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

[JM]

Clocks and time are not the same thing. Time is a property of nature observable as changes. Day turns to night, summer follows winter, and rivers flow downstream. Clocks, however, are man made objects that do what we tell them to do. For everyday use we regulate clocks to match the noon-to-noon interval,we use stop watches to compare race contestants, and for rocket launches the clocks count backwards.
Asserting the relation for synchronizing clocks ( Einstein, 1905) t_{a1+ta2 =2 tb represents the use of the isotropic nature of light to tell the clocks a and b how to relate to each other. Taking t(X,Y,Z,T) as the time of the moving frame, and entering the coordinates of the stationaryframe X,Y,Z,T for the light emission at ta1,the reflection at tb, and the return at ta2 (taking account of the Postulate of Constant Light Speed ) leads to the transform relation t = (T-vX/c²)/√(1-v²/c²). When entering values of X and T as independent variablesthe dimensions of t are the same as those of T. The analysis as a whole suggests that t is a time assigned to the clocks of the moving frame by the properties of light, and is not related to any everyday time the clocks might have. SR time is more like a stopwatdh, measuring t and T as each frame has its own view of the light wave.
For me, these ideas greatly clarify SR clocks, anyone else?}

 

[lugita15]

You're right that Einstein synchronization convention, like all conventions, is an arbitrary choice we make, not a property of time itself. And the fact that the one-way speed of light is constant according to it is just a logical consequence of our arbitrary definition, not something fundamental to nature. But as this long thread will attest, the story arguably doesn't end there. There are other synchronization conventions like slow clock transport, just as arbitrary as Einstein's, but having the property that the value of the one-way speed of light according to them is an empirical property of the universe, not just a trivial consequence of our definition.

 

[pervect]

Clocks and time are not the same thing. Time is a property of nature observable as changes. Day turns to night, summer follows winter, and rivers flow downstream. Clocks, however, are man made objects that do what we tell them to do. For everyday use we regulate clocks to match the noon-to-noon interval,we use stop watches to compare race contestants, and for rocket launches the clocks count backwards.

As far as science goes, we aactually need something to measure time, before we can do much with it.

Abstract discussions of what time is, in a philosophical sense, really belong in the philosophy forum.

And on a philosophical level, I'd personally disagree - time IS what we measure with clocks, at least this works better than some of your other suggestions. For instance, the seasons - which change in length, as the Earth's rotation gradually slows. Modern clocks actually measure something more fundamental than the cycle of the seasons.

But this is ultimately philosophy, and one can debate it endlessly -because it has no experimentally testable consequences.

Meanwhile, science can tell us a lot of usefull things about how clocks behave - and about how the seasons behave as well - because it's about things we can actually measure, and questions can actually be settled by pointing to experiment, rather than debated without end and without any resolution.

 

[Dale]

Clocks and time are not the same thing.

time IS what we measure with clocks

Note the subtle difference in these two statements that actually makes them compatible. JM's statement that clocks and time are not the same thing is a strawman caricature of pervect's statement. You can do the same thing with other measurable quantities.

"Mass is a scale" vs. "mass is what we measure with a scale"
"Temperature is a thermometer" vs. "temperature is what we measure with a thermometer"

People who like to agonize and philosophize over time like to attack the first statement while scientists like to make the second statement. It explains partly why scientists have so little patience for the discussion.

 

[JM]

. There are other synchronization conventions like slow clock transport, just as arbitrary as Einstein's, but having the property that the value of the one-way speed of light according to them is an empirical property of the universe, not just a trivial consequence of our definition.

The key to Einsteins theory is his Light Postulate. As I understand it, he based it on emperical evidence, Michaelson et al. OK?

 

[JM]

Abstract discussions of what time is, in a philosophical sense, really belong in the philosophy forum.

Meanwhile, science can tell us a lot of usefull things about how clocks behave - and about how the seasons behave as well - because it's about things we can actually measure, and questions can actually be settled by pointing to experiment, rather than debated without end and without any resolution.

I hadnt intended a philosopical discussion. But rather an attempt to understand Einsteins relativity. He places clocks in his two frames without stating basis or calibration ( re day, no.of seconds per day). There has resulted endless debate about the relation of the two sets of clocks. My conclusion that the time of the moving frame is assigned, and that the everyday time of the moving frame has no bearing ( when viewing from the stationary frame ) seems to me to resolve these discussions.
If clocks are man made, what does science say about their behavior?

 

[ghwellsjr]

The key to Einsteins theory is his Light Postulate. As I understand it, he based it on emperical evidence, Michaelson et al. OK?

And his Principle of Relativity Postulate. It's no wonder you have issues with your next post if you leave that out.

