What does "invariance of proper time" mean?

[SreenathSkr]

It would be better with a clear example to understand

 

[Mentz114]

Proper time is defined by the metric $c^2d\tau^2=c^2dt^2 -dx^2-dy^2-dz^2$. When the differentials are transformed with a Lorentz transformation $d\tau^2$ comes out the same.

$(\gamma dt-\gamma\beta dx)^2 - (\gamma dx-\gamma\beta dt)^2 = dt^2-dx^2$ because $\gamma^2(1-\beta^2) = 1$.

 

[ghwellsjr]

It would be better with a clear example to understand

Here is an example of the twin paradox. The blue twin remains inertial while the red twin is not inertial. At the beginning (at the bottom of the diagram) their clocks (their Proper Times) are the same. Each dot marks off one year of Proper Time. They separate and after 8 years, the red twin accelerates to come back to the blue twin. It takes him 8 more years and when they reunite, the blue twin has aged by 20 years. Count the dots along each straight line segment to see how many years they age for each of the three segments:

Now we can transform the coordinates of all the dots to a frame in which the red twin is stationary just after the twins separate:

If you count the dots marking off the accumulation of Proper Time for each of the three segments, you will see that they are the same as in the first diagram.

Here is another frame in which the red twin is stationary just before the twins reunite:

Again, if you count the dots for each segment, they are the same. This is an example of how the Proper Time remains the same no matter what frame we transform the scenario in to.

Furthermore, you can drawn in any number of lines along the 45-degree diagonals to show the propagation of light signals between the twins and they will start and stop at the same Proper Time for each twin in all the frames. For example, when the red twin accelerates, the time he sees on the blue twin's clock is shown by the thin blue line and the blue twin sees the red twin accelerate as shown by the thin red line. In the first frame, this looks like:

As you can see, when the red twin turns around, he sees the blue twin's time at 4 years and when the blue twin is at 16 years, he sees the red twin turn around at 8 years.

We can do the same thing in the second frame with the same results:

And finally results are the same in the third frame:

Do these examples help you understand the invariance of Proper Time?

 

[ChrisVer]

Invariance means invariance. That is it does not vary... with respect to what? in this case with respect to Lorentz transformations...
That means for example that if an observer measures it to be X in his reference frame, then any other observer related to the first by a Lorentz transformation will measure it X [and not X' ]...

 

[SreenathSkr]

Thanks to all. Better explanations

 

[Matterwave]

I think the other posters have given you adequate answers; however, they are a bit mathematical/formal in nature. I will give you an answer which is perhaps more physical and easier for you to understand. The proper time between two events A and B is the time ticked off by a clock in a frame for which events A and B happened at the same place. So let's say I am conducting an experiment in my laboratory with a flashlight. The flashlight flashes once, and a short time later flashes again. The first flash we call event A, and the second flash we call event B. The flashlight has not moved. The proper time between event A and B is the time ticked off by a clock that I am carrying. If you are moving with respect to me, then the time between event A and B for YOU is different than it was for me (you see the time between them as being longer, in other words, your clock will have ticked more times than mine did, which will make you think my clock is running slow -> time dilation), and, in addition, (very importantly) A and B happened in different places for you. But if I asked you "what is the proper time between events A and B" I am really asking you "what did MY clock read between events A and B" and so this obviously has only 1 answer which does not depend on how fast YOU are moving relative to me.

 

[ghwellsjr]

I think the other posters have given you adequate answers; however, they are a bit mathematical/formal in nature. I will give you an answer which is perhaps more physical and easier for you to understand. The proper time between two events A and B is the time ticked off by a clock in a frame for which events A and B happened at the same place.

This statement is at best misleading because it implies that the Proper Time between two events is dependent on a frame and that they have to occur at the same place. All that matters is that a clock, inertial or not, be present at both events. Clearly, different clocks will measure different Proper Times between those two events as I pointed out in my diagrams where the two events of interest are at the bottom and top of the diagrams. The blue twin measures 20 years of Proper Time while the red twin measures 16 years of Proper Time. There is not a single answer to the question of how much Proper Time is there between two events.

So let's say I am conducting an experiment in my laboratory with a flashlight. The flashlight flashes once, and a short time later flashes again. The first flash we call event A, and the second flash we call event B. The flashlight has not moved. The proper time between event A and B is the time ticked off by a clock that I am carrying.

That may be the Proper Time for the clock that you are carrying, provided of course that it is collocated with the flashlight when the two flashes occur but it doesn't matter what happens to the clock in between. Proper Time is associated with clocks, not just with events.

If you are moving with respect to me, then the time between event A and B for YOU is different than it was for me (you see the time between them as being longer, in other words, your clock will have ticked more times than mine did, which will make you think my clock is running slow -> time dilation), and, in addition, (very importantly) A and B happened in different places for you.

My motion with respect to you is irrelevant. What is important if I'm going to use my clock to measure a Proper Time between A and B is that it must be present at both events. I can move my clock differently than you move your clock between those two events. If your clock is inertial between those two events and mine is not, then my clock will measure a shorter Proper Time interval than yours will. This has nothing to do with Time Dilation which is not invariant. We're talking about the invariance of Proper Time.

But if I asked you "what is the proper time between events A and B" I am really asking you "what did MY clock read between events A and B" and so this obviously has only 1 answer which does not depend on how fast YOU are moving relative to me.

What you are talking about is called the spacetime interval or the Lorentz interval (and several other terms) which has only 1 answer. (It is also the Proper Time on an inertial clock that is present at both events but you are being too restrictive.)

