A.5 Galilean and special relativity (HL only)

Classical (Newtonian) mechanics assumes that time is absolute , a universal clock ticking at the same rate for everyone, everywhere. Einstein shattered this assumption in 1905 with his theory of special relativity, built on just two postulates:

1. The laws of physics are the same in all inertial (non-accelerating) reference frames.
2. The speed of light in a vacuum, $c\approx 3\times 10^8{\text{ m s}}^{-1}$, is the same for all inertial observers, regardless of the motion of the source or observer.

The second postulate is the truly radical one. In Newtonian mechanics, velocities add simply: a ball thrown forward on a moving train has its speed added to the train's speed. Light does not behave this way. This constancy of $c$ forces us to abandon the idea that all observers measure time and space the same way.

Before Einstein, the Galilean transformation described how to convert measurements between two frames. If frame $S^{\prime }$ moves at velocity $v$ along the $x$-axis relative to frame $S$, the Galilean transformation gives:

$x^{\prime }=x-vt,y^{\prime }=y,z^{\prime }=z,t^{\prime }=t$

Notice that time is the same in both frames: $t^{\prime }=t$. This embeds the assumption of absolute time.

The Lorentz transformation replaces Galilean transformation when speeds are comparable to $c$:

$x^{\prime }=\gamma (x-vt),t^{\prime }=\gamma \left(t-\frac{vx}{c^2}\right)$

where the Lorentz factor is:

$\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

Lorentz Factor ($\gamma$): A dimensionless quantity $\gamma =\frac{1}{\sqrt{1-v^2/c^2}}$ that appears in all relativistic transformations. It equals 1 when $v=0$ and increases without bound as $v\to c$. It is always $\ge 1$.