Inertial Frames distinguished by proper times

[yogi]

A first spaceship S1 departs from Earth and quickly accelerates to a velocity V = c/2. S1 travels the shortest path (dead heading) toward a distant planet Alpha so that it arrives in 20 years as measured by a clock on S1. One year after S1 is launched from Earth as measured by a clock on the earth, a second spaceship S2 is launched along the same trajectory (we assume neither Earth nor alpha have moved during this experiment). S2 quickly accelerates to a greater velocity than c/2 and at some point S2 overtakes S1. When S2 arrives at Alpha, the clock aboard S2 will read a lapse time of 10 years since it departed from earth.

Both S1 and S2 move at uniform velocity, so each is a valid inertial system, but the proper clock rate of the S2 clock is 1/2 the proper clock rate of the S1 clock. Based upon the difference in proper rates - if observers on each spaceship take the measure of the other as S2 passes S1, will an observer in S1 arrive at the same conclusion about contraction and clock rate in the S2 frame as S1 determines about contraction and clock rate in the S2 frame?

 

[Ich]

The proper clock rate in S1 and S2 is 1s/s. Proper (German: "eigen") means belonging to itself, contrary to "with respect to". Try to remember this before posting.

 

[pervect]

S2 quickly accelerates to a greater velocity than c/2 and at some point S2 overtakes S1. When S2 arrives at Alpha, the clock aboard S2 will read a lapse time of 10 years since it departed from earth.

Both S1 and S2 move at uniform velocity, so each is a valid inertial system

S2 is in at least two different inertial systems, because it accelerates. It's unclear if S1 breaks to a stop or not - if S1 does break to a stop, it accelerates as well.

Accelearting clocks are not inertial clocks by defintion.

 

[robphy]

Is this a fair rephrasing of the problem?

The Earth and alpha are inertial and at rest with respect to each other (so, they have parallel worldlines, vertical in their frame).

Ignoring the time when S1 was on the Earth awaiting launch... at event P, S1 travels inertially toward alpha at speed c/2 and arrives at alpha at event P', after 20 years on S1's clock.

Ignoring the time when S2 was on the Earth awaiting launch... at event Q, one year after P according to the Earth's clocks, S2 travels inertially at toward alpha some speed greater than c/2 so as to overtake S1. That is, their worldlines cross before alpha. S2 arrives on alpha at event Q', after 10 years on S2's clock.

Is this the situation?

If so, then

Both S1 and S2 move at uniform velocity, so each is a valid inertial system,

true

but the proper clock rate of the S2 clock is 1/2 the proper clock rate of the S1 clock.

false, as Ich says. The "proper clock rate of S1" is the clock rate of S1 measured by clock S1. Likewise, for S2.

Based upon the difference in proper rates - if observers on each spaceship take the measure of the other as S2 passes S1, will an observer in S1 arrive at the same conclusion about contraction and clock rate in the S2 frame as S1 determines about contraction and clock rate in the S2 frame?

What it all boils down to is that you have two intersecting inertial worldlines.

S1 and S2 are inertial observers, between P and P' and between Q and Q', respectively. So, identical experiments they perform on each other (e.g. comparing doppler effects or time dilation effects) will agree.

In fact, their relative-velocity can be computed from the data you've given. I get 0.41c. (S2's velocity must be 0.755c and alpha is 11.5 light years away.) I'm too lazy right now to calculate the intersection event... but according to the earth, S1 will arrive after 23 years, and S2 after 16 years (including the 1 year wait). I hope I didn't make any mistakes.

You can figure out when and where they intersect according to the Earth frame and what each clock reads at that meeting.

 

[yogi]

Robphy:

"Is this a fair rephrasing of the problem?

The Earth and alpha are inertial and at rest with respect to each other (so, they have parallel worldlines, vertical in their frame).

Ignoring the time when S1 was on the Earth awaiting launch... at event P, S1 travels inertially toward alpha at speed c/2 and arrives at alpha at event P', after 20 years on S1's clock.

Ignoring the time when S2 was on the Earth awaiting launch... at event Q, one year after P according to the Earth's clocks, S2 travels inertially at toward alpha some speed greater than c/2 so as to overtake S1. That is, their worldlines cross before alpha. S2 arrives on alpha at event Q', after 10 years on S2's clock.

Is this the situation?"

Yes - that is a correct. We have two intersecting world lines, and S1 and S2 are both inertial observers whose local clocks are running at different rates. And yes the relative velocity can be computed - but the actual numbers are immaterial, as is the location where S2 overtakes S1. The issue is whether there is any experiment that can be performed by the spaceships themselves to detect the difference between the intrinsic rate of S1 and S2. Between the two spatial points Earth and alpha the S1 clock records 20 years, the S2 clock records 10 years. The proper time interval is the time recorded by a clock attached to the observed body. While it is true that an observer in SI will always measure the S1 clock to be runiing at one minute per minute, and likewise an observer in S2 will measure the S2 rate as one minute per minute, there is nontheless a difference in how the proper rates get established ...the ratio of 20 years per 20 years and 10 years per 10 years. Is there a way to recover by experiment the difference between the numbers that formed the ratios in the two systems?

 

[RandallB]

Assuming “quickly accelerates” = instant acceleration to use straight SR no GR.
I get pretty much what robphy gets:

S1 v= 0.5c
S2 v= 0.756
Distance to travel 11.58 LY
S1 23.16 Earth time (20 S1 time)
S2 16.32 Earth time (10 S2 time 11 including wait)

S1 time .866 of earth
S2 time .655 of Earth

Relative S1-S2 v= 0.411
S2 time .912 of S1 (& S2 time .912 of S1 of course)

They meet at 1.475 LY from earth
2.95 Earth time
2.6 s1 time
2.27 S2 time (including the 1 y wait on earth)
1.27 S2 elapsed time
(note: S2 had two different speeds to build 2.27 time since being together with S1)

Classical SR math.