I hadnt intended a philosopical discussion. But rather an attempt to understand Einsteins relativity. He places clocks in his two frames without stating basis or calibration ( re day, no.of seconds per day). There has resulted endless debate about the relation of the two sets of clocks. My conclusion that the time of the moving frame is assigned, and that the everyday time of the moving frame has no bearing ( when viewing from the stationary frame ) seems to me to resolve these discussions.
If clocks are man made, what does science say about their behavior?

If you want to understand Einstein's Special Relativity, you need to accept the answers that you are given by people who already understand it instead of putting up resistance to them.

Do you know what Einstein means by a Frame of Reference? Do you know how he defines it? Do you understand that it is based on, not just man-made clocks, but man-made rulers and man-made protractors and man-made structures? A Frame of Reference is entirely man-made and so is all of science. Why this fixation on just the clocks?

And most importantly, how can you tell which is the moving frame that has no bearing with regard to time because it is assigned and which is the stationary frame for which time has bearing and for which time is not assigned?

 

[lugita15]

The key to Einsteins theory is his Light Postulate. As I understand it, he based it on emperical evidence, Michaelson et al. OK?

Well before Einstein, theory of relativity came out, Lorentz explained the results of the Michelson-Morley experiment in terms of aether. He said that objects moving with respect to the aether are contracted by the aether, and clocks moving with respect to the aether slow down. Thus Lorentz could explain why the speed of light seemed to be the same in all frames: it was because the rulers and clocks used to measure the speed of light were inaccurate because of length contraction and time dilation. But then it was found (not by Einstein, I might add) that the Lorentz transformation related not only the aether to other frames, but also other frames to each other. Thus Einstein's theory of relativity distinguishes itself from Lorentz's theory by treating all frames as equal, and thus saying that the speed of light really is the same in all frames.

 

[JM]

And his Principle of Relativity Postulate. It's no wonder you have issues with your next post if you leave that out.That postulate is dealt with in Part 2 of his paper. It doesn't enter Part 1 with the clocks,does it?
If you want to understand Einstein's Special Relativity, you need to accept the answers that you are given by people who already understand it instead of putting up resistance to them.My post relates to the relation of the clocks of the two frames. What is your understanding of this relation?
Why this fixation on just the clocks?Because my interest is the relation between the clocks, a subject of many threads.
And most importantly, how can you tell which is the moving frame that has no bearing with regard to time because it is assigned and which is the stationary frame for which time has bearing and for which time is not assigned?

The two frames are equivalent. One could choose either one and the properties would be the same. I chose to 'view from the stationary frame' as Einstein named it. The moving frame is the one with velocity v in the X direction of the staationary frame.

 

[JM]

and thus saying that the speed of light really is the same in all frames.

So, do you think that he based his light postulate on empirical results?

My apologies for not knowing how to work the 'quotes'.
JM

 

[ghwellsjr]

The two frames are equivalent. One could choose either one and the properties would be the same. I chose to 'view from the stationary frame' as Einstein named it. The moving frame is the one with velocity v in the X direction of the staationary frame.

OK, fine. Now do you accept Einstein's calculation for τ (tau), the rate at which a moving clock ticks in your stationary frame as a function of t, the rate at which the stationary coordinate clocks tick in your stationary frame and v, the velocity of the moving clock?

τ = t√(1-v²/c²)​

 

[lugita15]

So, do you think that he based his light postulate on empirical results?

Maybe I wasn't clear enough. Lorentz had a theory which said that things moving with respect to the aether experience length contraction and time dilation. Einstein was able to make a simpler theory which explained the same empirical results, but discarded the need for an aether by saying that the speed of light is genuinely the same in all inertial reference frames.

 

[JM]

George, Here is a more considered response to your post #7.

If you want to understand Einstein's Special Relativity, you need to accept the answers that you are given by people who already understand it instead of putting up resistance to them.

I have read a lot of books and papers on SR and have not found the answers to my questions. The authors are not available to me so I am trying on my own to understand what it is that they are saying. I hope this forum can help. The question under discussion in this thread is 'what is the relation between the clocks of the stationary and moving frames?' My understanding is given in post #1 with the following addition.
A light ray starts at the origin of the stationary frame K and reaches the location X at time T. During this time the origin of the moving frame advances a distance vT, leaving the distance X-vT for the light to travel to reach X. The light postulate says that the light travels at speed c in both frames, so X=cT, and X-vT=ct. Combining these equations leads to t=T-vX/c<sup>2. This is exactly the approximation of the Lorentz transform for v/c<<1. This analysis did not make any use of the clocks of the moving frame.
My conclusion is that the time of the moving frame is determined (assigned ) by the light postulate, and is independent of any'everyday' time the moving clocks might have.

Do you agree?</sup>

 

[GeorgeDishman]

I hadnt intended a philosopical discussion. But rather an attempt to understand Einsteins relativity. ... what does science say about their behavior?

