Is the Changing Clock Rate in Relativity Directionally Dependent?

[RandallB]

No, I meant what I said. Check out for instance MTW's "Gravitation", pg 168, section $6.3 entitled "Constraints on the size of an accelerated frame".

I took a look at the book and yes the plots should cross as they do for SR frames, where one has been accelerated to a fixed higher speed. (The SR frame work, is the best to be working with yogi on, but if you think you can help him in a truly GR environment go ahead you have 100’s of posts to go)

The issue MTW are dealing with is an accelerating frame for the moving point. And yes those lines should not cross except that they should overlap as they do at t=0.

But why Kip has a problem with seeing an overlap at g-1 I don’t understand. By simply recognizing “simultaneity” (A simple SR issue) and applying it to this accelerating frame it is clear that the “time” at this distance is in the past for this “accelerating frame”. Therefore it has a time of t<0 where the speed and this line are the same and parallel with the original stating line for the point g-0 at t=0.
Thus the correct lines will obviously progress with curves to the left that go to some limit parallel to the original horizontal line off set up somewhat.

Likewise the lines to the right represent points at “future distances” and again by “simultaneity” rule those times will be in the future. Here the speeds are higher for the accelerating frame therefore the slope needs to be progressively steeper and curving the line forward. This Projects an expectation fitting with their other graphs. But they do repeat the concern of the g-1 point again. I’m just an independent non-pro but if I ever meet Kip again and have the chance maybe I’ll bring it up to him. I’m sure the book having been so long ago he’d be allowed some revisions to his old judgments.
RB

 

[yogi]

Jesse - Do i believe in Lorentz invarience - yes and no - how is that for a fence sitter. If you ask - do I believe in the invariance of the interval between two spacetime events in two different frames in relative uniform motion - the answer is yes until some experiment yet to be performed castrs doubt upon it - but there is no experiment that tests the transforms completely - when we take two space time points in two frames and derive the interval, the xv/c^2 term cancels - this has not been verified by experiment - so yes - I will withhold judgement at this point as to the universal validity of the transforms.

Is it always permissible to shift from the frame in which the clocks were syncronized to make interrogations of the orbiting clocks? I think not.
The satellite clocks will not be seen as running at a uniform rate in another frame in motion wrt to the non-rotating Earth centered reference frame.

 

[yogi]

pervect - your post 104 - I see no ambiguity either - don't know where you could have acquired the idea that i did - in fact it is what I have been trying to get across - earlier I drew an analogy to GR - the clock at the greater G potential runs faster - that clock at a lower potential runs slower - things are not reciprocal in GR nor are they in SR when only one clock has been accelerated into orbit

 

[yogi]

Randall - you are obviously going to have a difficult time undertanding the Gravity" book you borrowed - you are having a difficult time with my several statements to the effect that the resolution of the question doesn't involve anything but SR - first you accuse me of saying Relativity is Broken - title of another post - they you allege I am making a claim about GR - my reference to GR is strictly an analogy - nothing to do with it other than the fact that SR has in common with GR the fact that in some experiments things are not reciprocal.

 

[JesseM]

Jesse - Do i believe in Lorentz invarience - yes and no - how is that for a fence sitter.

I don't think you understand, yogi. "Lorentz-invariance" is just a mathematical property of certain equations, deciding whether or not a given equation shows Lorentz-invariance is as straightforward as deciding whether it's a polynomial.

Let's first consider the related concept of "Galilei-invariance", which is a bit simpler mathematically. The Galilei transform for transforming between different frames in Newtonian mechanics looks like this:

$$x' = x - vt$$
$$y' = y$$
$$z' = z$$
$$t' = t$$

and

$$x = x' + vt'$$
$$y = y'$$
$$z = z'$$
$$t = t'$$

To say a certain physical equation is "Galilei-invariant" just means the form of the equation is unchanged if you make these substitutions. For example, suppose at time t you have a mass $$m_1$$ at position $$(x_1 , y_1 , z_1)$$ and another mass $$m_2$$ at position $$(x_2 , y_2 , z_2 )$$ in your reference frame. Then the Newtonian equation for the gravitational force between them would be:

$$F = \frac{G m_1 m_2}{(x_1 - x_2 )^2 + (y_1 - y_2 )^2 + (z_1 - z_2 )^2}$$

Now, suppose we want to transform into a new coordinate system moving at velocity v along the x-axis of the first one. In this coordinate system, at time t' the mass $$m_1$$ has coordinates $$(x'_1 , y'_1 , z'_1)$$ and the mass $$m_2$$ has coordinates $$(x'_2 , y'_2 , z'_2 )$$. Using the Galilei transformation, we can figure how the force would look in this new coordinate system, by substituting in $$x_1 = x'_1 + v t'$$, $$x_2 = x'_2 + v t'$$, $$y_1 = y'_1$$, $$y_2 = y'_2$$, and so forth. With these substitutions, the above equation becomes:

$$F = \frac{G m_1 m_2 }{(x'_1 + vt' - (x'_2 + vt'))^2 + (y'_1 - y'_2 )^2 + (z'_1 - z'_2 )^2}$$

and you can see that this simplifies to:

$$F = \frac{G m_1 m_2 }{(x'_1 - x'_2 )^2 + (y'_1 - y'_2 )^2 + (z'_1 - z'_2 )^2}$$

