Lorentz Constant and Some Algebra
With the advent of the Michelson and Morley experiment in 1891, physics was elevated to a new scientific paradigm, new aspects of nature were revealed leading to what is now called modern physics.
The constant search for experiments and results that confirm hypotheses is one of the paths used by the sciences. And so we can say that science is like an ongoing process whose product is constantly being modernized.
In this supposed process there are some pieces that are sui-generis, as if it were a separate character. In the case of physics, we highlight the Lorentz constant, which we will algebraically describe below as a way of glimpsing this fragment of explicit knowledge, which is still not commonplace
The demonstration we want to show here requires the reader to be aware that there are three basic elements in this context:
- a stationary inertial frame of reference, S;
- an inertial frame of reference in motion with respect to the stationary frame, S′;
- a well-determined physical event.
The event is being monitored by observers, one in each reference system. The moving system has a constant velocity v.
Thus, from the origin of each system we can write, based on the justifications for the result of the Michelson and Morley experiment, respectively:
We will do the demo for the X axis.
From Galileo’s axis transformation formula, we will apply the multiplicative factor γ to it:
Thus, substituting this result in the previous corresponding equation, we have
or
Applying this methodology to the time variable (Lorentz transformation):
and so we have
which also substituting in the corresponding equation, will be
This expression will have a quadratic term of x
that must necessarily have a unitary coefficient to meet the equation initially written for the stationary system, that is:
Simplifying and equating the coefficient to one, we have
solving step by step
and behold