 

[Matterwave]

I can understand your objection. Between time-like separated events A and B there are many time-like world lines which pass through both (i.e. for which both are happening at the same place), but only one of those will be inertial. But I think it is a pretty standard definition (at least in Special Relativity) that the proper time between two time-like separated events A and B IS the spacetime interval between events A and B and IS the time ticked off by an inertial clock for which events A and B happen at the same place. I should have added the word "inertial" in front of clock in my previous post.

 

[ghwellsjr]

I can understand your objection. Between time-like separated events A and B there are many time-like world lines which pass through both (i.e. for which both are happening at the same place), but only one of those will be inertial. But I think it is a pretty standard definition that the proper time between two time-like separated events A and B IS the spacetime interval between events A and B and IS the time ticked off by an inertial clock for which events A and B happen at the same place. I should have added the word "inertial" in front of clock in my previous post.

Yours is not the definition that wikipedia gives for Proper Time. Can you provide a reference that supports your claim?

And why are you including "at the same place"? That is not a requirement for the spacetime interval.

 

[Matterwave]

Yours is not the definition that wikipedia gives for Proper Time. Can you provide a reference that supports your claim?

Well, you might look at MTW Section 1.4 wherein they talk about the "Interval = proper distance/proper time" between two close by events. I quote "In spacetime the intervals ("proper distance," "proper time") between event and event satisfy the corresponding theorems of Lorentz-Minkowski geometry..." "...From any event A to any other nearby event B, there is a proper distance, or proper time, given in suitable (local Lorentz) coordinates by [defines spacetime interval]" These authors seemed to have conflated spacetime intervals with proper distance and proper time.

And why are you including "at the same place"? That is not a requirement for the spacetime interval.

What do you mean? If events A and B do not happen at the same place in your (inertial) frame, can you still say that the proper time (read: space-time interval) between them is the time ticked off by your clock?

EDIT: Maybe I should make an analogy to see if you agree that our disagreement is analogous to this:

Say I draw 2 dots on a piece of paper A and B and you ask "what is the distance between A and B?". My answer would be "The distance between A and B is given by the straight-edge that I place between A and B. In other words, I draw a straight line between A and B and measure the length of that line, that is the distance between A and B". Your answer would be "The distance between A and B depends on the path you take between A and B, there are many lines that connect A and B and different lines have different lengths, so there isn't one distance between A and B".

Of course your answer would be right in some contexts, and mine would be the assumed one in others. For example, if we are talking about distances between cities, we may not always talk about the straight-line distance since most people travel between cities driving cars which have to follow roads. Even without considerations for roads, we usually don't consider the straight-line distance through the Earth, but rather the distance along the surface. However, if we are talking about say the distance between my ceiling and my floor, we are usually talking about a straight-line distance.

Would you say this is a fair analogy of our disagreement?

 

[Fredrik]

Matterwave, what you call the proper time between two timelike separated events A and B is the proper time of the geodesic from A to B, i.e. the curve from A to B that has the minimum maximum proper time. The definition of proper time assigns a number (the proper time of the curve) to each timelike curve that satisfies some technical requirements, not just to each pair of timelike separated events.

Edit: I changed minimum to maximum after Matterwave's correction below.

 

[Matterwave]

Matterwave, what you call the proper time between two timelike separated events A and B is the proper time of the geodesic from A to B, i.e. the curve from A to B that has the minimum proper time. The definition of proper time assigns a number (the proper time of the curve) to each timelike curve that satisfies some technical requirements, not just to each pair of timelike separated events.

I think you mean maximum proper time...?

As far as I know, in this thread we are working in flat space-time. As such, between any two time-like separated events A and B there is exactly 1 inertial reference frame for which A and B occur at the same place. This is the "straight line" path from A to B. All other reference frames for which A and B occur at the same place must be non-inertial at some point, (if A and B don't occur at the same place, we can't use just a clock to tick off the space-time interval between A and B) these paths are the non "straight line" paths from A to B. I don't think it is uncommon that we take the "straight line path" as THE definition of the proper time between events A and B.

Please see the edit in my previous post.

I don't know, maybe there is a large majority who do not work with the same terminology that I work with. But I have basically always worked with "when the space-time interval is negative (depending on your signature convention), we call it a proper time, when the space-time interval is positive, we call it a proper distance".

 

[Fredrik]

I think you mean maximum proper time...?

Oops, yes.

As far as I know, in this thread we are working in flat space-time.

Yes, but we're not only working with geodesics. For example, the path of the astronaut twin in the standard twin paradox scenario is not a geodesic. It consists of two geodesics joined together, so it can still be discussed in terms of inertial coordinate systems. You just need two of them instead of one. But if we change the scenario a bit, e.g. by considering constant acceleration during the turnaround phase instead of infinite acceleration at a single event, we no longer have that option.

I don't think it is uncommon that we take the "straight line path" as THE definition of the proper time between events A and B.

I don't know how common it is, but I don't think it's a good definition. (It's certainly not THE definition). What makes proper time such a useful concept is that it can be assigned to any massive object's world line. But sure, if you really want to associate a proper time with a pair of timelike separated events (instead of with a timelike curve), then your way is the best way to do it.

I don't know, maybe there is a large majority who do not work with the same terminology that I work with. But I have basically always worked with "when the space-time interval is negative (depending on your signature convention), we call it a proper time, when the space-time interval is positive, we call it a proper distance".

I think this way too, but only about infinitesimal segments of curves.

