Introduction to Special Relativity

Introduction to Special Relativity
Key concepts, historical background, and fundamental results

1. Historical Context

2. Fundamental Postulates

1. Principle of Relativity

   • The laws of physics are identical in all inertial (non‑accelerating) reference frames.
   • No inertial frame is physically privileged.

2. Constancy of the Speed of Light

   • In vacuum, light propagates at a constant speed c = 299 792 458 m s⁻¹, independent of the motion of the source or the observer.
   • This speed is the same in every inertial frame.

3. Core Concepts

3.1. Inertial Frames & Coordinates

• An inertial frame is one in which a free particle moves with constant velocity (Newton’s first law holds).
• Coordinates are usually denoted ((t, x, y, z)) in one frame (S) and ((t', x', y', z')) in another frame (S′) moving at velocity v relative to S.

3.2. Lorentz Transformation

For motion along the x‑axis:

[ \begin{aligned} t' &= \gamma \left(t - \frac{v x}{c^{2}}\right) \ x' &= \gamma \left(x - v t\right) \ y' &= y \ z' &= z \end{aligned} ]

where

[ \gamma = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} \quad (\text{Lorentz factor}) ]

• Reduces to the Galilean transformation when (v \ll c) ((\gamma \approx 1)).
• Guarantees that the spacetime interval remains invariant.

3.3. Spacetime Interval (Invariant)

[ \Delta s^{2}=c^{2}\Delta t^{2}-\Delta x^{2}-\Delta y^{2}-\Delta z^{2} ]

• Timelike ((\Delta s^{2}>0)): events can be causally connected; a frame exists where they occur at the same spatial point.
• Spacelike ((\Delta s^{2}<0)): no causal influence; there is a frame where they are simultaneous.
• Lightlike ((\Delta s^{2}=0)): separation exactly on the light cone; describes the path of a photon.

3.4. Time Dilation

A moving clock runs slower as seen from a stationary frame:

[ \Delta t = \gamma ,\Delta \tau ]

• (\Delta \tau) = proper time (time measured by a clock at rest relative to the event).
• Example: muons produced in the upper atmosphere (proper