Measuring distance, speed and clock

[Stephanus]

Thank you mentors and readers for your responses.
I'm sorry, if I haven't got the chance to study all of them. Someone in this forum gives me a very good software to draw space time diagram.
So, I can draw ST less than 5 minutes compared to using Excel in 1 hour. So I can draw ST anytime I want.
I still study the software.
I don't even have the chance to study
HarryLyn - #147
HarryLyn - #16
Pervect - #17
And surely the next posts.
But thanks anyway.
I will response later.

 

[ghwellsjr]

Rather than say that distance "doesn't exist", I find it more expedient to say that it's a confusing concept because it's observer dependent. I spend a fair amount of time trying to explain to non-physicists why, in the context of special relativity, the observer independent Lorentz interval is actually simpler than the observer-dependent notion of distance, with rather limited success.

I don't think it's confusing at all. Nugatory gave a very simple explanation in the very first response on this thread:

Distance to an object at rest relative to us is easy - bounce a light signal off it, divide the round-trip time by two, multiply by c, and we have the distance. If we get the same answer every time we know the object is at rest relative to us - if not, we know the distance to the object at the moment that the reflection happened.

And in fact, that's exactly the same process that could be used for a space-like Lorentz interval where an inertial observer passes through one of the two events in question and the "moment" of the reflection is midway between the sending and receiving of the light signals.

But in reality, the Lorentz interval is not the same kind of a thing as distance because usually when we are talking about distance, we don't mean the distance between two events but rather the distance between two objects at a particular time, which of course, will be different events depending on the selected reference frame. In Nugatory's definition, the assumed reference frame is the rest frame of the observer (who is also one of the objects) but unless we are willing to teach, and have those non-physicists learn, that distance is frame-dependent (not necessarily observer-dependent), then we haven't taught, and they haven't learned, Special Relativity.

 

[Stephanus]

[..]Do you understand how to measure the distance to an object by bouncing a signal off it ? Nugatory has explained this above very clearly.

If the object is moving towards or away from you, you will stll measure the ditance between the emission and the reflection events.

YES!. That absolutely makes sense! Even if the object has moved several cm or distance away, all we KNOW from the radar information is the time when we emit the signal and the time WHEN we RECEIVE the signal. Not the position of the object 'now'.

 

[Stephanus]

You will not stand still long enough to learn anything. You asked about measuring but you seem to ignore the answers and change the subject when people try to explain.

Sorry, I tought you were referring this

https://www.physicsforums.com/threa...n-but-velocity-invariant.819113/#post-5142973
Good question. I think some spacetime diagrams may help to answer your question.

Let's take a scenario where an observer sends out a light pulse to a reflector 6 feet away and measures with his clock how long it takes for the reflection to get back to him. Since light travels at 1 foot per nanosecond, it will take 12 nsecs for him to see the reflection and he will validate that the light is traveling at 1 foot per nsec for the 12 feet of the round trip that the light takes. If he is following the precepts of Special Relativity, he will define the time at which the reflection took place at 6 nsecs according to his clock and I have made that dot black:

This is what you actually referred, Nugatory's answer.

However, given proper time we can get all the other measurements. Distance to an object at rest relative to us is easy - bounce a light signal off it, divide the round-trip time by two, multiply by c, and we have the distance. If we get the same answer every time we know the object is at rest relative to us - if not, we know the distance to the object at the moment that the reflection happened.

 

[Stephanus]

I know, it's been a week since this post. But I'd like to ask anyway. I'm new in SR, so there are many symbols which I don't recognize. What is the meaning of these symbols?

Basically, if you have two observers, one of which is moving relative to the other, who synchronize their clocks such that they both read zero when they are colocated, you can write a very simple relationship between the proper time of emission for one observer, and the proper time of reception for the other:

$t_{r} = k t_{e}$

where $t_r$ is the time of reception, and $t_e$ is time time of transmission.

What does $t_r\text{ time of the reception}$ mean?
Which one is correct?
A: t_r = January 1st 2015, 18:00; t_e = January 20th 2015, 19:30, or something like
B: t_r = 20 seconds, t_e = 30 seconds
It's likely B, but if it's B then, what is t? The reverse of frequency? Needs confirmation here.
Thanks a lot.

 

[harrylin]

I know, it's been a week since this post. But I'd like to ask anyway. I'm new in SR, so there are many symbols which I don't recognize. What is the meaning of these symbols?

