[Halc]

What I still find awkward is trying to use the correct terms as my natural language is not scientific and in such areas, as in these forums, I often received replies that are, not criticizing exactly, but often correcting my use of terms that have precise scientific meanings rather than addressing the point I am making.

Fair enough. Keep in mind that science is a precise language and meanings change when wordings change.

The time of a moving clock is dilated - there is more of it.

Dilated means there is less of it.

Yes I can see that is how it is taken, yet it seems to me that if  frame A is moving relative to frame B, then if clock A ticks 5 times between event 1 and event 2, then being in frame A the difference will be time only

By this I presume that clock A is stationary in frame A, and ditto for B, and both are inertial. If clock A is present at events E1 and E2, then clock B is present at only one of them at best, and thus hasn't measured the inertial temporal separation between those two events.  So OK, let's say the two events are separated by 5 seconds. (I am assuming units/ticks to be one second each)

Proper time, the time displayed on the clock, 5 units

All clocks read proper time, so yes.  Clock A displays 5 seconds at event E2 assuming zero at E1.

While observed from frame B, each tick will be longer by the Lorentz factor, so after 5 ticks the time B will measure = γ5 units.

From frame B, seconds on clock B are the same length, 1 second each. But it is present only at event E1 I presume, so it reaches event E3 and E4 when clock B measures 2.5 and 10 seconds respectively (assuming γ is 2).  In frame B, events E4 and E2 are simultaneous, and in frame A, events E3 and E2 are simultaneous.  The difference is due to the fact that after E1, the two clocks are no longer in each other's presence, and hence the ordering of all these events is frame dependent.

So more time has passed between events 1 & 2 for a clock in Frame B, when it is measured from Frame A relative to which frame B is moving,

OK, I see what you're saying.  It reads 10 seconds at the event simultaneous with E2 in frame B, yes.

Yes, but why then do we talk about clocks slowing – including Einstein who wrote:

As a consequence of its motion the clock goes more slowly than when at rest.

Quote granted.  Einstein uses that wording it seems when speaking of dilation. The comment sans context lacks frame references, which are there (frame K) if context is restored.  It 'goes more slowly' in a frame in which it has more motion than does a clock which, in that frame, has less or none.

If the moving clock is keeping exact time, as is the resting clock, how is it that "As a consequence of its motion the clock goes more slowly than when at rest"?

It keeps exact proper time, not exact time, since there is no such thing.  I meant to convey that.  Under GR there is sort of an exact time, but no clock rate seems to ever be compared to it.  Nobody can build a clock that compensates and thus reads actual time, if there is such a thing.

it is the correct time for that clock as an inertial clock;

They're both inertial in your example

Indeed they are  but why was that worth commenting on?

Making sure you know the difference between the term inertial and stationary. Only the latter is a relation. You used the term in a context which seemed to imply it being the stationary one.

My point was that as an inertial clock the first postulate was relevant and the clock therefore had to be displaying proper time.

OK, but even a non-inertial clock will display proper time.  A non-inertial clock might be present at the same two events as an inertial clock but read a different duration between those two events due to a different temporal length of the two respective worldlines.

OK. So rates are only relevant to frames or reference, not observers within those frames?

The rates are relevant to observers, but those rates don't reflect what those observers will observe.  A moving clock might run slow in my frame, but if it is coming towards me, I will observe it running faster than my local clock.  If a ship a lightyear away departs for Earth at 0.9c, the trip will take 1.11 years in my frame, but 40 days to an observer because the light from the departure event takes a year to get here.  The clock on the ship will log 159 days to make the trip, so if that clock is observed from Earth, it will appear to run 159 days in 40 days, or about 4x faster than an Earth clock.  That's what I mean by the difference between how a frame orders events and how events are observed

Both the resting frame and the moving frame are inertial frames - (I know you agree with that )

Yes, but I might have issue with any frame being called 'the resting frame'.

So both have the same time rate.
The observer in Frame A measures proper time along its world line where only time is changing.
The observer in Frame B measures dilated time along A's world line because, seen from B's frame, both the time and location of clock A are changing.

See, that's why I don't like observers.  The observer in one frame observes a faster rate for the incoming moving clock, which is quite different than saying that the moving clock is dilated (slower) in that frame.  He can of course compute how far away each event is and compensate, but then it becomes a computation, not an observation. I just find it easier to skip observation and say that time is dilated for objects moving relative to a given frame.

Now it seems to me that the rate observer B, in frame B, measures for Clock A, in frame A, is different from both the rate in frame B measured from frame B and the rate in frame A measured from frame A.

