The interaction between two electriferous particles

[dreamsfly]

There are two electriferous paticles,moving in the same v&direction,then in our opinion,there will be an magnetic effect,but in particles' opinion,there will not be,how to explain it in relativity?

Thanks!

 

[pervect]

What does "electriferous" mean? It's not familiar, nor in the dictionary.

If we imagine two charged particles, stationary with each other, they repel each other with a certain force F in their rest frame. The force represents a rate of change of momentum with time, ie F = dp/dt

If we use "proper time" instead of coordinate time, the force becomes a 4-force, which is the same for all obsesrvers. I.e the 4-force = dp/dtau. "proper" time is just time as measured by a clock on the particle itself.

When we observe two charged particles from a moving frame, the 4-force between them stays constant. When we break this force down into components, though, we find that this constant 4-force is described differently by the two observers. One observer attributed the 4-force entirely to electric fields, the other observer sees both electric and magnetic fields. Both observers agree on the value of the 4-force, though.

 

[lazycai]

I think I can understand this for EM forces.
However, how about forces due to gravity?
In the proper frame of reference, both particles' masses are equal to their rest masses;
in the lab frame of reference, their masses are greater than their rest masses.
Therefore, according to Newton's law of gravitation F=GMm/r^2,
accelerations due to gravity appears to be different in the two inertial frames of reference.

I am not sure whether this problem arised from these two reasons below:
1. The local frame of each particle is not an inertial frame due to the presence of the gravitational force between them;
2. Newton's law of gravity does not apply here.

So, is one of them right or there are other reasons to explain it?
Thanks!

 

[Ich]

It´s similar. Look up http://en.wikipedia.org/wiki/Gravitomagnetism"

 

[Meir Achuz]

Both observers agree on the value of the 4-force, though.

No. the Minkowski force (4-force) is a 4-vector, so its components are different in different Lorentz systems.

 

[Meir Achuz]

There are two electriferous paticles,moving in the same v&direction,then in our opinion,there will be an magnetic effect,but in particles' opinion,there will not be,how to explain it in relativity?

Thanks!

There is a magnetic effect in one Lorentz system that is not present in another. In general, the "Lorentz force" q(E+ vXB) will be different in the two systems.
However, F=ma doesn't hold in special relativity.
If the acceleration is related properly to q(E+vXB), the acceleration
is always directly from one particle to the other in any system.
This is treated in Sec. 15.3.2 of "Classical Electromagnetism" by Franklin.
I have not seen it discussed in other texts.

 

[pervect]

No. the Minkowski force (4-force) is a 4-vector, so its components are different in different Lorentz systems.

4-vetors are generally regarded as being frame-independent objects, though their components can (and indeed must) vary with different coordinate systems.

For the purposes of this thread, what is important is that we know that 4-vectors representing dynamical quantites such as forces must transform in an identical manner that 4-vectors representing kinematical quantities such as distance. All 4-vectors transform in exactly the same way, regardless of their origin as dynamical or kinematical quantities.

Thus if we know the 4-force on an object in one coordinate system at a specific point in space-time, we know what the 4-force is on that object in any coordinate system at that same point, via the Lorentz transforms.

An analogy might help:

If we have two particles at rest relative to each other at some specific coordinates, they have a geometric relationship.

When we view the same two particles from different frames of reference, the space and time coordinates get "mixed together" by the Lorentz transform. Two particles that are separated only by space in one coordinate system are separated by both space and time in another, moving coordinate system. Space and time are not independent, but part of a unified entity known as space-time.

In a similar manner, E and B fields between the particles, like the coordinates of the particles, "mix together" in different frames of reference. Moving E fields create B fields, and moving B fields create E fields. E and B fields are thus not independent, but part of a larger unified entity - which has a name, the Faraday tensor.

 

[dreamsfly]

Then it's true that it will be different in different frame?That is to say ,in some frame,there may be Lorentz force,in some other there may not?

 

[pervect]

Then it's true that it will be different in different frame?That is to say ,in some frame,there may be Lorentz force,in some other there may not?

It's a fact that in some frame there will be a Lorentz force, and in another frame there will not be.

Another standard example is a charge moving in a magnetic field. If you transform to the frame of the charge, the charge appears to be stationary. There is still, however, a force on the charge - but this force is due to the electric field.

As I mentioned previously, this is why we talk about the "electromagnetic" field as a unified entity. They really are not separate, they are deeply connected, in much the same way that space and time are connected.

This is analogous to the way we talk about "space-time" as a unified entity.

 

[dreamsfly]

Thank you very much for your answer!

 

[Meir Achuz]

It's a fact that in some frame there will be a Lorentz force, and in another frame there will not be.

The Lorentz force F is related to the relativistic 4-vector (sometimes called the "Minkowski Force") (\gamma v.F;\gamma F).
If a 4-vector=0 in one Lorentz frame, it must be zero in all Lorentz frames.