Question

The four-velocity(world-velocty) is defined by : u^μ =d x^μ /dτ ,where τ is the proper time of the object. I don't understand why it's defined with respect to the proper time but not the time of reference frame of the observer.

I understand that the proper time is a Lorentz invariant, but if that's the purpose of the definition which is making the four-velocity independent of the reference frame, the numerator of the derivative (dxμ)is still not a Lorentz invariant, so the four-velocity is not a Lorentz invariant after all. It is challenging to conceptualize how the balloon displaces the air. Think about what happens when you blow up a balloon

Answer 1

The four-velocity is defined to be invariant across all reference frames by using proper time (τ), despite its numerators (coordinate differentials) seeming reference frame dependent. This Lorentz invariance is crucial in the domain of special relativity, which deals with high velocities nearing the speed of light. Conventional velocity addition fails here, and relativistic transformations contextualize velocity among frames in relative motion.

Step-by-step explanation:

The four-velocity or world-velocity in physics is an object's velocity in spacetime, defined concerning an object's own proper time (τ), which is invariant of different reference frames. While it may seem that including the coordinate differentials in the numerator would make the velocity dependent on the reference frame, the use of proper time for normalization ensures Lorentz invariance. This is important because when performing measurements in different reference frames, especially in the realm of special relativity, we need quantities that remain consistent despite these differences.

Relativistic transformations of velocity, which are derived from the Lorentz transformation, demonstrate that velocities do not simply add up as in classical mechanics (Galilean relativity). Instead, the velocities of objects as measured by two observers in relative motion depend on the relative velocity of the observers themselves. At high speeds, close to the speed of light, the relativistic effects cannot be ignored and the classical summing of velocities fails to provide accurate predictions. Einstein's theory of relativity corrects for this by introducing a way to transform velocities between relatively moving frames that considers the finite and constant speed of light.