Newtonian vs Relativistic Mechanics

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No, this is not correct. Look at the coordinates I gave for events. Alice, Bob, and Bob's light ray all start at (t, x, y) = (0, 0, 0) in Alice's frame. After 1 unit of time in Alice's frame--"time" meaning coordinate time in that frame--Alice is at event (1, 0, 0); Bob is at event (1, 0.6, 0); and Bob's light ray is at event (1, 0.6, 0.8). All three of these events have t = 1, i.e., they are at coordinate time 1. Alice's proper time between the two events is 1; Bob's is 0.8; and Bob's light ray has zero spacetime interval, which strictly speaking should not even be called its "proper time" since that term only applies to a timelike interval, not a null interval.
Thank you, thank you! I think the light is dawning! (I know that must be hard to believe, hehehe!)
It is all down to semantics - understanding the words in the right way.
Let me see if I am getting it now. Proper time and coordinate time are not different ways of measuring the time. They are not measuring time against different time scales. They are descriptions of what is being measured. Proper time is the label applied to time measured on a worldline. Coordinate time is the label applied to times that are 'coordinated' by being measured in one frame by a single observer.
None of these involve "measurement of time in another frame". They all involve coordinate times in Alice's frame.
Yes, when Bob's light has travelled 0.8 light units in Bob's clock, in Bob's frame, Alice measures the light to have travelled 1 unit in Alice's frame, because the light has travelled 1 unit in her frame.
It is not Alice reading Bob's measurement differently, it is Alice making her own measurement of the time in her frame.
 
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Perhaps it will help to clarify what I'm asking if I add this: your discussion talks about the representation of the same spacetime interval in different frames. But time dilation involves the comparison of two different spacetime intervals.
Intervals are invariant, so you can do the comparison in a single frame; no transformation between frames is needed. But you need to compare different intervals--in this case, an interval along Bob's worldline with an interval along Alice's worldline. How would you make such a comparison to show time the dilation of Bob relative to Alice?
I am sorry I am not sure what you are asking; on one level I could say that the events for Alice would be similar to those for Bob; she could be using her own light clock to time the events (which is what is at the heart of my two clock comparison) - yet I am sure there will be some reason why you don't like that idea...
You say:
...] time dilation involves the comparison of two different spacetime intervals.
yet I thought we were talking about one spacetime interval, and the different ways it is measured in two frames. It is the interval between two events: the emission of the light in Bob's clock and that light travelling 0.8 of the distance to the mirror in that clock. Two different measurements of the interval (that is the difference between the relevant coordinates in two different frames of reference) between those two spacetime events.
The Spacetime Interval in Bob's frame, S = t (the proper time for Bob between the light being emitted in the clock he is holding on to and the light travelling 0.8 of the distance to the mirror) = 0.8.
The Spacetime Interval in Alice's frame S = √(t2 - vt2) = √(1 - 0.36) = 0.8
Which is the invariant spacetime interval between two spacetime events, one at rest and one moving.
So I am confused again now that you say they are two Spacetime Intervals.
Where am I going wrong?

Pervect: I believe that one can measure time in a frame using synchronised clocks at rest in that frame.
 
  • #128
PeterDonis
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Proper time is the label applied to time measured on a worldline.
Time measured by an observer following that worldline, yes.

Coordinate time is the label applied to times that are 'coordinated' by being measured in one frame by a single observer.
Sort of. As long as we are talking about inertial coordinates in SR (i.e., flat spacetime), this view works ok, because you can think of the coordinates as corresponding to measurements made by a fleet of observers with measuring rods and clocks, all having the same state of motion as the "reference" observer (Alice or Bob or whoever). Note that even here, a single observer isn't making all the measurements, because observers can only make measurements at events on their worldlines, and one observer can't be on multiple worldlines. But in any case, as soon as you try to use non-inertial coordinates, or any coordinates in curved spacetime (i.e., when gravity is present), this no longer works.

