Special Relativity in two pages. Just maths, no verbiage, to show how simple Mother Nature keeps things.

1. Special Relativity in two pages (simple maths, but no verbiage)
©Russell John Childs, PhD - 2017-08-01

2. Part I - Kinematics
Invariant interval
Maximum velocity, , cannot be infinity, since objects could duplicate themselves, by occupying multiple positions
simultaneously. must be a universal constant, otherwise observers would disagree on the maximum. Thus:
This generalises Pythagoras to a 4D vector with invariant length.
Lorentz transformation
Derivation 1
Derivation 2
General Lorentz:
Einstein summation: or
Four-velocity:
Four-gradient: .
Four-momentum:
Addition of velocities
Four-frequency and Relativistic Doppler Effect:

3. Part II - "Paradoxes"
Twin:
Let be the age of the shipborne twin.
Let be the age of the Earthbound twin as seen by the shipborne twin.
During deceleration, ,of the ship to , the shipborne twin sees:
After deceleration to
Thus, initially, the Earthbound and shipborne twin both see time running slower for the other. During
deceleration, the shipborne twin sees time accelerate for the Earthborne twin, who suddenly becomes older.
Pole in barn:
A pole, of length , moves at . A barn is long. The back and front of the pole are contained within the
barn at points and .
In the frame of the pole, however, these are: and , which
are not simultaneous, so the pole can disaagree that it fits in the barn.
Part III - Electromagnetism
Four-potential:
Electromagnetic four-momentum:
Electromagnetic tensor and Lorentz force:
Four-current:
Lorenz gauge:
Maxwell's equations: