Meaning of Proper time

Question

Sorry for a bit of a basic question, but want to clarify things in my head. Is proper time quantified by the amount of physical process that an object, or physical system undergoes, for example the decay of an unstable element? And is the reason why proper time is independent of the frame of reference because such a physical process occurs without needing to introduce any coordinate system, thus regardless of the reference frame that you measure the process occurring in, it will occur at the same rate in the same manner?

For example, consider two observers equipped with identical clocks. One of the observers is considered at rest on Earth and the other is in a spaceship moving at a considerable fraction of the speed of light. Now is the notion of proper time that each observer, in their own reference frame will observe their clock to tick (cycle) at the same rate (however they will not observe each others clocks to tick at the same rate, as coordinate time is not frame-independent. Indeed from their own reference frame they will observe the clock in the other reference frame, moving relative to them, to tick at a different rate, as they are using their own coordinate time to measure the process). The point being is that if they both have identical clocks, then if the clock given to the observer on Earth ticks in a certain way for that observer in their rest frame on Earth, then the clock given to the observer in the fast moving spaceship will tick in exactly the same way in their frame of reference on the spacecraft (i.e. it won't suddenly change how it ticks to being aperiodic or non-linear, etc.)

Answer 1

Proper time of an observer is time as measured by the observer's own clocks. So it's obviously frame-independent because calculating proper time of a given observer requires to use his own frame of reference.

Answer 2

I think you are mixing up two different concepts, which is muddying the waters.

Firstly, relativity (both special and general) is a geometrical theory and the proper time for an observer has a precise definition as the length of a world line along which the observer travels (give or take a factor of $c$). This length is calculated using the metric. As Titus says, the proper time is equal to the time experienced by the observer, and it's easy to see why this is. In the observer's rest frame they aren't moving ($dx = dy = dz = 0$) so the world line is a straight line up the time axis. The length of this line is obviously equal to the elapsed time experienced by the observer.

It is an assumption in relativity that the proper time is an invariant i.e. any observer in any frame, accelerating, in a gravity well or whatever, will calculate the same proper time. To see how this assumption produces all the weird effects like time dilation or length contraction see How do I derive the Lorentz contraction from the invariant interval?, and many other related questions on this site.

There is a second (but related) fundamental principle in relativity that a local experiment will always produce the same result i.e. that if we put you in a sealed box in free fall you cannot tell how the box is moving by a local experiment. So if your experiment is to observe some clock mechanism, whether it's a caesium atom or a clockwork watch, it will always tick in the same way. But this process of timing is not what determines the proper time.