Short Summary of Special Relativity

Special Relativity is based on two principles:

1. The laws of physics are the same in all inertial frames.

2. The speed of light (c) is constant for all observers.

From these follow the key effects:

• Time dilation: moving clocks tick more slowly.

• Length contraction: moving objects shrink along the direction of motion.

• Relativity of simultaneity: simultaneity depends on the observer.

Space and time form a unified space-time.
Events are separated by an invariant interval:

[
ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2.
]

Motion in space-time is represented by worldlines.
The light cone divides space-time into past, future, and elsewhere.

Relativistic dynamics is governed by four-vectors, such as:

• position (x^\mu),

• four-momentum (p^\mu = (E/c, \vec{p})),

• four-current (j^\mu),

• electromagnetic four-potential (A^\mu).

Energy and momentum satisfy:

[
E^2 = p^2 c^2 + m^2 c^4.
]

Proper time (d\tau) is the time measured by the moving particle:

[
d\tau = \frac{dt}{\gamma}.
]