The Lorentz factor, denoted by γ, is indeed defined in terms of the square of the velocity relative to the speed of light, c. It is given by the equation:
γ = 1 / sqrt(1 - (v^2 / c^2))
where v is the magnitude of the velocity.
The Lorentz factor appears in the equations of special relativity and is associated with time dilation, length contraction, and relativistic mass increase. It relates the time and space coordinates of events observed in different inertial reference frames.
In special relativity, the concept of velocity is still meaningful and includes both magnitude and direction. The Lorentz factor captures the relativistic effects that arise when objects move at speeds close to the speed of light. It is not solely about speed (magnitude of velocity), but it also accounts for the direction of motion.
The inertial reference framework, described by special relativity, is indeed independent of the direction of travel. The laws of physics and the mathematical formulas remain consistent regardless of the direction of motion. However, this does not mean that velocity is reduced to speed alone. Velocity still encompasses both magnitude and direction, and the Lorentz factor accounts for the effects of both in the context of special relativity.