Why Lorentz length contraction happens only along the line of propagation of light (the direction of motion) and not perpendicular to the direction of propagation of light ?

Why Lorentz length contraction happens only along the line of propagation of light (the direction of motion) and not perpendicular to the direction of propagation of light ?                                                      Lorentz length contraction states that an object appears shorter to an observer when it is in motion (at high speed approaching the speed of light) relative to the observer.Lorentz length contraction occurs  along the line of propagation of light (the direction of motion) and not perpendicular to it is due to :            the nature of spacetime and the Lorentz transformation. In special relativity, spacetime is described using four-dimensional coordinates (x, y, z, and t).When an object moves at relativistic speeds along the x-axis (the direction of propagation of light), its length contraction occurs only along the x-axis, as the object's length in the y- and z-directions (perpendicular to the direction of motion) remains unchanged.

The mathematical explanation for this lies in the Lorentz transformation equations, which describe how the coordinates change when an object moves at relativistic speeds. The Lorentz transformation equations describe how space and time coordinates are transformed from one inertial frame of reference to another. Here are the equations:     For an object moving with velocity v along the x-axis, the Lorentz transformation equations are:

x' = γ(x - vt)

y' = y

z' = z

t' = γ(t - vx/c^2)                                where:

 x, y, z, and t are the space and time coordinates in the stationary frame (S)

- x', y', z', and t' are the space and time coordinates in the moving frame (S')

- v is the relative velocity between the two frames

- c is the speed of light

- γ is the Lorentz factor, given by:

γ = 1 / sqrt(1 - v^2/c^2)

These equations describe how the space and time coordinates are transformed from one inertial frame to another. Transformation equations for the x-coordinate (along the line of propagation) involve the Lorentz factor (γ) which depends on the object's velocity.Lorentz factor (γ) is responsible to the contraction of length along the x-axis.The transformation equations for the y- and z-coordinates (perpendicular to the direction of motion) do not involve the Lorentz factor (γ). As a result, the length of the object in these directions remains unchanged.