How does the eigenvector interpretation of Lorentz transformations generalise to higher dimensions?

Question

When I came across this YouTube video, I immediately realised how Lorentz transform corresponds to a linear transformation where the vectors representing the speed of light ($\vec{c}$ and $-\vec{c}$) are in the only directions that do not change, thus being eigenvectors. After working out the math for a 2D spacetime

$$ L\vec{v} = \beta \hat{t}$$ $$ L\hat{t} = \gamma (-\vec{v})$$ $$ L\vec{c} = \lambda_1 \vec{c}$$ $$ L(-\vec{c}) = \lambda_2 (-\vec{c})$$

I reached the Lorentz matrix

$$ L = \gamma \begin{bmatrix}1&-v\\-v/c^2&1\end{bmatrix}$$

where $ \gamma $ is the Lorentz factor. I later realised the eigenvalues must be related to the relativistic doppler effect, as one of the values was $ \lambda = \sqrt{\frac{1-\beta}{1+\beta}}$. From my limited understanding, there seems to be a clear correlation between these linear algebra properties and the consequences of Lorentz transformations; please correct me if my interpretation is wrong.

How does the eigenvector interpretation generalise to higher dimensions? For example, if we have two dimensions (space and time), we may find two linearly independent eigenvectors whose direction (speed) does not change, $\vec{c}$ and $-\vec{c}$. So far so good. However, if we add another spatial dimension, introducing the famous light cones, what would the independent eigenvectors be? As far as I know, an $n$-dimensional space can only have $n$ linearly-independent eigenvectors (in this case $n=3$), which does not seem to make possible to represent the invariant light cones.

Answer 1

In the following I will set $c=1$. A 2D boost is given by the matrix: $$ \Lambda = \gamma\begin{pmatrix}1 & -v\\-v & 1\end{pmatrix} \\ \gamma = \frac1{\sqrt{1-v^2}} $$ Indeed, the eigenvectors are light like, with eigenvalues given by the Doppler shift: $$ \Lambda \begin{pmatrix}1\\\pm1\end{pmatrix}= \lambda_\pm\begin{pmatrix}1\\\pm1\end{pmatrix}\\ \lambda_\pm = \sqrt{\frac{1\mp v}{1\pm v}} $$ Actually, formulas are cleaner in terms of rapidity: $$ v = \tanh w \\ \Lambda = \begin{pmatrix}\cosh w & -\sinh w\\-\sinh w & \cosh w\end{pmatrix} \\ \lambda_\pm = e^{\pm w} $$ The link with Doppler shift is no coincidence. For a wave, the wave 4-vector: $$ K=\begin{pmatrix}\omega\\k\end{pmatrix} $$ also changes according to the Lorentz transformations. For light, from the dispersion relation: $$ \omega^2=k^2 $$ $K$ is light like (along the light cone) so gets multiplied by $\lambda_\pm$ under a boost.

By linearity, the generators lightcones are mapped to generators of the lightcone. However, in higher dimension, the lightcone is only globally preserved by a boost: the individual generators are typically sent to new generators. The notable exception are the ones along the directions of eigenvectors of the boost. In 1+2D, this gives only four invariant generators.

Physically, this is simply relativistic aberration. You typically visualize it by looking at how the celestial sphere is transformed. In 1+2D, you have a celestial circle, that you can parametric by the single azimuth $\phi$: $$ \begin{pmatrix}\cosh w & -\sinh w & 0 \\-\sinh w & \cosh w & 0\\ 0 & 0 & 1\end{pmatrix}\begin{pmatrix}1 \\ \cos\phi \\ \sin\phi \end{pmatrix} = (\cosh w-\sinh w\cos\phi)\begin{pmatrix}1 \\ \frac{-\sinh w+\cosh w\cos\phi}{\cosh w-\sinh w\cos\phi} \\ \frac{\sin\phi}{\cosh w-\sinh w\cos\phi} \end{pmatrix} $$ Qualitatively, you observe a bunching up in the direction of the boost and the four fixed points at $\phi=0,\pm\pi/2,\pi$. The prefactor will give you the additional Doppler shift.