 

[robphy]

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Here's my calculation using rapidities... analogous to angles in Euclidean trigonometry... in Maple code.

restart;
theta:=arctanh(1/2.);
                        theta := 0.5493061443
d:=20*sinh(theta);
                           d := 11.54700538
phi:=arcsinh(d/10.);
                         phi := 0.9866469608
tanh(phi);
                             0.7559289459
tanh(phi-theta);
                             0.4114378276
20*cosh(theta);
                             23.09401076
1+10*cosh(phi);
                             16.27525231
cosh(phi-theta);
                             1.097167541

with a little more trigonometry, you can get everything else you would want. (beta=(v/c)=tanh(rapidity), gamma=cosh(rapidity))

One can use radar methods to operationally measure all of the kinematic quantities.

 

[yogi]

RandallB/robphy - More than I expected - good tutorial(s). Thanks. My interest in posing the question relates the information which is lost (or apparently lost) if we simply consider the two spaceships S! and S2 passing each other. Without the the benefit of knowing the initial conditions, SR predicts that the S1 and S2 frames are equivalent (equally valid inertial frames as they pass one another with relative velocity v) - and accordingly each would arrrive at the same conclusion(s) regarding contraction and clock rate in the other frame (actually Einstein never really said this in his 1905 paper, but SR is generally given this interpretation). So even though the S1 clock and the S2 clock are running at their own proper rate of one second per second, a second in S1 is different than a second in S2 when both are observed in the earth-Alpha frame. So when an observer in S2 uses his watch to make a measurement of the apparent length of S1, he will be using a different time base than the S1 observer uses his watch to measure the apparent length of S2.

There is a common Gamma factor, but we have different times T1 and T2 to use as a bases for dilation. So is it correct to conclude that the S1 observer figures the T2 time dilation based upon the value of a second in the T1 frame, and the S2 observer figures the T1 time dilation based upon the value of a second in the S2 frame? If they transmit their results to the earth, their conclusions will not be the same because each is arriving at the value of a dilated second in the other frame in terms of his own time.

 

[robphy]

I'm not sure what the times T1 and T2 mean precisely.

Are they the elapsed times for each observer to complete their trips?
If so, just realize that these a proper times between two different sets of events... (it just so happens that the launch events occurred on the Earth's worldline and the arrival events occurred on alpha's worldline). So, there is nothing to be found in comparing T1 and T2.

If not,...
Are they the observed time-dilations in studying the duration of one tick of the clock? That is, is T1 the duration of 1 tick of S1's clock , as determined by S2? More precisely, let A and B be events on S1's worldline corresponding to two consecutive ticks of his clock... so, S1 says the "time-difference between A and B" is 1 tick. Is "T1 ticks" what S2 says is the "time-difference between A and B"?
If so, then (by the relativistic symmetry of these inertial observers.. which can be borne out by a radar experiment) T1=T2, which is essentially gamma, which is a function of the relative-speed between these inertial observers.

In any case, consider the following experiment.
One tick after their meeting (when S1 and S2 meet, as S2 overtakes S1), each observer sends off a light signal to the other. Each observer will measure the same elapsed time on his clock when that signal arrives from the other. So, their identical procedures yields identical measurements (which they can write down in their log books for later comparison)... reflecting their SR-symmetry. (In fact, if each observer understands that the received signal was sent after one tick according to the emitter, the ratio of these proper time-intervals (1 tick for the emitter, T1 ticks for the receiver) gives the Doppler factor, k, which can be used to determine their relative-speed.)

 

[yogi]

Thanks robphy - i want to ponder your response before replying

Yogi

 

[yogi]

T1 is simply an increment measured in the S1 frame - for example one second as measured by the clock in S1. Likewsie T2 is an increment measured in the S2 frame by the S2 clock - e.g., also one second as measured by the clock S2. If S1 sends out a light pulse every second as measured by his own clock, and S2 does the same as measured by his clock, an observer on earth, knowing the two recessional velocities, would be able to determine that one second as measured by the S1 clock is different than one second as measured by the S2 clock. In other words one second of proper time in S1 is different than one second of proper time in S2.

As I read your post 9 you are saying that, in accordance with SR, if S1 and S2 transmit back and forth radio signals they will arrive at reciprocal results - that is, as between S1 and S2 there is no significance to the fact that T1 seconds are not the same as T2 seconds, or in any event it cannot be determined by signals sent between S1 and S2 that there is a difference between T1 and T2 seconds, but only the relative velocity will be revealed.

But if S1 transmits a signal every second according to the S1 clock and S2 transmits a signal every second according to the S2 clock, then, since the relative velocity v between S1 and S2 is known, each can determine the extra delay caused by their relative recession velocity v (after S2 overtakes S1). Once the relative velocity is determined, either observer can compare his own one second clock ticks with the rate at which pulses are arriving from the other spaceship...After compensating for the changing distance, either observer will see that the received pulses are being sent at a different rate than one per second as measured by the clock in the receiving spaceship, and each will agree upon which spaceship has the longer seconds!

 

[jtbell]

But if S1 transmits a signal every second according to the S1 clock and S2 transmits a signal every second according to the S2 clock, then, since the relative velocity v between S1 and S2 is known, each can determine the extra delay caused by their relative recession velocity v (after S2 overtakes S1).

This is the relativistic Doppler effect. Both S1 and S2 observe the other's signals arriving at the rate

f_{observed} = f_{source}\sqrt{\frac{1+v/c}{1-v/c}}[/itex]<br /> <br /> where v is the relative velocity of the source and observer, and is positive if the source is approaching the observer. <br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> either observer will see that the received pulses are being sent at a different rate than one per second as measured by the clock in the receiving spaceship, and each will agree upon which spaceship has the longer seconds! </div> </div> </blockquote><br /> If S1 and S2 send signals at the same rate, they each receive the other&#039;s signals at the same rate. By the formula above, they each compute the same source frequency.

 

[Ich]

each will agree upon which spaceship has the longer seconds!

Yes. Always the other one.

 

[robphy]

T1 is simply an increment measured in the S1 frame - for example one second as measured by the clock in S1. Likewsie T2 is an increment measured in the S2 frame by the S2 clock - e.g., also one second as measured by the clock S2. If S1 sends out a light pulse every second as measured by his own clock, and S2 does the same as measured by his clock, an observer on earth, knowing the two recessional velocities, would be able to determine that one second as measured by the S1 clock is different than one second as measured by the S2 clock. In other words one second of proper time in S1 is different than one second of proper time in S2.