 

[ghwellsjr]

Yours is not the definition that wikipedia gives for Proper Time. Can you provide a reference that supports your claim?

Well, you might look at MTW Section 1.4 wherein they talk about the "Interval = proper distance/proper time" between two close by events. I quote "In spacetime the intervals ("proper distance," "proper time") between event and event satisfy the corresponding theorems of Lorentz-Minkowski geometry..." "...From any event A to any other nearby event B, there is a proper distance, or proper time, given in suitable (local Lorentz) coordinates by [defines spacetime interval]" These authors seemed to have conflated spacetime intervals with proper distance and proper time.

As I said earlier, you are talking about a [time-like] spacetime interval, which, of course, is the Proper Time on an inertial clock that is present at both events. MTW also state that it is the Coordinate Time interval in the frame in which the two events are at the same place. But these are definitions for the spacetime interval, not a definition of Proper Time.

And why are you including "at the same place"? That is not a requirement for the spacetime interval.

What do you mean? If events A and B do not happen at the same place in your (inertial) frame, can you still say that the proper time (read: space-time interval) between them is the time ticked off by your clock?

You are equating "your clock" with a clock that is at rest in "your (inertial) frame" and therefore cannot be a clock in motion in "your (inertial) frame". But, as I said before, you are being too restrictive. A and B do not have to be at the same place in "your (inertial) frame" or any other (inertial) frame. If you are trying to measure a timelike spacetime interval, all that matters is that an inertial clock be present at both events. If the events are not at the same place according to a particular frame, then the clock must be moving according to that frame, and if the clock moves inertially between those two events, then the Proper Time on that clock will measure the spacetime interval between those two events. If the clock is not moving inertially between those two events, then it is still the Proper Time for that clock but it is not the spacetime interval. Furthermore, the Proper Time on either of these two clocks (one inertial and one non-inertial) is invariant, all frames agree on the calculation or measurement of those time intervals.

EDIT: Maybe I should make an analogy to see if you agree that our disagreement is analogous to this:

Say I draw 2 dots on a piece of paper A and B and you ask "what is the distance between A and B?". My answer would be "The distance between A and B is given by the straight-edge that I place between A and B. In other words, I draw a straight line between A and B and measure the length of that line, that is the distance between A and B". Your answer would be "The distance between A and B depends on the path you take between A and B, there are many lines that connect A and B and different lines have different lengths, so there isn't one distance between A and B".

Of course your answer would be right in some contexts, and mine would be the assumed one in others. For example, if we are talking about distances between cities, we may not always talk about the straight-line distance since most people travel between cities driving cars which have to follow roads. Even without considerations for roads, we usually don't consider the straight-line distance through the Earth, but rather the distance along the surface. However, if we are talking about say the distance between my ceiling and my floor, we are usually talking about a straight-line distance.

Would you say this is a fair analogy of our disagreement?

Our disagreement is that you are claiming that the definition of Proper Time is the same as the definition of a timelike spacetime interval. If the OP had asked: what does "invariance of spacetime interval" mean?, then your answers would be on the right subject (although still not on target because you are restricting your self to one frame). But he didn't ask about the spacetime interval so your answers are not on the right subject.

 

[ChrisVer]

Again, what's wrong or misleading in restricting yourself in one particular RF when all the RF are connected via Lorentz Transformations?

 

[ghwellsjr]

As far as I know, in this thread we are working in flat space-time.

Agreed.

As such, between any two time-like separated events A and B there is exactly 1 inertial reference frame for which A and B occur at the same place. This is the "straight line" path from A to B. All other reference frames for which A and B occur at the same place must be non-inertial at some point,

As far as I know, in this thread we are working only with inertial frames.

(if A and B don't occur at the same place, we can't use just a clock to tick off the space-time interval between A and B)

Of course we can. A clock can move inertially between A and B (assuming a time-like separation) and it will tick off the space-time interval between A and B. And that will be the Proper Time on that clock.

these paths are the non "straight line" paths from A to B.

As long as the path is for an inertial clock, then it will be a "straight line" path from A to B even though A and B are not at the same place according to a particular inertial reference frame.

I don't think it is uncommon that we take the "straight line path" as THE definition of the proper time between events A and B.

That is the definition of the spacetime interval between events A and B which also happens to be the Proper Time interval of an inertial clock that passes through A and B. I asked you for a reference and you gave me MTW which I pointed out does not support your definition. If your definition is not uncommon, can you find a reference for the definition of Proper Time (not a definition for Spacetime Interval or Lorentz Interval or Invariant Interval) that defines it the way you do. I pointed you to the definition of Proper Time in wikipedia and it does not agree with yours.

Please see the edit in my previous post.

I don't know, maybe there is a large majority who do not work with the same terminology that I work with. But I have basically always worked with "when the space-time interval is negative (depending on your signature convention), we call it a proper time, when the space-time interval is positive, we call it a proper distance".

That's like saying when a chicken is male we call it a rooster and when a chicken is female we call it a hen, therefore all males are roosters and all females are hens.

 

[ghwellsjr]

Again, what's wrong or misleading in restricting yourself in one particular RF when all the RF are connected via Lorentz Transformations?

The question is about invariance. That doesn't make any sense if you are restricting yourself to one particular RF.

 

[ChrisVer]

Invariance means that a quantity Y(RF1)=Y(RF2) for any RF2 (Lorentz transformed)...of course it isn't necessary to restrict yourself, but exactly because we are talking about invariance, nobody stops you from doing so.
So if you choose RF1, and you can show that Y doesn't change for some arbitrary other RF2, then you can say Y is invariant.