Comparing this with the original equation, you can see the equation has exactly the same form in the primed coordinate system as in the unprimed coordinate system. This is what it means to be "Galilei invariant". More generally, if you have any physical equation which computes some quantity (say, force) as a function of various space and time coordinates, like $$f(x,y,z,t)$$ [of course it may have more than one of each coordinate, like the $$x_1$$ and $$x_2$$ above, and it may be a function of additional variables as well, like $$m_1$$ and $$m_2$$ above] then for this equation to be "Galilei invariant", it must satisfy:

$$f(x'+vt',y',z',t') = f(x',y',z',t')$$

So in the same way, if we look at the Lorentz transform:

$$x' = \gamma (x - vt)$$
$$y' = y$$
$$z' = z$$
$$t' = \gamma (t - vx/c^2)$$
where $$\gamma = 1/\sqrt{1 - v^2/c^2}$$

and

$$x = \gamma (x' + vt')$$
$$y = y'$$
$$z = z'$$
$$t = \gamma (t' + vx'/c^2)$$

Then all that is required for an equation to be "Lorentz-invariant" is that it satisfies:

$$f( \gamma (x' + vt' ), y' , z', \gamma (t' + vx' /c^2 ) ) = f(x' ,y' ,z' , t')$$

There may be some more sophisticated way of stating the meaning of Lorentz-invariance in terms of group theory or something, but if an equation is Lorentz-invariant, then it should certainly satisfy the condition above. Maxwell's laws of electromagnetism would satisfy it, for example. And it's pretty easy to see that if it satisfies this mathematical condition, then the equation must have the same form when you transform into a different inertial frame using the Lorentz transform. So this is enough to show beyond a shadow of a doubt that given Lorentz-invariant fundamental laws, all the fundamental laws must work the same in any inertial reference frame, and if you know the equation for a given law as expressed in some particular inertial frame (the rest frame of the center of the earth, for example) then it is a straightforward mathematical question as to whether or not this equation is Lorentz-invariant, it's not an experimental issue (the only experimental issue is whether that equation makes correct predictions in the first place). Do you disagree with any of this?

Is it always permissible to shift from the frame in which the clocks were syncronized to make interrogations of the orbiting clocks? I think not.

The satellite clocks will not be seen as running at a uniform rate in another frame in motion wrt to the non-rotating Earth centered reference frame.

Uh, why does this mean it's not "permissible" to shift into another frame? That's the whole point, that different frames disagree about whether a given set of clocks is running at a uniform rate. But all frames will agree on all physical questions like what two clocks will read at the moment they meet at a single location in space. You need to define what you mean by "permissible", when physicists use this term all it means is that you can use the same laws of physics in another frame and all your predictions about physical questions will still be accurate (that's why it's not 'permissible' to use the ordinary rules of SR for inertial frames in a non-inertial coordinate systems, because you would make wrong predictions if you did this). Given Lorentz-invariant laws, this is automatically going to be true for all inertial frames.

Also, the GPS clocks are programmed to adjust themselves so that they tick at a constant rate in the frame of the earth. My other point was that this is a completely arbitrary choice made by the designers, you could just as well design the orbiting GPS clocks to adjust themselves so that they tick at a constant rate in the frame of an inertial observer moving at 0.99c relative to the earth. Would you then say it is not "permissible" to analyze these clocks in the rest frame of the earth, since they would not be running at a uniform rate in the Earth's frame?

 

[pervect]

pervect - your post 104 - I see no ambiguity either - don't know where you could have acquired the idea that i did - in fact it is what I have been trying to get across - earlier I drew an analogy to GR - the clock at the greater G potential runs faster - that clock at a lower potential runs slower - things are not reciprocal in GR nor are they in SR when only one clock has been accelerated into orbit

Fine with me. Actually I think I need to tighten this up a little bit. The round trip time for light signals between two obserers is something that can be measured - A sends a signal #1 to B, B sends a signal #2 to A on recipt of A's signal. A measures the interval on his clock between the sending of the signal #1 and the receiving of signal #2 as the "round-trip time".

If this round-trip time is always constant, we can always compare the rate of two clocks unambiguously.

We may need to demand that the round-trip time is constant for both A and B before we can compare rates, but I think it is true that if A's round trip time is constant, so is B's. I'm relying on my intuition a bit here, though.

Of course A and B don't necessarily have to agree on the value of the round-trip time (and in general they won't) - they just have to agree that it's constant.

 

[Hurkyl]

but there is no experiment that tests the transforms completely

The transforms are mathematical things, not physical things -- there cannot be an experiment that tests them at all.

