JM said:
ghwellsjr said:
I don't know what you mean by a linked frame and I see no advantage or need for an accelerating frame when any inertial frame will do everything that needs to be done and so much more simply. So I'm not the one to ask about other types of frames but I see DaleSpam has provided an answer. It's a good bet to trust what he says.
Refer to 1905 section 4: "It is at once apparent that this result still holds good if the clock moves from A to B in any polygonal line, and also when the points A and B coincide."
It is not apparent to me. If it is to you,can you explain it to me?
Where are the points A and B in terms of x,y,z,t, and where is the polygonal line? The theory of section 3 refers to clocks moving parallel to x, so how to make a polygon out of that?
In section 3, the clocks were moving parallel to x because it is conventional in the standard configuration of the Lorentz Transformation to align the axes so that the motion is along the x-axis. It doesn't matter physically which direction the motion is in, we just assign the two co-ordinate systems so that the relative motion between them is along the x-axis. Remember, all frames are equally valid, including ones where the only difference is the orientation of their axes.
So once Einstein establishes that any clock that moves in a reference frame along the x-axis will tick at a slower rate than the co-ordinate clocks of that reference frame, he generalizes the concept to include any clock moving in any direction and he says to pick any two additional clocks, one at any point A and one at any other point B, not necessarily aligned along the x-axis, which had previously been synchronized with each other when at relative rest, and move the one at A to the position of the one at B at some relatively slow velocity v, then when it gets there, it will be slow by ½tv
2/c
2 compared to the clock at B. (Note that this formula is approximate and only applies to a slow-moving clock.)
Then he says that we can repeat the process, moving the A clock from the first B position to another B position in any other direction and we will get the same additional difference in clock time when it gets there. We can repeat the process as many times and in as many directions as we want, even to the point where we eventually return the A clock to its original location and the same formula applies if we take the total time t for the clock to make its round trip. This is what he means by the A and B points coinciding.
JM said:
The picture that sentence suggests to me is a series of stationary frames, each one aligned along one segment of the polygonal line, with an accompanying moving frame. The change of direction from one segment to another implies an acceleration of the clock. I don't see anything in section 3 about that.
You can do the analysis with multiple additional frames if you want, but it is just more complicated with no additional increase in knowledge.
JM said:
If one clock is on the equator and the other is at the pole then their positions will never coincide. So what is the explanation?
Prior to space travel (or sustained air travel), this was the only way to carry out the experiment. And it still would work, neglecting any effects from gravity, even if the clocks don't ever come to the same location because we are considering just one inertial rest frame, that of the clock at the pole. But of course nowadays, we just have the moving clock take off in a spaceship (or airplane, which has been done).
Don't be confused by the oft-repeated statement that clocks have to be co-located at the start and end of the journey of one of them to compare times. All frames will show that there is a difference in accumulated times, even if they don't agree on the absolute times on the two clocks (because of simultaneity issues).
JM said:
ghwellsjr said:
I'm sorry, I can't figure out what you mean here. What are the moving clock's roots in the stationary frame and what 'events' are you talking about.
See the posts on page one of this thread, and the one just above.
You repeated several times that man-made clocks do what we tell them to do but let's assume that they all have one thing in common, they tick once per second. Then the only issue is how many ticks have transpired between point A and point B, agreed? In this sense, we can treat them as stop watches, even if they actually display time as hours, minutes, and seconds or if they count backwards.
But the point is that no one makes a clock that is designed to tick slowly when it is traveling with respect to some rest frame--how in the world would they do that? And you overlook that fact that two identical clocks in inertial relative motion would each tick more slowly compared to its own tick rate. How do you design clocks to do that? No, it happens independently of any purposeful design, in fact if you tried to make it happen, it wouldn't be reciprocal.
JM said:
ghwellsjr said:
I thought we resolved that the confusion was over Taylor and Wheeler's restrictive definition of a 'proper clock' and you were fine (post #107) with the fact that any moving clock, inertial or not, will tick more slowly than the co-ordinate clocks in the frame in which it is moving
The confusion was about the meaning of the phrase 'moving clocks run slow'. It was cleared up with the qualifier that proper clocks run slow, not arbitrary coordinate clocks. We didn't get to what a correct definition of a proper clock is.
It wasn't cleared up by the qualifier that only "proper clocks" run slow and we did get to the correct definition of a "proper clock". But it is not a generally acknowledged definition. It is what I would call a private definition made by Taylor and Wheeler on page 10 of their book which you pointed out. No one else talks about a "proper clock". Instead, we keep repeating that all clocks keep track of "proper time". This applies to inertial clocks and non-inertial clocks, moving clocks, stationary clocks, accelerating clocks and co-ordinate clocks. All clocks keep track of their own proper time. They don't have any choice. Of course we are talking about carefully designed clocks that aren't affected by environmental effects, such as a pendulum clock.
JM said:
I am fine with inertial clocks being slow, but not non-inertial ones. As I noted above, I don't see how the inertial analysis of section 3 applies to non-inertial clocks.
Well, I hope you can see it now.
JM said:
ghwellsjr said:
The link to the book was provided by harrylin in post #18 and quoted by you in post #23 so I thought you had taken a look at it.
I have that book, and I have read it. I don't recall anything about clocks moving in various directions, or all clocks being proper. I was hoping that you could suggest a text better than Taylor.
Thanks again for your efforts.
JM
Here is the link to Einstein's 1920 book:
http://www.bartleby.com/173/.
Now if you look at the end of chapter 12, you will see this statement:
As a consequence of its motion the clock goes more slowly than when at rest.
Then if you look at chapter 23, you will see where Einstein once again discusses a clock moving in a circle with respect to a stationary clock.