# What are the eigenvalues of the Lorentz matrix?

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?)

• I recommend you try to answer and interpret the third one for yourself!
– user12029
Apr 15 '14 at 20:14
• Indeed, you can work it out for yourself. Remember to find the eigenvalues just solve $\mathrm{det} (A -\lambda \mathbb{I})=0$. Apr 15 '14 at 20:54
• what about the other 2 questions? Also I know the eigenvalues of the matrix corresponding to a boost in the z direction are the Doppler shifted frequencies or sth like that, but I wasn't sure how to interpret them physiscally. Also I was asking about the eigenvalues of the general (boost in a random direction) matrix Apr 18 '14 at 10:37
• If I remember correctly, the eigenvalues are simply $\exp(\pm \phi)$, where $\phi=\mathrm{arctanh} (v^2 / c^2)$, commonly referred to as the rapidity. It is a hyperbolic angle which allows one to interpret a Lorentz transformation as a rotation. Apr 18 '14 at 11:08

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.

• thanks; 1) so is there a matrix that would allow us to transform quantities between non-inertial frames? Relativistically I mean. 2) Does it make sense that the eigenvalues are only the photons travelling (anti)parallel to the boost? I mean isn't the speed of light the same no matter the direction of the boost? Apr 20 '14 at 21:40
• (1) Special relativity is a theory of inertial reference frames. Fully treating accelerated motion is done better in general relativity. (2) Photons traveling at some angle $\theta$ to the boost direction make a different angle $\theta'$ with that axis in the boosted frame. This is called "relativistic beaming." Since their direction changes, the paths of these photons are not eigenvectors of the boost.
– rob
Apr 20 '14 at 21:45
• @SuperCiocia You can have a transformation that allows you to transform quantities between non-inertial frames, but it wouldn't be a matrix (at least not globally), because matrices are representations of linear transformations, which means non-curvy worldlines. You can handle acceleration in special relativity, just not with linear algebra. May 25 '18 at 12:30

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.

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.