I'm asking this question because I'm feeling a bit confused about how Lorentz transformations relate to the electromagnetic tensor, and hope someone can help me clear out my possible misunderstandings. Please excuse me if the answer is obvious.
In special relativity, the EM field is described by the tensor
$$F^{\mu\nu} = \begin{pmatrix}0 & -E_{x} & -E_{y} & -E_{z}\\ E_{x} & 0 & -B_{z} & B_{y}\\ E_{y} & B_{z} & 0 & -B_{x}\\ E_{z} & -B_{y} & B_{x} & 0 \end{pmatrix}$$
which is an anti-symmetric matrix. Then, recalling the one-to-one correspondence between skew-symmetric matrices and orthogonal matrices established by Cayley’s transformation, one could view this tensor as an infinitesimal rotation matrix, that is, a generator of 4-dim pseudo-rotations. This seems at first natural: given that space-time 4-velocities and 4-momenta for a fixed mass particle have fixed 4-vector norms, all forces (including EM) and accelerations on the particle will be Lorentz transformations. However, this page is the unique reference I've found which states such relationship (and I don't fully understand the discussion which follows, which I find somewhat disconcerting).
- Is this line of reasoning correct?
On the other hand, according to Wikipedia, a general Lorentz transformation can be written as an exponential,
$$\mathbf \Lambda(\mathbf ζ,\mathbf θ) = e^{-\mathbf ζ \cdot \mathbf K + \mathbf θ \cdot \mathbf J}$$
where (I'm quoting) $\mathbf J$ are the rotation generators which correspond to angular momentum, $\mathbf K$ are the boost generators which correspond to the motion of the system in spacetime, and the axis-angle vector $\mathbf θ$ and rapidity vector $\mathbf ζ$ are altogether six continuous variables which make up the group parameters in this particular representation (here the group is the Lie group $SO^+(3,1)$). Then, the generator for a general Lorentz transformation can be written as $$-\mathbf ζ \cdot \mathbf K + \mathbf θ \cdot \mathbf J = -ζ_xK_x - ζ_yK_y - ζ_zK_z + θ_xJ_x + θ_yJ_y +θ_zJ_z = \begin{pmatrix}0&-\zeta_x&-\zeta_y&-\zeta_z\\ \zeta_x&0&-\theta_z&\theta_y\\ \zeta_y&\theta_z&0&-\theta_x\\ \zeta_z&-\theta_y&\theta_x&0\end{pmatrix}.$$
- How does this matrix relate with the EM tensor? By comparison between the two matrices, it would appear that the components of the electric and magnetic field ($\mathbf E$ and $\mathbf B$) should be linked, respectively, with $\mathbf ζ$ and $\mathbf θ$. I'm missing what the physical interpretation of this would be.
Answer
Physically, the only thing that the electromagnetic field tensor and a Lorentz transformation generator have in common is that they both happen to be antisymmetric rank 2 tensors. The link doesn't go any farther than that.
However, this coincidence does lead to a few analogies. For example, if you know about Lorentz transformations, then you know that an antisymmetric rank 2 tensor contains two three-vectors inside it, namely $\boldsymbol{\zeta}$ and $\mathbf{K}$. Then if somebody tells you the electromagnetic field is the same kind of tensor, you'll automatically know that it can be broken down into two three-vectors, namely the electric and magnetic fields. But this is a purely mathematical analogy.
A more physical result comes from the equation of motion $$\frac{d u_\mu}{d\tau} = (q/m) F_{\mu\nu} u^\nu.$$ where $u^\mu$ is the four-velocity; you can expand this in components to verify it's just the Lorentz force law. Now, comparing this with an infinitesimal (active) Lorentz transformation $$\Delta u_\mu = \Lambda_{\mu\nu} u^\nu$$ we see that the Lorentz force is equivalent to an active Lorentz transformation acting on the four-velocity, with generator $(q/m) F_{\mu\nu}$.
We can do some quick sanity checks:
- Magnetic fields cause rotations. If we start with a nonzero three-velocity and apply a magnetic field, the velocity spins around.
- Electric fields cause boosts. If we apply an electric field, the three-velocity grows in the direction of the field, just like it does in the direction of a boost.
Two caveats to this result:
- As stated in the link you gave, this result doesn't allow us to think of electromagnetism as a geometric phenomenon, because different particles have different values of $q/m$ and hence are acted on by different Lorentz transformations. It's just a nice heuristic.
- Be careful to distinguish between active and passive Lorentz transformations. Most of the ones you'll run into are passive (i.e. used to switch between coordinate systems), but as ACM points out, such transformations are described by matrices, not tensors. Above, I'm considering active rotations and boosts, and everything is taking place in a single coordinate system.
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