Special relativity (SR) is valid only in Inertial frames. Lorentz transformations (LT) includes both rotations as well as boosts. But rotations gives accelerated frames so how can we connect these SR and LT? In other words, how does this "rotation" give Lorentz Invariance?
Your problem, fortunately, is ony one of ambiguous words. In the context of Lorentz transformations 'rotation' means a single (not dependent on time) transformation. From one frame (or equivalent system of bodies) to a corresponding frame (or equivalent system of bodies) that only differes by the direction of axes, in a specific way however that garantees that orthogonal axes remain orthogonal and that their skrew sense remains the same (in order to exclude reflections, that, for good reasons, are not considered rotations).
You are right that rotational motion (which can be represented by rotations which depend on time - curves in the space of rotations - can, according to relativity, only happen in limited parts of space since sufficiently far away from the rotation center the velocity of body-fixed points would exceed c.
Ulrich proposed correct answer. Passive LTs are those that help understand how observers in different inertial frames observe same events. It does not mean, however, that any particular frame or object is rotated or boosted.
On the other hand, Active LT is something that happens to charge in electromagnetic field. The charge is rotated around the direction of the magnetic field (and the magnitude of rotation during small time delta t is proportional to the magnitude of the magnetic field) and boosted along the direction of the electric field (the acceleration is proportional to the magnitude of electric field). Due to special property of LTs - namely, conservation of the squared 4-momentum - electromagnetic field never affect the mass of the object. That is why the trace of EM field energy-momentum tensor is always zero, and photons are massless particles.
@ Murod Abdukhakimov & Ulrich Mutze: Would you explain me in a simpler way?
Maybe this would be helpful:
http://en.wikipedia.org/wiki/Inertial_frame
SR deals just with Galilean systems, those moving relative to each other at a constant speed with no rotation.
The transformation rules between these systems follow the Lorentz equations.
If the frames "are rotated at fixed angles" the Lorentz transformation will include this.
If the frames "are in rotation changing their angles" then SR and the Lorentz transforms will no longer hold valid and General Relativity dealing with accelerated systems must be used.
Take two identical houses. One has its front door facing north. The other's front door faces east. The two houses are related by a 90 degree rotation. It does not mean that either of the houses is spinning continuously. It simply means that to transform from one house to the other, you must execute a single 90-degree rotation once.
That said, it is a common misconception that special relativity cannot deal with accelerating (non-inertial) frames. This is not true! Take an accelerating object, for instance. At any point along its worldline, you can attach a tangent and affix an inertial frame to it. This is how special relativity can deal with things like the infamous "twin paradox": by attaching an inertial frame to the accelerating twin's worldline at every point and then integrating proper time along that worldline you can calculate exactly how much younger the traveling twin will be compared to his Earth-bound sibling. You don't need general relativity for this.
One possible reason for this misconception is that even in flat spacetime (which is what special relativity describes) one may use curved coordinates. These could be the coordinates of a rotating frame, an accelerating frame, or for that matter, something as mundane as a spherical coordinate system. The difference between special and general relativity is that in the presence of gravity, the use of curved coordinates is no longer optional. You _must_ use curved coordinates because the underlying spacetime is not flat anymore.
Yes, Lorentz Transformations apply to fixed rotated angles, even in the case of a rotation on the x1-x4 plane, which Sommerfeld treated in his "On the Composition of Velocities in RT" 1909 paper. But as I understood the original question of Prasanth deals with ROTATING angles, which in the x1-x4 plane would mean accelerated translation movement. How can we discribe that such an angle is changing with time, if time is already included in the coordinate axis with respect to which the angle rotates? The only way to handle this would be to reinstate Newtonian Time in M4 to be used as the requiered scalar parameter to make the angles open or close. This thema I explain in mathematical detail in the paper on "trajectories in M4" which you'll find under my publications.
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