Physical intuition for why proper time is an invariant quantity

Mathematically, I understand why proper time, $\tau$ is an invariant quantity, since it is defined in terms of the spacetime metric $d\tau=\sqrt{-ds^{2}}$ (using the signature $(-+++)$ and with $c=1$). More abstractly, $\tau$ simply parametrises the length between two points along a worldline and hence is "obviously" invariant in this sense.

However, putting this aside for a moment, intuitively I'm less certain how to provide an answer to the question: why it is the case that proper time is physically an invariant quantity?

Consider a particle in Minkowski spacetime. If two different observers, Alice and Bob, are moving at different velocities with respect to the particle and with respect to one another, and each measures the elapsed time for the particle to propagate from one point to another, then they will measure different time intervals to one another. However, they will both agree on the elapsed proper time of the particle. Is the reason why this is the case because the question, "what is the time 'experienced' by the particle?", is a frame independent question - the proper time is a measure of the amount of "physical process" that the particle undergoes as it "moves" along its worldline, and this is a physical (coordinate independent) phenomenon? If Alice and Bob disagreed on the amount of elapsed proper time then they would be disagreeing with the particle on how much time has elapsed for the particle which would be nonsense?!

Apologies for such a basic question, I'm hoping someone can clear up any confusion for me.

• I think the proper time needs to be measured between events to be comparable? Maybe? (Not sure if the event is when your machine detects it or when it is theorized to happen though)
– Emil
Aug 8, 2017 at 17:02
• The concept of proper time is hard for me to grasp too, it's a necessary definition based on which a lot of the relativistic kinematics in non-inertial frames is possible, I would say it has to be tied to the idea that there exist some sort of an invariant unit of time (like atom clocks or something) which every one agrees upon. Aug 8, 2017 at 17:09
• @Emil Yes, it needs to be measured between events. That is why in my example I discuss the elapsed proper time as the particle propagates between two points. Aug 8, 2017 at 17:12
• @DavidLeonardoRamos That is what I think, that proper time must be a measure of some kind of periodic "physical process", for example the transition of an electron from one orbital to another, which is clearly an observer independent. Aug 8, 2017 at 17:15
• @Countto10 Good point. I realise that "experienced" is a bit of a loose term here, but I wasn't quite sure how to best phrase this part of the question. Aug 8, 2017 at 17:16

Let's be precise here. The 'invariance' in question is invariance of the spacetime interval under Lorentz transformations. Lorentz transformations here relate the coordinates of an event as measured by Alice with that of Bob, where they have a boost velocity with respect to each other. As such, Alice measures some time $t_A$ and Bob $t_B$. The transformations do the job of taking you from one observer to the other to see what it's like on their side of the world; it's like saying that Alice puts herself in Bob's shoes or vice versa.

But when you are talking about proper time, you are, by definition, adhering to only one observer: the particle itself. It doesn't make sense to say that the particle is boosted with respect to itself. There is no ambiguity in choosing a reference frame before deciding on performing a measurement, because the reference frame/observer has been chosen, a priori.

• So is the point then that both observers are "asking the same question", i.e. what is the amount of time "measured" by the particle, which must have an invariant answer since the particle can not "measure" an amount of time other than the amount that it has "measured" (i.e. it can't be boosted with respect to itself). In other cases, there is an ambiguity since the observers are "asking different questions", e.g. how much time has elapsed for the particle relative to me, or how much time has the other observer measured for the elapsed time for the particle relative to them. Aug 8, 2017 at 17:39
• That is, all observers are boosting to the same reference frame - that of the particle, and hence they must all surely measure the same elapsed time in that frame. Aug 8, 2017 at 17:44
• I wouldn't bring in Alice and Bob into the picture to measure the time as experienced by the particle, because they can not do that; only the particle can. And that is the point. Once you specify that you want to know what proper time is, there is only one such observer that remains eligible to measure values: the particle. There's no way it can be boosted away to a new frame in the hope of getting a different value. Aug 8, 2017 at 17:50
• Sorry, I didn't phrase that very well. I meant that both Alice and Bob are asking the same question: "what is the elapsed time between two points along the trajectory of the particle, as measured by the particle?". Of course they should agree on the answer to this question as the particle measures a single value for the elapsed time (it would be impossible for it to boost with respect to itself and obtain multiple values). Aug 8, 2017 at 18:37
• Having thought about your answer a bit more, would it be correct to say that the proper time of an object is defined as the time measured by the object itself, i.e. in the rest frame of the object. There is only one frame in which the object will be at rest - all observers will agree on this fact - and as such, only one value of time that can be measured in which the object is at rest. Hence, all observers will agree on the proper time of the object precisely because there can only be one frame in which the object is at rest. Aug 9, 2017 at 18:38

