What are inertial systems of reference? Why are transformations between inertial systems of reference linear? A simple analysis of transformations between systems of reference.
We have two systems of reference, denoted by and . Each system has his own co-ordinates; where is the space-time co-ordinate for and is the space-time co-ordinate for . The general transformation from to can be written as
where , , and are unknown functions. It is convenient to use other symbols. We write , , and . The general transformation can then be written as
A special path is known as . We call a system an inertial frame of reference, IF a free moving body will follow the path defined by . Now
(3) , thus
Now IF and are both inertial systems of reference, then and , therefore we obtain
but if this is true for arbitrary values and , then we obtain
The transformation between two inertial systems of reference satisfies . Then it is clear that transformations between two inertial systems of reference are linear transformations. As , and , thus . It is convenient to set , then we can write . Transformations between inertial systems of reference are linear transformations.