The more general way to look at coordinates is that they are just assignments of unique sets of four numbers to each event in spacetime, plus some conditions on the assignments to make the numbers work the way we are used to having coordinates work (things like nearby events should have "nearby" coordinates, etc.). "Time" is just one of the four numbers (and even calling it "time" depends on some assumptions that might not be true for some choices of coordinates).

when Bob's light has travelled 0.8 light units in Bob's clock, in Bob's frame, Alice measures the light to have travelled 1 unit in Alice's frame, because the light has travelled 1 unit in her frame.
No, this is still confused. What are "light units"? What "units" does light travel in? The spacetime interval along a light ray's worldline is always zero. So you must be comparing some other pair of spacetime intervals to get these values of 0.8 and 1 and somehow show how they correspond to each other. How are you doing that?

I thought we were talking about one spacetime interval, and the different ways it is measured in two frames.
If you are trying to show how time dilation works, this is not correct. You have to compare two different intervals. Which ones, and how do you compare them?

Where am I going wrong?
Consider the following spacetime intervals (all coordinates of events are given in Alice's frame):

A) The interval between events (0, 0, 0) and (1, 0, 0). These are two events on Alice's worldline.

B) The interval between events (0, 0, 0) and (1, 0.6, 0). These are two events on Bob's worldline.

What are the values of the spacetime intervals A and B? What is the ratio between them? What picks out these two particular intervals as the right ones to use to show that Bob is time dilated relative to Alice?
 
  • #129
PeterDonis
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What picks out these two particular intervals as the right ones to use to show that Bob is time dilated relative to Alice?
Btw, if you're having trouble answering this question, consider an alternative pair of intervals:

a) The interval between events (0, 0, 0) and (1.25, 0, 0). These are also two events on Alice's worldline.

b) The interval between events (0, 0, 0) and (1.25, 0.75, 0). These are also two events on Bob's worldline.

You should be able to confirm that the ratio between these two intervals is the same as the ratio between intervals A and B from my previous post. What does that ratio represent?

To help in answering the above, you might also consider a third interval:

c) The interval between events (0, 0, 0) and (1.25, 0.75, 1). These are two events on the worldline of Bob's light ray, and it is easy to see that the spacetime interval between them is zero. The second event, as should be evident from its y coordinate, is the event at which the light ray reaches Bob's mirror (which is located 1 unit from Bob in the y direction). Now look at the x coordinate of this event: it is the same as that of event b. And the t coordinate of event c is the same as the t coordinate of both events a and b. What does that tell you? And how does it show that events a and b are good ones to use to show that Bob is time dilated relative to Alice?
 
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What are "light units"? What "units" does light travel in
I was using 'light units' as the equivalent of 'light seconds', or 'light years' in the same way we have been using 'units" or 'time units'. No more than that.
So 0.8 light units is no more than a distance.
 
  • #131
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In frame S', we have an emission event, and a reception event, but both events aren't located at the same spatial position. So if we assume that the emission event occurs at the origin of S', the reception event occurs at some location that is not the origin of S'. To measure the time t', we need to introduce some concept of clock synchronization, or simultaneity. There are several ways we could do this, the approach I would use is to use two clocks, one at the location of the emission event, one at the location of the reception event, and some means of synchronizing the clocks.
There is no need to add simultaneity or clock synchronization issues to the plethora of complications that have been forced onto this very simple problem.

Let the "train platform" observer simply drop a clock every 1mm along the platform. On the train, when the lamp switches on it squirts water onto the platform (instantaneously of course), and when the light reaches the target water is again squirted onto the platform.

The observer on the platform simply walks down the platform and subtracts the times between the wet clocks. This is his elapsed time between the events.

Later, over dinner, he compares his notes with the observer that was on the train, and they discover that the platform observer's elapsed time was more.
This is simply because the light traveled a greater distance according the the platform observer, there is nothing more to it.