What science tells us is that, if you draw a spacetime diagram of say the twin paradox, clocks measure time in the same way that an odometer would measure distance if the diagram showed two cars driving across a salt lake. Newton assumed they would work like altimeters in a balloon and a plane taking different paths to reach the same altitude. If you think of Einstein's statement that "time is what a clock measures" and then understand what that says about the nature of time itself, SR will start to make sense.

 

[Dale]

A light ray starts at the origin of the stationary frame K and reaches the location X at time T. During this time the origin of the moving frame advances a distance vT, leaving the distance X-vT for the light to travel to reach X. The light postulate says that the light travels at speed c in both frames, so X=cT, and X-vT=ct. Combining these equations leads to t=T-vX/c^{2. This is exactly the approximation of the Lorentz transform for v/c<<1. This analysis did not make any use of the clocks of the moving frame.}

^{I don't know why you would claim that. Isn't t the time according to clocks in the moving frame?}

 

[Mentz114]

So, do you think that he based his light postulate on empirical results?
JM

I think Einstein tried to imagine what would happen if one caught up with and passed a light pulse. This could happen if the speed of light was not the same for all observers. He found the idea had absurd consequences and did not fit in with Maxwell's equations. However, if SoL is the same in all frames, order is restored and Maxwell's equations are not violated. See "The Electrodynamics of Moving Bodies".

 

[harrylin]

Clocks and time are not the same thing. Time is a property of nature observable as changes. Day turns to night, summer follows winter, and rivers flow downstream.

Do you claim that the rotation of the Earth and the seasons do not function as natural clocks?
Another natural clock is set up with C14; the oldest clock is the rotating Earth (providing days), then refinements came in the form of sun dials and mechanical clocks.

[..] Asserting the relation for synchronizing clocks ( Einstein, 1905) t_{a1+ta2 =2 tb represents the use of the isotropic nature of light to tell the clocks a and b how to relate to each other.}

No, that would lead to self contradiction; light is simply defined to be isotropic wrt to the chosen reference system.

[..] The analysis as a whole suggests that t is a time assigned to the clocks of the moving frame by the properties of light, and is not related to any everyday time the clocks might have. SR time is more like a stopwatdh, measuring t and T as each frame has its own view of the light wave.
For me, these ideas greatly clarify SR clocks, anyone else?

I think that that is quite right.

 

[harrylin]

[..] The question under discussion in this thread is 'what is the relation between the clocks of the stationary and moving frames?' [..]
My conclusion is that the time of the moving frame is determined (assigned ) by the light postulate, and is independent of any'everyday' time the moving clocks might have.
Do you agree?

Nearly so: the light postulate commonly includes the synchronisation convention, however a convention is itself not a postulate and the light postulate doesn't prescribe what people should do*.
By the way, this convention was already in use before special relativity. For practical reasons astronomers had to assign times to distant events, and this was done by assuming the same speed of light in all directions. And note that the times of both frames (also the stationary frame) are assigned by the synchronisation convention.

* Einstein distinguished these things better in his formulation of the second postulate in 1907, as follows (emphasis mine):

"We [...] assume that the clocks can be adjusted in such a way that
the propagation velocity of every light ray in vacuum - measured by
means of these clocks - becomes everywhere equal to a universal
constant c, provided that the coordinate system is not accelerated.
[..this] "principle of the constancy of the velocity of light"

PS. I forgot to add this link of Einstein's illustration of the train and the embankment:
http://www.bartleby.com/173/9.html

 

[JM]

Einstein was able to make a simpler theory which explained the same empirical results, but discarded the need for an aether by saying that the speed of light is genuinely the same in all inertial reference frames.

lugita, thanks for the clarification. Apparently we agree that Einsteins theory is based on experience. Somewhere I got the idea that some people disagree.
JM

 

[JM]

DaleSpam:

I don't know why you would claim that. Isn't t the time according to clocks in the moving frame?

Einstein provided clocks that are stationary in each frame. Initially he said only that they are synchronized among themselves and with a "specified stationary clock'. The specified clock is not specified for the moving frame, so these clocks are, so to speak, idly waiting for instructions as to what clock they are to synchronize with. The analysis you quoted leads to a definition of the time of the moving frame t based on the X and T of the stationary frame. ( Actually it is Einstein's rigorous analysis, the quoted analysis is only to demonstrate principles.) The moving clocks can now synchroneze with t. In section 4 of part 1 Einstein identifies a clock "located at the origin of the co-ordinates of k ( the moving frame), and so adjusted that it marks the time t (my notation)". The adjustment appears to be the instruction to the moving clocks to use the values of X and T ( and v and c) and the relation for t = gamma(T-vX/c²) to produce a specific value of t.
I hope this clarifies my understanding.
JM

 

[JM]

Do you claim that the rotation of the Earth and the seasons do not function as natural clocks?

