Whose Clock Slows Down in Relativity?

[yuiop]

That all said, it would be improper in the case of relativity to say that he who travels the longer worldline path experiences the lesser time. Distance thru spacetime is nothing more than the accrued proper-time over a defined interval as reference, and the rate of proper-time is the very same for each and all. He who resides at both events always travels the shorter path, and thus ages the least. If both observers reside at both events, then the one who travels the shorter path ages the least.

Do you agree?

GrayGhost

I do not disagree with your definition of path length through Minkowski 4 space. I was simply pointing out that the particle that has the longest path length through Euclidean 3 space $\sqrt{(x^2+y^2+z^2)}$ has the shortest path length through 4 space. There are conditionals on this observation. (1)That when comparing elapsed proper times of two particles that they are both present at the initial and final events. (2)That the particles are constrained to move at the speed of light or less. (3)That the particles always travel forward in coordinate time. If we take the path length in 4 space as:

$$\tau = \sqrt{(ct)^2 - (x^2+y^2+z^2)}$$

then we can rewrite this as:

$$\tau = \sqrt{(ct)^2 - (\sqrt{x^2+y^2+z^2})^2}$$

$$\rightarrow \tau = \sqrt{(ct)^2 - (vt)^2}$$

Now vt is the path length through 3 space and the particle with the greatest vt is the particle with the least proper time interval in 4 space, because vt is subtracted from the coordinate time component ct.

I disagree that we have to restrict ourselves to one set of coordinates as long as we make it clear what we are doing. In GR when we analyse what happens at the event horizon in Schwarzschild coordinates we can draw interesting conclusions by transforming to a different set of coordinates such as Kruskal-Szekeres coordinates.

In other words, if we draw the paths of particles on a time space diagram and measure the total path lengths with a ruler, then the paths with longest ruler measured paths lengths is always the path with least accumulated proper time if the constraints I mentioned above for realistic particles are adhered to. If you can show a valid counter example for realistic particles, then I will withdraw my assertion.

 

[Dale]

There are conditionals on this observation. (1)That when comparing elapsed proper times of two particles that they are both present at the initial and final events. (2)That the particles are constrained to move at the speed of light or less. (3)That the particles always travel forward in coordinate time.

There is one more condition. Consider the twin's from the reference frame where the traveling twin is initially at rest. In that frame they both travel the same distance through Euclidean 3 space, just one does it over a shorter period of time. So you are also limited to the condition that you are using an inertial frame where they traveled different (Euclidean 3-) spatial distances. Just because that condition is met in one inertial frame does not mean that it is met in all inertial frames.

 

[yuiop]

There is one more condition. Consider the twin's from the reference frame where the traveling twin is initially at rest. In that frame they both travel the same distance through Euclidean 3 space, just one does it over a shorter period of time. So you are also limited to the condition that you are using an inertial frame where they traveled different (Euclidean 3-) spatial distances. Just because that condition is met in one inertial frame does not mean that it is met in all inertial frames.

Good observation Dale. To circumvent the need for this additional condition I would like to revert to my original version, where the particle that travels the greatest distance through Euclidean 4 space $$\left(\sqrt{(ct)^2 + x^2 + y^2 +z^2}\right)/c$$ has the shortest path length (proper time) in Minkowski 4 space $$\left( \sqrt{(ct)^2 - x^2 - y^2 - z^2}\right)/c$$. Does that seem reasonable with my original 3 conditions? I welcome any corrections to my terminology or conclusions here .

 

[PAllen]

Good observation Dale. To circumvent the need for this additional condition I would like to revert to my original version, where the particle that travels the greatest distance through Euclidean 4 space $$\left(\sqrt{(ct)^2 + x^2 + y^2 +z^2}\right)/c$$ has the shortest path length (proper time) in Minkowski 4 space $$\left( \sqrt{(ct)^2 - x^2 - y^2 - z^2}\right)/c$$. Does that seem reasonable with my original 3 conditions? I welcome any corrections to my terminology or conclusions here .

I have no disagreement with this as stated. Pedagogical value is a matter of judgment and also differs by individual. My concern is that since only some of Euclidean intuitions carry over, rather than having to define the precise limits on this, just learn to use non-Euclidean metrics directly.

The intuition you want to encourage is 'longer on paper is shorter in time', but you have to add all those qualifiers:

- only between a given pair of point, not for comparing paths between different points
- only if all path segments are less the 45 degrees to the t axis (assuming c=1)
- only if you can not move back in t (while you can move back in x, and in Euclidean geometry there is no 'special coordinate' where moving back is prohibited).

I honestly don't see a net pedagogical benefit by the time you add all the qualifiers.

 

[Dale]

Does that seem reasonable with my original 3 conditions?

Seems reasonable to me. I think there is some value to being able to look at a spacetime diagram and know which is the shortest interval, and this could help. I agree with PAllen though, that it is best to learn to work with the metric directly. Maybe this would be a useful "stepping stone" towards that goal.

 

[Mike_Fontenot]

The only data point that I wondered about, just shooting from the cuff, was when the traveler became 26 and his sister 17
[...]
what are the ages of both at the initial turnabout point when he drops momentarily back into the Earth frame, and (if you know) what was their separation then?

Good (and perceptive) questions.

Starting at a separation L of 40 ly, when he (the traveler) is 26, she (according to him) is 17, she (according to her) is 47.9, and velocity beta = +0.774, then:

After 1 year at -1g, you get beta = 0, he is 27, she (according to both siblings) is 49.1, and L = 40.5 ly.

