Regarding the "acceleration" of the EM field

[rumborak]

In the thread about EM waves, the EM wave equation

$$\left(c^2\nabla^2 - \frac{\partial^2}{\partial t^2} \right) E = 0$$

got me pondering. The term

$$\frac{\partial^2}{\partial t^2} E$$

is the second derivative of the field. In a solid-body wave equation, that same term (not with E, but the physical displacement instead) essentially comes down to the inertia of the local particle, through F=ma, a being the second derivative of displacement.

So, the pondering is: How come the EM field is able to work on this second-derivative level as well? Does the EM field have a sort of "inertia" (and even "speed" as the first derivative), just like mass does? Are those two connected on a fundamental level, e.g. "any property of spacetime can have acceleration-dependent effects"?

I hope this thought doesn't come across as too crazy :)

 

[blue_leaf77]

I will probably need somebody to correct me: mechanical inertia as implied in Newton's equations is expressed by the mass of the object in question, in slank language bigger mass bigger inertia. So why do we have to relate the inertia such defined with EM field?

(and even "speed" as the first derivative), just like mass does?

We identify that one in the mechanical wave as speed because the quantity that evolves in time is a displacement, while the first derivative of electric field? Maxwell's equation gives you the answer.

I hope this thought doesn't come across as too crazy :)

Don't worry, I think there will be 'physicists' who are able to explain your confusion and not avoid it, lol. youknowwhatimean.

 

[rumborak]

mechanical inertia as implied in Newton's equations is expressed by the mass of the object in question, in slank language bigger mass bigger inertia. So why do we have to relate the inertia such defined with EM field?

Hmm, I don't think I did a good job at explaining what I meant. The analogy to the solid-body wave equation, and thus the concepts of inertia etc, were really just as an analogy.
What I am curious about is purely about that second-derivative aspect of the EM field. That is, the EM field is a vector field in space, which in its most basic form just means there is a vector associated to every point in space (and I've never seen it being defined as anything but that). Now however, the wave equation above operates on the second time derivative of said field. Meaning, somehow there must be a physical aspect that encodes/carries the information that, *not only* is the E field currently at value XYZ, but it's first derivative is at ABC and its second derivative is DEF. At least in my (maybe simplistic) understanding, a field in its most basic form would really just have a value E(x,y,z).
So, it seems to me the EM field must have a certain "momentum" as well. That is, it can "accelerate" in its values.

Does that make it clearer?

 

[rumborak]

Hmm, thinking about it more, I guess whether it's a second derivative in time, or the Laplace in space, comes down to the same thing really for this purpose. But it's the same argument really; for a local theory of physics, somehow I would think any physical effect could only depend on its immediate neighbor in spacetime (i.e. the first derivative), but not the neighbor of the neighbor (i.e. the second derivative).

EDIT: Aaaaaactually, after all there are two directions for each dimension from a given point. So, a second derivative would combine the lower neighbors distance to the current value, and the higher neighbors distance to the current value. Hmm, maybe that's why second-order derivatives are possible, but nothing higher than that.

Thoughts?

 

[tech99]

Hmm, thinking about it more, I guess whether it's a second derivative in time, or the Laplace in space, comes down to the same thing really for this purpose. But it's the same argument really; for a local theory of physics, somehow I would think any physical effect could only depend on its immediate neighbor in spacetime (i.e. the first derivative), but not the neighbor of the neighbor (i.e. the second derivative).

EDIT: Aaaaaactually, after all there are two directions for each dimension from a given point. So, a second derivative would combine the lower neighbors distance to the current value, and the higher neighbors distance to the current value. Hmm, maybe that's why second-order derivatives are possible, but nothing higher than that.

Thoughts?

For the case where the wave is propagating in a dielectric, it displaces electrons, which have both inertia and a restraining force.

 

[rumborak]

I must be terrible at explaining what I mean. My argument/question is purely about the EM field. Adding particles with masses would just convolute things. The whole thing revolves around the EM wave equation in my original post, which needs no particles to work.

 

[blue_leaf77]

Well, I guess your question belongs to one of those "already-by-nature" types. The second derivative in the wave equation stems from Maxwell's equations which can't be derived by any mean. Giving argument about how its value at a certain point in spacetime be related with other immediate neighboring points by derivatives doesn't really answer the question because such behavior is not necessarily how nature works, for example in heat diffusion equation, the time derivative involved is of first order.

 

[rumborak]

Yeah,I think as nice an argument the "spatial neighborhoods" might be, it's not particularly based in observable evidence. Then again, supposedly a local theory is exactly a theory that supposes any effect can only depend on stuff happening in a point's vicinity.