I think putting real numbers in here might help clarify this:

On the train:
Distance between lamp and mirror is 1 meter. Light leaving lamp is event 1. Light arriving at mirror is event 2.
Elapsed time according to train observer: 3.3ns

Train is moving at 0.9c

On the platform:
Horizontal distance traveled by train between events: 2.05 meters.
Vertical distance traveled by light between events: 1 meter (same as train observer).
Total distance covered by light between events: 2.24meters
Elapsed time between events: 7.5ns

All the platform observer needs to do is drop clocks 2.24 meters apart, accurate to 1ns or so, to witness this time dilation. This is absolutely trivial and does not require any Poincaré-Einstein synchronization methods etc.
Where I work, we move atomic clocks around tens of thousands of meters and expect them to still be within nanoseconds of each other. They not moving anywhere near relativistic speeds and any shift due to SR time dilation is virtually zero. In other words, it is irrelevant for this problem exactly how the S' observer chooses to synchronize the clocks along his x-axis, it is enough to state that they can be synchronized.

This is not an apparent time difference caused by the platform observer trying to look at the train passenger's clock through binoculars as the train zooms by causing latency in observation. It is a real time difference between the clock on the train, and the clocks distributed along along the platform.
 
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  • #132
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Consider the following spacetime intervals (all coordinates of events are given in Alice's frame):

A) The interval between events (0, 0, 0) and (1, 0, 0). These are two events on Alice's worldline.

B) The interval between events (0, 0, 0) and (1, 0.6, 0). These are two events on Bob's worldline.

What are the values of the spacetime intervals A and B? What is the ratio between them? What picks out these two particular intervals as the right ones to use to show that Bob is time dilated relative to Alice?
A) The time interval is 1 time unit. The Spacetime Interval = t = 1.

B) The time interval is 1 time unit. The spacetime interval = √(t2 - x2) = 1- 0.36 = 0.8

c) The ratio is 1.25 : 1

Here I have to say that using diagrams isn't letting me explain what my problem is as I cannot seem to draw them so that my intent is clear, so I will try descriptively:

Yet the problem that is bothering me is that when Alice measures that Bob has travelled 0.6 units to arrive at (1, 0.6, ) (at a speed of 0.6c)
Alice must also measure that Bob's light, having travelled 1 unit at 'c' in the y direction, will be at point (1, 0.6, 1) having travelled 1.166 units in Alice frame in 1 time unit! Whereas travelling at 'c' it would have travelled 1 unit along the rotated path. At which time it would have travelled 0.6 units along the x axis and have arrived at point (1, 0.6, 0.8).
When Alice measures Bob's light has travelled 1 unit, she also measures this is only travelled 0.8 of the distance to his mirror; and can calculate it will therefore only have travelled 0.8 units in Bob's frame.
Now Alice's light is also travelling along the y axis at 'c', remember the two clock's are synchronised, therefore when Bob's light has travelled 0.8 units y-wards Alice's light will also have travelled the same distance along the y axis. And as her light is travelling at 'c', only 0.8 units of time can have passed in Alice's frame when Bob's light, measured by her in her frame, has travelled 1 unit.
So when t (the time coordinate in alice's stationary frame) = 0.8,
t' ( the time measured by Alice to have passed in Bob's moving frame) = 1 (Because that is how far she measures it to have travelled)

However, while the Spacetime interval for Alice's light to have travelled 0.8 units along the y axis
= t = 0.8,
that for Bob's light to have travelled 1 unit, that is 0.8 of the distance toward his mirror measured by Alice
= √(t'2 - x2) = √1 - 0.36 = 0.8
which to me smacks of the Spacetime interval being invariant whether measured for Alice's light to travel 0.8 units, her measurement of Bob's light travelling 1 unit, and even Bob's light travelling 0.8 units, measured in Bob's frame.

Btw, if you're having trouble answering this question, consider an alternative pair of intervals:

a) The interval between events (0, 0, 0) and (1.25, 0, 0). These are also two events on Alice's worldline.

b) The interval between events (0, 0, 0) and (1.25, 0.75, 0). These are also two events on Bob's worldline.

