Addition of ( Parallel ) Relativistic Velocities
"Any intelligent fool can make things bigger and more complex... It takes a touch of genius  and a lot of courage to move in the opposite direction"  Albert Einstein ( 1879  1955 )
§ The Problem :
At velocities approaching the speed of light, , mass  particles and other object bodies contract in the direction of motion as well as the fact that measurements of intervals of time dilate ( expand ) as seen by an outside ( relatively ) stationary observer. In fact, the speed of light itself determines the very upper limit of velocity at which any object body or mass  particle can attain because otherwise the frame of reference of such an object body ( or mass  particle ) would "outrun" any light propagation from itself. And this occurrence would thus violate the Lorentz Transformation Equations upon which all else has been derived in our discussion of special relativity by giving imaginary number results as can be seen directly from the Lorentz equations themselves:
However for velocities simply approaching the speed of light, nevertheless no simple Galilean addition of velocities of two or more frames of references of bodies will suffice because of [ physical body ] length contraction and time dilation effects. Remember that
and hence any distortions in either a body's distance in physical length or its time of travel due to relativistic effects of motion will have be accounted for, so to speak, by a "correction factor" which we will soon see.
How to solve this problem of Addition of Relativistic Velocities will now become the subject of the following discussion.
§ The Relative Motion of Frames of References of Moving Bodies:
§ Assume frames of reference for systems , , and with the following stipulations:
is relatively stationary
is moving away from with relative velocity
is moving away from with relative velocity .
§ We already know the following:
Lorentz transformation between and
and, hence, Lorentz transformation between
and
§ We therefore want to find the Lorentz Transformation Equations connecting system frame with :
In other words, any successive Lorentz transformations will be equivalent to one (1) Lorentz transformation and hence demonstrating the invariance of the Lorentz Transformation Equations as prescribed by Special Relativity [ see: The Relativity Postulates  The Principle of Relativity ].
Let's suppose that system is a space ship and that some body object residing inside is actually system .
Applying Lorentz transformations to the moving object inside space ship is
Our task is therefore to relate the position and time of the object inside the moving space ship ( system ) to a stationary observer ( system ) on the outside as follows:
Likewise, the y  displacement inside the space ship ,
becomes for the outside stationary observer in reference frame
And, the z  displacement inside space ship ,
gives
§ Now Notice Several Consequential Things:
1).
2).
that is, there is no Lorentz velocity or length contraction and everything goes over into classical Galilean Transformation Equations!
3a).
In other words, it is virtually impossible to combine several Lorentz transformations into one final transformed coordinate system where there will be a relative velocity greater than as long as in at least one inertial system no object body [ or mass  particle ] travels faster than . And, of course, it will always be possible to describe one inertial system in which a given body is traveling less than or equal to since The Principle of Relativity [ see: The Relativity Postulates ] is á priori always true.
3b).
3c). Let's try this again as follows:
4). since
gives an invariant Lorentz transformation for an object body [ or mass  particle ] in where the object body itself forms frame and traveling at velocity with respect to system which in turn is traveling at velocity with respect to system , and, hence, the object body in ( or ) is traveling at velocity with respect to ; and, therefore, we can finally write
§ In conclusion we can therefore state the following:
Relativistic Velocity Transformation Equations: A Summary
Relativistic Velocity Transformation Equations: An Example^{∗}
^{∗}note: this example is used in the future upcoming Relativity Science Calculator Mac application
But first,
§ Example 1).
Let be the velocity vector of a rocket as observed in relatively stationary traveling at, say, 0.6 with theta angle of, for example, 30°. Also, let move away from stationary at = 0.1.
We now calculate the velocity and angle of the rocket as observed in :
§ Example 2).
Let be the speed of light with a directed beam having theta angle of 30^{°}. Also, let move away from stationary at the same = 0.1. In other words, everything is the same as above but we now substitute the speed of light for the rocket and its velocity.
As before, we now calculate the speed of light and angle of the entering light as observed in .
Remember that since light is diffuse throughout a given volume of pure vacuum space devoid of matter, it does not possess a unique direction and therefore light only possesses speed and is not a vector. But in our situation, we are prescribing a directed beam such as a laser.
So, as far as answering the question as to the angle of entry of our directed light beam into system coming from stationary system , the answer to this is the same as in above for Example 1, namely
A Philosophic Question to Ponder
Aside from the philosophic questions of time and mass dilations arising from relativistic addition of velocities, we also observe from the above examples that there exists a "pseudo  rotation" in spacetime geometry of the velocities observed in as compared to relatively stationary . For this and other geometries being shown here, please go to Minkowski's "Light Cone" wherein it's also shown that we humans live in a "45^{°} physical reality of knowing" beyond which there is cosmological Elsewhere.
In other words, our human perceptions of external reality can tell us only a limited amount of truth regarding the external world. At a deeper level of understanding, therefore, it is only by means of philosophic and mathematical inquiry that an observer in and can relate their "experiences" to each other beyond what their respective naïve perceptions will tell them. To the observer in there is one sort of angle of velocity motion but to the other observer in there is another, but different, angle of velocity motion; whereas, hence, neither would realize that their respective angle of velocity motion differs from the other except by deeper philosophic and mathematical inquiry.
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