 

[ghwellsjr]

Invariance means that a quantity Y(RF1)=Y(RF2) for any RF2 (Lorentz transformed)...of course it isn't necessary to restrict yourself, but exactly because we are talking about invariance, nobody stops you from doing so.
So if you choose RF1, and you can show that Y doesn't change for some arbitrary other RF2, then you can say Y is invariant.

If you restrict yourself to one RF, such as the first one in post #3, you might believe that the blue twin's time of 10 years being simultaneous with the red twin's turnaround time was invariant but by showing two other RF's, it is obvious that it is not. On the other hand, after seeing the second and third RF's you might think that the blue twin's time of 16 years when he sees the red twin turning around in the fourth diagram could not be invariant until you can see it in the fifth and sixth diagrams.

 

[ChrisVer]

But you have two reference frames in each diagram...one moving and one stationary...
In any diagram the times are not invariant because the one [red or blue] line is a Lorentz transformed RF of the other [blue or red].
This is reflected in writing [stationary=1, moving=2]:
d \tau^2_1 = dt^2_1 = dt^2_2 - dx^2_2 = \frac{dt^2_2}{\gamma^2}= d \tau^2_2

 

[ghwellsjr]

But you have two reference frames in each diagram...one moving and one stationary...

I know that it is quite popular to make spacetime diagrams that include two reference frames but that's not the way I do mine. I think it is too confusing and totally unnecessary to mix two frames on to one diagram. Even if you do, it is not a requirement to make one moving and one stationary. A Loedel diagram is one in which the two frames are moving in equal but opposite directions.

But you are welcome to take any of my diagrams or make up your own and combine them into one diagram of the type that you are talking about and provide explanation to teach how to interpret them.

In any diagram the times are not invariant

I can't speak for other diagrams, but for mine, I mark off equal increments of Proper Times along the worldlines with dots and they are invariant as can be see by the different diagrams.

because the one [red or blue] line is a Lorentz transformed RF of the other [blue or red].

The thick lines on my diagrams are worldlines of massive objects representing observers and/or their clocks. They are not Reference Frames. The entire diagram with its grid lines and coordinates represents the Reference Frame.

This is reflected in writing [stationary=1, moving=2]:
d \tau^2_1 = dt^2_1 = dt^2_2 - dx^2_2 = \frac{dt^2_2}{\gamma^2}= d \tau^2_2

I don't know what you are trying to represent by these equations. You need more explanation.

 

[ChrisVer]

I am saying you have an observer on the blue line, that doesn't move wrt to it, so as the blue line progresses in spacetime, he is measuring the time dt_1 alone with the ticking of the clock (his dx=0),,, So your RF is that which moves together with the blue one...
Now your red observer is moving with respect to the blue. His measurements are thus not only d \tau_1=dt_1 (as you tried to note with the simultaneous point of acceleration) but he is also measuring dx . So the proper time he measures it
d \tau_2^2 = dt_2^2 - dx^2_2 = dt_2^2 (1 - \frac{dx_2^2}{dt_2^2}) = \frac{dt_2^2}{\gamma^2}

What you meant by the non matching simultaneous times is that you are trying to say that dt_1 =dt_2 which is obviously wrong since they are not at rest wrt each other.
What is though the same for both of them in any diagram, is that d \tau_1 = d \tau_2

 

[ghwellsjr]

I am saying you have an observer on the blue line, that doesn't move wrt to it, so as the blue line progresses in spacetime, he is measuring the time dt_1 alone with the ticking of the clock (his dx=0),,, So your RF is that which moves together with the blue one...
Now your red observer is moving with respect to the blue. His measurements are thus not only d \tau_1=dt_1 (as you tried to note with the simultaneous point of acceleration) but he is also measuring dx . So the proper time he measures it
d \tau_2^2 = dt_2^2 - dx^2_2 = dt_2^2 (1 - \frac{dx_2^2}{dt_2^2}) = \frac{dt_2^2}{\gamma^2}

What you meant by the non matching simultaneous times is that you are trying to say that dt_1 =dt_2 which is obviously wrong since they are not at rest wrt each other.
What is though the same for both of them in any diagram, is that d \tau_1 = d \tau_2

I still have no idea what you are talking about. Would you please define what all the terms are referring to? And would you please state which of my diagrams you are referencing off of, if any? Are you referring to the dots on the diagrams or the grid lines or what? And any time you are talking about an observer measuring something, would you please state how he is making that measurement. Keep in mind that this thread is talking about the invariance of Proper Time so please try to make it clear how your discussion relates to that topic.

 

[ChrisVer]

define what? the coordinate time dt_i as observed by the i twin in each of your diagrams? or the proper time tau_i?
Take whichever diagram you want (of course from yours, I am not going to invent any diagram) but in particular I wrote about the 1st diagram...
I am referring to the wordlines of your diagrams...
I don't find any reason to state how the measurement is done... but for the blue line in your first diagram, the measurement is done by the blue line's clock, whereas the measurement of the red should be done by ruler and clock (trivial SR stuff)

 

[Matterwave]

Well, to me to say "the proper time between events A and B" in special relativity it is natural to assume that it is the time as measured by the inertial frame for which events A and B occur at the same place. This is as natural to me as saying the "distance between points A and B is measured by the straight line distance between them". But if everybody is not happy with that, I can change the language I use to "spacetime interval" with no problems. I don't see what's the point of allowing the clock you use to measure time to move within your inertial reference frame though...if the clock is not co-moving with you, then that clock is not a good one for you to make measurements with. It would be like using a ruler that isn't straight...

