# Deriving the Lorentz Force

1. Dec 5, 2006

### Aero

I read somewhere that the whole of magnetism, and in particular the lorentz force, can be found merely by applying the lorentz transformation to transform the coordinates of the electric field of a charged particle from the frame in which the particle is at rest to a frame in which the particle is moving.

I am finding it difficult to do this, mostly because I thought that the lorentz transformation transforms the coordinates of spacetime events and not vector fields. How can you fit vector fields into a four-dimensional spacetime?

Thanks

2. Dec 5, 2006

### robphy

You mean COMPONENTS of the electric field.

Lorentz Transformations applied to the spacetime of Special Relativity do transform coordinates of spactime events. However, Lorentz Transformations can also be applied to the "tangent [vector] space" [also a Minkowski vector space] of an event, which transforms components of vectors and tensors based at that event.

The spatial vector fields you seek are components of an antisymmetric tensor $$F_{ab}$$.
Depending on sign conventions, given an observer with 4-velocity $$u^a$$, the electric field according to that observer is $$u^aF_{ab}$$ and the magnetic field according to that observer is $$u^a \frac{1}{2}\epsilon_{ab}{}^{cd}F_{cd}$$. Note that each field [co]vector is spacelike, in fact, purely-spatial [i.e. orthogonal] to $$u^a$$

Last edited: Dec 5, 2006
3. Dec 5, 2006

### Meir Achuz

The E and B fields are components of a second rank tensor in a Lorentz transformation. You just have to study a good book on SR.