After 2 years at -1g, you get beta = -0.774, he is 28, she (according to him) is 81.2, she (according to her) is 50.3, and L = 40 ly.

(And for the following 2 years of +1g, you get similar behavior at the midpoint, with beta = 0, L = 39.4 ly, and her age (according to both siblings) being 51.5 when he is 29.)

Mike Fontenot

 

[GrayGhost]

yuiop,

While it is true that the longer pathlength thru euclidean 4-space (ie paper-length hypothenuse) is actually a shorter pathlength thru Minkowskian 4-space (ie < ict), I personally would not recommend presenting it that way in a beginners relativity classroom. We refer to "worldline length" very often in discussions of relativity theory, and in the case of the spacetime interval, it's meaning is global as opposed to POV-restricted. Folks would not (always) qualify "Euclidean vs Minkowskian" every time they spoke of "the length of the interval". So although what you say is correct, IMO this would likely lead to student confusion. Maybe not in a perfect world, but in reality, there'd likely be needless routine debates of "longer vs shorter".

Per the Minkowski metric, no qualification is necessary, because s is numerically nothing more than the moving observer's proper time readout t' ... where t' < t ... which is true for all. In fact, when we talk about worldline lengths in general, we always refer to the comparison of durations experienced by both clocks. The length ict' under a euclidean metric does not represent time readings of moving clocks. So my main concern would be ensuring folks do not think that the length of ict' in a euclidean space is the length of the spacetime interval s as defined by Minkowski, because folks will refer to the ict' length "as the spacetime interval". Yes?

Gray Ghost

 

[GrayGhost]

Mike Fontenot,

OK. I was interested in knowing the numbers before the brother is 26. Basically, he accelerates off from earth, goes inertial, then (reverses thrusters at) -1g to gradually commence the direction reverseal. At some point thereafter, he (for the first time) arrives back in the Earth frame at the turnabout point ...

what was the separation & both ages at the moment he first attained a state of rest with his sister?

GrayGhost

 

[Mike_Fontenot]

I was interested in knowing the numbers before the brother is 26.
[...]
what was the separation & both ages at the moment he first attained a state of rest with his sister?

Starting at a separation L of 37.6 ly, when he (the traveler) is 21, she (according to him) is 4.2, she (according to her) is 40.6, and velocity beta = +0.968, then:

After 1 year at -1g, you get beta = +0.774, he is 22, she (according to him) is 12.3, she(according to her) is 43.2, and L = 40 ly.

After 2 years at -1g, you get beta = 0, he is 23, she (according to both siblings) is 44.4, and L = 40.5 ly.

After 3 years at -1g, you get beta = -0.774, he is 24, she (according to him) is 76.5, she(according to her) is 45.6, and L = 40 ly.

After 1 year at +1g, you get beta = 0, he is 25, she (according to both siblings) is 46.8, and L = 39.4 ly.

After 2 years at +1g, you get beta = +0.774, he is 26, she (according to him) is 17, she(according to her) is 47.9, and L = 40 ly.

Mike Fontenot

 

[GrayGhost]

Mike Fontenot,

OK then. Thanx for the extra data. I just wanted to make sure that the sister was reasonably younger than age 17 at the initial turnabout point, because you showed her at age 17 later on during a reversal on the return leg. That said, on the surface, I don't see anything fishy with your age declarations at each event. I'll assume that you integrated the proper time correctly for the accelerating observer. The inertial parts are fine, and generally speaking, your accelerating intervals look in order.

Have you ever tried plotting the data on graph from spreadsheet?

GrayGhost

 

[GrayGhost]

...the particle that travels the greatest distance through Euclidean 4 space ( [ (ct²+x²+y²+z² ]^1/2 )/c has the shortest path length (proper time)
in Minkowski 4 space ( [ (ct²-x²-y²-z² ]^1/2 )/c. Does that seem reasonable with my original 3 conditions? I welcome any corrections to my terminology or conclusions here .

yuiop,

I been looking over your posts again on this matter, and something has occurred to me.

Since we do not begin with an all-real euclidean system, I don't believe you can say that (ct')²=(ct)²+(vt)². All you can say is that (ict')²=(ict)²+(vt)². They are not the same thing. Applying Pathagorus' theorem to the complex system, "requires" that distances follow the Minkowski metric, because imaginary axes are rotated (90 deg) causing the hypothenuse to become a non-hypothenuse. IOWs, the system we start with is not a real euclidean system, it's instead a complex euclidean system with imaginaries. This complex euclidean system IS the Minkowskian system from the get-go.

So ...

(ict')²+(vt)² = -(ct)²+(vt)²​

where timelike is negative since v<c
where spacelike is positive since v>c​

It cannot be said that ...

(ict)²+(vt)² = (ct)²+(vt)²​

That said, IMO it does not seem that you can ever say that the length of ict' on a spacetime diagram is the longest path, even though "it appears as such visually". It would be in an all-real euclidean space, but it's not, it's a complex system with imaginaries.

You disagree?

GrayGhost

 

[GrayGhost]

In the context of the twins paradox we might explain it like this. Calculate the proper time that elapses for each segment of the each twins journey. Add up the elapsed proper times of the segments. The twin that has the least total of proper times, will as a general rule, be the twin that has experienced the least proper time. This is a true and correct statement, but it is not very helpful to anyone. It amounts to this. Question: Which twin experiences the least proper time? Answer: The twin that experiences the least proper time.