 

[Fredrik]

I don't see what's the point of allowing the clock you use to measure time to move within your inertial reference frame though...if the clock is not co-moving with you, then that clock is not a good one for you to make measurements with. It would be like using a ruler that isn't straight...

Is anyone saying that the coordinate time of an inertial coordinate system should be measured by a clock that isn't stationary in that coordinate system? I haven't read every post, but I doubt that anyone here would make such a claim.

I don't know why you're so focused on inertial coordinate systems. The OP didn't ask specifically about them. There are lots of scenarios that involve accelerating clocks, in particular the twin paradox. The definition of "proper time" can be stated without even mentioning a coordinate system, inertial or non-inertial.

 

[Matterwave]

Is anyone saying that the coordinate time of an inertial coordinate system should be measured by a clock that isn't stationary in that coordinate system? I haven't read every post, but I doubt that anyone here would make such a claim.

If I was reading ghwellsjr's responses correctly he is saying I am too restrictive when I specify that in order for the Space-time interval between two time-like separated events to be properly measured by a clock (and just a clock), one must do so in an inertial reference frame where the two events happen at the same place. He seems to say that simply the clock must be in inertial motion and be at the two events, and my "inertial reference frame" need not have the two events happening at the same place. That is how I read his responses to my post in his post #16.

I don't know why you're so focused on inertial coordinate systems. The OP didn't ask specifically about them. There are lots of scenarios that involve accelerating clocks, in particular the twin paradox. The definition of "proper time" can be stated without even mentioning a coordinate system, inertial or non-inertial.

The OP asked why Proper time is an invariant. I interpreted that statement as asking why the space-time interval for two time-like separated events is an invariant, and I wanted to make things more physical for him by specifying an inertial reference frame from which to work.

 

[PeterDonis]

The OP asked why Proper time is an invariant. I interpreted that statement as asking why the space-time interval for two time-like separated events is an invariant

I think the question can be interpreted more generally than that (though I don't know if the OP intended it that way). The proper time along any timelike curve, geodesic or not, between two events is an invariant. The reason is simple geometry; the proper time is just the geometric length of the curve.

 

[Matterwave]

I think the question can be interpreted more generally than that (though I don't know if the OP intended it that way). The proper time along any timelike curve, geodesic or not, between two events is an invariant. The reason is simple geometry; the proper time is just the geometric length of the curve.

Ok, fair enough.

 

[Saw]

Just wanted to say that I like Matterwave's approach. Of course all other mathematical / geometrical concepts that have been mentioned are good and necessary, but if we want to be didactic and pedagogical, one has to unveil the phsyical background which those concepts "represent". Thus in terms of measurement, I think it is clear in the literature that proper time between events A and B is what a watch being present at both events measures, ie its ticks between A and B. Hence we are talking, yes, about timelike events, since a material and hence infra-luminal object (the clock in question) has "moved" in spacetime from one to the other. In the frame where such clock is at rest, the same has only moved in time. In other frames, it has moved both along space and time. And when all frames calculate the spacetime interval between the two events, by plugging into an identical formula [sqr (∆t2 - ∆x2)] their respective time and distance measurements, they all get the same value, that is why it is invariant.

But I would go one step further. This measurement is invariant because it reflects a physical happening, which is often the key to solving a problem. (You know, reality is usually invariant...) The measurement of the watch present at both events, its chronological ticks are the direct image or reflection of a phsyical issue that may be the object of a practical problem. For example, you may want to know if Mr X, who has witnessed event A, will survive until event B, knowing that his health will only allow 20 beats more of his heart (20 biological ticks). Or you may want to know if his clock itself will survive to event B, in view of its mechanical state.

 

[Fredrik]

I think it is clear in the literature that proper time between events A and B is what a watch being present at both events measures, ie its ticks between A and B.

Only if it's moving at constant velocity from A to B. If it accelerates at all, the number of ticks will not match "the proper time between A and B". It will match the proper time of the clock's world line from A to B.

 

[ghwellsjr]

I am saying you have an observer on the blue line, that doesn't move wrt to it, so as the blue line progresses in spacetime, he is measuring the time dt_1 alone with the ticking of the clock (his dx=0),,, So your RF is that which moves together with the blue one...
Now your red observer is moving with respect to the blue. His measurements are thus not only d \tau_1=dt_1 (as you tried to note with the simultaneous point of acceleration)

I never tried to note this. I have no idea what you are talking about. The adjacent red dots indicating Proper Time are spaced farther apart than the Coordinate Time which shows Time Dilation.

but he is also measuring dx . So the proper time he measures it
d \tau_2^2 = dt_2^2 - dx^2_2 = dt_2^2 (1 - \frac{dx_2^2}{dt_2^2}) = \frac{dt_2^2}{\gamma^2}

What you meant by the non matching simultaneous times is that you are trying to say that dt_1 =dt_2 which is obviously wrong since they are not at rest wrt each other.

I never meant what you are claiming. In each of my diagrams, there is only one set of coordinates. There are not separate coordinates for each observer. The differential form of your equation has nothing to do with simultaneous times. Again, I have no idea what you are talking about.

What is though the same for both of them in any diagram, is that d \tau_1 = d \tau_2

No, they are not the same in any of the diagrams I drew. I have no idea what you are talking about. Here's the diagram. Can you show any of the claims that you are making?

