Velocity-addition formula

In classical physics velocities are added vectorially. As in the special theory of relativity against each other moving inertial frames are related to each other by Lorentz transformations, here are two speeds are different than in classical physics together with the total speed.

Definition

An observer is moving towards the observer with the velocity in the direction of the axis. For the observer, a body moving with velocity u ' Then this body for the observer has the velocity u with the components

With

• The speed of light and
• The Lorentz factor

Interpretation

If the speeds are involved very small compared to the speed of light

As the denominator (and the term under the square root in the numerator ) hardly differs from one

And it results in a good approximation, the usual non-relativistic velocity addition:

Example: in a moving train with a person runs with respect to the train in the direction of travel. The measured by a stationary observer on the embankment speed of the person is just around 0.17 nm / h slower than those obtained with simple addition. For comparison, the diameter of an atom is of the order of 0.1 nm That is, the " Zugläufer " comes less far than one would expect from nonrelativistic calculation in the hour nearly two atomic diameter - which, at a distance traveled of 205 mi sure is negligible.

For velocities near the speed of light, however, there are clear deviations from the non-relativistic addition rule, see the following examples.

Conclusions

As a result of the addition can also by superimposing two speeds not to exceed the speed of light.

Example 1

It should be

Then

And not about 1.5 c.

Example 2

Is the speed U ' of the observer is equal to the speed of light, then it is also the observer

Is, for example,

Then

Thus in particular

Derivation

To keep the formula image, we use as unit of length the distance that light traveled in one second, and call it a light second. Then, time and length have the same units and the dimensionless speed of light is investigations in other systems of units do not bring deeper insights.

From the inverse Lorentz transformation (replacement of by - )

Follows that the transformation is linear, the differential

Thus it follows for the speeds which determines the observer,

Conversely ( replacing by -, with all factors )