How to perform lorentz transformation on Cartesian coordinates $x= [x_1,x_2,x_3,x_4]$

Question

Can any one explain me what is lorentz transformation and how is this different from orthogonal transformation. How to perform lorentz transformation on cartesian co-ordinates ?

Answer 1

In the Cartesian plane with coordinates $(x, y)$, and with $\theta$ denoting a real number, a rotation has the form $$ R_{\theta}\left[\begin{array}{@{}c@{}} x \\ y \\ \end{array}\right] = \left[\begin{array}{@{}rr@{}} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{array}\right]\left[\begin{array}{@{}c@{}} x \\ y \\ \end{array}\right] = \left[\begin{array}{@{}c@{}} x\cos\theta - y\sin\theta \\ x\sin\theta + y\cos\theta \\ \end{array}\right], $$ while a boost has the form $$ B_{\theta}\left[\begin{array}{@{}c@{}} x \\ y \\ \end{array}\right] = \left[\begin{array}{@{}rr@{}} \cosh\theta & \sinh\theta \\ \sinh\theta & \cosh\theta \\ \end{array}\right]\left[\begin{array}{@{}c@{}} x \\ y \\ \end{array}\right] = \left[\begin{array}{@{}c@{}} x\cosh\theta + y\sinh\theta \\ x\sinh\theta + y\cosh\theta \\ \end{array}\right]. $$ A rotation preserves the Euclidean metric $dx^{2} + dy^{2}$, while a boost preserves the Lorentz metric $-dx^{2} + dy^{2}$.

In four-dimensional space, analogously, a rotation preserves the Euclidean metric $$ dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2} + dx_{4}^{2}, $$ while (modulo the choice of timelike coordinate) a Lorentz transformation preserves the metric $$ -dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2} + dx_{4}^{2}. $$ Writing down a general rotation or Lorentz transformation is not as pleasant as in the plane, but the situation in the plane conveys the general flavor of the distinction.

Answer 2

Special Relativity says that when objects are in motion space gets stretched and time gets shortened (from what it usually would be if it was at rest). Given four coordinates, a transformation (or a linear map) between any other 4 coordinates can be thought of as a 4x4 matrix. Given a speed that your reference frame is moving (with respect to your rest frame) and a direction that your frame is moving in, the Lorentz transformation is a 4x4 matrix which you can multiply your coordinates by to get the elongated or shortened coordinates viewed from a rest frame.

An orthogonal map is a matrix that just rotates the coordinates but does not change anything else unlike a Lorentz transformation because Lorentz transformations change the size of an object in the direction of motion more than in other directions.

Answer 3

An example of a Lorentz transformation for two inertial frames with relative velocity $v$ in the $x$ direction is: \begin{equation*} \left[ \begin{matrix} x' \\ y' \\ z' \\ ct' \\ \end{matrix} \right] = \left[ \begin{matrix} \cosh\theta & 0 & 0 & \sinh\theta \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \sinh\theta & 0 & 0 & \cosh\theta \end{matrix} \right] \left[ \begin{matrix} x \\ y \\ z \\ ct \end{matrix} \right] \end{equation*} where $\theta = \tanh^{-1}(v/c)$.

Superficially, this looks almost like a rotation matrix, except that it uses hyperbolic trig functions instead of normal trig functions. From that analogy, you might also be led to rediscover the invariance of $c^2 t^2 - x^2 - y^2 - z^2$ under Lorentz transformations - as opposed to the invariance of $x_1^2 + x_2^2 + x_3^2 + x_4^2$ under orthogonal transformations.