Well I'd say it a little differently, and in doing so, I do not believe that problem arises. We cannot measure space as the other moving fellow does. We each take our own measurements. Assuming each twin travels a different pathlength between the 2 events, then the only proof of this is the accrued proper time of each observer. We got to look at clocks, and compare. Given light speed invariant, the rate at which time passes by me per me is the same rate it passes by you per you. Therefore, he who travels the longest path must age the most, because he must accrue the most proper time over the interval.

GrayGhost

 

[Dale]

FYI GrayGhost, the use of ict with a (++++) metric fell out of favor several decades ago. You can certainly find it in many older texts, but it is quite outdated. Now everybody uses ct or even t with a (-+++) metric or a (+---) metric. It doesn't make your comments wrong, just dated.

 

[Dale]

Well I'd say it a little differently, and in doing so, I do not believe that problem arises.

No, I agree with yuiop. What you are saying is true, but it is a tautology.

If 2 observers reside at both events, eg in the twins scenario, then the one who travels the shorter path thru the continuum ages less over the interval, because he accrues less proper time.

Here "path through the continuum" means "spacetime interval", and "ages less" means "experiences less proper time", and "proper time" is "timelike spacetime interval". So your earlier statement is:

"If 2 observers reside at both events, eg in the twins scenario, then the one who travels the shorter spacetime interval experiences less timelike spacetime interval over the interval, because he accrues less timelike spacetime interval".

Which is tautologically true. Similarly, this statement is tautological:

"Therefore, he who travels the longest spacetime interval must experience the most timelike spacetime interval, because he must accrue the most timelike spacetime interval over the interval"

 

[Mike_Fontenot]

Have you ever tried plotting the data on graph from spreadsheet?

No, I never use spreadsheet software. I do rough sketches of a lot of the numerical data, just to see the overall general picture. When more precision is needed, I use my own homemade plotting programs.

In my paper, I have a precise plot of her (the home sibling's) age, versus his (the traveling sibling's) age (according to him), for his age segment from 26 to 30. I was interested in knowing the exact shape of the simultaneity curve during one of the complete "yoyo-type" cycles. The plot confirms what can be shown analytically: that during a segment of constant acceleration, the most rapid relative (positive or negative) ageing by the home sibling (according to the accelerating sibling) occurs as the velocity is passing through zero (in either direction).

If anyone would like to play around with these types of scenarios, I would be happy to email you the executable of my "CADO" program. I have an executable for Microsoft machines, and also for Linux machines. I don't have one for Macs.

If you want the program, send me an email, including a statement that you won't use (or knowingly allow the use of) the program for any commercial purposes. Also specify Microsoft or Linux.

To email me, remove the de-spamming animal name in my address: mlfasfzebra@comcast.net .

Mike Fontenot

 

[yuiop]

I have no disagreement with this as stated. Pedagogical value is a matter of judgment and also differs by individual. My concern is that since only some of Euclidean intuitions carry over, rather than having to define the precise limits on this, just learn to use non-Euclidean metrics directly.

The intuition you want to encourage is 'longer on paper is shorter in time', but you have to add all those qualifiers:

- only between a given pair of point, not for comparing paths between different points
- only if all path segments are less the 45 degrees to the t axis (assuming c=1)
- only if you can not move back in t (while you can move back in x, and in Euclidean geometry there is no 'special coordinate' where moving back is prohibited).

I honestly don't see a net pedagogical benefit by the time you add all the qualifiers.

I would like to take the qualifiers one at a time.

- only if all path segments are less the 45 degrees to the t axis (assuming c=1)

In an introduction to SR and spacetime diagrams, this is as good a time as any, to demonstrate that velocities of real particles or clocks that are greater than the speed of light is incompatible with SR and causes contradictions.

- only if you can not move back in t (while you can move back in x, and in Euclidean geometry there is no 'special coordinate' where moving back is prohibited).

That you can only move forward in coordinate time is a very natural intuition for any lay person that has not not yet been introduced to relativity. If a tutor wrote this problem on the board:

My route to work is 45 kilometers in a straight line. I leave home at 9.00 am. I arrive at the paper shop 30 kilometers down the road at 9.30 am and buy a newspaper. I leave the paper shop at 9.35 am and arrive at work at 8.35 am the same day. What is my average speed?

The students would immediately object to your traveling from the paper shop to your work in minus one hour. The concept that coordinate time only advances is entirely natural and we have never measured anything that contradicts this and the thermodynamic arrow of time generally supports this.

- only between a given pair of point, not for comparing paths between different points

I am not quite sure what you mean by this last qualifier. If by "comparing paths between different points" you mean comparing elapsed proper times between clocks that are initially co-located and finally co-located, then this is as good as time as any to introduce the students to the concept that you can not compare elapsed proper times of spatially separated clocks in meaningful invariant way. If two clocks are spatially separated, then different observers will disagree on which clock is ticking slower. Until the clocks are re-co-located any comparison of elapsed proper times is just a matter of opinion and is observer dependent.

To GrayGhost. I am not ignoring you. It is just that I believe that Dalespam is doing a better job in responding to your objections than I ever possibly could.