You should be able to confirm that the ratio between these two intervals is the same as the ratio between intervals A and B from my previous post. What does that ratio represent?
a) time interval 1.25 units. Spacetime interval = t = 1.25

b) time interval 1.25 units. Spacetime interval = √(1.252 - 0.752) = 1

c) the ratio is 1.25 : 1

d) the ratio is the Lorentz factor.

But are you saying that the time dilation is something that happens to the invariant Spacetime interval rather than the time in the moving clock increasing as Einstein described here:
As judged from K, the clock is moving with the velocity v; as judged from this reference-body, the time which elapses between two strokes of the clock is not one second, but
M5.GIF


seconds, i.e. a somewhat larger time.
I am understanding what you are telling me, yet am struggling to fit it to what Einstein is describing. After all he makes no mention of Spacetime intervals...
 
  • #133
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Sort of. As long as we are talking about inertial coordinates in SR (i.e., flat spacetime), this view works ok, because you can think of the coordinates as corresponding to measurements made by a fleet of observers with measuring rods and clocks, all having the same state of motion as the "reference" observer (Alice or Bob or whoever). Note that even here, a single observer isn't making all the measurements, because observers can only make measurements at events on their worldlines, and one observer can't be on multiple worldlines. But in any case, as soon as you try to use non-inertial coordinates, or any coordinates in curved spacetime (i.e., when gravity is present), this no longer works.

The more general way to look at coordinates is that they are just assignments of unique sets of four numbers to each event in spacetime, plus some conditions on the assignments to make the numbers work the way we are used to having coordinates work (things like nearby events should have "nearby" coordinates, etc.). "Time" is just one of the four numbers (and even calling it "time" depends on some assumptions that might not be true for some choices of coordinates).
Thank you, but please remember I was educated to High School Level - I had to leave university after 1 term for health reasons. So I am trying to get to grips with Special Relativity. There seems little point in discussing anything to de with General Relativity until I have grasped this.
 
  • #134
PeterDonis
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A) The time interval is 1 time unit. The Spacetime Interval = t = 1.
Yes.

B) The time interval is 1 time unit. The spacetime interval = √(t2 - x2) = √(1- 0.36) = 0.8
Yes.

c) The ratio is 1.25 : 1
The ratio of Alice's interval to Bob's interval, yes. Which means the ratio of Bob's interval to Alice's interval is the reciprocal of that, or 0.8. Which is also Bob's time dilation factor, relative to Alice. Or, if you want to use the SR symbols, the ratio 1.25 is ##\gamma##, and the ratio 0.8 is ##1 / \gamma##.

when Alice measures that Bob has travelled 0.6 units to arrive at (1, 0.6, ) (at a speed of 0.6c)
Alice must also measure that Bob's light, having travelled 1 unit at 'c' in the y direction, will be at point (1, 0.6, 1)
No, it won't; it will be at event (1, 0.6, 0.8). Light doesn't travel at speed 1 in the y direction; it travels at speed 1 overall.

The rest of this section of your post just compounds your error here; you need to rethink it.

a) time interval 1.25 units. Spacetime interval = t = 1.25

b) time interval 1.25 units. Spacetime interval = √(1.252 - 0.752) = 1

c) the ratio is 1.25 : 1

d) the ratio is the Lorentz factor.
All correct.

are you saying that the time dilation is something that happens to the invariant Spacetime interval
No. Please read carefully. I am saying that the term "time dilation" is just a way of describing the fact that the ratio between the two intervals--Alice's to Bob's--is the ##\gamma## factor, 1.25. What picks out those two intervals? The fact that they both have the same difference in the ##t## coordinate in Alice's frame (0 to 1.25 in the case of the pair just above). In other words, the starting and ending events for both intervals happen at the same time according to Alice. And happening at the same time according to Alice is the key criterion for picking out events on different worldlines that "correspond" to each other with respect to Alice.

So what we are saying when we say that Bob's clock is "time dilated" relative to Alice's is that, if we pick an interval on Bob's worldline that "corresponds" to a particular interval on Alice's worldline, the ratio of the two intervals (Alice's to Bob's) will be the ##\gamma## factor. Bob's interval will be shorter, so his clock is "running slow" relative to Alice.

