# What does "invariance of proper time" mean?

It would be better with a clear example to understand

## Answers and Replies

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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
Gold Member
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
Gold Member
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' ]...

Thanks to all. Better explanations

Matterwave
Gold Member
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
Gold Member
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
Gold Member
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.

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ghwellsjr
Gold Member
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
Gold Member
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?

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Fredrik
Staff Emeritus
Gold Member
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.

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

ghwellsjr
Gold Member
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
Gold Member
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
Gold Member
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
Gold Member
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
Gold Member
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]:
$d \tau^2_1 = dt^2_1 = dt^2_2 - dx^2_2 = \frac{dt^2_2}{\gamma^2}= d \tau^2_2$

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ghwellsjr
Gold Member
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 in to 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
Gold Member
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
Gold Member
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
Gold Member
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 refering 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