 

[GrayGhost]

FYI GrayGhost, the use of ict with a (++++) metric fell out of favor several decades ago. You can certainly find it in many older texts, but it is quite outdated. Now everybody uses ct or even t with a (-+++) metric or a (+---) metric. It doesn't make your comments wrong, just dated.

Maybe you misunderstood my prior post? Your response suggested that I may not have known that the Minkowski metrics (-+++) and (+---) are both used. I must say though, I was unaware indeed that any (++++) Minkowski metric was ever used, and find that quite suprising ...

Here's what I was saying ...

Yuiop suggested the 4d euclidean length of the path along ict' would be ... ct' = sqrt[ (ct)²+(vt)² ], ie the longest path, which stemmed from yuiop's assumption that (ct')²=(ct)²+(vt)². That's a (++++) euclidean metric. This would be true if the time axes were not imaginary, however they are.

So I merely pointed out that applying Pathagorus' theorem to our particular complex systems cannot produce yuiop's (++++) metric, they can only produce the Minkowskian (-+++ or +---) metrics. Which itself means that one cannot say (ct')²=(ct)²+(vt)² in the very first place, or equivalently ... one cannot say the longest path is ict'.

EDITED: On the other hand, if we ignore the fact that time is represented as imaginary, one can easily say (ct')²=(ct)²+(vt)² resulting in the euclidean metric (++++). But then that is mathemaically improper. Nonetheless, this makes the paper length of ict' the longest, as yuiop pointed out. Are there more benefits to doing this, than not doing this? IMO, I don't think so. I suppose one could always try, see if they can find ways of avoiding any resultant confusion maybe. I have witnessed countless debates over many years as to whether the longest worldline length is the shortest proper time experienced, versus whether the shortest worldline length is the shortest proper time experienced. Few surrender their position. However, I'd also have to submit that most do not qualify as to whether the system is euclidean vs Minkowskian. IOM though the "complex" euclidean system we begin with IS a Minkowskian system from the start.

Am I incorrect on this reasoning?

GrayGhost

 

[GrayGhost]

To GrayGhost. I am not ignoring you. It is just that I believe that Dalespam is doing a better job in responding to your objections than I ever possibly could.

Well, we'll see. No problem though. Thanx for the notice yuiop :)

GrayGhost

 

[yuiop]

Maybe you misunderstood my prior post? Your response suggested that I may not have known that the Minkowski metrics (-+++) and (+---) are both used. I must say though, I was unaware indeed that any (++++) Minkowski metric was ever used, and find that quite suprising ...

Here's what I was saying ...

Yuiop suggested the 4d euclidean length of the path along ict' would be ... ct' = sqrt[ (ct)²+(vt)² ], ie the longest path, which stemmed from yuiop's assumption that (ct')²=(ct)²+(vt)². That's a (++++) euclidean metric. This would be true if the time axes were not imaginary, however they are.

I think what Dalespam was getting at is that the Minkowski (-+++) metric:

$$s^2 = - (ct)^2 +x^2 + y^2 + z^2$$

can be written as a Minkowski (++++) metric like this:

$$s^2 = + (ict)^2 +x^2 + y^2 + z^2$$

which is no different to the first form that does not include the imaginary term. This seems to be your preferred form.
I am not sure why you are insisting that the time axes should be imaginary. I can write the Minkowski metric as:

$$c^2(\Delta \tau)^2 = (\Delta t)^2 - (\Delta x)^2 - (\Delta y)^2 - (\Delta z)^2$$

which is perfectly valid and does not involve any imaginary quantities at all. For any particle moving at the less than the speed of light, $$(\Delta \tau)$$ is always real for any real value of $$(\Delta t)$$.

 

[GrayGhost]

yuiop,

Thanx, for the response. I have to run at this instant, so I'll respond a little later. I see what you're saying. I'll try to cut to the chase. thanx.

GrayGhost

 

[Dale]

Maybe you misunderstood my prior post? Your response suggested that I may not have known that the Minkowski metrics (-+++) and (+---) are both used. I must say though, I was unaware indeed that any (++++) Minkowski metric was ever used, and find that quite suprising ...

Here's what I was saying ...

Yuiop suggested the 4d euclidean length of the path along ict' would be ... ct' = sqrt[ (ct)²+(vt)² ], ie the longest path, which stemmed from yuiop's assumption that (ct')²=(ct)²+(vt)². That's a (++++) euclidean metric. This would be true if the time axes were not imaginary, however they are.

So I merely pointed out that applying Pathagorus' theorem to our particular complex systems cannot produce yuiop's (++++) metric, they can only produce the Minkowskian (-+++ or +---) metrics. Which itself means that one cannot say (ct')²=(ct)²+(vt)² in the very first place, or equivalently ... one cannot say the longest path is ict'.

EDITED: On the other hand, if we ignore the fact that time is represented as imaginary, one can easily say (ct')²=(ct)²+(vt)² resulting in the euclidean metric (++++). But then that is mathemaically improper. Nonetheless, this makes the paper length of ict' the longest, as yuiop pointed out. Are there more benefits to doing this, than not doing this? IMO, I don't think so. I suppose one could always try, see if they can find ways of avoiding any resultant confusion maybe. I have witness countless debates over many years as to whether the longest worldline length is the shortest proper time experienced, versus whether the shortet worldline length is the shortest proper time experienced. Few surrender their position. However, I'd also have to submit that most do not qualify as to whether the system is euclidean vs Minkowskian. IOM though the "complex" euclidean system we begin with IS a Minkowskian system from the start.

