Components of a vector in Hyperbolic Geometry

Question

I was recently studying "The classical theory of field' by L.Landau. In the introduction the book introduced theory of relativity. (I know I should be asking such questions on physics stack exchange, but I wanted the mathematical rigor). he derived Lorentz transformations which are: $$ x'=\gamma (x+vt) $$ $$ y'=y $$ $$ z'=z $$ $$ t'=\gamma \left(t-\frac{vx}{c^2}\right) $$ For the cases where the two observes $x$, $y$ and $z$ Axis are parallel to each other and the motion is along $x$ axis only. Landau book derive Lorentz transformations with the help of components of the four vector in the hyperbolic space and arguing that the length of four vector is invariant under transformations in inertial frame. Then it says that: \begin{align} x'&=x\cosh(\psi)+ct \sinh(\psi) \\ ct'&=-x\sinh(\psi)+ct\cosh(\psi) \end{align} These transformations are only applicable for rotation of axes(of $tx$ plane). My question is how he came up to this? I want a more general approach regarding this derivation because the book just said that the formula is same as that of rotation,We only have to replace the trigonometric part by the hyperbolic part.

Answer 1

Starting with constancy of th speed of light $c$ the under a linear Lorentz boost with velocity $v$, the straight lines $$x = ct$$ of movement at speed of light remain constant as geometric lines. This is accomplished by the coordinate transformation $$ x' = \alpha x \pm \beta ct$$ $$ct' = \gamma x \pm \delta ct$$ of both coordinates $x, ct = c * t$

Then the lines $x=\pm ct$ transform to $$\frac{x'}{ct'} = \frac{\alpha ct \pm \beta ct}{\pm \gamma ct + \delta ct}= \frac{\alpha \pm \beta }{ \pm \gamma + \delta} =\pm 1 $$ yielding $$\frac{\alpha^2 - \beta ^2 }{- \gamma^2 + \delta^2} = 1 $$

So up to a common scaling factor, we can set $$\alpha = \delta =\text{cosh} u, \quad \beta =\delta=\text{sinh} u$$ because any two expressions with constant difference of two coordinates squared can be parametrized in the same way by hyperbolic functions as for the sum by parametrization of the trigonmetric function $$\text{sin}^2 \varphi + \text{cos}^2 \varphi = 1 ,-\pi\lt \varphi\le \pi , \qquad \text{cosh}^2 u - \text{sinh}^2 u = 1, -\infty \lt u \le \infty $$ and the classical velocity is the velocity of the point $$x'=0 \longrightarrow v= c *\text{tanh} u$$.

Conundrum solved: The relativistic velocity parameter $u$ is not the quotient of two coordinates $x/t$ (not a scalar quantity, not additive, limited by $\pm c$) but an uniform group parameter (uniform scalar, additive, unlimited) like the angle $\varphi$ on the unit circle. The circle is replaced by the unit hyperbolas $$ct^2 -x^2 =\pm 1$$, which are invariant curves under Lorentz transformations like the unit circle under rotations.

The classical Einstein parametrization by v in his clumsy notation is $$\cosh u = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}, \sinh u = \frac{\frac{v}{c}}{\sqrt{1-\frac{v^2}{c^2}}}, $$ that lost its impenetrable barrier of understanding in a world of classical Newtonian trajectories $$t\to \vec x(t)$$ evolving in 3-space with an universal celestial time parameter $t$, after Minkowskis creation of the space time continuum, only, after 1909 and the proper time as world line length parameter in space-time.