Just to clarify...
Technically speaking, "proper-time" is akin to an arc-length... it is a number (as the result of an integral or sum)... and it is invariant (all observers will agree on it). So, one second of proper time in S1 is THE SAME AS one second of proper time in S2. What is different, however, is that each clock has a different spacetime displacement-vector between successive ticks of his clock [i.e., each has a different unit timelike vector tangent to his inertial clock worldline]. In general, an observer will say that the time-components of these vectors (i.e. the apparent durations between ticks for each clock) are unequal. This is time-dilation,

As I read your post 9 you are saying that, in accordance with SR, if S1 and S2 transmit back and forth radio signals they will arrive at reciprocal results - that is, as between S1 and S2 there is no significance to the fact that T1 seconds are not the same as T2 seconds, or in any event it cannot be determined by signals sent between S1 and S2 that there is a difference between T1 and T2 seconds, but only the relative velocity will be revealed.

yes... regarding T1 and T2 to be the vectors descrbed above.

Conerning that last part of your post, jtbell and Ich addressed that. (I was too slow working on the first part )

 

[jtbell]

Conerning that last part of your post, jtbell and Ich addressed that. (I was too slow working on the first part )

Call it parallel processing on a small scale.

 

[yogi]

Granted, if the S1 signals are being sent at the same frequency as the S2 signals, each will draw the same conclusion. But the pulse rate frequencies are not the same. Each is using his own clock to measure one second; the S1 clock is running at a faster rate than the S2 clock.

Let me analogize to a GPS situation. One clock is considered to be synchronized with a point at the center of the Earth (the non rotating Earth centered reference frame) we call it E clock. A second clock C2 is first synchronized with E and then put into circular Earth orbit.

C2 will run at a different rate because of two factors (the height and the velocity). If C2 is corrected for altitude, what is left is an orbiting clock C2 that runs a slower rate than E clock.
In this case the distance does not vary - so the only correction would be a transverse Doppler - What is observed is C2 running slower than E. Seconds take longer in the Satellite frame than in the ECRF.. If both C2 and E are used to transmit pulses at one second intervals, the pulses received from C2 will have a slower frequency than one pulse per second when received by E and the pulses received from E will have a higher frequency than one pulse per second when measured by C2. C2 must be compensated to make the seconds equal. There is no ambiguity as to which clock is running slower and which clock must be compensated.

 

[yogi]

One more example using a linear experiment rather than a satellite clock in free fall - a pion created in the lab and qucikly accelerated to near c velocity. Here the pion clock can be considered to have sent two pulses, one at the time of creation and one at the time of disintegration. If the lifetime of an at rest pion (approx 0.02 usec) is extended by a factor of 20 because of its high velocity wrt the lab, the lab detector will receive 2 pulses in say 20 x 0.02usec and the lab observer will conclude that the pion clock is running slower than the lab clock. But if a lab transmitter is sending signals every 0.02 usec, then during the extended lifetime of the high velocity pion, 20 pulses would have been transmitted. If there is a hypothetical observer P attached to the pion, how could P reach the conclusion that the Earth clock is running slower than the pion clock?

 

[robphy]

One more example using a linear experiment rather than a satellite clock in free fall - a pion created in the lab and qucikly accelerated to near c velocity. Here the pion clock can be considered to have sent two pulses, one at the time of creation and one at the time of disintegration. If the lifetime of an at rest pion (approx 0.02 usec) is extended by a factor of 20 because of its high velocity wrt the lab, the lab detector will receive 2 pulses in say 20 x 0.02usec and the lab observer will conclude that the pion clock is running slower than the lab clock. But if a lab transmitter is sending signals every 0.02 usec, then during the extended lifetime of the high velocity pion, 20 pulses would have been transmitted. If there is a hypothetical observer P attached to the pion, how could P reach the conclusion that the Earth clock is running slower than the pion clock?

Using your data...
T=pion lifetime (0.02 us)
gamma=20
k=Doppler factor= gamma+sqrt(gamma+1)*sqrt(gamma-1)=39.97
(another formula: k=exp(arccosh(gamma)))

Note that gamma*T="20 x 0.02usec" is the "lab's apparent -duration of the period of the pion's signal-emissions". It is NOT the "lab's reception-period of the pion's signal-emissions", which is kT="39.97T".

Note further that, while the lab may emit 20 regular signals (at 0.02 us intervals according to the lab) in the lab-time gamma*T, the pion receives only the signal at the meeting... it does not receive any other. (To reach the pion before it dies, the lab must have sent a signal at T/k=0.02usec/39.97. Waiting for 0.02us is too long.) If the pion lived long enough, the pion would have received the lab's signal [sent at 0.02 us after meeting] when the pion's clock read kT=39.97*0.02 us... just like the lab's result.Using a notational variant of the diagram from https://www.physicsforums.com/showthread.php?t=113915

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[yogi]

robphy - You are assuming the lab transmitter and receiver to be at the starting point. We can place the lab transmitter and receiver midway between the starting point of the pion and the point of disintegration (for v=0.99c the disintegration length is only about 6 meters). With the lab transmitter and recorder so located, the pion will receive more pulses during the experiment than the lab recorder.

 

[robphy]

robphy - You are assuming the lab transmitter and receiver to be at the starting point. We can place the lab transmitter and receiver midway between the starting point of the pion and the point of disintegration (for v=0.99c the disintegration length is only about 6 meters). With the lab transmitter and recorder so located, the pion will receive more pulses during the experiment than the lab recorder.

I'll try to produce a diagram of what I think you are describing.
However, after a sketch I made on paper, the following issue presented itself to me.

The pion's worldline is a segment with a finite proper-time of 0.02us. With the lab "midway between the starting point of the pion and the point of disintegration", the lab worldline bisects the pion's worldline-segment. In order to make their experiments "identical", we should only consider the analogously bisected 0.02us-segment of the lab's worldline. Without that concession, the symmetry between the inertial observers is broken... one is allowed to live a longer proper-time than the other.