My point is that we can do little to change the things in nature that awaken us to the existence of time. But we can do what we like with the clocks we make, and what we do is arbitrary and suited to our particular need.

No, that would lead to self contradiction; light is simply defined to be isotropic wrt to the chosen reference system.

You are right, Einstein defined the light speed to be the same in both directions. Then the relation between the clocks A and B became the starting point for his derivation of the transforms.

I think that that is quite right.

See also my clarification in a later post.
JM

 

[lugita15]

lugita, thanks for the clarification. Apparently we agree that Einsteins theory is based on experience. Somewhere I got the idea that some people disagree.
JM

JM, the point I was trying to make is that Einstein's theory is not the only theory that explains the experimental facts at hand. Lorentz's theory is also based on these same facts, but Einstein's is simpler, so it's preferred.

 

[JM]

* Einstein distinguished these things better in his formulation of the second postulate in 1907, as follows (emphasis mine):

"We [...] assume that the clocks can be adjusted in such a way that
the propagation velocity of every light ray in vacuum - measured by
means of these clocks - becomes everywhere equal to a universal
constant c, provided that the coordinate system is not accelerated.
[..this] "principle of the constancy of the velocity of light"

PS. I forgot to add this link of Einstein's illustration of the train and the embankment:
http://www.bartleby.com/173/9.html

Thanks for your comments. Can you tell me where the 1907 info is published?
JM

 

[JM]

JM, the point I was trying to make is that Einstein's theory is not the only theory that explains the experimental facts at hand. Lorentz's theory is also based on these same facts, but Einstein's is simpler, so it's preferred.

I accept this. I like Einsteins theory because the math is tractable and the physical principles are clear.
JM

 

[GeorgeDishman]

JM, the point I was trying to make is that Einstein's theory is not the only theory that explains the experimental facts at hand. Lorentz's theory is also based on these same facts, but Einstein's is simpler, so it's preferred.

Perhaps more importantly, GR is (AFAIK) the only working theory of gravity and it reduces to the Minkowski Metric as mass tends to zero. There is no equivalent approach that allows one to derive LET.

 

[JM]

OK, fine. Now do you accept Einstein's calculation for τ (tau), the rate at which a moving clock ticks in your stationary frame as a function of t, the rate at which the stationary coordinate clocks tick in your stationary frame and v, the velocity of the moving clock?

τ = t√(1-v²/c²)​

George: This formula comes directly from the transform equation t = ( T-vX/c²)/√(1-v²/c²), with the assumption that X=vT.
But is the formula intended as an example, or as a universal truth? Consider...
All theclocks are synchronized, so that all clocks of K read T, not just the clock at X=vT,and all the clocks of k read t, not just the one at x=0. If X=0, thus 'pointing at the origin of K', t = T/√ ( 1- v²/c²), and t>T. Because of synch. this result applies also to the theclocks at the origin of k. The conclusion, seemingly, is that the moving clock at the origin of k can run slow or fast depending on the value of X.
So is 'slow clocks' a universal truth? I'm looking for help, yes or no, and why.
JM

 

[Mentz114]

lugita, thanks for the clarification. Apparently we agree that Einsteins theory is based on experience. Somewhere I got the idea that some people disagree.
JM

Can JM ( or Lugita if he agrees with the above) state what experience it was that convinced Einstein that the speed of light must be constant ?

 

[ghwellsjr]

OK, fine. Now do you accept Einstein's calculation for τ (tau), the rate at which a moving clock ticks in your stationary frame as a function of t, the rate at which the stationary coordinate clocks tick in your stationary frame and v, the velocity of the moving clock?

τ = t√(1-v²/c²)​

George: This formula comes directly from the transform equation t = ( T-vX/c²)/√(1-v²/c²), with the assumption that X=vT.
But is the formula intended as an example, or as a universal truth? Consider...
All theclocks are synchronized, so that all clocks of K read T, not just the clock at X=vT,and all the clocks of k read t, not just the one at x=0. If X=0, thus 'pointing at the origin of K', t = T/√ ( 1- v²/c²), and t>T. Because of synch. this result applies also to the theclocks at the origin of k. The conclusion, seemingly, is that the moving clock at the origin of k can run slow or fast depending on the value of X.
So is 'slow clocks' a universal truth? I'm looking for help, yes or no, and why.
JM

Yes, slow clocks is a universal truth in Special Relativity.

Einstein's derivation of the Proper Time on a clock moving at speed v as a function of t, Coordinate Time, in a frame comes from section 4 of his 1905 paper. Remember, Coordinate clocks always remain fixed at the locations at which they were synchronized within a particular Frame of Reference. If you look at his derivation, he starts off talking about "one of the clocks which are qualified to mark time t when at rest relatively to the stationary system". What he means is that there is a second synchronized clock located at the spatial origin of one reference frame prior to time zero which then becomes stationary in a second reference frame moving at v with respect to the first reference frame after their mutual time zero. He asks the question, "What is the rate of this clock, when viewed from the stationary system?"

