How does special relativity account for the time on a single moving clock?

[JM]

The expression that all clocks tick at at a rate of one second per second relates to a comparison of one ideal clock to another ideal clock that are adjacent to each other and stationary with respect to each other.

HI yuiop, thanks for checking in. My latest thoughts on tick rate are in Post 140. What do you think of them?
JM

 

[JM]

Pauli's textbook on SR and Minkowski's EM (plus the basics of GR) is pretty decent IMO. Though it is oriented towards serious physics or engineering students. It's inexpensive and was written in the 1920's so it doesn't have newer, more abstract, more exotic developments in the field.

What are the titles and publisners ( or other source) for Pauli and Mink?
JM

 

[JM]

Without following that discussion, I saw a later post by you from which it appears that it's still not clear to you. Maybe useful if someone else gives a try?

- A and B are different "stationary" points (xA, yA, zA) and (xB, yB, zB).

- the straight constant speed trajectory AB is one line of the polygon.

Thanks, harrylin. I follow your ideas. My need is for the math that connects the time t'' of moving rotated frame with the time t of the original stationary frame K. Assume the points A, (0,0,0 ) and B, ( 1,1,0) wrt K. I accept that the relation between the frames with their respective 'x' axes aligned with these points is the same as the relation between the original K and K' axes. But a clock moving at v along the line between A and B moves at only vcos45 wrt K, for example. So what is the math that connects t'' with t?
JM

 

[JM]

I have no idea what you are talking about here but I don't think it can be related to what Einstein said with regard to inertial coordinate systems in Special Relativity, which is what this thread is about.

Right on George!

 

[Dale]

What would you call the clocks in the latticework described by Kip Thorne on page 3 of his upcoming http://www.pma.caltech.edu/Courses/ph136/yr2011/1102.2.K.pdf?

I would call them "clocks".

 

[Dale]

My answer is t' = [1/m + v²/c² - uv/c^{2t.
Whats yours?}

^{What is the question and what are m, u, and v?}

 

[harrylin]

Thanks, harrylin. I follow your ideas. My need is for the math that connects the time t'' of moving rotated frame with the time t of the original stationary frame K. Assume the points A, (0,0,0 ) and B, ( 1,1,0) wrt K. I accept that the relation between the frames with their respective 'x' axes aligned with these points is the same as the relation between the original K and K' axes. But a clock moving at v along the line between A and B moves at only vcos45 wrt K, for example. So what is the math that connects t'' with t?
JM

It looks as if my next remark didn't reach:
"
- the straight same constant speed trajectory BC is another line of the polygon. The Lorentz transformation relating "time" in a co-moving frame along AB is identical to that along BC: x is by definition the direction of motion. And obviously from point B the clock has to continue its counting from where it was the moment before - that's just common sense.
"
I'll try again. For the first leg, the X-axis of K and K' is by definition chosen along the line AB. It is you who draws the lines and defines the frames for the calculation. Thus you give A and B the same Y and Z coordinate (in this case you can keep them both 0), and v along x is simply v. That's how the Lorentz transformations are defined. And how the math between the polygon lines is connected I explained next. So, I'm afraid that you could not follow me.
To elaborate: you choose for the calculation for BC new reference frames with X and X' oriented along BC.

 

[ghwellsjr]

What would you call the clocks in the latticework described by Kip Thorne on page 3 of his upcoming http://www.pma.caltech.edu/Courses/ph136/yr2011/1102.2.K.pdf?

I would call them "clocks".

Then what would you do to JesseM's quote to make it satisfactory to you?

 

[JM]

What is the question and what are m, u, and v?

m is the coefficient in the LT often referred to as gamma. v is the speed of the frame moving in the x direction of the stationary frame K. u is the speed of a single clock moving in the + x direction of K. The question is 'what is the time on the single moving clock'?

 

[JM]

It looks as if my next remark didn't reach:

I think you didn't understand my question.
Section 4 of 1905 envisions a single clock moving along a polygon path wrt a stationary frame K. The clock starts at a point of K and returns to the same point of K. What you have described is the time of the clock wrt the polygon path. What you have not described is the time of the clock wrt the original frame K. This is the time that is required in order to make a valid comparison with the K time at the end of the path.

 

[GAsahi]

I think you didn't understand my question.
Section 4 of 1905 envisions a single clock moving along a polygon path wrt a stationary frame K. The clock starts at a point of K and returns to the same point of K. What you have described is the time of the clock wrt the polygon path. What you have not described is the time of the clock wrt the original frame K. This is the time that is required in order to make a valid comparison with the K time at the end of the path.

If the time on the clock at rest wrt the polygon is $T$ and the speed of the second observer is $v$, then the clock for the observer moving wrt the polygon will show $t=T \sqrt{1-(v/c)^2}$ when the observers are reunited so they can compare clocks.
Does this answer your question?

 

[Dale]

m is the coefficient in the LT often referred to as gamma. v is the speed of the frame moving in the x direction of the stationary frame K. u is the speed of a single clock moving in the + x direction of K. The question is 'what is the time on the single moving clock'?

For approximately the 100th time I refer you to the formula I posted back in post 36. The time displayed on the clock is:

$$\tau = \int \sqrt{1-v(t)^2/c^2} dt$$
So in frame K
$$\tau = t \sqrt{1-u^2/c^2}+C_1$$
And in the other frame
$$\tau = t \sqrt{1-\frac{(u+v)^2}{c^2(1+uv/c^2)^2}}+C_2$$