I am understanding what you are telling me, yet am struggling to fit it to what Einstein is describing. After all he makes no mention of Spacetime intervals...
IIRC he does later in the same book; but in any event, Einstein's book is not a textbook. A good textbook on the subject is Taylor & Wheeler's Spacetime Physics; it introduces the spacetime interval very early, precisely because it has been found (over the decades since Einstein wrote his book) that the spacetime interval, and spacetime geometry, provides a good way of conceptualizing the key aspects of relativity.

To briefly explain what Einstein was saying in more modern terms: the "somewhat larger time" he refers to as elapsing "as judged from this reference-body" between two strokes of the moving clock is Alice's spacetime interval--which of course is the same as the change in coordinate time in Alice's frame (as can easily be seen from the numbers given above). The time elapsed on the moving clock is Bob's spacetime interval. And the ratio of the two, Alice's to Bob's, is ##\gamma##, which is the expression that Einstein wrote down.

One other thing to keep in mind when reading Einstein's book is that he originally wrote it in German, and you are reading an English translation. In some ways this is unfortunate, since some of the wording in the translation is not really what a native English speaker would have written to describe the same concepts--still more so for a native English speaker today vs. one a century ago.
 
  • #135
pervect
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There is no need to add simultaneity or clock synchronization issues to the plethora of complications that have been forced onto this very simple problem.
I have to disagree, unfortunately. The OP seems to think that the time interval t is "the same" as the time interval t', even though they have different numerical values. This seems to be confusing him greatly - which it should, if it were in fact true, it would be a logical contradiction.

I'm pointing out that the time interval t is not "the same" as the time interval t'. In the jargon of SR, one is a proper time interval, the other is not a proper time interval. Thus, they cannot be "the same" interval. Which is what one would expect, they have different numerical values, which should be a very big clue they are not "the same".

But the OP isn't familiar with the jargon, so this short answer isn't helpful. Hence, the longer answer. Additionally, I can't help but point out that the question revolves about comparing two time intervals - time intervals that are different, but the OP doesn't see why they are different, he thinks they are the same. The comparision process to illustrate why they are different is complicated by the fact that none of the diagrams even includes time, hence the diagrams are not so helpful as they might be in figuring out why the proper time interval t is different from the non-proper time interval t'. You say there is "no need" to draw a Minkowskii space time diagram, but it seems to me that it basically is needed, as the OP is claiming that two different things are the same, when they are in fact different. One approach to illustrate they are different is to draw a diagram to illustrate the difference, but for this to be effective, the diagram needs to actually show what is being compared - and in this case what is being compared are the two the time intervals, t and t'. But to compare them effectively, we need to illustrate these intervals on the diagram - and the current set of diagrams don't even show time at all, so it's just not a good tool for answering the question.
 
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  • #138
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I have to disagree, unfortunately. The OP seems to think that the time interval t is "the same" as the time interval t', even though they have different numerical values. This seems to be confusing him greatly - which it should, if it were in fact true, it would be a logical contradiction.

Please allow me to explain what I think and how it works.