So τ is the Proper Time of a single clock put in motion at time zero compared to the infinite number of Coordinate Clocks that remain stationary. We are comparing the time on this moving clock to the times on the adjacent clocks as it passes by them. The moving clock will always run slower than the stationary clocks. But remember, we are comparing one clock to a bunch of different clocks that have been previously synchronized.

So we always know the tick rate of a clock moving in a Frame of Reference by the simple formula expressed above.

 

[lugita15]

Perhaps more importantly, GR is (AFAIK) the only working theory of gravity and it reduces to the Minkowski Metric as mass tends to zero. There is no equivalent approach that allows one to derive LET.

Yes, I was just talking about the facts underlying special relativity.

 

[Dale]

The analysis you quoted leads to a definition of the time of the moving frame t based on the X and T of the stationary frame.

So then the analysis clearly did make use of clocks in both the moving and stationary frames.

 

[harrylin]

See also my clarification in a later post.
JM

I saw all your later posts, but no clarification that would affect my comments (or that of others).

Thanks for your comments. Can you tell me where the 1907 info is published?
JM

Sure, you can find it here: https://www.physicsforums.com/showthread.php?t=575526
With an illegal copy of a not-so-perfect translation linked in post #2.

 

[JM]

So then the analysis clearly did make use of clocks in both the moving and stationary frames.

Are we talking semantics here?
Einsteins mathematical derivation of the transform of time begins with the equation defining the synchronization of the moving clocks. The light postulate is then used with geometry to develop the transform. The only feature of the moving clocks used in this analysis is the synch. relation. He later speaks of clocks qualified to mark the time of the moving frame. Since the time of the moving frame is defined by the transform, doesn't 'qualified' mean the clocks display the time given by the transform?
At this point there seems to be a choice. Either the clocks originally placed in the moving frame are 'programed' to display t, or additional clocks are supplied ( as we would use stop watches) to display t.
JM

 

[JM]

Yes, slow clocks is a universal truth in Special Relativity.
So we always know the tick rate of a clock moving in a Frame of Reference by the simple formula expressed above.

George: I believe I understand your explanation. I have studied section 4 and my questions are:
Einstein refers to a clock qualified to mark the time t (my notation) when at rest relativily to the moving system and so adjusted that it marks the time t. This adjustment seems to mean that the moving clock displays the time t given by the transform, doesn't it?
Then he says "Between the quantities x,t,and τ, which refer to the position of the clock,..." (his notation), x and t being the coordinates of the stationary frame and τ being the time of the moving frame. By what justification does x refer to the position of the clock? In the transforms, as they are usually viewed, x and t are independent variables allowed over the range -∞ to +∞. If slow clocks is universal then x must be permanently restricted to the values x=vt. If x is an independent variable then there is no significance to where x is 'pointing' because all clocks read the same value wherever located.

Thanks for your participation.
JM

 

[Dale]

Since the time of the moving frame is defined by the transform, doesn't 'qualified' mean the clocks display the time given by the transform?

"Qualified" just means that the clock keeps correct time. I.e. if it is at rest wrt some process that takes exactly 10 s then it measures 10 s as opposed to something like 11 s or 9 s.

 

[JM]

George and DaleSpam,
Are we done here?

The discussion here leads to the suggestion that the 'slow clock' idea, with τ<t, is only one possible result for the moving clocks. Choices of x other than x=vt lead to different relations, such as τ>t for x=0. Synchronization means that the result applies to all the moving clocks, including the one at the origin of the moving frame.

Do you all accept this idea, or do we have some more to talk about?

JM

 

[Dale]

Do you all accept this idea, or do we have some more to talk about?

No, I don't accept the idea.

If you have a clock which is moving in an arbitrary fashion (including, but not limited to, x=vt) you use the following formula to calculate the time displayed on the clock:
\tau = \int \sqrt{1-v(t)^2/c^2} dt
http://en.wikipedia.org/wiki/Proper_time#In_special_relativity

The integrand is always less than or equal to 1, so you never get d\tau&gt;dt where t is the time coordinate in an inertial frame and v is the clock's velocity in that frame.

 

[ghwellsjr]

Yes, slow clocks is a universal truth in Special Relativity.
So we always know the tick rate of a clock moving in a Frame of Reference by the simple formula expressed above.

George: I believe I understand your explanation. I have studied section 4 and my questions are:
Einstein refers to a clock qualified to mark the time t (my notation) when at rest relativily to the moving system and so adjusted that it marks the time t. This adjustment seems to mean that the moving clock displays the time t given by the transform, doesn't it?