 

[RandallB]

The satellite clocks will not be seen as running at a uniform rate in another frame in motion wrt to the non-rotating Earth centered reference frame.

This statement alone stands as a claim the Einstein’s version of relativity is broken so quit complaining.

What I’d wish you’d do tell us what your trying to do in your posts, they seem to flip from one objective to a the other.
Are you:
A) trying to convince others that the standard view of SR (and extending into GR as well) is somehow wrong or incomplete. And your 'LR-like' view or something else must be better.
OR
B) sincerely trying to learn SR completely, to filling the gaps of information about it, that leave you unable to see concepts in books you’ve had for 10 years.

For the sake of all those that are trying to respond to you, be clear on this; are you arguing a point of view; or trying to learn something. And please refrain on saying “yes & no” or “both”, do one or the other.

 

[yogi]

I have no agenda Randall - my interest in SR goes back many years - likely before you were born - I pose questions that come to mind - if those questions lead to a different view ...one I have overlooked - great. Usually what happens on these boards is an attack - either I don't understand it or I am not qualified to question it - but for a few posters, the attitude is always one of condesention.

 

[yogi]

If anyone thinks it is easy to synchronize GPS satellite clocks in some frame than the non-rotating Earth centered system -I would like to see how you would go about doing it.

 

[yogi]

Hurkyl - The transforms relate time and distances - they are not abstract mathematical artifact - these are physical things - my point is with you and Jesse - the mathematical relationships (LT) have been confirmed in certain experiments - but those experiments do not test the fundamental premise upon which Einstein's derivation was based - perhaps w/o complete justification, I do have a stong conviction that the spacetime interval is invarient.

 

[yogi]

pervect - i would agree that any interrogation must depend upon the constancy of the round trip time - and since the GPS clock will always be found to be running at a uniform rate relative to the ground station - we have a convincing demonstration of the constancy of c - at least in the Earth centered frame

 

[yogi]

Jesse - your post 110 - last Paragraph: Transforming to another frame - what I have been trying to discuss was the non reciprocal reality of time dilation in certain experiments - if we have a GPS clock that was originally at the top of the tower and launched into orbit from there with no velocity correction - we have a situation where the ground clocks run consistently fast wrt to the GPS satellite clock and the GPS satellite clock always runs slow wrt to the ground clocks and we can verify this non symmetrical situation by interrogating the GPS clock with radio signals - that is the experiment. To inquire as to transforming to a frame in high speed motion wrt to the Earth simply subverts the objective. Such a transformation of course is possible, but you have missed the whole point -we are now back in a situation where whatever is measured is apparent - the two frames have not been synchronized - Einstein only makes a prediction about real time dilation when one of two originally synchronized clocks is accelerated to a uniform velocity wrt the other. This is the subject of the thread - which i would like to explore further.

 

[JesseM]

If anyone thinks it is easy to synchronize GPS satellite clocks in some frame than the non-rotating Earth centered system -I would like to see how you would go about doing it.

If the satellites can figure out their velocity relative to the center-of-earth frame and adjust their clock rates accordingly, it is trivial to figure out their velocity relative to any other inertial frame and adjust their clock rates to be constant in that frame instead. Do you doubt that if I know my velocity in Earth's frame and I know the Earth's velocity in frame X, I can easily figure out my velocity in frame X, and from this figure out how much my clocks would be slowed down in frame X?

 

[JesseM]

Jesse - your post 110 - last Paragraph: Transforming to another frame - what I have been trying to discuss was the non reciprocal reality of time dilation in certain experiments

More ill-defined terminology...what does "non-reciprocal" mean in yogi-speak? Surely you don't mean "non-reciprocal in the way the laws of physics work in different frames", do you? Please address the main part of post 110 and not the last paragraph, where I explained that "Lorentz-invariance" is simply a mathematical property of certain equations, and that given laws of physics whose equations in our own inertial frame have this property, it is automatically going to be true that the laws of physics will obey the same equations in all other inertial frames. Do you deny this or not?

if we have a GPS clock that was originally at the top of the tower and launched into orbit from there with no velocity correction - we have a situation where the ground clocks run consistently fast wrt to the GPS satellite clock and the GPS satellite clock always runs slow wrt to the ground clocks and we can verify this non symmetrical situation by interrogating the GPS clock with radio signals - that is the experiment.

It is symmetric in how the laws of physics work in different inertial frames, which is all that most physicists would mean by "symmetric" in the context of special relativity. If you have your own idiosyncratic definition of "symmetrical", please present it.

