How to resolve the contradiction in twin clocks?

[xinhangshen]

I could be mistaken, but on reading this it sounds as if you have not yet followed through on my suggestion that you try to understand exactly why Einstein said "Time is what a clock measures".

A and B are both measuring time by the distance (or angle) that their ball moves on the rod. Those are their clocks.

Because A and B are separated by .6 light-seconds (still using my example from a few posts back) the only way that A can know what B's clock reads at the moment that A's clock reads one second is to wait until A's clock reads 1.6 seconds so that the light hitting his eyes was emitted from B when A's clock read one second. The theory of special relativity and a respectable amount of experimental evidence tell us that B's clock will read .8 when the light that reaches A when A's clock reads 1.6 leaves B.
(and likewise if we switch A and B - the situation is completely symmetrical).

You are right when you set the speed of the ball of clock B at 1/$\gamma$ m/sec in A's reference frame.

Now if I set the speed of the ball of both clocks to 1 m/sec in A's reference frame, then observer at A will see the position of the ball of clock B is at 1 meter in A's reference frame after 1.6 seconds. Deducting the time for light to travel, it perfectly matches the position of the ball of clock A. Now let's transform the time-space point of the ball of B from A's reference frame to B's reference frame, we will get

(t, xB, yB) => (tB', xB', yB')
xB' = $\gamma$(xB - vtB)
tB' = $\gamma$(tB - vxB/c2)
yB' = yB

if t = 1 sec and $\gamma$ = 1.25, then xB = 0.6c, yB = ut = 1x1 = 1 m which corresponds to

xB' = 0
tB' = t/$\gamma$ = 0.8 second
yB' = 1 m = u'tB' (i.e., u' = 1.25 m/sec)

In this case, you will see the speed of the ball of B is increased by a factor of $\gamma$ while time is decreased by the same factor $\gamma$. Now let's have a look at a real clock. We can only use the position of the pointer of a clock to tell the time, not the time itself. If we are on a rocket and people ask you what time it is now, you will just use the angle of the pointer of your watch to tell the time. If the hourly arm has an angle of 30 degrees, you will say it's 1 o'clock, 60 degrees, 2 o'clock, etc. You will never ask people, "Wait, please tell me the speed of the rocket first as I have to calculate the new speed of the arm before I can tell you the time."

That is, in the physical world, we always use the position to represent time, while the position is the multiplication of speed and time. If the speed increased by a factor $\gamma$ and time decrease by the same factor $\gamma$, the clock will never notice the change. Then, using such a clock, we will never notice any effects of special relativity.

I would like you to rethink of it deeply with a completely open mind!

 

[Nugatory]

Now if I set the speed of the ball of both clocks to 1 m/sec in A's reference frame then...

That is physically impossible if the two clocks are identically constructed. If they are identically constructed, then the speed of B's ball in a frame in which B and his clock are at rest must be the same as the speed of A's ball in a frame in which A is at rest.

 

[pervect]

I really don't understand what you think your point is.

That is, in the physical world, we always use the position to represent time. While the position is the multiplication of speed and time. If the speed increased by a factor γ and time decrease by the same factor γ, the clock will never notice the change. Then, using such a clock, we will never notice any effects of special relativity.

Your statement isn't literally true - we could use a number of things besides position to represent time if we wanted to. In fact it's routine to see clocks with LED readouts that represent time by electronic states. You probalby own several of them.

I'm not sure why you are limiting yourself to saying that only position can possibly represent time. It's not true - it's not a good sign that your argument starts with an untrue point :-(.

Focusing on measuring time only via position makes the problem only slightly more complex but apparently it makes it just complex enough that you can confuse yourself :-(

If you can imagine a radio receiver and an electronic clock, that can encode the current time and broadcast it via a radio signal (such a clock is encoding time by a means other than position, obviously), then we can make the problem so simple that it would be more difficult to get the wrong answer.

We have two observers, A and B, moving by each other. At time T=0, both are at the same place, and we reset both their clocks.

At time T A emits a timestamped radio signal that encodes the message:"Clock A, time=T". At some time k*T, k being the doppler shift factor, B receives the signal and broadcasts a reply, which says. "Received signal from clock A=T. Time of reception B=k*T"

Because of relativity, the doppler factor k for sending a signal from A to B is the same as the doppler factor k' for sending a signal from B to A

Thus, at a time k^2, clock A receives the above signal from B

Using the fact that the speed of light is a constant, "c", knowing that the signal was sent at T and that the echo/retransmission arrived at k^2*T, A concludes that the time in A"s frame at which the rebroadcast occurred is (1+k^2)*T/2, exactly halfway between the time of transmission and the time of reception.

