What are the eigenvalues of the Lorentz matrix?

Question

The Lorentz matrix defines the transformation of a four-vector between different frames of reference, such that $$ p^{'\mu} = \Lambda^{\mu}_{\ \ \nu}p^{\nu} $$ where in this example $p^{\mu}$ is the four-momentum.

1) Are Lorentz transformations of this form only valid for constant (not changing in magnitude) velocities?

I guess so, since $\gamma$ is a function of $v^2$. How can we transform between accelerating frames?

2) Is Lorentz invariance a law of nature?

Which physical quantities should we expect to be invariant (forces? charge?)?

3) What are the eigenvectors and the eigenvalues of the general Lorentz matrix?

I mean what is their physical significance? They do not change under Lorentz transformations?

(I know the ones for the boost in the z direction are something like the Doppler shifted frequencies, but what does this mean? They are the same in all frames? What about the eigenvalues for the boost in a random directiom matrix?)

Answer 1

1. A Lorentz transformation lets you compute an object's properties in one inertial frame, given its properties in another inertial frame. Inertial frames, by definition, do not accelerate. An accelerating object is always instantaneously at rest in some inertial frame.

2. Whether such-and-such is a law of nature is an experimental question. We have no evidence that Lorentz invariance is broken, but people are looking. You might look at the participants in this conference to get an idea of the field.

3. The most general Lorentz matrix is a product of three rotations and three boosts. For pure rotations we may always choose our coordinate system so that the Lorentz matrix has the form $$\left(\array{ 1\\ &1\\ &&\cos\theta & -\sin\theta \\ &&\sin\theta & \cos\theta \\ }\right).$$ A timelike vector, or a vector along the rotation axis, has eigenvalue 1, since they are not affected by rotations. In the plane of rotation the eigenvectors are $(1,\pm i)$; all real vectors in the plane of rotation get rotated. The corresponding eigenvalues are $e^{±i\theta}$.
    
   Similarly, we may always choose our axes so that a boost is written $$ \left(\array{ \gamma & -\gamma\beta \\ -\gamma\beta & \gamma \\&&1\\&&&1}\right) .$$ Some algebra shows that the non-unity eigenvalues of this matrix are $$ \gamma (1\pm\beta) = \sqrt\frac{1\pm\beta}{1\mp\beta} $$ which is, as you say, the relativistic Doppler shift between an observer at rest and an emitter in the boosted frame. You can verify by hand that the corresponding eigenvectors are $(1,\pm1,0,0)$. These are the light-like worldlines on a Minkowski diagram: the paths taken by photons which would be found later to have the associated Doppler shifts.
    
   A boost in a random direction would have the same four eigenvalues: $\gamma(1\pm\beta)$ for light-like vectors parallel and antiparallel to the boost, and unity for vectors in the spacelike plane perpendicular to the boost.

Answer 2

Concerning 3) What are the eigenvectors and the eigenvalues of the general Lorentz matrix?

This is a good question, which I’m just currently researching. In general, a composition of rotation and Lorentz boost has at least two eigenvectors. With a limited rotation angle, this composition has four eigenvectors. In a particular case, the composition of two Lorentz boosts has a quartet of eigenvectors. As for the eigenvalues, it’s fair that they are mutually inverse.

Answer 3

1. You may define homogeneous Lorentz transformations of vectors in spacetime between any two frames, accelerated or rotating. However the transformations will depend on where the vector you are transforming is applied in spacetime. Coordinate transformations for accelerated observers, instead, are non linear.

2. So far, Lorentz invariance seems to be a fundamental symmetry of Nature. But it is good to keep the door a little open.

3. Generally, two of the eigenvalues of a proper orthochronous Lorentz transformation are real, positive, and reciprocal $\exp[\pm \Theta]$, while the other two are complex conjugated phases $\exp[\pm i \Phi]$. The corresponding eigenvectors are all light-like. For special transformations, as boosts and rotations, some eigenvalues may be +1 or -1. For some very special transformations, all the eigenvalues are +1, but the matrix is not the identity, it is defective. Only if some eigenvalues are +1 or -1 the eigenvectors may not be light-like. The same goes for proper antichronous transformations, but the real eigenvalues get negative $-\exp[\pm \Theta]$. For improper transformations, an eigenvalue is -1, another is +1, and the remaining two eigenvalues are complex conjugated phases $\exp[\pm i \Phi]$ or. reciprocal real, positive $\exp[\pm \Theta]$ for orthochronous transformations and negative $-\exp[\pm \Theta]$ for antichronous ones.