Do you agree that if the GPS clock is orbiting the Earth at a constant speed in the center-of-the-earth frame, that means that in other inertial frames the speed of the GPS clock is not constant? (This would be just as true in Newtonian mechanics as in relativity, of course.) Do you agree that if each inertial frame assumes the same relationship between instantaneous speed in that frame and instantaneous rate of ticking (ie that if the clock is moving at speed v in that frame it will be slowed down by a factor of $$\sqrt{1 - v^2/c^2}$$), then different inertial frames will disagree about the relative rate of the tower clock and the orbiting clock at a given moment (with 'given moment' meaning something different in different frames too, due to different definitions of simultaneity), yet they will all make the same prediction about how far behind the orbiting clock will be at the moment it completes an orbit and reunites with the tower clock at a single point in space? Please, please give me direct answers to this question, when I ask you questions in my posts they are not meant to be rhetorical, and it's incredibly frustrating when I ask you questions that I hope will help pin down your nebulous comments and you just ignore them and comment on a single statement in my post.

To inquire as to transforming to a frame in high speed motion wrt to the Earth simply subverts the objective. Such a transformation of course is possible, but you have missed the whole point -we are now back in a situation where whatever is measured is apparent - the two frames have not been synchronized - Einstein only makes a prediction about real time dilation when one of two originally synchronized clocks is accelerated to a uniform velocity wrt the other. This is the subject of the thread - which i would like to explore further.

There could only be a "real time dilation" in the sense that all frames would agree on how much time elapsed on two clocks between two points in time in a case where the clocks started at the same location and ended at the same location. Einstein would certainly never say that in a case where two clocks started at separate locations and then one accelerated towards another, there is any "real" (frame-independent) truth about which clock was ticking faster or slower. Different frames would disagree about this, and there is no physical reason to prefer one frame's analysis to another. If you disagree with this, then it's important that we first see if you agree with my statements above that any laws of physics which have the mathematical property of Lorentz-invariance will automatically work the same way in all different frames, and whether you also agree that all frames will make the same prediction about all physical questions like what two clocks will read at a moment when they are at a single location in space. If you don't agree with this, then you're expressing some basic ignorance about purely mathematical issues in relativity which needs to be corrected. If you do agree with this, yet still feel that there is some other reason to "prefer" one frame's analysis of the situation to another's, you need to explain what sort of aesthetic criteria you are using here to prefer one over the other despite the fact that all will see the same laws of physics and make the same physical predictions. And if you also want to continue to defend the absurd proposition that Einstein would agree with you about preferring one frame's analysis over another's, you need to provide the quotes that you think support this interpretation.

 

[Hurkyl]

Hurkyl - The transforms relate time and distances

No they don't!

The transforms relate (inertial) coordinate systems, which most certainly are "abstract mathematical artifact".

E.g., IIRC, Einstein made a big deal about showing how to construct what he called an inertial reference frame, via a hypothetical network of clocks, each of them "at rest" with a given inertial observer, and "synchronized" with his clock. I put those terms in quotes because those are also terms in need of construction.

If coordinate systems were physical things, Einstein would have just said "measure it".

 

[RandallB]

I have no agenda Randall - my interest in SR goes back many years - I pose questions that come to mind - if those questions lead to a different view ...one I have overlooked - great. Usually what happens on these boards is an attack - either I don't understand it or I am not qualified to question it - but for a few posters, the attitude is always one of condesention.

Fine – I’ll let your condescension towards me pass as a result of the above.

I understand I don’t have the “pro” tag, banner or ribbon of the mentors & advisors on this board - so you don’t have to take my encouragements to heart.

As to reciprocal views being relative – Based on how you pose those questions as conclusions, from the frame of reference of many trying to respond, it can often be seen as attacks on SR as they know it to be. So that part seems to be the same from both views.

 

[Hurkyl]

No they don't!

The transforms relate (inertial) coordinate systems, which most certainly are "abstract mathematical artifact".

I guess I should qualify this...

The transforms do relate coordiante position, and coordinate time, so in some sense they can be said to relate times and distances.

But my point still stands -- coordinate position and coordinate time are derived from coordinates, which are not physical things.



Case in point: the Julian and Gregorian calendars are not the same -- they¹ assign different time coordinates to events. Would you say that the transformation between these calendars is a physical thing?


1: I'm assuming any usual² method of identifying spatial position. A calendar, by itself, is unable to assign time coordinates to almost every event!


2: Notice I said usual, and nothing such as "intrinsic" or "determined by reality" -- it's a convention we use as humans. And it's even changed over time, such as when time zones were instituted, and when calendars were changed!

 

[pervect]

Randall - you are obviously going to have a difficult time undertanding the Gravity" book you borrowed - you are having a difficult time with my several statements to the effect that the resolution of the question doesn't involve anything but SR - first you accuse me of saying Relativity is Broken - title of another post - they you allege I am making a claim about GR - my reference to GR is strictly an analogy - nothing to do with it other than the fact that SR has in common with GR the fact that in some experiments things are not reciprocal.

Since I've had some run-ins with RandallB in the past, I didn't and don't consider it an enjoyable use of my time to respond to his response.

Hopefully the point I wanted to make has been made, though.

