Relativistic journey to a star

We're going to study the classic example of the relativistic journey to a star. This example will enable us to extend what we've already seen in the diagram of reciprocal time dilation with another of relativity phenomena: length contraction. In the time axis we'll measure year units, therefore the units on the x-axis will be light-years.

Diagram 14

We start from the origin event (0, 0) on earth where the clocks of the inertial observers A (grey) and B (blue) are synchronized.
Observer A remains on earth.
Observer B undertakes a journey to a star 3 light-years away from Earth in a spaceship traveling at a speed v = 3/5 c.
We'll analyze the distances and times measured by observers A and B when the spacecraft reaches the star (we'll call it event E1).

Diagram 15: earth view

In A's frame of reference, the coordinates of the arrival event are: E1_A (5, 3).
In B's frame of reference, the coordinates of the arrival event are: E1_B (4, 0).
For observer A on earth, observer B takes 5 years to reach the star and travels a distance of 3 light-years.
Observer B on the spacecraft measures that it's taken 4 years to reach the star and has traveled a distance of 2.4 light-years (length contraction).
During the journey, A observes that time slows for B. He calculates that for B only 4 years have elapsed.
During the journey, B observes that time slows for A. He calculates that for A only 3.2 years have elapsed.

Surely the most difficult point to grasp is that for Observer B on the spaceship, only 3.2 years have elapsed on Earth when he reaches the star. The answer lies in the reciprocal time dilation.

Diagram 16: spaceship view

To help understand what we've seen in Diagram 15, let's see the point of view of the spaceship:

Reciprocal length contraction

Along with time dilation, length contraction is another consequence of the relativity of simultaneity (Diagram 9). In a similar way as we've alredy seen in reciprocal time dilation, each observer measures the distance between two points in the space or "world" formed by the set of simultaneous events in his frame of reference at a certain instant T. And just as happens with time dilation, this leads to reciprocal length contraction between two frames in relative motion.

In the same way that we've seen in the mathematical derivation of time dilation, the length contraction in the direction of movement corresponds to the Lorentz factor: $\boxed{L'=Lγ}$

Proper length is the length of an object in an inertial reference frame in which it is at rest.

We have two inertial observers A (grey color) and B (blue color) in relative motion. Taking observer A as the reference origin, observer B moves on a 3-meter bar (within its frame of reference) at the speed v = 3/5 c. The Lorentz factor for this velocity is γ = 1.25, therefore for observer A, the bar measures 2.4 meters.

Events E1 and E2 are the ends of the bar (simultaneous events) at time t = T₁ for observer A.
Events E3 and E4 are the ends of the bar (simultaneous events) at time t = T₂ for observer A.
Events E1 and E4 are the ends of the bar (simultaneous events) at time t '= T' for observer B.
Observer A at both T₁ and T₂ measures that the length of the bar is L = 2.4 meters.
Observer B at T' measures that the length of the bar is L' = 3 meters.
The bar is at rest in the reference frame of observer B, therefore observer B measures the proper length of the bar.
The contraction of the length is not an optical effect, the bar actually measures L = 2.4 meters in the "worlds" of simultaneous events of observer A.