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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 ?

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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.

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    @edenster can you give me the simple example for transformation – 2017-01-21
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    Let matrix in this question gives a transformation for velocity in the x-direction. http://physics.stackexchange.com/questions/30166/what-is-a-lorentz-boost-and-how-to-calculate-it where v is the velocity of your reference frame. To transform your coordinates simply multiply your 4d vector by this matrix. – 2017-01-21
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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.

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    how can I relate my Cartesian coordinates with your coordinates and is there a general 4X4 lorentz transformation matrix – 2017-01-21
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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.

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    what does d stands in your answer – 2017-01-21
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    The $d$ stands for a "differential". The "coordinate differential" $dx_{i}$ is shorthand for the function that returns the $i$th component of a vector, e.g., if $v = (v_1, v_2, v_3, v_4)$ is a vector, then $dx_{3}(v) = v_3$. The two "metrics" I wrote are "fields of inner products"; if $v$ and $w$ are vectors, then$$g(v, w) = v_1 w_1 + v_2 w_2 + v_3 w_3 + v_4 w_4$$in the Euclidean case, and$$g(v, w) = -v_1 w_1 + v_2 w_2 + v_3 w_3 + v_4 w_4$$in the Lorentzian case. – 2017-01-21
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    Incidentally, Daniel Schepler's (+1) choice of signs is conventional in relativity. Mathematically, any "isometry" of the Lorentz inner product as I've written it is also an isometry of the Lorentz inner product with the opposite sign. – 2017-01-21
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    I was looking for a generalized 4x4 lorentz transformation matrix so that I can multiply my Cartesian coordinates with the matrix – 2017-01-22
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    The Lorentz group is six-dimensional; if we call the coordinates $(x, y, z, t)$, then rotations in the $(x, y)$-plane, in the $(x, z)$-plane, and in the $(y, z)$-plane, and the boosts in the $(x, t)$-plane, in the $(y, t)$-plane, and in the $(z, t)$-plane generate. I believe it's possible to write out a general Lorentz transformation as an explicit $4 \times 4$ matrix whose entries are functions of six variables, but is that really what you want/need? It may be sufficient just to write $A = [a_{ij}]$. – 2017-01-22
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    I am looking for 4 x 4 general lorentz transformation matrix such that the below distance formula is invariant even after the various lorentz transformation. http://math.stackexchange.com/questions/1368443/whats-the-right-way-to-calculate-hyperbolic-distance-on-the-hyperboloid-model?rq=1 – 2017-01-22