In the nomenclature of the transform that Einstein developed in section 3 of his 1905 paper, he uses τ (tau), not t or t', as the time on the moving clock. [NOTE: in his version of the LT, he uses β, beta, as the Lorentz factor instead γ, gamma, which is in common usage today. We now use β to mean v/c. Also, we commonly use t' to refer to the transformed time. Just don't get confused by this difference in nomenclature.]

In any case, I explained what Einstein means in the part of my quote from post #28 that you left out: t is the time on a clock that was at rest in the stationary frame prior to t=0 and then at t=0 it instantly accelerates to velocity v and so becomes at rest in the frame moving at v where the transformed time is represented by τ. For any given time t in the stationary frame, you can calculate the time τ on a clock at the spatial origin of the moving frame using the simple formula τ=t√(1-v²/c²).

Then he says "Between the quantities x,t,and τ, which refer to the position of the clock,..." (his notation), x and t being the coordinates of the stationary frame and τ being the time of the moving frame. By what justification does x refer to the position of the clock?

It is justified because he takes a clock from the stationary frame located at the origin (t=0 and x=0) and puts it at the origin of the moving frame (τ=0 and ζ=0) and the origin of the moving frame is defined as x=vt according to the stationary frame. This is simply what using the Lorentz Transformation is all about. Since we know where the origin of the moving frame is at any time in the stationary frame, we also know where the moving clock is since it is at rest at the origin of the moving frame.

In the transforms, as they are usually viewed, x and t are independent variables allowed over the range -∞ to +∞. If slow clocks is universal then x must be permanently restricted to the values x=vt. If x is an independent variable then there is no significance to where x is 'pointing' because all clocks read the same value wherever located.

The whole point of Einstein's derivation is to eliminate x from the equation but if you want, you can include it and say for any given x we can calculate both t and τ. Or you could say that for any given t we can calculate x and τ. We are of course assuming that v is constant and that we only care about t≥0, τ≥0 and x≥0.

After having developed the relationship between the time on a moving clock relative to the times on the stationary coordinate clocks, we extrapolate to the more general case of delta times so that we don't have to be restricted to the origin of a specific frame or even a specific speed and we can determine the instantaneous tick rate of an accelerating and moving clock compared to the coordinate time.

But because √(1-v²/c²) can only be a number less than 1, we know that a moving clock will tick slower than the stationary coordinate clocks and that's why we say "slow clocks is a universal truth in Special Relativity".

Thanks for your participation.
JM

You're welcome and I apologize for taking so long to respond to your questions--I just don't recall seeing your post until now.

 

[Dale]

In the nomenclature of the transform that Einstein developed in section 3 of his 1905 paper, he uses τ (tau), not t or t', as the time on the moving clock.

I hope I didn't cause any confusion. I was using τ as proper time. I don't know what usage JM intended.

 

[ghwellsjr]

I hope I didn't cause any confusion. I was using τ as proper time. I don't know what usage JM intended.

I was showing Einstein's derivation of Proper Time in post # 28 so it shouldn't have been confusing.

 

[JM]

George and DaleSpam,
I have read your last replies. I don't understand clearly the reasoning of 1905 where he starts a clock from rest to develop the standard formula for 'slow clock'. Let me state in detail my reasoning.
1. Stay within SR, ie all moving clocks are stationary in the frame moving toward +x at speed v.
2.The transform equation relating the time τ of the moving frame to the coordinates x and t of the stationary frame is τ = ( t - vx/c2)/√(1-v2/c²)
3. x and t are independent variables, ie they can take on any values both + and -.
4.For example let x=0.5ct (v/c = 0.8) in the equation to find τ =t.
5. Since the moving clocks are synchronized this result applies to all the moving clocks, including the one at the origin of the moving frame.
6. Therefore the moving clocks are not always slow.
Comments?

​

Is there a relation between the above and the section of 1905 on 'slow clocks'?
JM

 

[PAllen]

George and DaleSpam,
I have read your last replies. I don't understand clearly the reasoning of 1905 where he starts a clock from rest to develop the standard formula for 'slow clock'. Let me state in detail my reasoning.
1. Stay within SR, ie all moving clocks are stationary in the frame moving toward +x at speed v.
2.The transform equation relating the time τ of the moving frame to the coordinates x and t of the stationary frame is τ = ( t - vx/c2)/√(1-v2/c²)
3. x and t are independent variables, ie they can take on any values both + and -.
4.For example let x=0.5ct (v/c = 0.8) in the equation to find τ =t.
5. Since the moving clocks are synchronized this result applies to all the moving clocks, including the one at the origin of the moving frame.
6. Therefore the moving clocks are not always slow.
Comments?