Of course.  One's own proper rate is just '1', a comparison between two identical things. The rate of A in B's frame (at least under SR) is the same as the rate of B is A's frame.  Not so under GR where two clocks can remain at a constant separation and yet have their time diverge, with one clock clearly racking up more time than the other no matter the frame.

The language gremlin bites again!
Yes I know that you are right for Special Relativity was deduced from Einstein's two Postulates.
If it is wrong to say that Einstein used the Lorentz Translation Equations to shew that a clock goes more slowly,

I think it is since the transform is the conclusion of the argument (the prediction), not one of the premises from which the behavior was derived.

it is certainly true that he used them to argue that point in chapter XII. He made no reference as far as I can see to the constant speed of light in that chapter.

Agree, the effect was already established at that point and the paper is turning to quantification.  Here is how to compute exactly what clocks will do given this or that motion.  From that, experiments like Hafele–Keating could be performed with a range of expectations that would verify or falsify this new theory.
The Lorentz transformation was obviously already around at the time, else it would have been named after Einstein. The theory was a cumulation and generalization of the work of several contributors.

[Colin2B]

STR and GR are mathematical models of reality that do not get into the mechanics of phenomena. It would take a new theory to explain mechanics of gravity or time dilation. But vibrations in a clock are subject to external forces like gravity.

This is incorrect. For example, SR very clearly describes the mechanism of time dilation.

[PmbPhy]

I created a webpage to provide the derivation. See: http://www.newenglandphysics.org/physics_world/sr/time_dilation.htm

[Fussball]

For example, SR very clearly describes the mechanism of time dilation.

Where does SR get into the mechanics of a clock or its vibrations? Unless you're talking about a light clock. I agree that a horizontal light clock, moving horizontally, has its tick rate slowed due to motion. I am skeptical of the vertical light clock, where the theory of aberration applies.

[Fussball]

A moving clock slows down or does it tick normally because in its frame it is always stationary. Therefore, a moving clock will tick normally (without slowing). Does my question make sense?

[pzkpfw]

A moving clock slows down or does it tick normally because in its frame it is always stationary. Therefore, a moving clock will tick normally (without slowing). Does my question make sense?

Say you and a friend are drifting in space. You are both wearing watches.

The distance between you is changing, so there is relative motion.

There is no absolute concept of "moving" or "not moving", so both of you can consider yourself as at rest, and the other is the one who is moving.

For both of you, your own watch (no matter what kind of clock it is) is ticking at 1 second per second.

For both of you, the watch on the other is slow.

[Fussball]

For both of you, your own watch (no matter what kind of clock it is) is ticking at 1 second per second.

For both of you, the watch on the other is slow.

That's not possible. It either ticks slow for all or it ticks normally for all. I say that it ticks normally for all, because a clock ticks according to its own frame. And in its own frame it is always at rest and therefore it ticks normally for all. Where is the possibility of recording the slowing down of a clock when in its own frame it is always at rest?

[Colin2B]

Where does SR get into the mechanics of a clock or its vibrations?

If you think SR has to do with the mechanics of clocks then you have completely misunderstood relativity and ought not to be answering questions in this section

A moving clock slows down or does it tick normally because in its frame it is always stationary. Therefore, a moving clock will tick normally (without slowing). Does my question make sense?

Within its own frame a clock is ticking normally, but other clocks moving relative to it will be measured as slow

[Fussball]

Within its own frame a clock is ticking normally, but other clocks moving relative to it will be measured as slow

If all clocks are at rest in their own frame, where does the slowing down take place?

[Colin2B]

That's not possible. It either ticks slow for all or it ticks normally for all. I say that it ticks normally for all, because a clock ticks according to its own frame.

Then you are again mistaken.
I suggest you stop posting in topics you know nothing about.

If all clocks are at rest in their own frame, where does the slowing down take place?

It takes place in the measurements between frames.
I suggest you start by reading up about Galilean relativity and then try to understand the difference with SR.

Until you have done that, please don’t participate in this thread as your comments are only confusing the OP’s questions.

[TyroJack (OP)]

I created a webpage to provide the derivation. See: http://www.newenglandphysics.org/physics_world/sr/time_dilation.htm [nofollow]

Thank you, that is an excellent derivation - it is exactly how I think and wanted to write it down! Great!
There is one further point though, that I believe is worth making using your diagram:
From S', the 1 second  measured to have passed in S is measured to have taken 7.09 seconds. The clock is measured to take longer to tick in frame S.
Yet an identical light clock in frame S' would also have ticked in 1 second, measured in S', while S' measured the greater time to pass in S.