 

[yogi]

Pervect - I appreciate your comments and frequently read your posts although a cannot always make a sensible response. See Below

 

[yogi]

Pervect - I appreciate your comments and frequently read your posts although often cannot make a worthwhile response.

I do recall at one time in another thread you made a statement that the time discrepancy is physically explainable - but then the direction of the thread veered off along other lines as is often the case - so i wasn't able to get back to the subject

In my previous posts on this thread I have tried to target the situation Einstein created when he combined the relationships that were obtained from observations in another frame - specifically to my way of thinking there is a quantum jump from the notion of two observers in relative motion drawing identical conclusions about lengths and clock rates in the other frame - as opposed to the idea that one of two synchronized clocks put in motion will run slower than the one that was not moved. There is a real loss of time between the clocks when they are later compared - in other words, did Einstein get the right answer for the wrong reason - or by intuition. He does not attempt to justify or even rationalize this - it works to explain the experiments - but it doesn't follow from the initial postulates of SR - in one sense it seems to require its own special postulate and - i guess what i am saying is that something is missing.

 

[yogi]

So to pick up the thread - it seems there is a discontinuity in the development of the notion of real time dilation - for example if we explore what Einstein asserts in part 4 we could imagine the following scenerio - we plan a trip to Altair but first we draw a line connecting the Earth and Altair and at the center point of this line we place a clock C - we also place a clock E on the Earth and a clock A on Altair. All are initially synchronized. Now from the center point (location of clock C) we draw a large circle that intersects both the Earth and altare. This is the trajectory the spaceship will follow. The spaceship has a clock R also initially synced with A, E and C. The spaceship is quickly accelerated to its crusing velocity v wrt Earth and steered to maintain a uniform velocity tangent to the circular path it will follow during the entire voyage (there will of course need to be some lateral thrust to correctly follow the path - this radial acceleration however does not enter into the accumulated time on clock R because acceleration per se does not affect clock rates as has been determined many times in centrifuge experiments). Einstein tells us that the R clock will read less than the A clock when R passes by A, and that it makes no difference what path is followed so long as the velocity remains constant - this is simply the first half of the twin thing - and since the second half is dynamically the same as the first half, half the time is lost in arriving at Altair and half is lost on the return trip.

The three clocks A, E and C remain in sync since they are not moved - if the gamma factor is 0.5 the R clock should read half that of the A clock at the time R passes A - during the return trip the R clock will again lose the same amount of time - so it will read half the E clock time when R returns to earth.

The issue is whether in real time dilation experiments there is symmetry - either the R clock runs at the same rate as A,C and E or it runs at a different rate - and if it runs at a different rate how does it know physically at what rate to move its hands. We can interrogte the R clock from C at each and every mile along the way just as we can interrogate the GPS clock from the Earth tower. We know from GPS data, unless a GPS clock is corrected for its velocity relative to the tower clock, the Earth clock and the satellite clock will run at different rates at all times while they are in relative motion.

A possible query is whether the R clock is a good reference frame - true it has slight radial component - but not one that would influence its rate in a measurable manner - so if we assume R is at rest in an approximate inertial platform - would we expect measurements made in the spaceship frame using R to determine the rate of time passage on clocks , C, E and A to be the same as measurments made by C, E and A to measure R. In other words, if we sent interrogation radio signals from R to C, what conclusion could R arrive at from the signals received from C

 

[yogi]

lets take it a step further - we will have C send out interrogation signals every hour - they are received by tansceiver on the spaceship and they immediately transmit the time reading on clock R - now we presume c is constant and we have defined the radial distance to be always constant - so we know the over and back time is always the same - so after each transmission is received by C, C should be able to say that R is always running at 1/2 the rate of C (for gamma = 0.5). In a like manner, R can send interrogations to C and R should be able to determine from what it receives from C that C is running at double the rate of R.

If this really occurs - then do we not have a reference frame problem - if there is an intrinsic difference in the rate of uniformly moving clocks - it cannot be said that two clocks meeting in empty space with uniform relative motion will each judge the other clock to be reciprocally slow as measured by a local clock at rest. How does one distinguish between two clocks meeting in empty space and R flying by A after several thousand years of travel in deep space?

 

[Hurkyl]

Let's take it one step further: let's have R periodically send interrogations to E. (instead of to C)

Everything stated below is as seen by R.

(I'm using units in which the distance between A and E is 2, and c = 1)

I've written a program to run this experiment -- R is sending a signal every 0.01 units of time. E receives the signal and immediately broadcasts its time. R receives the return signal. R then takes the average of the send & receive times, by his clock, and compares it with the transmitted E-time.