​

Is there a relation between the above and the section of 1905 on 'slow clocks'?
JM

^{For one, you are confusing coordinate time and proper time. Your equation in (1) relates coordinate times of separated clocks. More precisely, it relates: if observer A synchronizes distant clocks using Einstein synchronization, how will observer B (moving relative to A) describe the results if they also use Einstein synchronization between clockes. Proper time (tau) is a completely different animal. It is only defined along the history of a single clock. As shown in Einstein's 1905 paper, every observer perceives every clock (moving or not), to go either the same rate as theirs (if not moving relative to said observer), or slower than said observer's clock. Note, especially, that if A interprets B's clock as slow, then B interprets A's clock as slow.}

 

[ghwellsjr]

George and DaleSpam,
I have read your last replies. I don't understand clearly the reasoning of 1905 where he starts a clock from rest to develop the standard formula for 'slow clock'. Let me state in detail my reasoning.
1. Stay within SR, ie all moving clocks are stationary in the frame moving toward +x at speed v.
2.The transform equation relating the time τ of the moving frame to the coordinates x and t of the stationary frame is τ = ( t - vx/c²)/√(1-v²/c²)
3. x and t are independent variables, ie they can take on any values both + and -.
4.For example let x=0.5ct (v/c = 0.8) in the equation to find τ =t.
5. Since the moving clocks are synchronized this result applies to all the moving clocks, including the one at the origin of the moving frame.
6. Therefore the moving clocks are not always slow.
Comments?

​

Is there a relation between the above and the section of 1905 on 'slow clocks'?
JM

Based on your post #26, I can see that you know how Einstein got from this:

to the first part of this:

and he did it by replacing x with vt but remember, there is more to the Lorentz Transformation than just the formula for τ. There are also the formulas for the spatial co-ordinates and if we plug x=vt into x'=γ(x-vt) we get:

x'=γ(vt-vt)=0

So this tells us that it is not all the clocks in the moving frame that the time co-ordinate applies to but only the one at the spatial origin of the moving frame which is where the moving clock that we are considering is located.

Now, concerning 3, x and t are not independent of each other in this situation, they are related by x=vt, as you pointed out in your post #26.

Concerning 4, I can't tell what you are doing, can you provide more detailed steps?

Points 5 and 6 were covered in my earlier comments.

 

[Dale]

2.The transform equation relating the time τ of the moving frame to the coordinates x and t of the stationary frame is τ = ( t - vx/c<sup>2)/√(1-v2/c2)
3. x and t are independent variables, ie they can take on any values both + and -.
4.For example let x=0.5ct (v/c = 0.8) in the equation to find τ =t.</sup>

^{OK, so x=0.5ct is the worldline of a clock which is moving at 0.5c in the +x direction in the stationary frame. Boosting by 0.8c gives you a clock which is moving at 0.5c in the -x direction in the moving frame. So yes, you have correctly determined that a clock which is moving at .5c in the +x direction in the stationary frame is slowed by the same amount as a clock which is moving at .5c in the -x direction in the moving frame.}

 

[morrobay]

No, I don't accept the idea.

If you have a clock which is moving in an arbitrary fashion (including, but not limited to, x=vt) you use the following formula to calculate the time displayed on the clock:
\tau = \int \sqrt{1-v(t)^2/c^2} dt
http://en.wikipedia.org/wiki/Proper_time#In_special_relativity

The integrand is always less than or equal to 1, so you never get d\tau&gt;dt where t is the time coordinate in an inertial frame and v is the clock's velocity in that frame.

I just posted a numerical proper time - velocity vs acceleration - problem
in the Homework Intro Physics section (page one) see update
With the evaluation of the above integral and it was not a good answer since the
recorded proper time of the (constant) accelerating clock ( with respect to Earth clock )
was greater than Earth clock ? So I have questions on that integral.
Once again , discussing proper time is confusing , so numerical problems might help

 

[Dale]

I checked your math, and it seems all right, but your answer was wrong. I don't know if you accidentally plugged it into the integrator wrong or if the integrator had some numerical problems.

 

[yuiop]

1. Stay within SR, ie all moving clocks are stationary in the frame moving toward +x at speed v.

Slightly odd way of expressing things, but otherwise OK. Let us try and define things a little more clearly. We have two reference frames S and S' in the standard configuration, which have a relative speed v in the x direction. Clocks at rest in S' have have a velocity of +v in the +x direction as measured in S, and clocks at rest in S appear to be moving in the -x' direction as measured in S'. Time measured by clocks at rest in S' are denoted by primed variables such as t'. The v mentioned in the standard Lorentz transforms is always the relative velocity of the the two reference frame as measured in S.

2.The transform equation relating the time τ of the moving frame to the coordinates x and t of the stationary frame is τ = ( t - vx/c²)/√(1-v²/c²)

You are using the symbol tau which normally stands for proper time, but the symbol on the left of that equation is actually a coordinate time as measured in the in frame S'. The equation is beter expressed as:

t' = ( t - vx)/√(1-v²)

where I am using units such that c=1 to make things more manageable.

