Relativity Calculator - Addition of Relativistic Velocities

Addition of Relativistic Velocities

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 measurement of time dilates ( contracts ) 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 would thus violate the Lorentz Transformation Equations upon which all has been derived up to this point in our discussion of Special Relativity by giving imaginary number results as can be viewed 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 distance contraction and time dilation effects. Remember that

Pasted Graphic 4.pict

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 form the following text.

The Relative Motion of Frames of References of Moving Bodies:

addition of velocities.png

§ Assume frames of reference for systems S, S', and S'' with the following stipulations:

S is relatively stationary

S′ is moving away from S with relative velocity ν

S″ is moving away from S′ with relative velocity ω.

§ We already know the following:

Lorentz transformation between S and S′

Lorentz Equations in S and S'.png

and hence

Lorentz transformation between S′ and S″

Lorentz Equations in S' and S''.png .

§ We therefore want to find the Lorentz Transformation Equations connecting system frame S″ with S:

relating S and S''.png

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 ].

The Proof:

Let's suppose that system S′ is a space ship and that some body object inside S′ is actually system S″.

Applying Lorentz transformations to the moving object inside space ship S′ is

Pasted Graphic 1.pict .

Our task is therefore to relate the position and time of the object inside the moving space ship ( system S′ ) to a stationary observer ( system S ) on the outside as follows:

space ship velocities 1.png

Likewise, the y-displacement inside the space ship S′,

becomes for the outside stationary observer in reference frame S

And, the z-displacement inside space ship S′,

Pasted Graphic 3.pict

gives

z-direction velocity.png

Now Notice Several Consequential Things:

1).

consequence 1.png

2).

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

vector addition of velocities.png

3a).

nothing faster than c.png

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 c as long as in at least one inertial system no object body [ or mass-particle ] travels faster than c . 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 c since The Principle of Relativity [ see: The Relativity Postulates ] is á priori always true.

3b).

universal c.png

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

universal c.png

4). since

addition of relativistic velocities

gives an invariant Lorentz transformation for an object body [ or mass-particle ] in S'' where the object body itself forms frame S'' and traveling at velocity ω with respect to system S' which is in turn traveling at velocity ν with respect to system S, and, hence, the object body in S'' ( or S' ) is traveling at velocity u with respect to S, therefore we can write

inverse addition of velocities.png

§ In conclusion we therefore state the following:

• every material body is itself an inertial frame of reference
• every material body will have a velocity less than or equal to c in every inertial system if its velocity in any one inertial system is also less than or equal to c

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