Proper Time and Time Dilatation

[LCSphysicist]

Homework Statement

I am a little confused with the concept of proper time

Relevant Equations

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I am a little confused with the concept of proper time: Using the invariance of the distance square in the Minkowski space, we can get the expression $d \tau = \frac{d t}{ \gamma}$ Now the problem:

Aren't the proper time the time measured by a moving clock? That is, if i am moving with speed v and carries with me a clock, "my" proper time is the time elapsed in my clock, right?

But, to the best of my knowledge, the time elapsed in a moving frame shouldn't be dilated? The so called time dilatation, so why the proper time in the expression above is lesser than the coordinate time?

The expression above has a special frame?

 

[Orodruin]

In your rest frame, you are not moving, and thus $\gamma = 1$ and $d\tau = dt$. There is no such thing as an inherently "moving frame", frames can only move relative to each other and you have to be careful in how you reference this.

In the frame where you move with speed $v$, you indeed have $d\tau = dt/\gamma$ with $dt$ being the coordinate time differential in that frame.