R-time : E-time
---------
0.0 : 0.0
0.073 : 0.037
0.141 : 0.074
0.202 : 0.111
0.256 : 0.149
...
1.8 : 3.585
1.81 : 3.616
1.82 : 3.647
1.83 : 3.678
1.84 : 3.708
...
3.602 : 7.242
3.607 : 7.245
3.612 : 7.247
3.618 : 7.25
3.623 : 7.253
3.645 : 7.264

The first group is, of course, as R is leaving Earth.
The second group is when R passes Altair. (at 1.813 R-time)
The third group is when R passes by Earth again. (at 3.628 R-time)

Compare the rates of the R and E clocks by this reckoning:

As R leaves Earth, R-time is running twice as fast as E-time.
As R passes by Earth again, R-time is once again running twice as fast as E-time.

In other words, it is exactly what one would expect from "reciprocosity".


Also, note that R is passing by Altair that E-time is running three times as fast as R-time. (As opposed to C-time, which is only running twice as fast as R-time)

 

[yogi]

Hurkyl - Thanks for your interest and the time taken. Your approach is however, what I endeavored to avoid - I cannot help but feel that to find an explanation in the symmetry of the transforms is a bootstrap argument - it leads to a nummerical solution (albeit a correct one as far as the numbers go) but what is missing is an appreciation of what is really going on in the moving frame. Treating the clock frames of R and E as equal casuses a shift from apparent observations to an actual time dilation - I am not sure Einstein was not guilty of the same methodology - What I tried to do was set up the simpliest scenerio possible - with a central clock C that is always the same distance from R ...the three clocks in the earth-Altare (ECA clocks) frame will always read the same - so if you are getting different relationships between R and E at different points in the journey - they can only be apparent because the C clock always runs at a constant rate and since the R clock is following a circle - it too must run at the same rate for the entire trip - so any statements to the effect that the R clock observes E to be running fast at the beginning and ending, and slow in the middle, must of necessity be apparent. What I am trying to do is disect the problem by proposing the simpliest geometry possible ... one which is free from any challenges that could be raised as to changing conditions (e.g., accelerations, turn around stress etc) as is frequently done depending upon how the twin paradox is rationalized to a non-paradox). Run your programe using only signals between C and R and vice versa - if you arrive at a time discrepancy when R returns to E, tell where and when the R clock runs fast wrt to C and where it runs slow.

 

[Hurkyl]

I didn't use the transforms at all, actually.

I did the entire problem in the coordinates of the ECA-rest frame. I placed Earth at spatial coordinates (1, 0), and Altair at spatial coordinates (-1, 0).

Since R travels with uniform speed in a circle (as measured by the ECA frame -- most inertial frames would disagree on both counts), its worldline has to look something like:

r(s) = <ks, cos s, sin s>

for some k. (The parameters are time, x, y) You gave the condition that the rate of the R-clock should be one-half that of its ECA-time coordinate. I used that to solve for k, and then I reparametrized the worldline by R's proper time:

$$\vec{r}(\tau) = \langle 2 \tau, \cos (\tau \sqrt{3}), \sin (\tau \sqrt{3}) \rangle$$

So, I'm now armed with the knowledge of how to find the ships ECA-coordinates if I know what R's clock reads.

Using this, if I decree that the ship emits a signal at R-time $\tau$, I could determine the ships ECA-coordinates when the signal is emitted. I can then compute when the signal arrives on Earth. I then numerically solve for the R-time $\tau'$ when R receives Earth's reply.

That is how I generated the chart.


I certainly agree that if I add a C-column via the above methodology, the C-times will always read exactly twice that of the R-times.


I also certainly disagree that the problem is free of acceleration. :tongue: While the ship may be accelerating gently, we are also looking at the problem over very large scales -- small * large = indeterminate, so we cannot brush away the effects of acceleration as being irrelevant.

 

[JesseM]

If this really occurs - then do we not have a reference frame problem - if there is an intrinsic difference in the rate of uniformly moving clocks

Why do you think your example shows an intrinsic difference in the rate of uniformly moving clocks"? "Uniformly moving" means uniform velocity, not just uniform speed, so R doesn't qualify.

Any thoughts on the questions I asked in my previous post?

 

[Hurkyl]

I think one of the important lessons that was learned from Einstein is that a concept is only relevant if we can determine it experimentally.

The big, overarching point is that you cannot talk about what time something happened way over there -- what you can talk about are the results of some sort of "timing experiment".

One way to give meaning to a phrase like "E observes that R's clock is running half as fast as his own" is to say that E is continually performing timing experiments to assign E-times to the readouts of R's clock. The results of these experiments yield E-times that are increasing twice as fast as the R-times to which they're assigned.


It is somewhat of a miracle (or more accurately, an assumption) that it is possible for E to perform this timing experiment on C, and find that the assigned E-times are equal to the readouts on C's clock. And conversely, that C performs this timing experiment on E to find that the assigned C-times are equal to the readouts on E's clock. The concept of a reference frame is an abstraction of this miracle.


When R performs this timing experiment on E's clock, and finds that the result is not that the assigned R-times are always half as much as E's clock, it shouldn't be a surprise.