What does $t_r\text{ time of the reception}$ mean?
Which one is correct?
A: t_r = January 1st 2015, 18:00; t_e = January 20th 2015, 19:30, or something like
B: t_r = 20 seconds, t_e = 30 seconds
It's likely B, but if it's B then, what is t? The reverse of frequency? Needs confirmation here.
Thanks a lot.

Time of reception is the instant that the signal is received (detected), and t in discussions usually means clock time, such as 18:00:00.
The time of reception is always later than the time of emission; the difference is the time for the signal to travel. First a signal is emitted by for example a flashlight at t_e = January 20th 2015, 19:30:00, and next that signal is received by for example your eyes at t_r = January 20th 2015, 19:30:01.
But in the comment that you refer to, the starting time is set at 00 seconds and the counting is in seconds.

 

[pervect]

I know, it's been a week since this post. But I'd like to ask anyway. I'm new in SR, so there are many symbols which I don't recognize. What is the meaning of these symbols?

What does $t_r\text{ time of the reception}$ mean?
Which one is correct?
A: t_r = January 1st 2015, 18:00; t_e = January 20th 2015, 19:30, or something like
B: t_r = 20 seconds, t_e = 30 seconds
It's likely B, but if it's B then, what is t? The reverse of frequency? Needs confirmation here.
Thanks a lot.

If I'm understanding the question correctly, B is correct. A is an expression of coordinate time, while B appears to be expression of proper time.

See wiki https://en.wikipedia.org/wiki/Proper_time

In relativity, proper time along a timelike world line is defined as the time as measured by a clock following that line.

So proper time is a time interval. It can be measured with a single clock, between any two events. In the post you took this snippet from, I specify that one uses the meeting point of A and B as one of the events, then you can give times after the meeting a proper time interval of "ten seconds after the meeting" or "10 seconds before the meeting". You also have to specify which clock you are referring to, i.e. A's watch reading or B's watch reading.

And a brief look at wiki's article on coordinate time, such as the ever-popular UTC, might also be helpful as a contrast, though you don't need to study all the details. Kknowing enough to distinguish a coordinate time from proper time would be very helpful though. https://en.wikipedia.org/wiki/Coordinated_Universal_Time

 

[Stephanus]

Thanks, I just tried to ask another question when your correction arrive. I suspect it's B. But I need to confirmation.
$\beta$ and $\gamma$ I already know. But $tr$ and $te$ is new for me.

 

[Stephanus]

Thanks Harrylin for the answer. Now I can read pervect answer unobstacled.

I would recommend the method used by Bondi in "Relativity and common sense". I believe I've seen it online, but I'll let you google for it. to avoid possible issues with the links being less than legitimate...

 

[Stephanus]

If I'm understanding the question correctly, B is correct.

You should be! That was your answer after all.

A is an expression of coordinate time, while B appears to be expression of proper time.

Coordinate time and proper time. Thanks a lot. That makes many things clear for me.

 

[Stephanus]

Basically, if you have two observers, one of which is moving relative to the other, who synchronize their clocks such that they both read zero when they are colocated, you can write a very simple relationship between the proper time of emission for one observer, and the proper time of reception for the other:

$t_{r} = k t_{e}$

where $t_r$ is the time of reception, and $t_e$ is time time of transmission. If you insist on synchronizing your clocks differently , then you'd need to rewrite this equation as

$(t_r - c_r) = k \, (t_e - c_e)$ and set the values of $c_r$ and $c_e$ such that the receiving clock reads $c_r$ when the transmitting clock reads $c_e$ at the moment when the two clocks are colocated.

Dear pervect, dear PF forum. I'd like to ask a question here.

..when the transmitting clock reads $c_e$ at the moment when [..]

I think $c_e$ is coordinate time, then.
So
A: Are $c_e$ and $c_r$ coordinate times?
B: Is $k$ a contant?
C: If $k$ is a contant and $t_e$ is proper time, then $kt_e$ is proper time. Is this true?
Supposed p is proper time and c is a coordinate times. Just like in vectors
Can I ask simple question?

p+p ->p
p-p -> p
c+p -> c
c-p -> c
c-c -> p
is this true?I don't think there's p-c.

what is $\text{proper time } - \text{ coordinate time}$?
What is the result of ${t_r - c_r}$
D: proper time?
E: coordinate time?