Am I incorrect on this reasoning?

Sorry about the confusion. Hopefully I can clear it up.

We want to obtain the line element from the coordinates and the metric as follows:

$$ds^2=-c^2dt^2+dx^2+dy^2+dz^2=g_{\mu\nu}x^{\mu}x^{\nu}$$

To do this we can adopt one of three conventions:

$$x^{\mu}=(ict,x,y,z)$$ and $$g_{\mu\nu}=\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}$$

or

$$x^{\mu}=(ct,x,y,z)$$ and $$g_{\mu\nu}=\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}$$

or

$$x^{\mu}=(t,x,y,z)$$ and $$g_{\mu\nu}=\begin{pmatrix}-c^2&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}$$

I was merely pointing out that the first convention, ict, fell out of use decades ago.

 

[PAllen]

I would like to take the qualifiers one at a time.
In an introduction to SR and spacetime diagrams, this is as good a time as any, to demonstrate that velocities of real particles or clocks that are greater than the speed of light is incompatible with SR and causes contradictions.
That you can only move forward in coordinate time is a very natural intuition for any lay person that has not not yet been introduced to relativity. If a tutor wrote this problem on the board: The students would immediately object to your traveling from the paper shop to your work in minus one hour. The concept that coordinate time only advances is entirely natural and we have never measured anything that contradicts this and the thermodynamic arrow of time generally supports this.
I am not quite sure what you mean by this last qualifier. If by "comparing paths between different points" you mean comparing elapsed proper times between clocks that are initially co-located and finally co-located, then this is as good as time as any to introduce the students to the concept that you can not compare elapsed proper times of spatially separated clocks in meaningful invariant way. If two clocks are spatially separated, then different observers will disagree on which clock is ticking slower. Until the clocks are re-co-located any comparison of elapsed proper times is just a matter of opinion and is observer dependent.

To GrayGhost. I am not ignoring you. It is just that I believe that Dalespam is doing a better job in responding to your objections than I ever possibly could.

Fine, but this still all boils down to 'you can use your Euclidean intuition except for all the ways SR is not Euclidean, and timelike motion is very special (in Newtownian physics, time is not included in the metric at all; when you bring it into the SR metric, you then say: use some Euclidean intuition except for all these gotchas). You could use each gotcha as above to teach a 'why', as suggested above. I prefer, instead, to make analogies to non-Euclidean geometry and highlight how it is risky to rely on Euclidean intuition.

Note, it is not so trivial to banish forward and back in time. If you are dealing with specelike paths, many non-inertial frames have paths of simultaneity that move forward and backward in time viewed in an inertial frame.

Anyway, I don't see that we disagree on an any facts or interpretations. We have a different judgment on the value of a heuristic that can be used with the appropriate rules.

 

[PAllen]

No, I agree with yuiop. What you are saying is true, but it is a tautology.

Here "path through the continuum" means "spacetime interval", and "ages less" means "experiences less proper time", and "proper time" is "timelike spacetime interval". So your earlier statement is:

"If 2 observers reside at both events, eg in the twins scenario, then the one who travels the shorter spacetime interval experiences less timelike spacetime interval over the interval, because he accrues less timelike spacetime interval".

Which is tautologically true. Similarly, this statement is tautological:

"Therefore, he who travels the longest spacetime interval must experience the most timelike spacetime interval, because he must accrue the most timelike spacetime interval over the interval"

I might say that that these statements are applications of definitions made to look tautological. I'm sure you agree that defintion of the invariants and their physical interpretation is perhaps the most essential part of SR.

Meanwhile, yuiop's proposed substitute Euclidean metric that can, with appropriate restrictions, allow one to read certain strictly qualitative results from a diagram, has no physical validity in the theory, and therefore can even less explain anything than a definition. You can't attach any meaning at all to the numeric value of interval based on dt^2 + dx^2. You can attach definite, invariant meaning to proper time.

 

[Dale]

I might say that that these statements are applications of definitions made to look tautological.

Do you know any other definitions for the terms I mentioned?

 

[PAllen]

I think I want to make more specific statement here, while still noting I don't think we disagree on anything except 'teaching approach'.

I would like to take the qualifiers one at a time.
In an introduction to SR and spacetime diagrams, this is as good a time as any, to demonstrate that velocities of real particles or clocks that are greater than the speed of light is incompatible with SR and causes contradictions.

When you go beyond world lines to spacelike paths, then both the prior and succeeding restriction go away (due to relativity of simultaneity). Then you are confronted with saying the Euclidean analogy loses its meaning. My preference is just to get right to the issue of defining timelike invariant (physical time along a world line) and spacelike invariant (an invariant that has the character of length). Timelike geodesics (inertial paths) maximize proper time. Spacelike geodesics minimize distance in an inertial frame in which the endpoints are simultaneous.

That you can only move forward in coordinate time is a very natural intuition for any lay person that has not not yet been introduced to relativity. If a tutor wrote this problem on the board: The students would immediately object to your traveling from the paper shop to your work in minus one hour. The concept that coordinate time only advances is entirely natural and we have never measured anything that contradicts this and the thermodynamic arrow of time generally supports this.
I am not quite sure what you mean by this last qualifier. If by "comparing paths between different points" you mean comparing elapsed proper times between clocks that are initially co-located and finally co-located, then this is as good as time as any to introduce the students to the concept that you can not compare elapsed proper times of spatially separated clocks in meaningful invariant way. If two clocks are spatially separated, then different observers will disagree on which clock is ticking slower. Until the clocks are re-co-located any comparison of elapsed proper times is just a matter of opinion and is observer dependent.