 

[ChrisVer]

1. In this diagram that you are showing, we are in the RF that is moving along with the blue line (the object following the blue line trajectory is at rest in this RF).
2. The proper time the object in blue measures is given by d \tau_1^2 = dt^2. Since it's not moving there is no displacement $dx$ you can measure with the ruler. What you measure is the time dt of its clock.
3. The red line is an object moving with respect to your particular RF. That means that you can measure distances dx it covers with a ruler of your RF, and also the time dt with your clock. The proper time it measures is given by d \tau_2^2 = dt'^2 - dx'^2 = \frac{dt'^2}{\gamma^2} (you need primed coordinate times since the red object is boosted wrt the blue object)
4. The invariance of proper time says that d\tau_1 = d \tau_2

 

[A.T.]

The invariance of proper time says that d\tau_1 = d \tau_2

No, the invariance of proper time means that \Delta\tau_1is the same in every RF and that \Delta\tau_2 is the same in every RF.

 

[Fredrik]

No, The invariance of proper time means that \Delta\tau_1is the same in every RF and that \Delta\tau_2 is the same in every RF.

I think that what he meant by $d\tau_1$ and $d\tau_2$ is $\sqrt{dt^2-dx^2}$ and $\sqrt{dt'^2-dx'^2}$.

If 1 and 2 was meant to denote different line segments and $d\tau_1$ and $d\tau_2$ are their proper times, then I agree with you of course.

 

[ghwellsjr]

I think that what he meant by $d\tau_1$ and $d\tau_2$ is $\sqrt{dt^2-dx^2}$ and $\sqrt{dt'^2-dx'^2}$.

No, that's not what he meant, at least that not what he said. He said:

d \tau_1^2 = dt^2

and:

d \tau_2^2 = dt'^2 - dx'^2 = \frac{dt'^2}{\gamma^2}

And I'm not just referring to the trivial squared/square root difference. Look also at his post number #20.

 

[ChrisVer]

It's the same thing...
$$d \tau_1^2 = dt^2 - dx^2= dt^2 (dx=0)$$ or if you like write it $$d \tau_1 = \sqrt{dt^2 - dx^2}=dt$$
for the same reason...
On the other hand:

$$d \tau_2^2 = dt'^2 - dx'^2= dt'^2 (1 + \frac{dx'^2}{dt'^2}) = \frac{dt'^2}{\gamma^2}$$ or else:

$$d \tau_2 = \sqrt{dt'^2 - dx'^2}= \frac{dt'}{\gamma}$$ ...

I don't understand what you are trying to say...

 

[ghwellsjr]

Just wanted to say that I like Matterwave's approach. Of course all other mathematical / geometrical concepts that have been mentioned are good and necessary, but if we want to be didactic and pedagogical, one has to unveil the phsyical background which those concepts "represent". Thus in terms of measurement, I think it is clear in the literature that proper time between events A and B is what a watch being present at both events measures, ie its ticks between A and B. Hence we are talking, yes, about timelike events, since a material and hence infra-luminal object (the clock in question) has "moved" in spacetime from one to the other. In the frame where such clock is at rest, the same has only moved in time. In other frames, it has moved both along space and time. And when all frames calculate the spacetime interval between the two events, by plugging into an identical formula [sqr (∆t2 - ∆x2)] their respective time and distance measurements, they all get the same value, that is why it is invariant.

But I would go one step further. This measurement is invariant because it reflects a physical happening, which is often the key to solving a problem. (You know, reality is usually invariant...) The measurement of the watch present at both events, its chronological ticks are the direct image or reflection of a phsyical issue that may be the object of a practical problem. For example, you may want to know if Mr X, who has witnessed event A, will survive until event B, knowing that his health will only allow 20 beats more of his heart (20 biological ticks). Or you may want to know if his clock itself will survive to event B, in view of its mechanical state.

If Mr X. wants to get to event B after being at event A, as long as they are time-like separated, he can get there before his heart gives out, no matter how soon that is. But not if he thinks that he has to be limited to the spacetime interval between those two events.

For example, here is a spacetime diagram showing Mr X reaching event A and wanting to get to event B before 20 more beats of his heart:

He can calculate the spacetime interval to event B as 24 beats so if he makes a bee-line to it, he will succumb before getting there as shown here:

However, if he just continues for a while longer without accelerating until sometime later, he can get to event B before his 20th beat:

Now the point of this exercise is to address the OP's question which is about the invariance of Proper Time. The spacetime interval is one example of an invariant Proper Time where an inertial clock goes between the two events in question. Note that these two events are not at the same Coordinate Distance in this RF. But another example is the second non-inertial path which has a different Proper Time interval between the two events but is also invariant. My point is that we don't want to be mislead that it is only an inertial clock that measures an invariant Proper Time interval, the same is true for any clock that passes through both events.

 

[ghwellsjr]

It's the same thing...
$$d \tau_1^2 = dt^2 - dx^2= dt^2 (dx=0)$$ or if you like write it $$d \tau_1 = \sqrt{dt^2 - dx^2}=dt$$
for the same reason...
On the other hand:

$$d \tau_2^2 = dt'^2 - dx'^2= dt'^2 (1 + \frac{dx'^2}{dt'^2}) = \frac{dt'^2}{\gamma^2}$$ or else:

$$d \tau_2 = \sqrt{dt'^2 - dx'^2}= \frac{dt'}{\gamma}$$ ...
Sure, if you're going to drop the subscripts and merely point out that we use the same formula to calculate the Time dilation of each observer, of course there's no problem with that. That's what I programmed into my application that draws my diagrams. But you started with this from post #20:

But you have two reference frames in each diagram...one moving and one stationary...
In any diagram the times are not invariant because the one [red or blue] line is a Lorentz transformed RF of the other [blue or red].
This is reflected in writing [stationary=1, moving=2]:
d \tau^2_1 = dt^2_1 = dt^2_2 - dx^2_2 = \frac{dt^2_2}{\gamma^2}= d \tau^2_2

That's what was confusing, especially when you said there were two reference frames in each of my diagrams and that the times are not invariant. I still don't know why you would write something like that.