I'm pointing out that the time interval t is not "the same" as the time interval t'. In the jargon of SR, one is a proper time interval, the other is not a proper time interval. Thus, they cannot be "the same" interval. Which is what one would expect, they have different numerical values, which should be a very big clue they are not "the same".
Thank you. I know what is causing confusion here - it is indeed the jargon of SR. Unfortunately; I am afraid that unless I am careful I tend to employ 'interval' with its literal meaning rather than as SR jargon, as you put it.
https://ac0077b2-a-62cb3a1a-s-sites.googlegroups.com/site/specialrelativitysimplified/home-1/minkowski-diagrams/Lorentz%20factor.png?attachauth=ANoY7crGGigryP97e5eUZ51gcU-xKwyFXvA10ufVcgTOOVVdazJYzRAeHVE1KNSJHNOgpYuLYXsKRsC3cT6rxQ3dS2w_1obZPyfcKuOGg6jjrE87IzRquRr-n6TYn42u-TluP4ADVwuuT9BP811xAHK8tNdAYuV71hFY1j7WhGycSKQDeeBtRUaUIISVoNrA9tkgLXkdYLe0jCDhj9jpxmRAldZkQG0BsRciIbYVv2srxis3M9xVigNPlh-YUKydLHr6wODeRWQR6WkattjKdJPr1HrghXUDdw%3D%3D&attredirects=0 [Broken]
depicted in green, ct is the time axis of the resting observer, Alice whose frame is.
depicted in red, ct' is the time axis of Bob, as seen by Alice. Rotated by Bob's movement along Alice's x axis. Alice measures this, as can be seen in the diagram as reaching point (0.6,0.8) in time t - it lies on the 0.8 coordinate of her time axis using orthogonal cartesian coordinates.
ct' is the time that Alice measures for Bob to reach that point: the time that Bob measures, converted by the Lorentz Transformation Equation. Which converts that measurement to be relative to Alice. Is that not the point of the Lorentz transformations?
To take a measurement in one inertial Frame and to make it relative to an observer in another inertial frame moving with respect to the former frame.
The moving observer has the additional movement between the frames as an added factor in calculating the measurement relative to the moving observer. So the measurement relative to the moving observer will always be greater by the Lorentz factor, γ.

Alice does measure the Spacetime interval (0, 0) to (0.6, 0.8) in the diagram. That is √(t'2 - x2) = √(1 - 0.36) = 0.8, the same as the spacetime interval between (0, 0) and (0.0, 0.8), the proper time for Alice.

Note that (me being pedantic here?) in a Spacetime diagram, which I believe this is, as we are plotting space - x, against time - ct, then after 0.8 seconds it is Alice that is at point (0,1), and it is Bob, not Bob's light that is at point (0.6, 0.8) - because we are plotting x against ct it is Bob and Alice who are moving along the time axis and Bob is also moving along the x axis. Light doesn't come into it. It is Bob who has moved in Space and time. So we are measuring Spacetime Intervals and the two here referred to are the same - the invariant spacetime interval.

The difference is between time t - the proper time, measured by Alice on her time axis, and the coordinate time measured by Alice on Bob's rotated time axis.

The actual Spacetime interval measured, and experienced by Alice is 0.8 along her time axis, which is also her proper time.
Her measurement on the Spacetime Interval for Bob, who also is displace laterally, also calculates out to 0.8, which seems right to me, for that means that by taking into account Bob's physical movement, she can calculate that the Spacetime Interval (which is a measure of the time elapsed, having subtracted the effect of any distance moved), yes, the Spacetime interval for Bob has the same value = 0.8.

Which seems to me to be entirely reasonable that two clocks that are synchronised are measured to have the same Spacetime intervals between the emission of their light pulses and their reflections in their respective mirrors.

 
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  • #139
PeterDonis
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depicted in green, ct is the time axis of the resting observer, Alice whose frame is.
No, it isn't. Your diagram is wrong, as is your analysis.

You need to go back and re-read, carefully, the posts where I gave you the coordinates of the events in question. There are four events that a correct diagram needs to depict. Each event has three coordinates of interest, not two; your diagram depicts only two, and you are mixing up which ones they are. You can't represent those events correctly in a Euclidean diagram because the geometry of spacetime (not space!) is not Euclidean; it's Minkowskian.

Also, I would recommend that you not even think about Lorentz transformations at all at this point. You have not even reached a correct understanding of how spacetime events are represented in a single frame. You need to do that first, before trying to understand how the representations in different frames are related, which is what Lorentz transformations are about.

Here are the correct coordinates of the four events, in Alice's rest frame. All coordinates are given as triples, (t, x, y).

O) The origin. This is at coordinates (0, 0, 0).

A) The event at which Alice is located at coordinate time 1 unit. This is at coordinates (1, 0, 0).

B) The event at which Bob is located at coordinate time 1 unit. This is at coordinates (1, 0.6, 0).