If this isn't immediately obvious, drawing a space-time diagram can help.

This is obviously different than the reading of B's clock, which was k*T. Hence we know that A's clock and B's clock cannot keep time at the same rate. This is independent of exactly HOW we encode time, whether we do it electronically, with an analogue readout, or via any other means.

This is a short outline of the derivation of the Lorentz transforation using Bondi's K-calcululs approach. One source of this is Bondi's book "Relativity and common sense". It's one of the simplest approaches to SR, requiring only high school algebra.

 

[Mentz114]

We can only use the position of the pointer of a clock to tell the time, not the time itself.

What does this mean ?

You fail to understand that relativity is not about clocks, but time itself. We define a clock to be that which measures the proper interval along a worldline, like an odometer measures the spatial interval.

If your ball 'clock' does not measure the proper interval it is not a clock, however you care to present it.

I would like you to rethink of it deeply with a completely open mind!

Your argument is based on the conviction that SR is wrong or irrelevant, together with ignorance of the meaning of SR. I suggest you do some learning before trying to do something that many great minds have failed to do.

 

[ghwellsjr]

I am pretty confused in the following situation:

Two identical clocks moving at a constant speed v from each other in x-direction. If each clock is made up of a ball moving at a constant speed of 1 on a ruler in y-direction, then the position of the ball of a clock is the time of the clock. According to special relativity, y' = y no matter at what speed the two inertial reference frames move away from each other. Thus, the two clocks will always have the same time in both reference frames if they start from the same time at the same position, which contradicts the time conversion formula in the Lorentz Transformation.

Can anybody give me an explanation how to resolve the contradiction?

Since you are not being persuaded by any of the many excellent answers that you have been given, I would like to try a different tact which is to recast your scenario into a much simpler, but maybe equivalent scenario in hopes that it might get down to the core of your issue.

You have constructed a clock defined from the point of view of a stationary ruler and against which a ball moves at a constant speed. The time is read off by the markings on the ruler adjacent to the ball.

Now I would like you to consider a similar clock from the point of a stationary ball and against which a ruler moves at a constant speed. The time is again read off by the markings on the ruler adjacent to the ball.

We are not concerned with the issue of whether these two clocks tick at the same rate, only that they are physically equivalent clocks, working on the same physical mechanism. Agreed?

Now I'd like to consider that we have two of our ruler/ball clocks oriented identically and traveling towards each other at some arbitrary but constant speed. As they pass each other, an observer located at the conjunction of the two balls notes that they display the same time on their respective rulers.

Now the question is: will the observer continue to actually see the times on the two ruler/ball clocks remaining equal to each other? What would Newton say? What do you say? And why?

 

[xinhangshen]

Since you are not being persuaded by any of the many excellent answers that you have been given, I would like to try a different tact which is to recast your scenario into a much simpler, but maybe equivalent scenario in hopes that it might get down to the core of your issue.

You have constructed a clock defined from the point of view of a stationary ruler and against which a ball moves at a constant speed. The time is read off by the markings on the ruler adjacent to the ball.

Now I would like you to consider a similar clock from the point of a stationary ball and against which a ruler moves at a constant speed. The time is again read off by the markings on the ruler adjacent to the ball.

We are not concerned with the issue of whether these two clocks tick at the same rate, only that they are physically equivalent clocks, working on the same physical mechanism. Agreed?

Now I'd like to consider that we have two of our ruler/ball clocks oriented identically and traveling towards each other at some arbitrary but constant speed. As they pass each other, an observer located at the conjunction of the two balls notes that they display the same time on their respective rulers.

Now the question is: will the observer continue to actually see the times on the two ruler/ball clocks remaining equal to each other? What would Newton say? What do you say? And why?

In this situation, the observer standing at the middle of the two clocks will always see the two clocks have exact the same time no matter whether you use special relativity or Newtonian mechanics because of the symmetry.

Here, your clocks are completely equivalent to my clocks in telling time. Actually, Nugatory's circle rod clock is also equivalent to our clocks. Nugatory's clock is actually the traditional mechanical clock. Since special relativity says that the speed of the hourly arm will increase by the factor of $\gamma$ if the clock moves, then the mark pointed by this arm will represents different time when the speed at which the clock moves is different. That means, the time the clock shows is incorrect once the clock moves. Therefore, special relativity leads to the conclusion that all mechanical clocks can't work correctly if they are moving.

 

[Nugatory]

Since special relativitysays that the speed of the hourly arm will increase by the factor of $\gamma$ if it moves, then the mark pointed by this arm will represents different time when its speed is different. That means, they are incorrect once it moves. Therefore, according to special relativity, all mechanical clocks can't work correctly if they are moving.