 

[Ich]

Without that concession, the symmetry between the inertial observers is broken... one is allowed to live a longer proper-time than the other.

Yes, that´s the point. The pion will do nothing but collect a blue-shifted series of pulses that have been emitted "long" before its creation. No way to claim that the time between the emission of the first and the last collected pulse corresponds to 0.02 µs in the lab frame.

 

[Ich]

To add another point of view:
Yogi, it seems like you´re still mixing up observers and reference frames. I guess that this creates the paradox you can´t resolve.
Classical example:
A (very long, very fast, gamma=2) train drives past your window. They switch the cabin lights on and off in intervals of 1 s.
My guess is that you´d expext your room to be lit in 2 s intervals.

 

[yogi]

Yes, that´s the point. The pion will do nothing but collect a blue-shifted series of pulses that have been emitted "long" before its creation. No way to claim that the time between the emission of the first and the last collected pulse corresponds to 0.02 µs in the lab frame.

The pion gets synchronized at its creation with the Earth clock - there can be another clock at the X origin in sync with the pion clock and the lab transmitter clock- nothing is transmitted before the pion is created - the lab transmiter is located at some point x=3 along the +x axis... the pion will intercept all of the pulses that were transmitted during the time it travels from the origin until it reaches x = 3. Let it be assumed that during the time the pion travels from x=0 to x=3, 10 pulses were transmitted by the lab clock located at 3. To make things simplier we will assume the pion generates a signal as it passes x=3
--this corresponds to 0.01usec of time on the pion clock. The pion would have received 10 signals and the lab would have received 2. How can an observer attached to the pion conclude that the Earth clock is running slower than the pion clock?

 

[yogi]

Ich - your post #22 - With a distributed light source the problem is not the same - if you have a single light source point, then I would expect to see flashes at a different frequency than what I would be transmitting if in fact the two trains had been originally synchronized, and it was the other train that had been accelerated (i.e., the Gamma factor of 2 arose from the fact that I was not occupying the train that got the acceleration). In that case, yes, I would expect to receive less flashes per minute than I would be sending - but correction would be required for the changing distance however. To eliminate the changing distance, I proposed the orbiting satellite (post 16) version of the same problem. In actuality, this is the only example to date that is actually carried out. The result is unambiguous - the clock that received the acceleration runs slow relative to the Earth clock - and the Earth clock (ECRF) runs fast relative to the orbiting clock (ignoring height factors). We never do the linear experiment, but if we ever did, it should conform with Einstein's original development of actual time dilation - specifically, when one of two synchronized clocks is accelerated, it will run slower than the non-accelerated clock. Although every refernce frame is equivalent for the purpose of making internal experiments, and they all measure c as constant, they are not equivalent from the standpoint of making measurments in another frame that is in motion relative thereto.

 

[Ich]

The pion gets synchronized at its creation with the Earth clock - there can be another clock at the X origin in sync with the pion clock and the lab transmitter clock- nothing is transmitted before the pion is created - the lab transmiter is located at some point x=3 along the +x axis...

When you start the emission of pulses as you described (synchronized in the lab frame), the clock will tick for a proper time of 0.01 µs - quite longer than the lifetime of the pion. You should not wonder why it generates more ticks when you compare different timespans. It´s the same old synchronization problem as ever.

 

[Ich]

We never do the linear experiment, but if we ever did, it should conform with Einstein's original development of actual time dilation - specifically, when one of two synchronized clocks is accelerated, it will run slower than the non-accelerated clock. Although every refernce frame is equivalent for the purpose of making internal experiments, and they all measure c as constant, they are not equivalent from the standpoint of making measurments in another frame that is in motion relative thereto.

I´m not sure whether you´re talking about SR here. If you did, you´re wrong. SR says the following:
Acceleration has nothing to do with it. The result is the same when you think of your room being a cabin on a initially accelerated train, or when both you and the train were accelerated by the same amount. The result is symmetric.
You will see you room be lit in 0.5 s intervals. At your location, you will see the train time going twice as fast as yours. And the train observers will see your time going half as fast than theirs. Both agree.
But the same is of course valid for a fixed point on the train. Again both will agree, but this time your time (measured at a fixed point on the train) will run fast.
It´s exactly this effect that you mistake for "different proper time rates" in your example of an orbiting satellite: The satellites trajectory is curved such that the Earth center clock always remains at the same x-position (x being in the direction of relative movement). The Earth center of course sees the satellite move along its x-axis. That´s the assymetry: you compare times at a fixed point in the satellite system, not in the Earth center system. Acceleration in the direction of movement has nothing to do with it.

 

[yogi]

Ich: Which clock experienced acceleration (changed its speed after clocks were synchronized in the same frame) has everything to do with it. I am not talking about the (v^2)/R acceleration of the orbiting satellite - the satellite clock is in free fall inertial frame, and because of its velocity v it runs at a slower rate than the clock in the ECRF (Ignoring altitude). One second pulses sent by a transmitter in sync with the ECRF will be received at shorter than one second intervals by the satellite receiver - signals sent from the satellite at one second intervals as measured by the satellite clock will be received at longer than one second intervals in the ECRF. Based upon these signals, each will conclude that the satellite clock is running slower than the clock in the ECRF.

 

[yogi]

Ich - as to your post 26, If two trains are separated by some significant distance and both initially at rest in the Earth frame - and each is equally accelerated toward the other until they reach a velocity v wrt the Earth frame, then as they pass one another, things will be symmetrical - each will see the other clock to be APPARENTLY running slower by the Gamma factor that relates their relative velocity. In actuality however, both clocks on the trains will be running at the same rate - what is observed is an apparent slowing of the other guys clock. As each train passes the other and continues to the point where the other train started, the actual amount of slowing as measured by an Earth frame clock at that point will be the same for both trains. This is a symmetrical situation. But because actual clock rate differences depend upon which train has accelerated (or which has accelerated the most) as per Einstein's 1905 paper, it is improper to interpret the relativistic formula T1 = Gamma(T2) as being reciprocal to situations involving actual time dilation.