Thanks

 

[Mentz114]

Dear pervect, dear PF forum. I'd like to ask a question here.
I think $c_e$ is coordinate time, then.
So
A: Are $c_e$ and $c_r$ coordinate times?
B: Is $k$ a contant?
C: If $k$ is a contant and $t_e$ is proper time, then $kt_e$ is proper time. Is this true?
Supposed p is proper time and c is a coordinate times. Just like in vectors
Can I ask simple question?

p+p ->p
p-p -> p
c+p -> c
c-p -> c
c-c -> p
is this true?I don't think there's p-c.

what is $\text{proper time } - \text{ coordinate time}$?
What is the result of ${t_r - c_r}$
D: proper time?
E: coordinate time?

Thanks

Proper time is the Lorentz length of the worldline $\sqrt{\Delta t^2-\Delta x^2}$. If the worldline is brought to rest ( becomes vertical, v=0) the proper time = t (coordinate time).

See https://en.wikipedia.org/wiki/Proper_time

 

[pervect]

Dear pervect, dear PF forum. I'd like to ask a question here.
I think $c_e$ is coordinate time, then.
So
A: Are $c_e$ and $c_r$ coordinate times?

Thinking it over, yes, $t_e - c_e$ is a proper time, because it's an interval, making both $t_e$ and $c_e$ coordinate times, because neither $t_e$ or $c_e$ in isolation is an interval. However, the expression $t_e - c_e$ , intended to be "simple" and intuitive, as written is actually not a coordinate independent expression for the underlying concept of proper time $\tau_e$. The concept of the proper time interval, $\tau_e$ is a coordinate independent concept, which is the same for all observers. The expression I which I wrote down, $t_e - c_e$ is a coordinate dependent expression that is equal to the proper time interval $\tau_e$ only in a particular coordinate system, the coordinate system associated with the emitter.

B: Is $k$ a contant?

I'd call k a parameter, which depends on the relative velocity between the transmitter and receiver. If the relative velocity is constant, which is the case I was addressing, then k is constant.

C: If $k$ is a contant and $t_e$ is proper time, then $kt_e$ is proper time. Is this true?

$t_e - c_e$ is a proper time, so that $k \, (t_e - c_e)$ is a proper time. Similarly $t_r - c_r$ is a proper time. The $\tau_e = k \tau_r$ is the true coordinate independent relationship between proper intervals. My use of $t_e$ and $c_e$ was an afterthought, intended to show you one simple way to calculate the proper time in the relevant coordinate system, the one associated with the emitter. But it only works properly in that particular coordinate system, it's not a general definition of the concept of proper time.

In general it doesn't make sense to add proper time to coordinate time, though of course the difference between coordinate times is equal to the proper time if you choose the right coordinate system. But it doesn't work for arbitrary coordinates.

 

[Stephanus]

Dear pervect, dear PF forum

Basically, if you have two observers, [..]you can write a very simple relationship between the proper time of emission for one observer, and the proper time of reception for the other:

$t_{r} = k t_{e}$

The $\tau_e = k \tau_r$ is the true coordinate independent relationship between proper intervals

Did you mean
"Basically, if you have two observers, you can write a very simple relationship between the proper time of emission for one observer, and the proper time of reception for the other:"
A: $\tau_{e} = k \tau_{r}$, or
B: $\tau_{r} = k \tau_{e}$, or
C: $t_{r} = k t_{e}$
Sorry, I'm new here. So any mistake will lead me lost very far away. I think "B". Because if t is coordinate time, then I don't think "coordinate time" can be multiplied. It can be added, substracted, but not multiplied.
Thanks for any respond.

 

[pervect]

Dear pervect, dear PF forum
Did you mean
"Basically, if you have two observers, you can write a very simple relationship between the proper time of emission for one observer, and the proper time of reception for the other:"
A: $\tau_{e} = k \tau_{r}$, or
B: $\tau_{r} = k \tau_{e}$, or
C: $t_{r} = k t_{e}$
Sorry, I'm new here. So any mistake will lead me lost very far away. I think "B". Because if t is coordinate time, then I don't think "coordinate time" can be multiplied. It can be added, substracted, but not multiplied.
Thanks for any respond.

$\tau_e$ and $\tau_r$ are both proper times, not coordinate times. I'm not sure why you think otherwise. However, if we assign the coordinate value of 0 to the point where the emitter and receiver are co-located, so that the time coordinate where the meet is given the value of zero, and if we also assign the time coordinate $t_e$ by the reading of the emitter clock (and assign the coordinate $t_r$ similarly to the reading on the receiver clock), we can also write $\tau_e = t_e$ and $\tau_r = t_r$ , i.e. the proper times are equal to the coordinate times when we choose our coordinate system and origin correctly. So the distinction isn't too critical if we make the right coordinate choices, but if we want to deal with general coordinates we have to be more careful.