In Euclidean geometry, you can compare lengths of any curves on the plane. In SR you can compare invariant interval along any timelike or spacelike paths. In your 'heuristic Euclidean metric' any conclusion you draw for curves that don't begin and end at the same points are likely wrong. This is an opportunity for major confusion, despite the 'teaching opportunity you describe.

 

[PAllen]

Do you know any other definitions for the terms I mentioned?

I'm trying to say that defining timelike interval in a certain way (mathematically) coupled with the (physical) interpretation that it represents physical time along a world line is not tautological at all. It is a very strong prediction to be proven true or false. And that often there is little more you can say about 'why' you age differently on one path than another than the definition proper time.

I am often bothered by 'explanations' of things with limited validity. For example, in the twin paradox, that the 'missing aging' is associated with some particular part of the path. Any of these break down with a different variant, and all are based on simultaneity conventions which are purely conventions (as you have explained in many other posts). In this discussion, the meaningless explanation I see is 'longer Euclidean path is shorter proper time'. Except this doesn't explain anything and is generally false. While interval is a coordinate independent quantity of any path (except you might exclude mixed timelike and spacelike), that all agree on, the Euclidean analogy is only good (qualitatively, not quantitavely) for certain pairs of timelike paths - and even then it doesn't explain anything, and certainly doesn't 'get around' any dissatisfaction with statements like 'you aged less because you took a path with less time on it'. That is no different than accepting, in Euclidean geometry, that it takes more string along path A than path B because A is longer than B.

 

[GrayGhost]

DaleSpam & yuiop,

Thanx for your prior responses. I haven't had any problem (or issue) with spacetime interval formulae. The one thing I was unaware of, was that (ict)²+(vt)² was an outdated former Minkowski metric. However, that doesn't effect any of the points I've been raising here.

Yuiop asked for opinions on the matter of defining a new spacetime interval length per the stationary POV, a euclidean 4d length ... (ct)²+(vt)². The hope was that something new could be used in classroom. The desire was to expedite the determination as to who aged more or less over the defined interval, or make the learning of relativity theory easier via convenience.

It seems inappropriate IMO, that within the same theory we might consider axes as imaginary in some circumstances and real in others. The imaginary axes are required for transformation (ie rotation) between the 2 systems, given the 2 postulates true. However, let's assume that there exist instances whereby we desire calculations that require no transformations between the 2 systems ... eg the one yuiop desires. OK, so we ignore the fact that time axes are imaginary, and inquire as to the length of the spacetime interval, per the stationary POV. We then attain this metric ... (ct)²+(vt)².

Now, we inquire as to the physical meaning of (ct)²+(vt)². The soln would represent the distance traversed thru euclidean 4-space, per only the stationary POV.

Next we ask, of what benefit might this be? I can think of none, personally. Yuiop has pointed out that it may make the determination as to who aged the least (vs most) more convenient.

Are there any disadvantages ... The magnitude of this 4d distance vector would pertain only to the stationary POV, it cannot be measured, and can only be predicted. There would exist no way of confirming the prediction correct, nor would we ever expect it to. It surely does not represent clock readings of either system. We'd now have a new (extra) description of the spacetime interval, a frame dependent solution, which would differ from the related invariant soln of Minkowski. Lengths of worldlines would no longer (numerically) equal the proper time experienced in all cases, and the meaning of a worldline length would likely become confused.

That said, although I respect any goal to add convenience to education, I venture that goal would not be met in this case. I could always always be merely mistaken :)

GrayGhost

 

[GrayGhost]

I am often bothered by 'explanations' of things with limited validity. For example, in the twin paradox, that the 'missing aging' is associated with some particular part of the path.

PAllen,

I was unaware of a "missing aging" crisis. I'm just curious ... are you referring to 'ole instantaneous velocity changes per the non-inertial POV maybe?

GrayGhost

 

[Dale]

The one thing I was unaware of, was that (ict)²+(vt)² was an outdated former Minkowski metric. However, that doesn't effect any of the points I've been raising here.

Correct, I was not intending it as a rebuttal of any of your points. I was simply mentioning that the convention is not commonly used any more. I don't know why it has fallen out of use, except that the others are more "GR-ish" (especially the last).

 

[PAllen]

PAllen,

I was unaware of a "missing aging" crisis. I'm just curious ... are you referring to 'ole instantaneous velocity changes per the non-inertial POV maybe?

GrayGhost

This refers to many recent threads on the twin paradox. There is no crisis. Just that some people put forward arguments that the 'missing aging' of the twin who ages less 'should' be attributed to one part of the path or another. Dalespam and I have argued that these attempts are misguided because:

- for some variation on the twin scenario (changing timing, smoothness of acceleration, changing whether there is any coasting or not), you find they break down

- more fundamentally, they are really making a statement about distant simulataneity between the longer aging world line and the shorter aging world line, and any such choice of simultaneity convention has a substantially arbitrary character (within certain fundamental limits: you can't consider two causally connected events to be simultaneous).