 

[Ich]

NO
From your previous posts I got the impression that you really try to understand SR, that´s why I joined the discussion. And now its the same old story: You promote your views of SR and don´t listen when people try to explain the real thing.
So, again, for the record:
Your understanding of SR in general and Einsteins paper in special is plain wrong. Its not a different point of view, its WRONG.
And your basic mistake is

to interpret the relativistic formula T1 = Gamma(T2)

There is no such formula in the lorentz equations, and you will never understand SR by interpreting time dilation. Even worse, your formula gives t2>t1, the contrary of time dilation.
The correct formula is gamma*(t1-vx). Only when you set x=v*t1 you get t2=(1/gamma)*t1, that is, time dilation.
You can derive all the results I discussed, and also what Einstein said in his paper, simply by applying the correct formula. No need for additional information like who has been accelerated and such things. Everything is unambiguous.
My advice: Sit down, try to understand what I posted about the train, calculate it yourself. Read my post about your satellites. Calculate the example yourself, using the correct formula. Derive why v²/r(*r) is important, not any initial accelerations. All the paradoxes and especially the real/apparent nonsense will vanish into thin air.
If you encounter any difficulties on your way, ask the people here.

 

[yogi]

No yourself - again you, like many, do not understand the difference between real time dilation (that shows up and is measurable in both one way and round trip experiments), and the apparent time dilation that results from the blind application of the LT transforms to a situation where you have failed to take into account the intrinsic asymmetry of the frames

The relationship between two spacetime events in the "at rest" system and the same two spacetime events in the moving system is directly determined by the invariance of the interval.

Your statement is:

"Only when you set x=v*t1 you get t2=(1/gamma)*t1, that is, time dilation.

I didn't specify which frame was T1 and T2 - its the same formula I gave - its the direct result of the invariance of the interval.

Read again what Einstein said in 1905 - when two clocks are synchronized in one frame and one of them is put in motion, the one put in motion will run at a lower rate than the one which is not put in motion. Ignoring oblateness - a clock at the equator will run faster than a clock at the North pole (directly from Einstein). Einstein never says the situation is reciprocal - it can't be where time dilation is actual - and that is the only situation to which I am referring -

Do some reading yourself - you will see that SR is explained in different ways by different authors - some of the explanations are mutually exclusive - but however analysed, the experimental results show that the clock put in motion runs slower - and that the clock which remains at rest runs faster relative to the clock put in motion - the situation is not reciprocal and Einstein clearly asserts this. Too bad you didn't read what he was saying before you took up preaching.

 

[robphy]

No yourself - again you, like many, do not understand the difference between real time dilation (that shows up and is measurable in both one way and round trip experiments), and the apparent time dilation that results from the blind application of the LT transforms to a situation where you have failed to take into account the intrinsic asymmetry of the frames

The relationship between two spacetime events in the "at rest" system and the same two spacetime events in the moving system is directly determined by the invariance of the interval.

I need a clear definition (or better, a spacetime diagram) to continue in this discussion. (In addition, a response to my last post would be nice.)

I understand "time dilation" to mean the following.
An observer A computes the following ratio:
"elapsed-proper-time-on-A's-watch from O to T (events on A's inertial worldline)"
divided by
"elapsed-proper-time-on-B's-watch from O to T' (events on B's inertial worldline)"
where observer A regards T and T' as simultaneous.
That ratio is symbolized by gamma, the "time-dilation factor".

Please clearly compare and contrast this definition with your terms
"real time dilation"
and
"apparent time dilation".
It would help me if you first state (either "SAME" or "DIFFERENT") then use a similar spacetime language (e.g. events, readings on watches, relationships between those events (on the same inertial worldine? simultaneous? connected by a light signal?, etc...) as in my defintion.) A spacetime diagram would be fabulous.

 

[yogi]

robphy Sorry - I did not mean to ignor your post 20 - let me see if I can get across the concept by sticking to the simple case of a satellite in circular orbit about the Earth (whenever I do this I get a response from someone that the satellite is not an inertial system, and therefore SR isn't applicable ...and the thread gets sidetracked on that issue)..So whatever your views are on that, there is abundant authority that a clock in orbit is a free falling inertial system i.e., it is as good as any other inertial frame for the purposes of SR.

Now we sync two clocks S2 and S3 with an Earth clock E at the center of the earth. The two clocks S2 and S3 are put in orbit and both are compensated for the altitude. If all clocks transmit a pulse every second according to their local time, then the pulses transmitted by S2 and S3 will arrive at E at more than one second intervals as measured by the E clock. The pulses sent by E will arrive at S2 and S3 in shorter intervals than one second. This is actual time dilation - The two clocks in orbit have been accelerated to orbital velocity after they were synchronzed with the E clock that measures time in the ECRF. The orbital radius remains constant - there is no question as to which clock is running fast.

Now we correct the S3 clock as we do in GPS systems so that it runs at a faster rate - it now transmits signals that are received by E once every second. The S2 clock still runs slow

If S3 is used as a base for measuring time dilation between E and S3, none will be detected - they run in sync. If the S2 clock is used, S2 concludes that it receives pulses at a greater rate than it is transmitting, and concludes that E is running faster. Likewsie, E concludes from the slower pulse rate received by the signals sent from S2, that S2 is running slower - the situation is not reciprocal - in real time dilation there cannot be reciprocity.

Now we correct both S2 and S3 to run at the same speed so that each transmits pulses that are received by E at one second intervals - but instead of putting them in the same satellite, we put one in a polar orbit and the other in an equitorial orbit at the same altitude...so they will have varying relative velocity between one another at different places in their orbits. As measured by the Earth clock, both S2 and S3 are running at the same speed - but as each views the other, they will determine that the other clock is running slow - here there is reciprocity, but the measurments are apparent - the clocks run at same corrected speed during their respective orbits - but they measure an apparent slowing of the clock in the other satellite.

I hope this clarifies what i mean by apparent and actual time measurements

 

[robphy]

robphy Sorry - I did not mean to ignor your post 20 - let me see if I can get across the concept by sticking to the simple case of a satellite in circular orbit about the Earth (whenever I do this I get a response from someone that the satellite is not an inertial system, and therefore SR isn't applicable ...and the thread gets sidetracked on that issue)..So whatever your views are on that, there is abundant authority that a clock in orbit is a free falling inertial system i.e., it is as good as any other inertial frame for the purposes of SR.

