Proper Time & 4-vectors: Clarification Needed

[e2m2a]

TL;DR Summary

Is proper time frame invariant?

I am confused. My understanding is that proper time is used in 4 vectors analysis because proper time is frame invariant. Every other inertial frame will agree on the same time increment if they use the proper time of that one reference frame. But when you do the Lorentz transformation, the proper time of the same reference frame is different for different inertial reference frames. Need clarification on why proper time of a given inertial reference frame is called invariant with respect to all other different inertial frames.

 

[Ibix]

I suspect you're misunderstanding what "proper time" is. It's the elapsed time along a worldline as measured by a clock following that worldline. For example, in a twin paradox scenario, the stay-at-home's proper time between departure and return is the time measured by the stay-at-home's wristwatch and the traveller's proper time between departure and return is the time measured by the traveller's wristwatch. These are different, but frame invariant - everyone will agree on the values.

A frame doesn't have a proper time. One could speak of the proper time of a clock at rest in an inertial frame. Everyone would agree on the elapsed time the clock would show between two events, but would not necessarily measure the same time with their own clocks (possibly plus synchronisation convention).

The Euclidean analogy for proper time is the length of an object, while coordinate time is its extent in the y direction. Everyone agrees on the length of a rod, but not necessarily its extent in the x- or y- direction, because the latter depends on the choice of how their axes are aligned. I suspect your problem is that you've aligned a rod parallel to the y-axis so that its length is equal to its extent in the y direction, and are expecting that everyone will therefore agree on its y extent. They won't - they'll all agree on its length.

 

[Orodruin]

I suspect you're misunderstanding what "proper time" is. It's the elapsed time along a worldline as measured by a clock following that worldline.

Pretty much this. It would be of interest to hear what OP has understood that proper time is as itmight help in disentangling any misconceptions.

 

[robphy]

The Euclidean analogy for proper time is the length of an object, while coordinate time is its extent in the y direction.

For better precision in the analogy,
I think
“the length of a given path between two points”
is better than “the length of an object”.

 

[cianfa72]

A frame doesn't have a proper time. One could speak of the proper time of a clock at rest in an inertial frame. Everyone would agree on the elapsed time the clock would show between two events, but would not necessarily measure the same time with their own clocks (possibly plus synchronisation convention.

Here I believe 'frame' actually means 'coordinate chart'. Btw in GR (i.e. curved spacetime) a global inertial frame does not exist at all.

 

[Ibix]

Here I believe 'frame' actually means 'coordinate chart'.

Well, l'm assuming we're talking SR here and using "frame" and "global inertial chart" more or less interchangeably.

Worth noting that "frames don't have a proper time" is true in any definition of frame, I think, because a proper time is defined along a worldline and no definition of "frame" is a single worldline.

 

[PeterDonis]

"frames don't have a proper time" is true in any definition of frame

Yes, that's correct. If a frame has a single "time" at all, it will be a coordinate time.

 

[Orodruin]

I mean, it is possible to construct a frame where the time coordinate corresponds to the proper time of a set of timelike curves (this is true for Minkowski coordinates in SR for example). However, this will always be a case if explicitly choosing the coordinates in a particular manner and as such is not really anything more than coincidence. Referring to proper time always assumes - whether explicitly or not - a selection of a particular world line.

 

[Ibix]

I mean, it is possible to construct a frame where the time coordinate corresponds to the proper time of a set of timelike curves (this is true for Minkowski coordinates in SR for example).

Indeed. And I suspect OP is confusing the proper time of a clock (possibly a notional one) following one of those fixed-spatial-coordinates lines with "the" proper time in a frame, which isn't really a helpful concept.

 

[Orodruin]

Indeed. And I suspect OP is confusing the proper time of a clock (possibly a notional one) following one of those fixed-spatial-coordinates lines with "the" proper time in a frame, which isn't really a helpful concept.

He would not be the first ... nor the last.

 

[Ibix]

nor the last

It's Christmas - we can hope.

 

[e2m2a]

Pretty much this. It would be of interest to hear what OP has understood that proper time is as itmight help in disentangling any misconceptions.

Ok this concept of proper time is obviously subtle. They say its the time of a moving clock along its worldline. But the velocity of this worldline, the ratio of its space/xt coordinates should differ with respect to different inertial observers. So wouldn't each inertial observers deduce a different proper time for the clock?

 

[Ibix]

So wouldn't each inertial observers deduce a different proper time for the clock?