What we see is that in each frame two times are measured - τ for the time measured on that frames clock and the longer time t measured for the tick of the clock in the other moving frame.

This is no surprise if we examine the invariant Spacetime interval between the two events.
Yes, the spacetime interval which is invariant whichever frame it is measured from.
Measured from the clocks frame the spacetime interval s² = c²τ² =1
Measured from a different frame from which the clock is moving, the spacetime interval s² = c²t² - v²t²
so    c²τ² = c²t² - v²t²
          τ² = t(c² - v²)
          τ = t(1 - v²/c²)^-½
or     τ = t/γ
So the time for a tick of the moving clock = γ seconds, measured from the frame relative to which it is moving.

[Colin2B]

I created a webpage to provide the derivation. See: http://www.newenglandphysics.org/physics_world/sr/time_dilation.htm

Thanks Pete, I was trying to find that for @TyroJack but couldn’t locate it.

There is also this http://www.newenglandphysics.org/physics_world/gr/grav_red_shift.htm
Which although it refers to gravitational redshift does give a useful diagram of the clock differences.

[Halc]

This is no surprise if we examine the invariant Spacetime interval between the two events.
Yes, the spacetime interval which is invariant whichever frame it is measured from.

Indeed, there is an invariant spacetime interval between any two events, but there are move than two events depicted in that example, even if none of them are explicitly labeled.  I count 6 events in all, but two of them (emit, detect) are called out, and yes, there is a frame independent interval between them.

Measured from the clocks frame the spacetime interval s² = c²τ² =1

That can't be.
τ is say a microsecond and x (distance) is 300 meters, so c²τ² - x² = 300m² - 300m² = 0, so s = √0 = 0, not 1.  The interval between any pair of events separated in a light-like manner is always zero.

Edit:  I was computing the interval between the bottom and top, not a detector at the bottom waiting for the reflected signal to return.  Yes, there is a nonzero interval between those two events, which in my example is 600m if the mirrors are 300m apart in the light clock.

[TyroJack (OP)]

Yes, the spacetime interval which is invariant whichever frame it is measured from.
Measured from the clocks frame the spacetime interval s² = c²τ² =1
Measured from a different frame from which the clock is moving, the spacetime interval s² = c²t² - v²t²
so    c²τ² = c²t² - v²t²
          τ² = t(c² - v²)
          τ = t(1 - v²/c²)^-½
or     τ = t/γ
So the time for a tick of the moving clock = γ seconds, measured from the frame relative to which it is moving.

c = 1
τ = 1 sec - given in the description.
t = 7.09 - given in the description.
v = 0.99c - given in the description.
γ = 7.09
¹/_γ = 0.141

Substituting for  τ = t/γ we get:    1 = 7.09 x 0.141   
                                                          1 = 0.99969

The clock in frame S ticks once a second.
The clock in frame S' ticks once every second.
The relative speed of the frames is 0.99c
Event 1.  The two clocks start their ticks when they are adjacent.
Event 2.  The clock in frame S completes its tick.
Event 3.  The clock in frame S' completes its tick.
The invariant spacetime interval between event 1 and event 2 is one second.
Relative to frame S event 1 and Event 2 are at the same location.
Relative to frame S' event 1 and event 2 are separated by 1 light second.

Relative to frame S' event 1 and event 3 are at the same location.
Relative to frame S event 1 and event 3 are are separated by 1 light second.

Relative the distance between event 3 and event 4 (that is the distance between the clocks after 1 second, measured on either clock) is 0.99 light second.

The invariant spacetime interval between event 1 and event 2 is 1 light second.
The invariant spacetime interval between event 1 and event 3 is 1 light second.

Ah! I can see what is leading you astray, Halc. Event 1, is the beginning of the tick and event two is the end of the tick, not emitting the light and receiving it - that is the mechanism of the clock. We could just as easily be talking about a pendulum clock!  For out purposes it is just a clock. After 1 second only the time coordinate has changed.
It is confusing because we use the light clock to shew how the moving clock has longer ticks. The light travel is also the time measured.  We are concerned with the movement of the clocks, not the lights within those clocks.

[Halc]

c = 1
τ = 1 sec - given in the description.
t = 7.09 - given in the description.
v = 0.99c - given in the description.
γ = 7.09
¹/_γ = 0.141

Substituting for  τ = t/γ we get:    1 = 7.09 x 0.141   
                                                          1 = 0.99969

No, 1=1.  There are only 3 digits of precision being used.

