I would like to know if there is any physical significance associated with the derivative of a quantity with respect to proper time or is it just a mathematical trick. Since proper time is measured in its "rest" frame of a moving particle, it seems to me that particle is not going through any dynamics and therefore time derivatives should be zero in the rest frame. I understand that we use derivatives with respect to proper time to keep things Lorentz invariant....but that sounds more like a mathematical requirement rather than something of physical significance. Example: what is the physical significance of four-velocity. I know four velocity is tangent to the worldline but I find it hard to remember through physical intuition. I always have to go through a book to find its definition.
Kindly excuse my ignorance.