All it means is that the amazing string of coincidences that allowed some of these experiments to agree with each other have finally come to an end. There's no inherent reason to think that any of these coincidences could be possible! In fact, according to GR, they are not: they can only be approximately possible on small scales. (Okay, I should point out that my confidence in this very last sentence isn't nearly as much as the rest of my post)

 

[yogi]

Jesse - Hurkyl Let me address part of what you have both said - it has to do with the propriety of R as a reference frame - here i will again direct you to Einstein's statement (1905 Part 4) - that it doesn't make any difference what path is followed - the equation for the time difference only concerns the relative velocity v between the clock that remains at rest after sync and the clock that is put in motion - I interpret this to mean that the moving clock (in our case R) could zig zag all over the place as long as its heading velocity remains v - if we believe Einstein's statement then should we not be able to disregard any radial acceleration that R incurs during the round trip flight - we all agree (I think) that we get the right answer by considering only C and R, and when we do that we arrive at the same result as Hurkyl gets using the chart methodology - so if arriving at the right numbers is the primary object - either approach is of equal utility.

Now if we follow hurkyl's logic that we will see R running at half the assigned C rate if we continue to use timing experiments between R and C at all points of the path, and vice versa, can we not conclude that the spaceship is a good inertial frame - it shouldn't really make a difference in the outcome if the path were comprised of many straight segments that approximate the circle and the readings were always taken at the midpoint of the seqments - we wouldn't really expect the total time difference between a circular path and a segmented path to be any different so long as the total distance traveled was the same - what I am trying to do is eliminate any affect of acceleration.

Now I do not see that this experiment leads to the notion of a universal time - but if both R and C run at the same speed at all times as a necessary consequence that they are not being acted upon by any forces or factors that would change the rate at which they run - then we do have a condition where they run at different rates as judged by each other - but unlike apparent time dilation - R judges C as fast and C judges R as slow ...in other words, they are running at intrinsically different rates, The question that follows is whether its permissible to ignor this difference when using the LT or some other methodolgy to make measurements in a ralativly moving frame. Again are we getting the right numerical answer for the wrong reason. Does not R have to take into account that it is running slow wrt to C when setting up its local clocks to make mesaurements in C's frame - for example let us say that prior to launch someone fiddles with the R clock so it doesn't keep the same time as C before launch - we would not expect that R would be able to get correct results when it is used to measure E and A rates.. More later

 

[yogi]

Jesse - I assume you were asking about the questions posed in your post 120 which referred to your Epistle posted in 110. If I have given the impression that I have any doubt about physical experiments producing different results in different frames - I did not intent to - Nor am I taking issue with the lorentz group of symmetries as a mathematical principle, although it is not always easy to see what is being conserved.

At this juncture however, I cannot say whether such mathematical constructs always represent the real world - we use them as though they were God given - just as we once used Newtonian mechanics w/o question.

 

[JesseM]

Jesse - Hurkyl Let me address part of what you have both said - it has to do with the propriety of R as a reference frame - here i will again direct you to Einstein's statement (1905 Part 4) - that it doesn't make any difference what path is followed - the equation for the time difference only concerns the relative velocity v between the clock that remains at rest after sync and the clock that is put in motion - I interpret this to mean that the moving clock (in our case R) could zig zag all over the place as long as its heading velocity remains v - if we believe Einstein's statement then should we not be able to disregard any radial acceleration that R incurs during the round trip flight

Are you referring to this part of section 4 of the 1905 paper?

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.

If we assume that the result proved for a polygonal line is also valid for a continuously curved line, we arrive at this result: If one of two synchronous clocks at A is moved in a closed curve with constant velocity until it returns to A, the journey lasting t seconds, then by the clock which has remained at rest the traveled clock on its arrival at A will be 1/2*tv^2/c^2 second slow. Thence we conclude that a balance-clock7 at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions.

If you're saying that this justifies treating the circular-moving clock as having its own rest frame, you misunderstand what he was saying there. Einstein was saying that time dilation as seen in an inertial reference frame is dependent only on velocity in that frame. So if something moves on a curved path at constant speed as seen in an inertial frame, you can calculate the time elapsed just by multiplying the coordinate time in that frame by $$\sqrt{1 - v^2/c^2}$$. But Einstein is not claiming there that the object moving on the curved path itself has its own rest frame where the usual rules of relativity apply; these rules only apply in frames "in uniform translatory motion", ie no acceleration (and changing direction always involves acceleration).

Now if we follow hurkyl's logic that we will see R running at half the assigned C rate if we continue to use timing experiments between R and C at all points of the path, and vice versa, can we not conclude that the spaceship is a good inertial frame - it shouldn't really make a difference in the outcome if the path were comprised of many straight segments that approximate the circle and the readings were always taken at the midpoint of the seqments

But making the path into a series of straight line segments wouldn't help either, because R would not have a single inertial rest frame throughout the journey, it would have a series of different inertial rest frames. This is just like the twin paradox, where for convenience you usually assume that the traveling twin's path is just made up of two straight line segments.