For now, I'll have to reserve comment on satellite/GPS issues... since this seems to complicate the situation beyond the originally posed question, which I tried to reformulate as a simple SR problem in my posts #4, #7, #9, #14, #18, and #20. (In SR [that is, flat spacetime on R⁴], I don't recall any need for a "compensation for altitude"... if one is needed, then it seems to me that the symmetry is already broken.) It seems to me that if your question can't be formulated as an SR problem, then it really isn't an SR problem. It may be a problem of inertial frames (i.e. geodesics) in a non-SR spacetime.

Now we sync two clocks S2 and S3 with an Earth clock E at the center of the earth. The two clocks S2 and S3 are put in orbit and both are compensated for the altitude. If all clocks transmit a pulse every second according to their local time, then the pulses transmitted by S2 and S3 will arrive at E at more than one second intervals as measured by the E clock. The pulses sent by E will arrive at S2 and S3 in shorter intervals than one second. This is actual time dilation - The two clocks in orbit have been accelerated to orbital velocity after they were synchronzed with the E clock that measures time in the ECRF. The orbital radius remains constant - there is no question as to which clock is running fast.

Now we correct the S3 clock as we do in GPS systems so that it runs at a faster rate - it now transmits signals that are received by E once every second. The S2 clock still runs slow

If S3 is used as a base for measuring time dilation between E and S3, none will be detected - they run in sync. If the S2 clock is used, S2 concludes that it receives pulses at a greater rate than it is transmitting, and concludes that E is running faster. Likewsie, E concludes from the slower pulse rate received by the signals sent from S2, that S2 is running slower - the situation is not reciprocal - in real time dilation there cannot be reciprocity.

Now we correct both S2 and S3 to run at the same speed so that each transmits pulses that are received by E at one second intervals - but instead of putting them in the same satellite, we put one in a polar orbit and the other in an equitorial orbit at the same altitude...so they will have varying relative velocity between one another at different places in their orbits. As measured by the Earth clock, both S2 and S3 are running at the same speed - but as each views the other, they will determine that the other clock is running slow - here there is reciprocity, but the measurments are apparent - the clocks run at same corrected speed during their respective orbits - but they measure an apparent slowing of the clock in the other satellite.

I hope this clarifies what i mean by apparent and actual time measurements

I'll have to ponder it... but I can't promise anything soon...
unless you'd like to continue the SR problem left off at post #20.
Are you trying to claim that "time dilation" (as I defined in my post #31) is not reciprocal between two inertial observers in SR [flat spacetime on R⁴]?

 

[yogi]

robphy: To my way of thinking, the GPS satellite analogy is the easy way to illustrate the notion of two relatively moving clocks - the corrections for altitude are put in because the G field is different at the height of the orbit. We can change this if you like to a thought experiment where the satellite orbits at sea level by constructing evacuated tunnels to circumscribe the Earth (an easy thought experiment but a difficult engineering task). With this model, we can forget about altitude compensation - the orbiting clock S2 does not run at the same rate as the E clock (the one that measures time in the non rotating Earth centered reference frame). The satellite clock S2 runs slower relative to the E clock and the velocity adjusted S3 clock runs faster than S2. If all three clocks transmit pulses at one second intervals as measured by their own clocks, the S2 pulses will be received by E at a slower frequency than S3 pulses, and the E transmitted pulses will be received in Sync with S3 but they will be blue shifted as to the uncompensated clock S2. As between S2 and E there is no ambiguity as to which clock is running faster (E) and which clock is running slower (S2)

I use this example because the distance between E and S2 remains the same - so there is no need to account for changes in length that arises in situations where clocks approach and recede from one another during the experiment. Nor is there any observed relativistic contraction in the radius of the Earth since the motion is transverse to the radial vector. There is a transverse Doppler effect - but it cannot account for the fact that when the satellite S2 clock is brought to rest in the Earth frame after many orbits, it will have logged less time than the E clock.

This is simply the orbiting case of the round trip clock described in part 4 of Einstein's 1905 paper. The difference is that we have, in the GPS satellite metaphore, a scenerio where each clock can continuously interrogate the rate of the other clock in relation to its own passage of time. Moreover, we have the data that confirm the difference in the actual rates at which satellite clocks run in relation to the E clock.

 

[robphy]

robphy: To my way of thinking, the GPS satellite analogy is the easy way to illustrate the notion of two relatively moving clocks - the corrections for altitude are put in because the G field is different at the height of the orbit.

This obviously renders it a non-SR [that is, it is not a problem in a flat R⁴ spacetime] problem, in spite what you claim in your post #32. If it involves geodesics, it is still a problem concerning inertial frames... just not in SR. While some aspects may be analogous to SR, some are not... for example, this correction.

I entered this thread because of the initial question you posed and my rephrasing of it (in post #4) as an SR problem, which you agreed to.
The issue in post #20 (an implicit question to you, awaiting your response) was left unresolved. Your tone changed (in general) in #30. The question changed in #32 to a non-SR problem, as expained above, while abandoning the original question. Finally, my question in #33 was left unanswered.

For these points, I will retire temporarily from this thread. Thanks for the conversation.

 

[yogi]

Likewise for the conversation Robphy- but post # 30 was not directed to you.

My intent was to try to nail down the fundamental difference between apparent time dilation and actual time dilation in one way travel experiments, and I argued that a clock in an orbiting satellite is a perfectly good inertial frame - just as is a clock that is put in uniform linear motion. But in the latter case one doesn't have a convenient experimental platform because pulses transmitted between linearly moving frames must be Doppler compensated, and the changing distance between sources and receivers must also be accounted for. So my shift was intended to get some resolution or agreement as to the relative rate of time passage in the case of GPS and then carry this over to flat spacetime as per my original post.