The expression $\tau_e = k \tau_r$ and $\tau_r = k \tau_e$ are both true. Note furthermore that the value of "k" depends only on the relative velocity between emitter and receiver, and neither "k" nor "v" changes in anyway if you switch the labels on the transmitter and receiver.

Try rerading Bondi's book "Relativity and Common Sense" if you can get a hold of it, he takes basically the same approach.

 

[Stephanus]

$\tau_e$ and $\tau_r$ are both proper times, not coordinate times. I'm not sure why you think otherwise. [..]
Try rerading Bondi's book "Relativity and Common Sense" if you can get a hold of it, he takes basically the same approach.

Thanks pervect, for your reply. Because you said in the previous post $t_{r} = kt_{e}$, but later you wrote $\tau_{r} = k\tau_{e}$. But thanks for your responds. Actually download Bondi ebook. I've read it at a glance. But, I've been reading your post, and I want to at least fully understand Post #17, before I go further.

 

[harrylin]

Thanks pervect, for your reply. Because you said in the previous post $t_{r} = kt_{e}$, but later you wrote $\tau_{r} = k\tau_{e}$. [..]

Stephanus, perhaps you overlooked or did not understand pervects answer that he gave already (slightly rearranged):

" when we choose our coordinate system and origin correctly, we can also write $\tau_e = t_e$ and $\tau_r = t_r$ , i.e. the proper times are equal to the coordinate times."

And that's, I think, just what he did in his first post on that sub topic.

 

[harrylin]

[..]
The expression $\tau_e = k \tau_r$ and $\tau_r = k \tau_e$ are both true. Note furthermore that the value of "k" depends only on the relative velocity between emitter and receiver, and neither "k" nor "v" changes in anyway if you switch the labels on the transmitter and receiver.

How can that be right? For sure those two k's cannot be the same! Those times are not "relative", they correspond to events. Thus I would think that if $\tau_e = k \tau_r$ then $\tau_r = \tau_e/k$

 

[Stephanus]

Stephanus, you overlooked or did not understand pervects answer that he gave already (slightly rearranged):

Thanks harrylin for your responds. But, first pervect said

Thinking it over, yes, $t_{e}−c_{e}$ is a proper time, because it's an interval, making both $t_{e}$ and $c_{e}$ coordinate times

I think by that he also mean $t_{r}-t_{r}$ is a proper time and $t_{r}$ and $c_{r}$ are coordinate times. How can we even multiply coordinate time? $kt$, and then he corrected it's not $kt_{e}$, but it's $k\tau_{e}$. Because he once made mistake, so I want confirmation because I'm afraid he made another mistake .

 

[Stephanus]

How can that be right? For sure those two k's cannot be the same! Those times are not "relative", they correspond to events. Thus I would think that if $\tau_e = k \tau_r$ then $\tau_r = \tau_e/k$

I was on the phone between your posts. I think if k>1 then $\tau_r = k \tau_e$, for a convenient way, we can choose $k\tau_r = \tau_e$ if k<1.
Supposed you $\tau_r = 14$ and $\tau_e=10$. We could write $\tau_r=1.4\tau_e$.
But if $\tau_r = 10$ and $\tau_e = 7$, we could write $0.7\tau_r = \tau_e$. But, I just let that pass, I want to learn the next paragraph in #17, perhaps I could glean some answer, before I ask pervect/this forum again.

 

[Stephanus]

What?? Post 50? And I am still grabbing Post 17?? I'm afraid the administrator will close this thread. Hmmh...

 

[harrylin]

Thanks harrylin for your responds. But [..] How can we even multiply coordinate time? $kt$, and then he corrected it's not $kt_{e}$, but it's $k\tau_{e}$. Because he once made mistake, so I want confirmation because I'm afraid he made another mistake .

That was not a correction, as he explained (and I repeated it); and in the Lorentz transformation you also multiply coordinate time. Just plug in some numbers and you'll see.

 

[Stephanus]

That was not a correction, as he explained (and I repeated it); and in the Lorentz transformation you also multiply coordinate time. Just plug in some numbers and you'll see.