Now I do not see that this experiment leads to the notion of a universal time - but if both R and C run at the same speed at all times as a necessary consequence that they are not being acted upon by any forces or factors that would change the rate at which they run - then we do have a condition where they run at different rates as judged by each other - but unlike apparent time dilation - R judges C as fast and C judges R as slow

Usually in the context of relativity, when people say things like "A judges B to be running slow" they just mean that in the inertial frame where A is at rest, B is running slow. But again, R does not have a single inertial rest frame throughout the journey. It is certainly true that in any frame where R is momentarily at rest, then in that frame C will be running slower than R at that moment, but then that frame will see R's velocity change throughout the journey and at some point it will have to see R running slower than C. And again, there is no physical reason to prefer one inertial frame's description of the whole situation to another's, so you cannot say that R was running slower than C at every moment, you can only say that R's average rate of ticking over an entire orbit was slower than C's (this will be true in every inertial frame).

Again are we getting the right numerical answer for the wrong reason.

WHY do you think it is wrong? You keep objecting to the idea that every frame is equally valid but you never spell out any reasons, and you seem to have agreed in your last post that as long as all the laws of physics have equations with the mathematical property of lorentz-invariance, this insures that every inertial frame will see the same laws of physics and will make the same predictions about all physical questions like what two clocks read at the moment they meet. If you agree with this, it seems that you must agree that no inertial frame's analysis is physically preferred over any other's, so all you're left with is some kind of aesthetic preference for one frame's description over all others, no different than someone who prefers cartesian coordinates to polar coordinates.

 

[Hurkyl]

in other words, they are running at intrinsically different rates

Well I have two responses:

(1) R interrogating C and R interrogating E are exactly the same experiment. What do you find so special about the former experiment that allows you to justify calling it "intrinsic" and the latter "apparent"?

(2) There is no "intrinsic" difference. The observer on R, using the R-clock to measure all relevant times, still finds that his heart is beating at 120 bpm, still finds that middle A is 440 Hz, still finds that the half-life of a pion is 26 ns, et cetera.

 

[jtbell]

it shouldn't really make a difference in the outcome if the path were comprised of many straight segments that approximate the circle and the readings were always taken at the midpoint of the seqments

But making the path into a series of straight line segments wouldn't help either, because R would not have a single inertial rest frame throughout the journey, it would have a series of different inertial rest frames.

In order to predict what C's (or E's or A's) clock reads in the inertial frame that R happens to be instantaneously at rest in, at any point along R's path, you need to perform a series of Lorentz transformations, from one straight segment to the next in yogi's approximation. Then, ideally, you would take the limit as the number of segments increases towards infinity and their length each approaches zero. Mathematically, this is of course an integral. But yogi's example is a two-(space)-dimensional problem so you need to use a two-dimensional version of the Lorentz transformation. :yuck:

 

[JesseM]

In order to predict what C's (or E's or A's) clock reads in the inertial frame that R happens to be instantaneously at rest in, at any point along R's path, you need to perform a series of Lorentz transformations, from one straight segment to the next in yogi's approximation. Then, ideally, you would take the limit as the number of segments increases towards infinity and their length each approaches zero. Mathematically, this is of course an integral. But yogi's example is a two-(space)-dimensional problem so you need to use a two-dimensional version of the Lorentz transformation. :yuck:

Well, if the speed as a function of time is v(t), the integral would be $$\int_{t_0}^{t_1} \sqrt{1 - v(t)^2/c^2} \, dt$$ regardless of how many dimensions the problem is, no? Anyway, yogi specified that R was moving at a constant speed in the rest frame of the other clocks, and in any frame where speed is constant you shouldn't need to do an integral.

 

[yogi]

My use of the word intrinsic is of course undefined in SR - - it is not the observed time - it is the difference between what two clocks that are in uniform relative motion will determine to have accrued on each clock when they are compared in the same frame. Let us proceed to use R and E as Hurkyl has chosen to illustrate - as long as they are in relative motion, we make measurements and get different results because relative velocity and distances are changing But what is being measured is a distortion - If R quickly stops halfway to Altare and interrogation signals are sent from E to R and from R to E - what do they report - after comparison R has fallen behind E by 1/2 - the comparison is being made in the same frame and there is no doubt that they have been physically running a different rate. But the result would be no different if we made the same inquiry while R is moving. But we must ask the question "What does your clock read" and not set up a Lorentz grid or some other formalism that can only lead to a distorted answer as to how fast time appears to be moving in the other frame. We can make this determination at any point we wish during the trip - and by asking the question: "What does your clock read" rather than: "how fast does time appear to be passing while we are in relative motion" we arrive at the correct physical reality...at all times during the voyage, proper time in the R frame accrues twice as fast as it does in the E, C, A frame. The result would be no different if R is in an orbit around some great attractor which C happended to be the center of - and it is no different on any other experiment - the invariance of the interval assures us that it is the high speed pion and not the Earth clock that runs slow - real time dilation is a one way thing.