This animation shows how the factor gamma, γ, would be determined by experiment. From the point of view of the observer in system A, system B is moving along the x axis at speed v. From the point of view of the observer in system B, system A is moving along the x axis at speed -v. Therefore it is natural to write:

$$\begin{array}{l}{x}_{A}=\gamma ({x}_{B}+v{t}_{B})\\ {x}_{B}=\gamma ({x}_{A}-v{t}_{A})\end{array}$$ |
(1.1) |

where t_{B} and t_{A}
are times measured in the A and B moving systems and γ is a factor independent of the xs and ts that
is to be determined. Equations 1 are the
same as Galilean transformations except for the factor γ. And the fact that t_{A} and t_{B}
may not be equal.

Thus x_{A}=0 when x_{B}=-vt_{B} and x_{B}=0
when x_{A}=vt_{A}. In
order to derive γ from these equations the
following experiment is performed. A
light pulse is allowed to propagate distance x_{A}=c*t_{A} in
system A and a similar light pulse is allowed to propagate a distance x_{B}=c*t_{B}
in system B. Of course, the speed of
propagation has to be c for both observer A and observer B whether the pulse observed
is in the same system as the observer or not. Then the equations above become:

$$\begin{array}{l}c{t}_{A}=\gamma {t}_{B}(c+v)\\ c{t}_{B}=\gamma {t}_{A}(c+v)\end{array}$$ |
(1.2) |

If we multiply the left sides and the
right sides of these equations we obtain a single equation:

$${c}^{2}{t}_{A}{t}_{B}={\gamma}^{2}{t}_{A}{t}_{B}({c}^{2}-{v}^{2})$$ |
(1.3) |

Dividing each side of this equation by t_{A}t_{B} we obtain:

$${c}^{2}={\gamma}^{2}({c}^{2}-{v}^{2})$$ |
(1.4) |

From which:

$$\gamma =\frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}$$ |
(1.5) |

Most importantly, notice that, if v is non-zero, γ cannot be 1 and therefore the simple Galilean
transformation which sets γ equal to 1 cannot be valid.