I apologize if anything I said that may have offended you - I have always found your posts to be cordial and well thought out

Regards

Yogi

 

[Ich]

Yogi,

even if my posts lack the cordiality you seem to expect they could help you if you tried to understand SR. I became more aggressive when I found that you exhibit more and more a crackpot attitude that doesn´t fit in with this forum.
You claimed in former threads that an orbiting frame is a perfect (SR) inertial frame. You have been told that you´re wrong. Still you come back here and claim the same.
Likewise, you have been told repeatedly that initial acceleration woud not decide who is ageing more slowly. Still you ignore that.
In this thread, I tried (not for the first time) to give you clues to understand how SR works. You (obviously) took it as a personal attack and repeated even more aggressive your false claims.
That´s not how it works.
Just TRY to listen to the people you are discussing with. Then they´ll stay in tune.

 

[robphy]

Yogi,
Yes, I know that #30 wasn't directed to me.
I wasn't offended in any way from anything you said.
It's just that I felt that this thread was veering off the initial question without resolving it... and it wasn't clear [to me] where it was going. So, I was going to take a break from it.

robphy

 

[yogi]

Yogi,

.
You claimed in former threads that an orbiting frame is a perfect (SR) inertial frame. You have been told that you´re wrong. Still you come back here and claim the same...

There are many references that support what I have said - take a look at Road to Reality p 394, or Spacetime physics pp 26-31 if you want some authority as to the validity of orbiting satellites as free float inertial frames. In fact, they are preferable to Earth in many respects - and we normally consider the Earth as a good inertial frame for making relativistic experiments. I know there are limitations - they are not good inertial frames on a large scale where tidal forces are significant - but for the small volume inside a GPS satellite - they are as good as need be. Until you can sight me some authority to the contrary, I think I will go along with Penrose and Wheeler


As for your comment re the initial acceleration, I have said this is a direct consequence of Einstein's explanation in part 4 of his 1905 paper - I urge you to read it ... when two spaced apart clocks are synchronized and one is quickly accelerated to a constant velocity and then travels to the location of the other clock, the one which has been put in motion will be out of sync (read less time) with the other when they are compared. But Einstein doesn't stop there - he gives several more examples - concluding that a clock at the equator will run faster than one at the pole.

I would like to engage in a discussion where the responses do not turn into a shouting match - if anything I have asserted is in violation of a confirmed experiment, please advise. If you have an explanation of why the two clocks in Einstein's illustration are reading different times at the end of the experiment, I would like to hear it. But remember, they are being compared in the same rest frame at the end of the experiment and the one which has been accelerated into uniform motion reads less (this is not a reciprocal situation just as the two readings on the twin's clocks (in a round trip journey) is not reciprocal.

 

[robphy]

There are many references that support what I have said - take a look at Road to Reality p 394, or Spacetime physics pp 26-31 if you want some authority as to the validity of orbiting satellites as free float inertial frames. In fact, they are preferable to Earth in many respects - and we normally consider the Earth as a good inertial frame for making relativistic experiments. I know there are limitations - they are not good inertial frames on a large scale where tidal forces are significant - but for the small volume inside a GPS satellite - they are as good as need be. Until you can sight me some authority to the contrary, I think I will go along with Penrose and Wheeler

I happen to have these on my shelf right now.
Certainly, as you say, it is valid to regard "orbiting satellites as free float inertial frames"... and yes, "they are preferable to Earth in many respects". In addition, I agree that "they are not good inertial frames on a large scale where tidal forces are significant - but for the small volume inside a GPS satellite - they are as good as need be."

So, for a single satellite, there's no problem... you can locally apply SR at any event.

For two satellites meeting at an event, there's no problem... you can locally apply SR at that meeting event.

The problem is that when you try to consider two satellites at different altitudes and you have to correct one because of its altitude, then you are now certainly outside "the small volume inside a GPS satellite". So, with the two satellites taken together in one frame, special relativity doesn't apply.

 

[yogi]

robphy, yes - what you say is quite correct- - I noticed after posting #39 that Rindler uses a strict definition of an ideal inertial frame as being one totally removed from all G fields (this may have been what hurkyl was referring to in a previous thread). Rindler, however, then goes on to say, as a practical matter we don't have access to this utopia so we do our experiments in a less than perfect environment. So with that understanding, perhaps we can ask the question of whether there is any difference between the principles involved in 1) the measured loss of time between a velocity uncompensated orbiting clock S2 and the ground E clock...and 2) the predicted time loss in the linear experiment where one clock is put in motion after being initially synchronized with a distant clock which remains at rest (as per Einstein part 4 of 1905).

 

[Ich]

Hi yogi, I´ve been away for two weeks.

I have one problem with our discussion: I already told you almost everything I have to tell, and you did ignore it. It´s ok for me to start again and discuss things until we come to a solution. But I will not tell everything a third time. So I expect that you address the points I make explicitly and that you tell me either that you agree or where you don´t (and why).

There are many references that support what I have said - take a look at Road to Reality p 394, or Spacetime physics pp 26-31 if you want some authority as to the validity of orbiting satellites as free float inertial frames. In fact, they are preferable to Earth in many respects - and we normally consider the Earth as a good inertial frame for making relativistic experiments. I know there are limitations - they are not good inertial frames on a large scale where tidal forces are significant - but for the small volume inside a GPS satellite - they are as good as need be. Until you can sight me some authority to the contrary, I think I will go along with Penrose and Wheeler

1. Nobody doubts that there is a local IF valid for the satellite. But it is never (not even for an infinitesimal time) valid at the center of earth. Therefore you can´t use SR to compare satellite and center time. GR will give you the correct result.
2. It is still possible and instructive to examine the problem in SR if you treat the satellite´s orbit as a polygon.

As for your comment re the initial acceleration, I have said this is a direct consequence of Einstein's explanation in part 4 of his 1905 paper - I urge you to read it ... when two spaced apart clocks are synchronized and one is quickly accelerated to a constant velocity and then travels to the location of the other clock, the one which has been put in motion will be out of sync (read less time) with the other when they are compared. But Einstein doesn't stop there - he gives several more examples - concluding that a clock at the equator will run faster than one at the pole.

I read and understood the paper. I agree with everything that Einstein said but not with your understanding that initial acceleration somehow decides which clock will read less time. An example:
3. Two synchronized clocks are accelerated by the same amount to +v and -v. After some time, clock 2 accelerates to +v. You bring both clocks slowly together and compare times: clock 2 shows less. So in this case it´s final acceleration which breaks the symmetry.