Dear harrylin, not that I want to argue with you, but how can we multiply coordinate time? As I understand it,
Coordinate time is, for example, July, 2nd 2015 18:00:00 and,
Proper time is, for example, 20 seconds.
What would be if you multiply July, 2nd 2015 18:00:00 by two? January 5,th 2031 12:00? Of course if you count start time is January 1st 1 CE 00:00. Correction it would be January 5,th 2030 12:00, because CE start at 1 not at 0 year. (or if we want the start time from the big bang 13 billion years ago, well...) Of course if you multiply July, 2nd 2015 by two FROM July 1st, 2015 than, it would be July 3rd 2015.
So, I think, just like vector and coordinate that I learned at high school (or junior high?) we can only multiply (July 2nd 2015 MINUS July 1st 2015) by two, then we can get answer. And the answer is proper time. And if we add that proper time to July 2nd, then again, we'll have coordinate time again.
And if I understand it correctly, if I treat proper time and coordinate time just like vector.
Is this right?
A. Proper Time + Proper Time -> Proper Time
B. Proper Time - Proper Time -> Proper Time
C. Proper Time * constant -> Proper Time
D. Proper Time / constant -> Proper Time
E. Coordinate Time + Proper Time -> Coordinate Time
F. Coordinate Time - Proper Time -> Coordinate Time
G. Coordinate Time + Coordinate Time -> ??
H. Coordinate Time - Coordinate Time -> Proper Time [EDIT: Coordinate Time]
I. Coordinate Time * constant -> ??
J. Coordinate Time / constant -> ??

 

[harrylin]

Dear harrylin, not that I want to argue with you, but how can we multiply coordinate time? As I understand it,
Coordinate time is, for example, July, 2nd 2015 18:00:00 and,
Proper time is, for example, 20 seconds. [..]

That understanding was wrong.
Coordinate time is the time (of a clock, or calculated) that is related to a reference frame. For example, July, 2nd 2015 18:00:00 GMT is a coordinate time of a time zone of the ECI frame. When you use it in equations, you simply count the number of days or seconds (or years) from a convenient reference time. Thus coordinate time is often handily chosen to be 0 seconds at the start time t₀, and after 20 seconds we then have t₁=20 s.
On the other hand, proper time is simply the "time" indicated by a clock in whatever state of motion.

See for example http://www.iep.utm.edu/proper-t/ , sections 3 (coordinate systems) and 15 (Time and Space Dilation). The Lorentz transformations compare time ("coordinate time") of two inertial reference systems in relative motion, and the proper time of a clock that is comoving with a reference system does not need to differ from the coordinate time of that reference system at that position.

 

[Stephanus]

That understanding was wrong.[..]
See for example http://www.iep.utm.edu/proper-t/ , sections 3 (coordinate systems) and 15 (Time and Space Dilation).[..]

Okay, okay. I click the link. Mentz114 has given me some link in Wiki, but I'm still studying Post 17, that's why I didn't click the link.

 

[Stephanus]

https://en.wikipedia.org/wiki/Proper_time
In relativity, proper time along a timelike (or lightlike) world line is defined as the time as measured by a clock following that line.

So, supposed if there is an astronout (A) wearing a red wrist watch, and he moves at 0.6c as shown in green world line.
And a rest observer (R) is staying in his room with a brown floor clock
So the proper time for (A) is shown by the red clock, because the red clock is moving with (A), if supposed (A) can see brown clock, it's not (A)'s proper time because brown clock ticks at different rate than the red clock. Is this something like that?

http://www.iep.utm.edu/proper-t/#H3The essence of the Special Theory of Relativity (STR) is that it connects three distinct quantities to each other: space, time, and proper time. ‘Time’ is also called coordinate time or real time, to distinguish it from ‘proper time’. Proper time is also called clock time, or process time, and it is a measure of the amount of physical process that a system undergoes. For example, proper time for an ordinary mechanical clock is recorded by the number of rotations of the hands of the clock. [..]
[EDIT: INSERT]
This invariance principle is fundamental to classical physics, and it means that in classical physics we can define: Coordinate time = Proper time for all natural systems. [..]

However, the distinction only gained real significance in the Special Theory of Relativity, which contradicts classical physics by predicting that the rate of proper time for a system varies with its velocity, or motion through space. The relationship is very
[EDIT]

[..]the faster a system travels through space, the slower its internal processes go. At the maximum possible speed, the speed of light, c, the internal processes in a physical system would stop completely. Indeed, for light itself, the rate of proper time is zero: there is no ‘internal process’ occurring in light. It is as if light is ‘frozen’ in a specific internal state.

So, I want to ask something here,
-"The faster a system travels, the slower its internal processes go."
-"Proper time is also called clock time, or process time, and it is a measure of the amount of physical process that a system undergoes"
But for the system itself, its one second is still one second right? All we know for 1 second is the tick of the second hand that moves 6⁰ at our desk, altough as I have often heard in this forum, we are traveling near the speed of light according to LHC.
Do I get it right?