The clock in frame S ticks once a second.
The clock in frame S' ticks once every second.

The relative speed of the frames is 0.99c
Event 1.  The two clocks start their ticks when they are adjacent.
Event 2.  The clock in frame S completes its tick.
Event 3.  The clock in frame S' completes its tick.

The invariant spacetime interval between event 1 and event 2 is one second.

c times one second actually, so a light second, not a second.  Seems to be a typo since you get it right everywhere else.

Relative to frame S event 1 and Event 2 are at the same location.
Relative to frame S' event 1 and event 2 are separated by 1 light second.

Relative to frame S' event 1 and event 3 are at the same location.
Relative to frame S event 1 and event 3 are are separated by 1 light second.

That's the interval, yes.  Agree with all of this.

Relative the distance between event 3 and event 4 (that is the distance between the clocks after 1 second, measured on either clock) is 0.99 light second.

There was an event 4?  None was defined.  Yes, in either frame, the distance between clocks 1 second after event 1 is .99 light seconds, but they're not the same two events being compared.

The invariant spacetime interval between event 1 and event 2 is 1 light second.
The invariant spacetime interval between event 1 and event 3 is 1 light second.

Ah! I can see what is leading you astray, Halc. Event 1, is the beginning of the tick and event two is the end of the tick, not emitting the light and receiving it - that is the mechanism of the clock.

I have no idea what you mean by this statement.  How are those different things?  Does not light get emitted at event 1?  At a different time or location?

We could just as easily be talking about a pendulum clock!  For out purposes it is just a clock. After 1 second only the time coordinate has changed.

I'm fine with that.  Event 1 is the start event of one second for two pendulum clocks, or maybe they're grenades thrown in simultaneously in different directions, set to explode in one second, which makes for an obvious event 2 and 3.  It changes none of the computations of intervals between events and such.

[TyroJack (OP)]

No, 1=1.  There are only 3 digits of precision being used.

Yes indeed 1=1 silly me!

c times one second actually, so a light second, not a second.  Seems to be a typo since you get it right everywhere else.

yes, thank you

There was an event 4?  None was defined. 

My mistake there is no need for an event 4; 0.99 is the distance between event 2 and event 3 - the distance frame S' has travelled from frame S.

Yes, in either frame, the distance between clocks 1 second after event 1 is .99 light seconds, but they're not the same two events being compared.

The invariant spacetime interval between event 1 and event 2 is 1 light second.
The invariant spacetime interval between event 1 and event 3 is 1 light second.

But I am not comparing the invariant spacetime interval between event 1 and event 2 with the invariant spacetime interval between event 1 and event 3.
Event 1 is the start of the ticks of each clock.
Events 2 and 3 are the ends of those ticks.
Each invariant spacetime interval can be measured two ways:
from S the interval between 1 and 2 is time only s²=(ct)² but measured from S' that interval s²=(ct')² - (vt')² . Similarly
from S' the interval between 1 and 3 is time only s²=(ct)² but measured from S that interval s²=(ct')² - (vt')²  (note: t is the time in the frame that is not moving and t' is the time in the moving frame in each scenario)
All straight forward and reciprocal as it should be...

Ah! I can see what is leading you astray, Halc. Event 1, is the beginning of the tick and event two is the end of the tick, not emitting the light and receiving it - that is the mechanism of the clock.

I have no idea what you mean by this statement.  How are those different things?  Does not light get emitted at event 1?  At a different time or location?[\quote]
I was responding to this:

Edit:  I was computing the interval between the bottom and top, not a detector at the bottom waiting for the reflected signal to return.  Yes, there is a nonzero interval between those two events, which in my example is 600m if the mirrors are 300m apart in the light clock.

If we were to look at the light in the clock it does indeed travel 600m as you say, but if we are examining the clocks then the clock in S does not move and the spacetime interval is the one second elapsed for that clock. s²=(ct)²

[Halc]

There was an event 4?  None was defined. 

My mistake there is no need for an event 4; 0.99 is the distance between event 2 and event 3 - the distance frame S' has travelled from frame S.

Frames don't have locations, so they don't travel.  I know what you mean by that however, and yes, an object moving at .99c in some frame is by definition .99 light seconds away after one second in that frame, but events 1 and 3 are more than 1 second apart in that frame.  The moving clock has been moving for 7.09 seconds in that frame, not 1.

Events 1 and 2 occur at the same location in space and event 3 happens 7.09 seconds after event 1, so events 2 and 3 are about 7 light seconds apart in space in either frame, but not necessarily in other frames.

I didn't respond to the rest because I don't disagree with any of it.