If you have an explanation of why the two clocks in Einstein's illustration are reading different times at the end of the experiment, I would like to hear it. But remember, they are being compared in the same rest frame at the end of the experiment and the one which has been accelerated into uniform motion reads less (this is not a reciprocal situation just as the two readings on the twin's clocks (in a round trip journey) is not reciprocal.

4. It´s instructive to see that even in a symmetric setup the time of the "moving" reference frame goes faster than your own, if you observe it at your position. I´m convinced that you are unaware of this fact.

 

[yogi]

Ich - your point 1 - a clock on a tower at the North pole can be used as one IRF, and a GPS satellite at the same orbital height as a second.

point 2 - you don't need to construct a polygon - the free falling orbiting frame works fine - it can be an ellipse with any eccentricity - it is a perfectly good IRF because for any experiment conducted therein inertia is isotropic

point 3 - I disagree - the situation is changed anytime any clock undergoes acceleration. But it is not the acceleration per se that affects the difference in time when the two clocks are later compared; acceleration is simply a means employed to change the speed between two clocks.

point 4 - I am fully convinced that observers in equivalent inertial frames will measure the apparent rate of the other guys clock to be running slower. But ...the whole post is not about apparent time dilation, it is about what happens when two at rest separated clocks are synchronized in the same frame, and one is put in motion (accelerated) and when it reaches the other clock, it reads less. Read part 4 of the 1905 paper again. In this description of what happens (I think Einstein used the word peculiar) there is only one acceleration - at the beginning. This is why I have consistently maintained over the course of numerous threads, you cannot sync two clocks on the fly - you can read the time of a passing clock by its hands when it is near - but you cannot be guaranteed that the other clock will be running at the same rate. At the end of the one way journey the two clocks read differently - the clock put in motion does not have to slow down at the end to be read - it can be read by the clock which has not moved and a comparison made as it passes by - but that is not true at the beginning - the initial conditions (namely synchronization followed by an acceleration) is vital to the conclusion reached by Einstien - namely, that one clock reads more than the other at the end of the one way voyage

 

[Ich]

1. Yes, that´s two IRFs. But one of those is only locally (and only for a short time) valid and does not include the north pole most of the time. What you try to do is to compare clocks in the two frames, using only SR. That is only possible if both clocks are always covered by both IRFs.
Relativity is a theory of relations (as the name implies). To calculate something, it is not sufficient that both observers are in a local IRF. You have to specify how these IRFs move relative to each other. And SR only deals with linear motion and global IFRs. Everything else is curved spacetime, and therefore you also start with a broken symmtery (see 2.).
2. Same problem. Gravity is not included in SR, and a IRF that travels in circles certainly does need gravity. To calculate "clock rates", you need GR, even if you can use weak field approximations. If you trie to get around that, you only confuse yourself. Why don´t you calculate circular motion the way SR allows it?
3. You disagree with what? I only said that the situation is symmetric until clock 2 accelerates. At this point you decide in which frame to compare clocks and which clock "really" loses time. Note: the decision is made as the last step of the experiment. Initial acceleration has nothing to do with it. Do you agree with that?
4.

the initial conditions (namely synchronization followed by an acceleration) is vital to the conclusion reached by Einstien - namely, that one clock reads more than the other at the end of the one way voyage

No, it is not. You can: a) sync two separated clocks at rest. b) sync a moving clock on the fly with the first clock when they meet. c) compare its reading with the second clock when they meet. The outcome is the same, and no acceleration is involved.

you cannot sync two clocks on the fly - you can read the time of a passing clock by its hands when it is near - but you cannot be guaranteed that the other clock will be running at the same rate.

Just to be sure we speak of the same thing: "sync" means that you set the clock to a certain value, it has nothing to do with clock rates.

 

[Hurkyl]

2. It is still possible and instructive to examine the problem in SR if you treat the satellite´s orbit as a polygon.

Minor nitpick -- "thou shalt not use calculus" is not one of the postulates of SR. There is no reason to restrict one's self to polygonal curves aside from computational simplicity.

 

[Ich]

No, but it is what Einstein proposed in the paper yogi is referring to, and you can learn a lot when you calculate the effect of small translations and rotations. The result is the same, but you can see how it is achieved.

 

[yogi]

Ich - We are using the word sync differently - both definitions are valid - I am using it mean: "to render synchronous in operation" That is why I am claiming, in general, two clocks cannot be synchronized "on the fly" They must both be in the same reference frame (at rest wrt each other). When one is accelerated after using Einstein synchronization, I claim they will not be running in sync thereafter.

 

[yogi]

Post 44 - item 1. Exactly - each clock is a valid local IRF. If the orbit bothers you - forget it - consider the case where two spaced apart clocks are synchronized (using my definition that they are running at the same rate) and one is later accelerated to a uniform velocity v wrt the other. I claim the two clocks are no longer in sync and that to this extent the fames are no longer equivalent. Inertial experiments are the same when carried out in each frame, but the clock in the frame that was put into motion runs at a slower rate than the clock which remained in the rest frame in which the clocks were originally synchronized. I am aware there is an alternative explanation of why the two clocks do not read the same when the moving clock reaches the stationary clock - but I think it is fallacious

 

[Ich]

#47: good to know; I already got the impression that we´re not talking the same things. I suggest that we use the definition I gave, because it´s highly unusual to fiddle with clock rates. From Wikipedia: The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom. There is nothing to tune.

 

[Ich]

#48: Wonderful, a precisely stated claim. Let´s discuss it.

To avoid misunderstandings. we´re talking about SR here and not some personal theory how things should be.

You claim that the accelerated clock ticks slower than the one staying "at rest". That means, the amount of time lag is proportional to the time you let the clocks fly until you compare them.
How do you deal then with the following prediction of SR:
If you accelerate the second clock to match the velocity of the first clock immediatly before you compare them, the second clock will read less exactly the amount that you think the first one would.
What SR says is that moving frames are equivalent (regardless which one accelerated) until you decide in which frame you want to compare the clocks. That contradicts your claim.