3. x and t are independent variables, ie they can take on any values both + and -.

Seems O.K.

4.For example let x=0.5ct (v/c = 0.8) in the equation to find τ =t.

Finding tau = t does not make much sense except in the case there is no relative motion. We can however find what the value of t' is when t=0, x = 0.5 and v=0.8 when the clock at the origin of S is next to the clock at the origin of S'.

t' = ( t - vx)/√(1-v²)

t' = ( 0 - 0.8*0.5)/√(1-0.8²)

t' = ( 0 - 0.4)/0.6 = -0.66666 seconds.

5. Since the moving clocks are synchronized this result applies to all the moving clocks, including the one at the origin of the moving frame.

No it does not. When t=0, x = 0 and v=0.8 when the clock at the origin of S is next to the clock at the origin of S':

t' = ( t - vx)/√(1-v²)

t' = ( 0 - 0.8*0)/√(1-0.8²)

t' = ( 0 - 0)/0.6 = 0 seconds.

There is a difference of -0.6666 seconds between the times of the clocks at rest in S' when the clocks at rest in S are all reading 0 according to the observers at rest in S.

6. Therefore the moving clocks are not always slow.

I don't think anyone knows how you arrived at this conclusion and you have not shown any algebraic or numerical examples of a situation where the clocks at rest in S' are not slower than clocks at rest in S. I think you are not clear in your mind about the differences between coordinate times that label events, elapsed times that measure the time interval between different events and the differences between proper times and coordinate times.

The equations shown so far only concern coordinate times that label events and says nothing about the relative rates at which clocks with relative motion run.

To obtain the elapsed time (t2-t1) in frame S, between two events when the elapsed time interval between those two events in frame S' is (t2'-t1') we use:

t2-t1 = \Delta t = ( t2&#039; + v*x2&#039;)/\sqrt{1-v^2} - (t1&#039; + v*x1&#039;)/\sqrt{1-v^2}

\Delta t = (\Delta t&#039; + v*x2&#039; - v*x1&#039;)/\sqrt{1-v^2}

\Delta t = (\Delta t&#039; + v* \Delta x&#039;)/\sqrt{1-v^2}

Since we after the proper time in the primed frame we only use a single clock at rest in that reference frame, so x2' must equal x1' so we can say:

\Delta t = \Delta t&#039; /\sqrt{1-v^2}

\Delta t&#039; = \Delta t * \sqrt{1-v^2}

\Delta \tau = \Delta t * \sqrt{1-v^2}

Now using the above equation can you find a single instance when \Delta \tau &gt; \Delta t * \sqrt{1-v^2}?

Maybe what you are getting at is that the coordinate time interval as measured in S' between between two events may be longer than the coordinate interval between those two events as measured in S? For example using:

\Delta t&#039; = \Delta t *\sqrt{1-v^2} - v* \Delta x&#039;

\Delta t&#039; can be greater than \Delta t if \Delta x&#039; is negative, but this does not mean individual clocks in S' are running faster than individual clocks in S according to observers at rest in S. It is simply a result of how clocks are synchronised and the relativity of simultaneity (What appears simultaneous in one rest frame is not simultaneous in another reference frame with relative motion).

 

[Dale]

1. Stay within SR, ie all moving clocks are stationary in the frame moving toward +x at speed v.
...
4.For example let x=0.5ct (v/c = 0.8) in the equation to find τ =t.

I just noticed this. These two conditions are mutually contradictory.

 

[yuiop]

I just noticed this. These two conditions are mutually contradictory.

Please explain. I am not seeing it.

 

[Dale]

Please explain. I am not seeing it.

If the clocks are stationary in the moving frame then their worldline is x=0.8ct, not x=0.5ct.

 

[yuiop]

If the clocks are stationary in the moving frame then their worldline is x=0.8ct, not x=0.5ct.

Still not seeing it. X is just a coordinate. If he said the clock started at the origin at (t,x) = (0,0) and specified a time duration of delta T = 1 then yes I would expect the clock to be at coordinates (1.0,0.8) but he did not specify a time duration. X depends on t.

Even if he specified a duration of 1 in S, the location of the moving clock is not necessarily 0.8 if the clock did not start at the origin, (which he did not specify). If he had made it clear that he meant \Delta x = 0.5 when \Delta t = 1.0 then there would be a contradiction when v=0.8, but he did not specify a time interval or a starting coordinate or that he talking about intervals (differences) rather than coordinates of individual events.

Perhaps you mean that the single statement:

4.For example let x=0.5ct (v/c = 0.8) in the equation to find τ =t.

is self contradictory if we interpret it to mean \Delta x = 0.5 c\Delta t \implies \Delta x / \Delta t = 0.5c \implies v/c = 0.5?