Well, let the thing that the two people observe be a clock -- a mechanical clock. If they disagree about the proper time on its worldline between two events, then they also disagree on its physical state at at least one of those events in general. That means that, say, if the two observers and the clock meet at some point on its worldline, then they will disagree about what time it says and all sorts of other details of its construction. That would be a disaster.

• That's kind of what I was trying to allude to in my OP. So is the point that proper time, in a sense, measures the amount of physical process that occurs. For example, the state of a particle will evolve in its own rest frame at a given rate that is intrinsic to the particle (and hence coordinate independent). This evolution is measured by the proper time of the particle, and if it wasn't an observer independent quantity, then different observers would disagree about the evolution of the particle, which is a frame independent phenomenon?! Aug 8, 2017 at 17:25
• Proper time is just the length of a curve. For different people to differ about it would be as absurd as different people disagreeing on how far it is from London to Paris because they are using different maps.
– user107153
Aug 8, 2017 at 17:49
• So would it be correct to say intuitively that as the proper time is the time measured by the particle in its rest frame, this must be an observer independent quantity - it would be absurd for two different observers to disagree on this value since then they would be disagreeing on the amount of time that has elapsed for the particle according to the particle. This would be tantamount to saying that the particle has evolved by different amounts in its own rest frame, which is of course, absurd. Aug 8, 2017 at 18:44
• @user35305, hmmm...appeal to ridicule? :) But of course we have to remain in agreement on the time continuum, or we won't remain in communication with each other and be coexistent within the same universe. Obviously that has little to do with physics, more into philosophy; but when you start talking about observing time you rapidly depart from physics. Take a small-town southerner to New York City and you'll see a discrepancy in "proper time." Assuming that clocks (objects) are the authority on time rather than life being authoritative is a circular argument; you prove clocks by clocks. Aug 8, 2017 at 20:12
• I think you have to be wary about saying things are 'absurd': QM is absurd, but correct, for instance. That said, the proper time along a path is indeed the length of time measured by something whose worldline is that path, and it's also the length of that path in spacetime (time is length). And the length of a path is a well-defined thing, just as the length of any other path is: it's not something that depends on how you describe it, and in particular it does not ever depend on the coordinate system you use.
– user107153
Aug 8, 2017 at 22:04

Yes, that is a valid way of understanding why the proper time is invariant; it does indeed come from an invariant question.

I am also a big fan of introducing Lorentz transforms by first looking at the transform for small $v$ which essentially sets $\gamma = 1$ and finding just $x' = x - v~t,~~t'=t - vx/c^2.$ This simplified view shows you that trying to compare time differences between things at two different positions is as frame-variant as trying to compare position differences between things at two different times: the distance between Kansas City and Washington is around $\text{1,500 km}$ if you are talking about at one instant of time, but if we're talking about the distance between where Kansas City is and where Washington D.C. is two-and-a-half hours later, then we need to know your reference frame because from my perspective they're still $\text{1,500 km}$ apart, but there's another perspective (an airplane flying from KCI to Dulles) for whom those locations are both "they're right here, outside my window!". This is to say nothing of two other reasonable perspectives, the geocentric non-rotating perspective by which Washington has moved I think $37.5^\circ$ to the East and therefore is $\text{4,700 km}$ away from where Kansas City was, or the heliocentric frame where those two locations differ by approximately $\text{270,000 km}$ apart or so, depending on what time of day it is. You have to be very careful to say "I want the distance between where these two things are right now," in classical physics, to have a number which all of these perspectives can agree on.1

Similarly as we move into relativity we have to become very careful to say "I want the time elapsed between these two events, in the inertial reference frame where they both occurred at the same location," so that both of those events happened "right here", otherwise we will be very confused. This time elapsed is the so-called "proper time" between the events.