No - they each get a different $\Delta x$ and they each get a different $\Delta t$ (coordinate time interval), but they are different in such a way that $c^2\Delta t^2-\Delta x^2$ is the same in each frame, and that is $c^2\Delta\tau^2$, the proper time interval squared (times $c^2$). Or, equivalently, they get different $\Delta t$ and different $\gamma$ but they are different such that $\Delta t/\gamma$ is the same in every frame, and that is $\Delta\tau$.

 

[Orodruin]

Ok this concept of proper time is obviously subtle. They say its the time of a moving clock along its worldline. But the velocity of this worldline, the ratio of its space/xt coordinates should differ with respect to different inertial observers. So wouldn't each inertial observers deduce a different proper time for the clock?

Relative to the coordinate times in their respective rest frames, yes. The point is that all observers will agree on the proper time between two events on the world line.

 

[e2m2a]

No - they each get a different $\Delta x$ and they each get a different $\Delta t$ (coordinate time interval), but they are different in such a way that $c^2\Delta t^2-\Delta x^2$ is the same in each frame, and that is $c^2\Delta\tau^2$, the proper time interval squared (times $c^2$). Or, equivalently, they get different $\Delta t$ and different $\gamma$ but they are different such that $\Delta t/\gamma$ is the same in every frame, and that is $\Delta\tau$.

Ok. That makes sense. Thanks.

 

[PeroK]

Ok this concept of proper time is obviously subtle. They say its the time of a moving clock along its worldline. But the velocity of this worldline, the ratio of its space/xt coordinates should differ with respect to different inertial observers. So wouldn't each inertial observers deduce a different proper time for the clock?

If we consider two events on this worldline. Every observer (inertial or not) must agree on what the clock reads at the first event and at the second event. For example, the clock could print out a piece of paper with its reading at each event. What's written on those pieces of paper is frame invariant.

Proper time in this case relates to that specific clock. Think of "proper" time as being tied to the readings on a specific clock. The events in question must be colocated with the clock.

Note that proper time relates to any particle/clock, whatever its state of motion. Think of a particle moving in a circle or ellipse or otherwise accelerating. Its proper time is what the clock reads; or, how much time is passing for the particle. Proper time is not tied to clocks at rest in some inertial reference frame.

You must not confuse the proper time of a clock with the coordinate time of a reference frame in which the clock is at rest. The subtlety is that a clock moving inertially, say, records the coordinate time for an inertial reference frame at only one particular fixed spatial coordinate in that frame. It does not record coordinate time at other points in that reference frame.

It's true that within an inertial reference frame all clocks may be syncronised. But, the different clocks are not synchronised in other frames. The synchonicity across these clocks is not frame invariant. That is a criticval point.

 

[vanhees71]

Ok this concept of proper time is obviously subtle. They say its the time of a moving clock along its worldline. But the velocity of this worldline, the ratio of its space/xt coordinates should differ with respect to different inertial observers. So wouldn't each inertial observers deduce a different proper time for the clock?

The problem with learning (special) relativity is that you have to relearn the concepts behind how to measure spatial distances and how to define time at different places.

On the other hand special-relativistic spacetime, the socalled Minkowski space has a lot in common with Galilei-Newton spacetime. First of all by assumption Newton's 1st law is valid in SR as well, i.e., there exists a class of inertial reference frames, where bodies which are not interacting with anything else, move in straight lines with constant velocity. Also by assumption for any observer at rest wrt. an inertial frame the geometry of space is Euclidean and there exist standard clocks, which are in practice realized with very high precision by atomic clocks (like the "cesium standard" defining the second as the unit of time via the frequency corresponding to a hyperfine transition of the Cs-133 atom).

The difference between Galilei-Newton spacetime and the spacetime model of special relativity then comes with Einstein's postulate about the "invariance of the speed of light in vacuum", i.e., the independence of the speed with which electromagnetic waves propagate in the vacuum from the velocity of the light source relative to an inertial observer.

