The four-velocity is a concept used in the theory of special relativity to describe the motion of an object in spacetime. It is a four-dimensional vector that combines both spatial and temporal components.
In special relativity, spacetime is treated as a unified entity, where time is combined with the three dimensions of space into a four-dimensional continuum. The four-velocity, denoted as u, is defined as the derivative of an object's four-position with respect to its proper time, which is the time experienced by the object itself.
The four-velocity vector has both magnitude and direction. The magnitude of the four-velocity is always equal to the speed of light, denoted as c, which is a fundamental constant in special relativity. The direction of the four-velocity vector represents the object's motion through spacetime.
The difference between the four-velocity and normal velocity (as commonly understood in classical physics) is that the four-velocity incorporates the effects of both time and space, while normal velocity only considers motion through space. The four-velocity takes into account the time dilation and length contraction effects of special relativity, which become significant at speeds approaching the speed of light.
The proper velocity is another term used in special relativity, which is the spatial component of the four-velocity. It represents the rate at which an object's position changes in space as experienced in its own frame of reference. Proper velocity is related to the object's normal velocity but accounts for the relativistic effects of time dilation and length contraction.
Observers in different frames of reference will measure different values for the four-velocity and proper velocity of an object. This is due to the relativistic effects of time dilation and length contraction, which cause a moving object's four-velocity and proper velocity to differ from those measured by an observer at rest relative to the object.
In summary, the four-velocity combines time and space components to describe an object's motion in spacetime. It incorporates relativistic effects such as time dilation and length contraction, and observers in different frames of reference will experience different values for the four-velocity and proper velocity of an object.