 

[yuiop]

In Euclidean geometry, you can compare lengths of any curves on the plane. In SR you can compare invariant interval along any timelike or spacelike paths. In your 'heuristic Euclidean metric' any conclusion you draw for curves that don't begin and end at the same points are likely wrong. This is an opportunity for major confusion, despite the 'teaching opportunity you describe.

I disagree that you can compare invariant intervals along any timelike paths. For example if we have one clock (A) that remains at rest in frame S and another clock (B) that moves at 0.8c relative to S, then according to an observer in S, less proper time has elapsed for clock B at any coordinate time t. Now if we switch to frame the rest frame of clock B, (S') then an observer at rest in frame S' will say less proper time has elapsed for clock A relative to clock B at any coordinate time t'. In other words asking "whose clock is really slower?" for two clocks that are not initially and finally co-located is meaningless even in the purely SR/Minkowski context. While B is still going away from A, which clock would end up with the least elapsed proper time when they come back together again, depends entirely upon which clock turns around and accelerates towards the other and without a crystal ball to predict the future, we do not know which clock that will be until they are actually alongside each other again. This is a very important point is the context of the main question originally posed in this thread.

 

[yuiop]

However, let's assume that there exist instances whereby we desire calculations that require no transformations between the 2 systems ... eg the one yuiop desires. OK, so we ignore the fact that time axes are imaginary, and inquire as to the length of the spacetime interval, per the stationary POV. We then attain this metric ... (ct)²+(vt)².

I am not sure why you are fixated on time axes being imaginary, when we can do the calculations for timelike paths without any imaginary or complex values. Sure, imaginary quantities can turn up when considering spacelike intervals, but if we restrict ourselves to the subject of this thread (clock rates) then this does not apply. I am also not sure why you think it is impossible to do transformations to different POVs. If we take two paths in a given reference frame where the path with the least proper time has the greatest Euclidean 4d length (ct)²+(vt)² and use the regular Lorentz transformation to obtain the POV of an observer at rest in a different reference frame, then it is still true that the path with the least proper time has the greatest Euclidean 4d length.

Now, we inquire as to the physical meaning of (ct)²+(vt)². The soln would represent the distance traversed thru euclidean 4-space, per only the stationary POV. Next we ask, of what benefit might this be? I can think of none, personally. Yuiop has pointed out that it may make the determination as to who aged the least (vs most) more convenient.

It is the POV of a single inertial observer, but as mentioned above we can transform to the POV of any inertial observer and the inequality still holds. Let us say we have clocks 1 and 2 with paths $(\Delta t1,\Delta x1)$ and $(\Delta t2,\Delta x2)$ respectively. According to any inertial observer, if the inequality $(\Delta t2)^2+(\Delta x2)^2 > (\Delta t1)^2+(\Delta x1)^2$ then it always holds that the inequality $\Delta \tau 2 > \Delta \tau 1$ is also true (given the 3 conditionals* that I gave earlier). Although the Euclidean 4d calculation is qualitative rather than quantitative (as PAllen pointed out) it is instantly visually obvious which is the path with the least proper time because any student already knows that the combined lengths of any two sides of a triangle is greater than the length of the third side and that the shortest spatial distance between any two points is a straight line. Here they can use their Euclidean intuition and then apply the relativistic "twist".

Are there any disadvantages ... The magnitude of this 4d distance vector would pertain only to the stationary POV, it cannot be measured, and can only be predicted. There would exist no way of confirming the prediction correct, nor would we ever expect it to. It surely does not represent clock readings of either system. We'd now have a new (extra) description of the spacetime interval, a frame dependent solution, which would differ from the related invariant soln of Minkowski. Lengths of worldlines would no longer (numerically) equal the proper time experienced in all cases, and the meaning of a worldline length would likely become confused.

Sure we would a different name for Euclidean 4d length other that worldline length or spacetime interval to avoid confusion. I disagree that it cannot be measured. Many so called measurements are not direct measurements but rather are calculations, e.g. velocity is not always a direct measurement but a calculation of distance versus time and kinetic energy etc. I agree that the Euclidean 4d distance is not frame invariant, but as an logical (true/false) inequality it is invariant.

It seems inappropriate IMO, that within the same theory we might consider axes as imaginary in some circumstances and real in others. The imaginary axes are required for transformation (ie rotation) between the 2 systems, given the 2 postulates true.

Again, I think you need to elucidate on why you think time is imaginary and why imaginary axes are required for transformation.

*As for the 3 conditionals I gave earlier, these are not over and above the conditionals required by SR for real clocks. (By "real clocks" I mean clocks with non zero real rest mass rather than imaginary or complex rest mass) These conditionals are required by SR too.

 

[PAllen]

I disagree that you can compare invariant intervals along any timelike paths. For example if we have one clock (A) that remains at rest in frame S and another clock (B) that moves at 0.8c relative to S, then according to an observer in S, less proper time has elapsed for clock B at any coordinate time t. Now if we switch to frame the rest frame of clock B, (S') then an observer at rest in frame S' will say less proper time has elapsed for clock A relative to clock B at any coordinate time t'. In other words asking "whose clock is really slower?" for two clocks that are not initially and finally co-located is meaningless even in the purely SR/Minkowski context. While B is still going away from A, which clock would end up with the least elapsed proper time when they come back together again, depends entirely upon which clock turns around and accelerates towards the other and without a crystal ball to predict the future, we do not know which clock that will be until they are actually alongside each other again. This is a very important point is the context of the main question originally posed in this thread.

