and their movement.