The concept of distance shrinking with increased speed is not directly related to time dilation. Instead, it is a consequence of another relativistic effect known as length contraction or Lorentz contraction.
According to the theory of relativity, as an object moves closer to the speed of light relative to an observer, its length along the direction of motion appears to contract or shrink from the observer's perspective. This effect is known as length contraction.
The magnitude of length contraction can be calculated using the Lorentz factor γ, which depends on the relative velocity (v) between the object and the observer, and the speed of light (c):
L' = L₀ / γ
In this equation, L₀ represents the proper or rest length of the object (i.e., its length as measured in its own reference frame), and L' represents the contracted length as measured by an observer in a different reference frame.
As the velocity (v) approaches the speed of light (c), the Lorentz factor γ becomes larger, causing the contracted length (L') to become smaller compared to the rest length (L₀). This phenomenon leads to the observation that objects moving at high velocities appear shorter along the direction of their motion.
It's important to note that length contraction is a relativistic effect that applies only to the dimensions parallel to the direction of motion. Perpendicular dimensions are unaffected by length contraction.
In summary, the perception of distance shrinking with increased speed is a consequence of length contraction, a relativistic effect predicted by the theory of relativity. It is distinct from time dilation, which relates to the different experience of time between observers in relative motion.