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I'm trying to find some intuition in Lorentz transformations. I understand that they are basically rotations by imaginary angle of vector of the form $\{ict,x\}$ (for $1+1$ space-time dimensions), and the result transformation is nothing other than hyperbolic rotation.

But unlike circular rotations, hyperbolic rotations are a bit weird, and I can't seem to find any everyday life example of such a type of rotation.

What could be an example of hyperbolic rotation from everyday life?

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The terminology "hyperbolic rotation" is somewhat ambiguous. Let me try to explain how a geometer thinks of this (which is probably different than how a physicist thinks of it).

In $n+1$ space time, Lorentz transformations can be understood geometrically as transformations of the subspace $t^2 - x_1^2 - \cdots - x_n^2 = 1$. The infinitesmal Lorentz metric, when restricted to that subspace, is isometric to the $n$-dimensional hyperbolic space $\mathbb{H}^n$; this is known as the hyperboloid model of $\mathbb{H}^n$.

The restriction of any Lorentz transformation is an isometry of $\mathbb{H}^n$, and any isometry of $\mathbb{H}^n$ is the restriction of a Lorentz transformation. So your question requires understanding the isometries of hyperbolic space.

The subgroup of isometries of $\mathbb{H}^n$ that fixes a point is actually isomorphic to the subgroup of isometries of Euclidean space $\mathbb{E}^n$ that fixes a point. To a geometer, these are what one ordinarily thinks of as "hyperbolic rotations". Infinitesmally, there is not difference between the two cases.

Regarding the example in your question, to a geometer this example is not a "hyperbolic rotation" but is instead a "hyperbolic translation". In the $1+1$ dimensional example you are talking about, $\mathbb{H}^1$ is just a line, and the action on this line is a translation, displacing all points on that line the same distance in the same direction. In $n+1$ dimensions, a "translation" of $\mathbb{H}^n$ has a unique line that is preserved, and it displaces points along that line the same distance and the same direction; other points, that do not lie on that line, will be moved a greater distance. This is unlike the situation of a translation of $\mathbb{E}^n$, in which all points are displaced in the same direciton and by the same distance.

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