Invariant means all observers agree on it, independent also of coordinate system. Otherwise it isn't invariant. All observer's agree on the interval along a world line: that a give particle decays or not, that a person dies of old age or not, what a clock reads along a world line. They may perceive that a clock on some world line goes a different rate than their own, they will radically disagree on which points of different world lines are 'simultaneous', but there is never any disagreement on what the clock on some world line does between two physical events. This is fundamental to both SR and GR.

Colocation is irrelevent. Suppose you are traveling at .99c from start s1 to s2. You send me a picture of yourself at s1, and send me another picture when you get to s2. However long it takes me to get these signals, I will agree on how much aging you will experience (as long as I know the distance from s1 to s1 as I would measure it, and your speed). You will experience the aging I compute. Further, if you know my start and end points as you would measure them, and my speed as you measure it, and compute the invariant interval along my path in these coordinates, you will get the same longer age that I experience. The explanation of why you agree on my invariant interval while still seeing my clock going slow is that you radically disagree on simultaneity. The event on my worldline I say is simultaneous to your passing s1 is not at all what you would say.

Let's make this last point more concrete. Let's say I blow up an H-bomb at t1 and t2 on earth. When I get your signals sent from s1 and s2, as I inerpret where s1 and s2 are, factoring in light delay, I find (miraculous coincidence) the you sent your singal from s1 at t1, and from s2 at t2. I compute both intervals and find you aged much less, consistent with the pictures you sent.

You do the same thing. Only you find that t1 and t2 are not remotely simultaneous with when you were at s1 and at s2. However, taking into account when they occurred as you see them, and where I was at t1 and t2, as you see it, would would compute the same age difference for me as I actually experienced.

 

[GrayGhost]

I am not sure why you are fixated on time axes being imaginary, when we can do the calculations for timelike paths without any imaginary or complex values. Sure, imaginary quantities can turn up when considering spacelike intervals, but if we restrict ourselves to the subject of this thread (clock rates) then this does not apply.

Hmmm. Well, we've been discussing the length of the spacetime interval as defined under the Minkowski model. OK then, are you suggesting here that imaginaries do not apply wrt clock rates?

Seems to me that you might not understand the meaning of complex systems. I do realize that Einstein did not use imaginaries in his OEMB. But Minkowski recognized that the Einstein model was equivalent to a relative rotation if the time axes were assumed orthogonal wrt 3-space. Remember, we begin with a spacetime illustration with imaginary time axes. Minkowski did not designate these as such for no good reason. They in fact were necessary, if Einstein's kinematic scenario is to be modeled consistent with his LT solns.

Where ict' = s, we start with this vector equation ...

(ict')² = (ict)²+(vt)²​

And we end up with this vector equation ...

(ct')² = (ct)²-(vt)²​

which reduces to ...

t' = t(1-v²/c²)^1/2​

If you think that the 2nd equation above exists regardless as to whether the i's exist in 1st eqn or not, you are mistaken. The 2nd eqn exists only after applying the Pathorean theorem, which itself requires the squaring of imaginary vectors (and where i² = -1) ... physically, the ict-axis (or likewise the ict'-axis) is rotated 90 deg into a real 3-space plane. It is this rotation that allows the LTs to result in the precise way they did.

That said, it is impossible to obtain this ...

(ct')² = (ct)²+(vt)²​

if starting from this ...

(ict')² = (ict)²+(vt)²​

On the other hand, one can obtain this last metric if one ignores the fact that the ict time axis is imaginary (as you wish to), because then no rotations/transformations are done. That is, ict' would not then be indicative of either the moving (or even the stationary) frame's measure of space or time. The resultant value of ict' no longer represents the time of anything real, and would represent a distance thru spacetime that cannot be measured or verified by anyone. I realize what you are trying to do yuiop, however if done, I see more losses than gains, IMO.

GrayGhost

 

[bobc2]

Hmmm. OK then, so you claim here that imaginaries do not apply wrt clock rates.

Seems to me that you may not understand the meaning of complex systems. Remember, we begin with a spacetime illustration with imaginary time axes. Minkowski did not designate these as such for no good reason. They in fact were necessary, if Einstein's kinematic scenario is to be modeled consistent with his LT solns.

GrayGhost, I always am impressed with your posts and do not wish to argue against the very good presentation you've made on this. But here is the derivation I've done before (sorry I don't remember the link). This is just to show there is an alternative derivation that begins with a 4-dimensional spacetime--just to show it is possible to develop the Minkowski metric without resorting to an imaginary axis. You can always put one in, just because any parameter or variable shown as a negative squared quantity, i.e., -X^2, can be represented with an imaginary number arbitralily inserted, i.e., -X^2 = (iX)^2. After developing the metric as shown below, you can of course do this in order to have an imaginary axis.

I derived the metric, then substituted in the ct without ever resorting to the imaginary i to obtain the Lorentz transformation for the double-bar time.

In this derivation we just use spatial coordinates throughout--just the spatial length of legs and hypotenuse of a right triangle. You can take any right triangle in normal X-Y space and then use the Pythagorean theorem--then solve for the length of one leg-- followed by a substitution of the imaginary number i in the negative hypotenuse squared term.

Some people object to this derivation claiming that the original diagram is arbitrarily contrived. But this is not the case. It is required if the speed of light is to be the same for all observers. Furthermore it is very general, because for any two observers moving relative to each other, you can always find a rest system that has the two movers going in opposite directions at the same speed.