The following derivations are based on the exposition by G. G. Lombardi here.
Time Dilation
A clock is made by sending a pulse of light toward a mirror at a distance L and back to a receiver. Each “tick” is a round-trip to the mirror and back. The clock is shown at rest in the “Lab” frame in Fig. 1a, or any time it is in its own rest frame. Consequently, it also represents the clock at rest in Rocket #1. Figure 1b is the way the clock looks in the Lab when the clock is at rest in Rocket #1, which is moving to the right with velocity v and lenticity u, hence speed v and pace u.
Some notation:
t = half of the time for the light to make one round-trip in the Lab or at rest in Rocket #1 (Fig. 1b)
t´ = half of the time for the light to make one round-trip in Rocket #1 in its own rest frame (Fig. 1a)
L = distance to mirror
So the times and distances are related as follows:
L = ct´ = t´/k and
ct² = L2 + vt² = L2 + t/u² = t/k²
where c is the speed of light, k is the pace of light, and the Pythagorean theorem is used.
Eliminate L from the equations:
ct = ct´ + vt = t´/k2 + t/u² = t/k²,
and solve for t´:
t´ = t √(1 − v²/c²) = t √(1 − k²/u²).
Since √(1 − v2/c2) = √(1 − k2/u2) < 1, the clock at rest in Rocket #1 appears slow to observers in the Lab.
Go back to Fig. 1c. Rocket #2 is moving with velocity v and lenticity u to the left. Those clocks would also appear slow to observers in the Lab. But it is also the way a clock at rest in the Lab appears in Rocket #1. So, Lab clocks also appear slow to observers in the Rockets. Relativity is symmetrical!
Length Contraction
As with the time dilation example, imagine a pulse of light reflects from a mirror back to a receiver. Light will be the reference used to measure the length of an object in the Lab and Rocket frames.
Some notation:
L´ = length of rod in the Rocket
L = length of rod in the Lab
t1 = the light pulse travel time to the end of the rod (Lab)
t2 = the light pulse travel time back from the end of the rod (Lab)
t = t1 + t2 = total travel time in the Lab
t´ = total travel time in the Rocket
u = pace of the Rocket in the Lab frame
v = speed of the Rocket in the Lab frame
In the Lab:
L + vt1 = ct1 and L − vt2 = ct2
or L + t1/u = t1 /k and L − t2/2 = t2/k
So t = t1 + t2 = (2L/c) / (1 − v2/c2) = (2Lç) / (1 − k2/u2).
In the Rocket:
2L´ = ct´ = t´/k or
t´ = 2L´/c = 2L´k.
We know from the time dilation formula that t´ = t √(1 − v2/c2) = t √(1 − k2/u2).
So 2L´/c = 2L´k = (2L/c) / √(1 − v2/c2) = (2Lk) / √(1 − k2/u2).
L´ = L / √(1 − v2/c2) = L / √(1 − k2/u2) or
L = L´ √(1 − v2/c2) = L´ √(1 − k2/u2).
The rod appears shorter in the Lab frame since √(1 − v2/c2) = √(1 − k2/u2) < 1.