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