 

[pervect]

View attachment 85807
So, supposed if there is an astronout (A) wearing a red wrist watch, and he moves at 0.6c as shown in green world line.
And a rest observer (R) is staying in his room with a brown floor clock
So the proper time for (A) is shown by the red clock, because the red clock is moving with (A), if supposed (A) can see brown clock, it's not (A)'s proper time because brown clock ticks at different rate than the red clock. Is this something like that?

Yes, that's the right idea.

So, I want to ask something here,
-"The faster a system travels, the slower its internal processes go."
-"Proper time is also called clock time, or process time, and it is a measure of the amount of physical process that a system undergoes"

This reference seems unclear and a bit muddled to me. I'd stick with the Wiki definition. It's also worth looking at the SI definition of the second, from NIST, http://physics.nist.gov/cuu/Units/current.html. The SI second measures proper time.

The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.

When you can carry out this definition precisely as written (well, you're actually allowed to ignore any/all of the quantum issues, SR is a classical theory), and actually count the number of vibration periods of some hypothetical cesium-133 atom undergoing the specified transition, you are measuring proper time.

Note that you need to specify what is called a worldline (a particular path through space-time), to measure the proper time. The "twin paradox" is an example of how two cesium atoms traveling diferent paths can start and end at the same pair of events, but experience different amounts of proper time on their journey.

It's also worth noting what proper time does not measure. Proper time has no concept of "now", it does not specify a mechanism of synchronizing clocks. Any measurement which requires clocks synchronization to be performed is not a measurement of proper time.

 

[Mentz114]

View attachment 85807
So, supposed if there is an astronout (A) wearing a red wrist watch, and he moves at 0.6c as shown in green world line.
And a rest observer (R) is staying in his room with a brown floor clock
So the proper time for (A) is shown by the red clock, because the red clock is moving with (A), if supposed (A) can see brown clock, it's not (A)'s proper time because brown clock ticks at different rate than the red clock. Is this something like that?

That is close. The brown clock is blue's proper time. Every clock shows the time along its own worldline.

Your diagram shows the proper times of both observers.

So, I want to ask something here,
-"The faster a system travels, the slower its internal processes go."
-"Proper time is also called clock time, or process time, and it is a measure of the amount of physical process that a system undergoes"
But for the system itself, its one second is still one second right? All we know for 1 second is the tick of the second hand that moves 6⁰ at our desk, altough as I have often heard in this forum, we are traveling near the speed of light according to LHC.
Do I get it right?

Locally one second is always one second and the speed of light is always c. No one notices any relativistic effects on their own clocks or rulers.

 

[Stephanus]

Dear pervect, dear PF Forum,
Thanks for you answer. I really appreciate it.

Yes, that's the right idea.

Finally...

-"The faster a system travels, the slower its internal processes go."
-"Proper time is also called clock time, or process time, and it is a measure of the amount of physical process that a system undergoes"

This reference seems unclear and a bit muddled to me. I'd stick with the Wiki definition. [..]

The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.

When you can carry out this definition [..]and actually count the number of vibration periods of some hypothetical cesium-133 atom undergoing the specified transition, you are measuring proper time.

So, what I mean is this.
Even if we travel fast and "The faster a system travels, the slower its internal processes go.", but 1 second for us is:
- "The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the Cs 133 atom.", or
- The movement of the second hand clock for 6⁰ clockwise (of course)
For a rest observer watching us moving, our clock is slower, but we still doesn't feel any different compared than we are at "rest", do we. The clock in our wristwatch, the atomic clock that (supposed) we carry, everything works "normal".

It's also worth noting what proper time does not measure. Proper time has no concept of "now", it does not specify a mechanism of synchronizing clocks. Any measurement which requires clocks synchronization to be performed is not a measurement of proper time.

I'll contemplate this.

Thanks for the answers.

 

[Stephanus]

That is close.

Finally...

The brown clock is blue's proper time. Every clock shows the time along its own worldline.

As long as the clock moves along with the observer, right?

Your diagram shows the proper times of both observers.

I want to ask something here.

Okay,... Blue and green move at the same velocity.
1. Can we say that blue and green are at the same frame of reference?
2. G2 will see Blue as B2, and B1 will see Green as G1, is this true?
3. If number 1 is true, can Blue use Green time as proper time?

Locally one second is always one second

Of course. I completely understand that.
Thanks.