Relativity Physics and Science Calculator - Addition of Relativistic Velocities .
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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, addition of ( parallel ) relativistic velocities , 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:

addition of ( parallel ) relativistic velocities

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 

addition of relativistic velocities

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:

addition of relativistic velocities

§ Assume frames of reference for systems  proper time,  mach's conjecture, and  speed of light with the following stipulations:

the calculus  is relatively stationary

kepler's 3rd law is moving away from  galileo-newton relativity  with relative velocity  kepler's 2nd law

kepler's 1st law is moving away from  early models of the universe with relative velocity  the conflict of religion and science.

§ We already know the following:

Lorentz transformation between  definition of parallax  and  earth-sun distance

Lorentz-FitZGerald Transformation Equations

and, hence, Lorentz transformation between

michelson-morley experiment and edward morley

Lorentz Equations

§ We therefore want to find the Lorentz Transformation Equations connecting system frame  1851 fizeau water experiment with  hippolyte fizeau:

augustin jean fresnel

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 Physics Postulates - The Principle of Relativity ].

§ The Proof:

Let's suppose that system star aberration is a space ship and that some body object residing inside  relativistic photon rocket is actually system  rocket equations.

Applying Lorentz transformations to the moving object inside space ship  konstantin tsiolkovsky-rocket man is 

wright brothers

Our task is therefore to relate the position and time of the object inside the moving space ship ( system  orville and wilbur wright ) to a stationary  observer ( system  doppler: sound and time dilation ) on the outside as follows:

relativistic photon space ship velocities

Likewise, the y - displacement inside the space ship  relativity of time,

philosophical relativity

becomes for the outside stationary observer in reference frame  hermann bondi

hermann minkowski

And, the z - displacement inside space ship  sphere of light,

pseudo euclidean geometry


max fleischer's einstein's theory of relativity silent film

§ Now Notice Several Consequential Things:


law of energy conservation


Galilean relativity

that is, there is no Lorentz velocity or length contraction and everything goes over into classical Galilean Transformation Equations!

vector addition of velocities


nothing travels faster than c

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 universal constant speed of light as long as in at least one inertial system no object body  [ or mass - particle ] travels faster than addition of relativistic velocities. 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  special relativity  since The Principle of Relativity [ see: The Relativity Postulates ] is á priori always true.


principle of relativity

3c). Let's try this again as follows:

universal c speed of light

4). since

addition of relativistic velocities

gives an invariant Lorentz transformation for an object body [ or mass - particle ] in body forms frame of reference where the object body itself forms frame galilean-newton relativity and traveling at velocity  time as 4th dimension  with respect to system maxwell equations which in turn is traveling at velocity e=mc2 with respect to system stationary vs. moving clocks, and, hence, the object body in  orbital angular momentum ( or  4-vector analysis of bodies of matter ) is traveling at velocity   with respect to  impulse and momentum; and, therefore, we can finally write

inverse addition of velocities

§ In conclusion we can therefore state the following:

Relativistic Velocity Transformation Equations: A Summary

relativistic transformation equations

Relativistic Velocity Transformation Equations: An Example

note: this example is used in the future upcoming Relativity Science Calculator Mac application

But first,

de broglie's quantum equations

§ Example 1).

Let doppler be the velocity vector of a rocket as observed in relatively stationary spacetime fabric  traveling at, say, 0.6einstein's 1905 relativisitic star aberration formula with theta angle james clerk maxwell of, for example, 30°. Also, let  newton relativity move away from stationary  relativistic momentum and energy  at  stationary frame of reference = 0.1moving frame of reference.

We now calculate the velocity classical momentum and angle  relativistic momentum of the rocket as observed in  relativisitic mass:

proper mass

§ Example 2).

Let energy-momentum vector be the speed of light  rest energy  with a directed beam having theta angle mass rest energy of 30°. Also, let  nothing faster than c move away from stationary  nothing faster than speed of light  at the same  frames of reference = 0.1inertial frame of reference. 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  quanta of light energy and angle  massless photon of the entering light as observed in  rest frame.

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  relative moving frame coming from stationary system  flash of light energy, the answer to this is the same as in above for Example 1, namely

momentum quanta light photon

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 coordinate time as compared to relatively stationary proper time. 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 non-relativistic velocities and fundamental law of the inertia of energy can relate their "experiences" to each other beyond what their respective naïve perceptions will tell them. To the observer in total relativistic energy there is one sort of angle of velocity motion but to the other observer in angle of velocity of motion 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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