And then you accused me of making false statements in post #22:

I am saying you have an observer on the blue line, that doesn't move wrt to it, so as the blue line progresses in spacetime, he is measuring the time dt_1 alone with the ticking of the clock (his dx=0),,, So your RF is that which moves together with the blue one...
Now your red observer is moving with respect to the blue. His measurements are thus not only d \tau_1=dt_1 (as you tried to note with the simultaneous point of acceleration) but he is also measuring dx . So the proper time he measures it
d \tau_2^2 = dt_2^2 - dx^2_2 = dt_2^2 (1 - \frac{dx_2^2}{dt_2^2}) = \frac{dt_2^2}{\gamma^2}

What you meant by the non matching simultaneous times is that you are trying to say that dt_1 =dt_2 which is obviously wrong since they are not at rest wrt each other.
What is though the same for both of them in any diagram, is that d \tau_1 = d \tau_2

I have asked for clarification. Or a retraction would do fine.

 

[Saw]

My point is that we don't want to be mislead that it is only an inertial clock that measures an invariant Proper Time interval, the same is true for any clock that passes through both events.

Ok, noted. If I understand it well, what several of you were pointing out is that what is technically called "Proper time" and is invariant is not only the time measured by a clock traveling inertially between the two events (which is the ST interval) but also the time measured by a clock present at both events after moving through some non-inertial path.

If so, ok, I guess that my attempt at "making the explanation more physical" should bear in mind that complication. That would require some rephrasing. But the basic idea is still valid, isn't it? The proper time is a direct and simple image of what happens in the example. You could learn it directly and simply by putting a watch around the wrist of Mr X, while he jumps from one RF to another (which also makes the watch change RFs). Or, if you are located in another RF which is less lucky, you will have to use the combination of several time and distance measurements so as to indirectly, with a little more complication, reach the same conclusion about the proper time between the two events and hence about what actually happened, which is obviously the same as for the first observer (ie, since proper time and the reality that it mirrors are invariant).

 

[ChrisVer]

From a prespective there are two RFs in each of your diagram... In the one we are talking about, the blue line observer is the rest RF and the red line observer is on a Lorentz boosted RF moving with u relative to the blue. Think about a muon and an observer on earth... if you are on the muon's rest RF then the observer is on a boosted frame wrt you..
by times I meant coordinate times. Obviously dt \ne dt'... the observer at rest will see that time to reach the acceleration point to be dt=10 ~yrs but the time for the moving one will be dt'=10 \gamma ~yrs (but the elapsed proper time will be the same for both)...

Then what I said about the simultaneous point of acceleration is that the proper time to reach acceleration, both of them measure, is the same. The coordinate time interval is not. Simultaneity is related to the coordinate time [but the clocks are not synchronized anymore] and not the proper time. In analogy to the muon example I mentioned above, the acceleration point can be replaced by some event- let's say the muon's decay- so the elapsed time the muon will measure is dt=2.2 ~\mu s whereas for the observer on Earth it can be much larger depending on their relative velocity by the gamma factor. Obviously in both the reference frames [muon, observer] the event of the decay doesn't happen simultaneously, but they both agree on some value- the muon's lifetime[ at rest]=proper time

 

[ghwellsjr]

My point is that we don't want to be mislead that it is only an inertial clock that measures an invariant Proper Time interval, the same is true for any clock that passes through both events.

Ok, noted. If I understand it well, what several of you were pointing out is that what is technically called "Proper time" and is invariant is not only the time measured by a clock traveling inertially between the two events (which is the ST interval) but also the time measured by a clock present at both events after moving through some non-inertial path.

Proper Time is the time measured by a clock no matter what it does. You don't have to stipulate anything about it being inertial or non-inertial. You don't have to limit it to any particular events. Every tick of the clock is an event. Every fraction of a tick of the clock is an event. Every interval between any of these events is invariant.

If so, ok, I guess that my attempt at "making the explanation more physical" should bear in mind that complication. That would require some rephrasing. But the basic idea is still valid, isn't it?

Yes, I can't think of anything more physical than the ticking of a clock. You don't have to work at making it more physical though. It's already there.

The proper time is a direct and simple image of what happens in the example. You could learn it directly and simply by putting a watch around the wrist of Mr X, while he jumps from one RF to another (which also makes the watch change RFs).

I cringe whenever I hear someone using this phraseology. Mr X doesn't jump frames just because he accelerates. Note that in my example in post #38, Mr X is never at rest. I'm sure what you mean is that he starts out at rest in one Inertial Reference Frame (IRF) and after he accelerates, he is at rest in another IRF. Although that is true, it is also true that he may start out at rest in one IRF and then he accelerates and is no longer at rest in that same IRF. But in my example, Mr X starts out at one speed according to the IRF in which I defined the scenario and after acceleration, ends up at a second speed according to the same IRF. In what sense do you consider him to jump or change frames?

Or, if you are located in another RF which is less lucky, you will have to use the combination of several time and distance measurements so as to indirectly, with a little more complication, reach the same conclusion about the proper time between the two events and hence about what actually happened, which is obviously the same as for the first observer (ie, since proper time and the reality that it mirrors are invariant).

