Twistor theory

PN is the space of real null rays in Minkowski space.

PT is the projective twistor space.

Z is a twistor, and Z2=Z3=0

You see Penrose tiling.

Minkowski space

Minkowski space is based on Lorentz transformation.

w = i ct

This is imaginary time.

You can't exceed the speed of light, so you must be inside of red lines.

Future ←

Past →

Lorentz transformation

You move from S(x,y,z) to S'(x',y',z') by uniform linear motion (v) on x.

This is called Galilean transformation.

c is the speed of light.

This is S'.

You see that the speed of light should be stable, so S' is the same.

You know Galilean transformation.

∴

Then you see S.

∴

A=E

∴

You know Lorentz factor.

Lorentz factor

When you move, your light clock is slower about T times. This is called Lorentz factor.

c is the speed of light.

2a/c=1

This is the light clock, so you always have 1 second.

When you move from A to C, your light clock move slower.

T=2L/c

T is time, and c is the speed of light.

You see Pythagorean theorem.

∴

You see 2a=c from the light clock.

This is called Lorentz factor.