Does superposition hold for relativistic particles?

[granpa]

the net electric field due to a collection of point charges is trivial to calculate because it is simply the superposition of the fields of the individual charges. but what if one of the charges is moving at relativistic speed and so its electric field is compressed. does superposition hold for relativistic particles?

 

[atyy]

Yes. Before special relativity, everything worked if we assumed that we were in absolute space and that Maxwell's equations were true, superposition and all. After relativity, the only difference is that every inertial frame became as good as absolute space, ie. Maxwell's equations were already relativistic, before the discovery of relativity.

 

[granpa]

well I can see that for a collection of particles all moving at nearly the same speed but I'm having trouble figuring out how that's possible when they arent.

 

[granpa]

2 particles moving at slightly different relativistic speeds pass one another. each particles field is compressed. how much is the field half way between them compressed?

 

[atyy]

well I can see that for a collection of particles all moving at nearly the same speed but I'm having trouble figuring out how that's possible when they arent.

Oh, I see. Coulomb's law is not consistent with special relativity. The correct generalization is Gauss's law, which is generally written as the first Maxwell equation, and which you will find written in integral or differential form. Most generally, Gauss's law alone is also not relativistic, and we need all 4 Maxwell equations and the Lorentz force law.

 

[atyy]

2 particles moving at slightly different relativistic speeds pass one another. each particles field is compressed. how much is the field half way between them compressed?

If the two particles are moving at constant speeds, you can calculate in a frame in which one is at rest to simplify things. Then use the Lorentz transform to switch to whatever frame you need after you have obtained the answer.

Edit: I should add that it matters greatly whether the charges are moving at constant speed or are accelerating. Superposition holds from very general considerations, but it's not intuitively obvious in many particular situations, so it's good (for you, I'm too lazy) to work these out to build physical intuition. This guy has some pretty cool stuff:http://www.cco.caltech.edu/~phys1/java/phys1/MovingCharge/MovingCharge.html.

 

[Meir Achuz]

2 particles moving at slightly different relativistic speeds pass one another. each particles field is compressed. how much is the field half way between them compressed?

"field is compressed" is an awkward way of describing it. The electric and magnetic fields of each charge moving at constant velocity can be written down explicitly, but they have a complicated dependence on velocity and direction. Superposition holds for the addition of these fields. If the particles are accelerating, the retarded time must be used and the problem is usually intractable.

 

[granpa]

well the reason superposition seems problematic for relativistic particles is this:

imagine that the compressed field from a particle extends beyond another charged particle at rest. how does the field on the other side of the stationary particle know to compress itself if all any given part of the field can see is the field immediately surrounding it? how does it know which particle it belongs to?

it occurs to me that rate of change of the field will be the same regardless of any intervening particles. maybe there is a force proportional to rate of change that compresses the field??