Now you've got me worried. Are you talking about your Mr X scenario? If so, who is the first observer?

 

[ghwellsjr]

From a prespective there are two RFs in each of your diagram... In the one we are talking about,

That would be this diagram:

the blue line observer is the rest RF

I have remade a diagram showing just the portion of interest for the rest RF of the blue line observer:

and the red line observer is on a Lorentz boosted RF moving with u relative to the blue.

To show this, we start with the red line observer in his rest RF:

And then we boost him using the Lorentz Transformation so that he is traveling at a speed of u=0.6c:

And then we combine the boosted red line observer's RF with the blue line observer's rest RF to get this:

Think about a muon and an observer on earth... if you are on the muon's rest RF then the observer is on a boosted frame wrt you..
by times I meant coordinate times. Obviously dt \ne dt'... the observer at rest will see that time to reach the acceleration point to be dt=10 ~yrs

Yes, the blue line observer at rest has a Proper Time equal to the Coordinate Time of 10 years.

but the time for the moving one will be dt'=10 \gamma ~yrs

Since gamma at u=0.5c is 1.25, this evaluates to 12.5 years but I don't see anything in any of the diagrams that corresponds to that coordinate time. It seems to me that you should have used dt'=8 \gamma ~yrs from the red observer's rest RF to his Lorentz boosted RF which evaluates to a Coordinate Time of 10 years. Please explain.

(but the elapsed proper time will be the same for both)...

This doesn't seem correct to me. The blue line observer's final Proper Time is 10 years and the red line observer's Proper Time is 8 years which are different, not the same. Please explain.

Then what I said about the simultaneous point of acceleration is that the proper time to reach acceleration, both of them measure, is the same.

Are you saying that the blue observer measures the Proper Time of the red observer at the point of acceleration to be 8 years, just like the red line observer measures? If so, what has that got to do with simultaneity?

The coordinate time interval is not. Simultaneity is related to the coordinate time [but the clocks are not synchronized anymore] and not the proper time.

Simultaneity is related to two different events and whether or not they have the same Coordinate Time. What two events are you talking about?

 

[georgir]

In Euclidean geometry, the distance between any points and the length of any line (curved or not) are invariant under rotation transformations. What does that mean?
It is something you would consider quite obvious so you might even say the question is stupid.
It means that even though coordinate values and coordinate differences can both change from the transformation, the sum of the squares of coordinate differences (and therefore its square root as well, which we call "distance") remains the same.

In relativity, space-time intervals between events ("distance") and proper time along any world-line ("length") are invariant under Lorentz transformation.
It means the same as above - specific coordinates and the differences between them can change, but the sum of the squares of the coordinate differences (with the distinction that time's square is taken to be negative) remains the same.

"Time" between two events is a difference in coordinates, and is changed by Lorentz transformations.
But "proper time along a specific worldline" is analogous to "length of a specific curve" in Euclidean geometry, and is not changed.
"Proper time" between two events in general is a bad phrase, just as "length" between two points - because there can be many lines connecting them. If someone uses it, they most likely mean "space-time interval", the analog of "distance", but the ambiguity is more dangerous in context of "invariance" discussions. Consider how "Length between two points is invariant" may be misinterpreted as the incorrect "The length of all curves between two points is equal" instead of the probably intended "The distance (length of the shortest line) between two points is invariant under rotation".

 

[ChrisVer]

This doesn't seem correct to me. The blue line observer's final Proper Time is 10 years and the red line observer's Proper Time is 8 years which are different, not the same. Please explain.

Then you are saying that a Lorentz transformation [Lorentz Boost] is going to change the proper time that we call invariant under Lorentz transformations? I am not the one who needs to explain but you. The problem is that the proper time is not that dt' but it will be the dt'/\gamma. So indeed you have to measure dt'=12.5 ~yr in order for the proper time to be invariant for both.
In order to see the 12.5yr in diagram you have to redo it in the red's rest reference frame... Then they will be equivalent but the red line will have to travel from t=0 to t=12.5 yrs...
The blue will measure d \tau = dt = 10~yr, so I got d \tau. Using the invariance of it, you can determine dt' = d \tau ~\gamma=10 \gamma

 

[ghwellsjr]

Then you are saying that a Lorentz transformation [Lorentz Boost] is going to change the proper time that we call invariant under Lorentz transformations?

No, I said the Proper Time of the turn around event for the red observer is 8 years in both his rest frame and in the frame in which he is traveling at 0.6c and I have shown this in several diagrams.

I am not the one who needs to explain but you. The problem is that the proper time is not that dt' but it will be the dt'/\gamma. So indeed you have to measure dt'=12.5 ~yr in order for the proper time to be invariant for both.
In order to see the 12.5yr in diagram you have to redo it in the red's rest reference frame... Then they will be equivalent but the red line will have to travel from t=0 to t=12.5 yrs...

OK, if you insist:

And now after boosting so that the red observer is traveling at 0.6c:

Is that what you want? As always, the Proper Time of the last (uppermost) dot is 12.5 years in both frames but I don't see how that relates to what you are explaining.

The blue will measure d \tau = dt = 10~yr, so I got d \tau. Using the invariance of it, you can determine dt' = d \tau ~\gamma=10 \gamma

Why are you talking about the blue observer or his Proper Time? We aren't boosting his rest RF. What has the blue observer or his Proper Time got to do with the red observer's Proper Time?