Now this can be used to construct spacetime measures by a convention, how to synchronize standard clocks, all of which are assumed to be at rest wrt. an inertial frame of reference at different places. With one standard clock, say located at the origin of the inertial frame in question, you can only measure the time at this one place, i.e., the duration between two events taking place at the origin of the reference frame. This was Einstein's great insight into the problems of electrodynamics with regard to the non-invariance under Galilei transformations. To synchronize clocks at different places you have to use the postulate of the invariance of the speed of light, and to define a "global" time in an inertial reference frame you need to choose some convention. The convention is that you consider all clocks to be at rest within the inertial frame under consideration and then they are tuned such as to be consistent with the invariance of the speed of light. This can be done by using one reference clock at the origin and sending a light signal to another clock at a given distance from the origin, letting it reflect back to the origin and measuring the time it takes for the signal to propagate to the distant clock and back and assume that the time it takes to move from the origin to the distant clock is half of the time measured for the light signal to go forth and back. This is in accordance with the assumed symmetries of the Euclidean space (isotropy and homogeneity) used to define the distance between the clocks. In this way you can prepare the clocks in advance, putting them to times $r/c$, where $r$ is the distance of the clock from the origin, and $c$ the speed of light in a vacuum, send a light signal from the origin to all the distant clocks, and start them when this light signal arrives at their corresponding places. Then by this convention (!) these clocks, all at rest within the inertial frame where this procedure is done, are all synchronized.

From this operation it is now clear that for any other inertial frame, where one has done the same synchronization procedure, that the clocks in different inertial frames are not synchronized relative to each other, but when transforming between different inertial frames, being in uniform motion relative to each other, you have to transform both the time, defined for each inertial frame by the synchronization procedure, and the spatial coordinates of "events" from one frame to the other. That results in the socalled Lorentz transformation, which substitutes the Galilei transformation of Newtonian mechanics, and with this transformation you can derive all the kinematic effects resulting from the symmetry assumptions of the Minkowski spacetime model, including the invariance of the speed of light in vacuum:

-the "relativity of simultaneity": To distant events which are at the same time as observed in one inertial frame are usually not simultaneous as observed in another inertial frame of reference moving with respect to the other.

-"time dilation": A standard clock moving wrt. an inertial frame with velocity $v$ "ticks slower" by a "Lorentz factor" $\gamma=1/\sqrt{1-v^2/c^2}$ compared to the time defined in this inertial frame (by the clock-synchronization convention explained above).

-"length contraction": When measuring the length of a rod, which is at rest wrt. an inertial frame $\Sigma$, $L_0$, the length of the as measured in another inertial frame $\Sigma'$ moving with velocity $v$ relative to $\Sigma$, is shorter by an inverse Lorentz factor, $L'=L_0/\gamma=L_0 \sqrt{1-v^2/c^2}$.

This results from the fact that the observer in $\Sigma'$ measures the length of the rod by marking the coordinates of its two end points simultaneously within his frame. These "measurement events" are not simultaneous when observed from the inertial frame $\Sigma$, and this is the origin of the length-contraction effect.

For the mathematical details of SR, see

https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf

although there the Lorentz transformation is derived in a somewhat different more formal way than using Einstein's clock-synchronization procedure. It is well worth to read the first paragraphs about this "kinematical part" of SR in the original paper:

https://en.wikisource.org/wiki/Translation:On_the_Electrodynamics_of_Moving_Bodies

 

[PeterDonis]

Moderator's note: A side discussion about synchronous frames/coordinate charts has been moved to a new thread:

https://www.physicsforums.com/threads/synchronous-frames-coordinate-charts.1010817/

 

[cianfa72]

You must not confuse the proper time of a clock with the coordinate time of a reference frame in which the clock is at rest. The subtlety is that a clock moving inertially, say, records the coordinate time for an inertial reference frame at only one particular fixed spatial coordinate in that frame. It does not record coordinate time at other points in that reference frame.

Yes, if the coordinate time defined for the inertial frame in which the clock moving inertially is at rest at a fixed spatial coordinate is actually its proper (recorded/elapsed) time.

 

[cianfa72]

Yes, if the coordinate time defined for the inertial frame in which the clock moving inertially is at rest at a fixed spatial coordinate is actually its proper (recorded/elapsed) time.

Thinking better about it, the coordinate time at the fixed spatial coordinate where the inertially moving clock being at rest in the given inertial frame, must be equals to the clock proper/elapsed time. Otherwise the frame would not be inertial !

 

[Mister T]

So wouldn't each inertial observers deduce a different proper time for the clock?

Of course not. If the clock reads noon, for example, all inertial observers will agree that it reads noon. It's a measurement, and all measurements are invariant. If you wait for that same clock to read 1:00 pm, again everyone will agree that it reads 1:00 pm. You could have the clock spit out a punch card with the time stamped on it. Every observer could, later on and at their leisure, examine the punch cards and verify that the first one reads noon and the second one reads 1:00 pm.

Now, and I think this is where your confusion arises, not all inertial observers will agree that an hour of time elapsed between the two readings.