1. As you can see this also becomes a little more difficult in relativity, as we start to disagree on when right now is at remote places. We technically have to say in the co-moving reference frame which sees both of them at rest for objects which exist over long periods of time, and we're permanently unsatisfied if they're not both at rest relative to each other -- or else we can talk about a proper distance between instantaneous events just like we do for the time-separation; then it's in the inertial reference frame where they both occurred at the same time.

In fact special relativity makes these two into disjoint circumstances: events which are objectively separated by distance generally admit reference frames which say "those both happened at the same time" whereas events which are objectively separated by time generally admit reference frames which say "those both happened right here, at the same place." The defining difference is whether light from one event could have reached the location of the other before it happened; and the only exceptions are the "null-separated" frames where light from the one event has just barely reached the other at the time when the new event has happened. These "null-separated" events form the third possibility, "one was objectively before the other and they were objectively not in the same place, but the time elapsed between them and the space difference between them can be brought arbitrarily close to 0 by selection of the right reference frames."

I think you answered your own question correctly, but you are close to conflating proper time and spacetime interval.

Proper time is invariant by definition. Proper time is the time elapsed in a frame where an object (or event) is at rest. In a frame where $dx = 0$ such that $ds = cdt := d\tau$.

For example, your stomach digested your lunch over a period of time today. It is, I guess, a postulate that you and all of the people sitting at your lunch table would measure the same amount of time that it took your stomach to digest your food.

The postulate is that some physical process happened in the universe and that this is an indisputible fact no matter your frame, no matter your coordinates in space or your velocity.

You, perhaps napping in a food coma, would measure what we call the proper time, proper because in your frame of reference your stomach is at rest.

What about moving frames? We postulate that the physical process of your stomach noshing, possibly on a turkey club, probably with cheese, should also be observable in any other frame of reference in the universe, and these frames of reference should be able to agree that, in our shared universe, your sandwich was digested with satisfaction. So, there must be some invariant between reference frames that all observers can agree on.

However, all of our stopwatches that timed your digestion will not agree. Measured time $dt$ is therefore not the invariant we are looking for.

But, if we also postulate a finite speed of light in all inertial reference frames, we conclude that space-time in our universe can be described 4D coordinate system with a metric signature, let's say (1,-1,-1,-1). The distance between two points in this coordinate system, $ds^2 = c^2dt^2 - dx^2$, does not depend on how your orient or translate your coordinates, and is the invariant we are after $ds^2 = ds'^2$.

Proper time is by definition the time elapsed in a frame where $dx = 0$, and can be agreed upon by any observer who measures the invariant interval $ds$ in their frame.

• Nice answer! Following your intuitive example, would it be correct to say that the proper time between two points for an object is observer independent since it is the time measured by the object itself in its rest frame. Hence, all other observers must agree on this value since they are all "asking" about the same quantity: "what is the time elapsed for the object relative to that object?" Which of course must have the same answer, otherwise one could have a scenario where the object has measured a proper time of $10s$ relative to itself (for example), and an observer... Aug 9, 2017 at 8:49
• ...(in a different reference frame) would claim that the object has measured a different time relative to itself, say $530s$. This would be contradictory and nonsense since it would imply that the object has "experienced" two different amounts of time at once in its own rest frame. Aug 9, 2017 at 8:50
• Exactly. We seem to live in a universe where there is an 'objective reality' that any observers should be able to agree on, no matter how they observed any two events. When talking about two events, not all observers will agree on stopwatch time or ruler distances. But, the something that they will agree on is the interval $ds^2=c^2dt^2-dx^2$. Some observers will be in a frame where $dx'=0$, so they will measure only $ds'=cdt'=cd\tau$. Because they are in a frame where they measured the two events without needing a ruler, only needing a stopwatch, we call their stopwatch time the proper time.
– well
Aug 9, 2017 at 9:57