C) The event at which Bob's light ray is located at coordinate time 1 unit. This is at coordinates (1, 0.6, 0.8).

There are three spacetime intervals of interest. They are:

O to A: Interval 1 unit. This represents Alice's proper time.

O to B: Interval 0.8 units. This represents Bob's proper time.

O to C: Interval 0 units. This represents the null interval of the light ray--all light rays have null intervals.

Your diagram and analysis does not correctly represent these events and intervals, even though I have described them to you several times now, and you have even calculated the intervals correctly. If you are inclined to dispute that point (which you did in your latest post), this should be a big red flag to you that you do not understand the correct representation of events in Alice's inertial frame. The coordinates that I have given above are correct; your objective should be to understand why they are correct, not to try to convince me that they are incorrect.

When we say that Bob is "time dilated" relative to Alice, we are comparing the intervals O to A and O to B, which are related by the factor ##\gamma##. Interval O to B is shorter; that's why we say Bob's clock "runs slow" relative to Alice. I've said this before as well, but it is still not reflected correctly in your analysis.

At this point I am closing the thread because we are going around in circles. If you have further questions, please PM me.
 
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  • #140
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Thank you. I know what is causing confusion here - it is indeed the jargon of SR. Unfortunately; I am afraid that unless I am careful I tend to employ 'interval' with its literal meaning rather than as SR jargon, as you put it.
https://ac0077b2-a-62cb3a1a-s-sites.googlegroups.com/site/specialrelativitysimplified/home-1/minkowski-diagrams/Lorentz%20factor.png?attachauth=ANoY7crGGigryP97e5eUZ51gcU-xKwyFXvA10ufVcgTOOVVdazJYzRAeHVE1KNSJHNOgpYuLYXsKRsC3cT6rxQ3dS2w_1obZPyfcKuOGg6jjrE87IzRquRr-n6TYn42u-TluP4ADVwuuT9BP811xAHK8tNdAYuV71hFY1j7WhGycSKQDeeBtRUaUIISVoNrA9tkgLXkdYLe0jCDhj9jpxmRAldZkQG0BsRciIbYVv2srxis3M9xVigNPlh-YUKydLHr6wODeRWQR6WkattjKdJPr1HrghXUDdw%3D%3D&attredirects=0 [Broken]
depicted in green, ct is the time axis of the resting observer, Alice whose frame is.
depicted in red, ct' is the time axis of Bob, as seen by Alice.

If the top line were green, and labelled vt, this diagram would be a correct Minkowskii diagram.

Note that on a Minkowskii diagram, the square of the hypotenuse is not the sum of the square of the other two sides, as it is in Euclidean geometry,. Rather, square of the hypotenuse is equal to the difference of the squares of the other two sides. In the jargon, the geometry is called a "Lorentzian" geometry.

Given this, we can write ##(ct)^2 - (vt)^2 = (ct')^2 ##, and we get ##t' ^2= [1-(v/c)^2 ]t^2## This is backwards from your result, but it says that the proper time interval of the moving observer is shorter than the improper time inverval of a stationary observer, which is the result we are looking for.

A quick (though not complete) way of partially justifying why it's the difference of the squares that is constant is this. We can write the equation of a light beam in the unprimed frame (t,x) as ##(ct)^2 - x^2 = 0##. In the primed frame (t', x'), the equation of a light beam is ##(c't')^2 - x'^2 = 0##. And we know that c' = c, so we can say that ##(ct)^2 - (vt)^2 = 0## implies that ##(ct')^2-x'^2=0##. It turns out that we can make a stronger statement than this, it turns out that ##(ct)^2 - x^2## is always equal to ##(ct')^2 - x'^2## even when the quantity is not zero. The jargon for this is that the quantity ##c^2t^2 - x^2## is given a name, the Lorentz interval or the space-time interval, and that the space-time interval has the property that it's value is independent of the choice of reference frame.


This implies that the time of the moving observer (in red), which is a proper time, is shorter than the time of the stationary observer (green). Which is as it should be.​
 
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