Special relativity has become a fundamental theory in the 20th century and is crucial for explaining many astrophysical phenomena. A central aspect of special relativity is the transformation from one reference frame to another. The following Lorentz transformation matrix gives the transformation from a frame at rest to a moving frame with velocity v along the z-axis:

γ 0 0 γβ
0 1 0 0
0 0 1 0
γβ 0 0 γ

where β = v/c with c being the speed of light in a vacuum, and γ is the Lorentz factor:
γ =
1
p
1 − β
2
(a) State and explain the two traditional postulates from which special relativity originates.
(b) Draw a plot of the Lorentz factor for 0 ≤ β ≤ 0.9 to see how its value changes.
One of the many exciting phenomena of special relativity is time dilation. Imagine astronauts
in a spaceship that is passing by the Earth with a high velocity.
(c) Are clocks ticking slower for the people on Earth or for the astronauts on the spaceship?
(d) How fast must the spaceship travel such that the clocks go twice as slow?