It's not just mechanical clocks, it's all time-dependent physical processes. Instead of the moving ball or the hands of a clock, we could use the melting of a block of ice or the evaporation of water in a bowl, the progressive decay of a sample of radioactive material...

But I think we may have found the source of your underlying confusion. Special relativity says that the physics must remain consistent whether we say that A is at rest while B is moving at a speed v relative to A; or B is at rest while A is moving in the other direction. Thus, SR does not allow us to say that a clock is right "until it starts moving" - every clock is always moving relative to some observers and at rest relative to others, always.

 

[Dale]

we always use the position to represent time, ...

I would like you to rethink of it deeply with a completely open mind!

I urge you to have an open mind also, particularly keep an open mind as you read through the experimental evidence which supports SR and contradicts Newtonian physics. Your own derivation shows that there are relativistic effects for the "y-axis" clock.

However, I take issue with your statement that we always use position to represent time. It is not correct, and even when it is correct it is always some sort of cyclical position. There is no example that I am aware of where a clock measures time in the way you have described. You are not making a general analysis of clocks here.

 

[xinhangshen]

Now Let us concentrate on resolving the above contradiction. As I mentioned, the clocks used in my thought experiment are just general physical clocks that can be most accurate atomic clocks but just have a special way to display the time (actually you can use the circular traditional display for the clocks as well but need two coordinates: y and z positions). Since we always have y = y' and z = z' in Lorentz Transformation, then we have the contradiction:

if you use the display (i.e. y position for my clocks or y and z positions for circular clocks) as the time in Special Relativity, then the displayed time is an invariant in Lorentz Transformation which contradicts the time conversion formula in Lorentz Transformation;

if you say that the display is not the time in Special Relativity because the ball moving speed or the arm rotating speed has been changed after Lorentz Transformation which makes the displayed time on general physical clocks incorrect in Special Relativity, then the time in Special Relativity becomes mysterious and Special Relativity is no longer a theory of physics.

DaleSpam, could you please give an explanation to resolve the contradiction?

 

[Nugatory]

if you use the display (i.e. y position for my clocks or y and z positions for circular clocks) as the time in Special Relativity, then the displayed time is an invariant in Lorentz Transformation which contradicts the time conversion formula in Lorentz Transformation;

You are (still) confusing two things:
- Proper time, which is invariant and what the position of the clock's hands (or the progress of any physical process: fraction of a radioactive sample that has decayed between two observations, number of oscillations of a cesium atom between two observations, number of my hairs which have turned gray between two observations) measures.
- Coordinate time, which is different for different observers using different coordinate systems (also known as "frames of reference"). The Lorentz transformations describe how to convert one observer's coordinates, including coordinate time, to another observer's coordinates in a way that preserves the laws of physics and especially ensures that the relationship between the position of the hands of the clock and each observers' coordinate time is consistent with the physical process moving the hands of the clock.

Edit: an exercise that you might find helpful would be to state what it means to say that two events at two different locations are simultaneous without using the words "at the same time". Once you do that, you can apply it to the two events "First clock's hand is pointing straight down" and "Second clock's hand is pointing straight down".

 

[harrylin]

Now Let us concentrate on resolving the above contradiction. As I mentioned, the clocks used in my thought experiment are just general physical clocks that can be most accurate atomic clocks but just have a special way to display the time (actually you can use the circular traditional display for the clocks as well but need two coordinates: y and z positions). Since we always have y = y' and z = z' in Lorentz Transformation, then we have the contradiction:

if you use the display (i.e. y position for my clocks or y and z positions for circular clocks) as the time in Special Relativity, then the displayed time is an invariant in Lorentz Transformation which contradicts the time conversion formula in Lorentz Transformation; [..]

That's erroneous, as I explained in post #13 already:
https://www.physicsforums.com/showthread.php?p=4439896

In a nutshell, the moving y positions do not correspond to the fixed y coordinates of the inertial frames of the Lorentz Transformation - it's that simple!

 

[Dale]

As I mentioned, the clocks used in my thought experiment are just general physical clocks

No, they are not. Clocks measure proper time along their worldline. The "clocks" used in your thought experiment do not. They also are not equivalent to any clock I am aware of, certainly they cannot be called "general physical clocks".

if you use the display (i.e. y position for my clocks or y and z positions for circular clocks) as the time in Special Relativity, then the displayed time is an invariant in Lorentz Transformation which contradicts the time conversion formula in Lorentz Transformation;

The measured value is an invariant on any measurement apparatus. For clocks, that means that the displayed time is invariant I.e. proper time is invariant. This is well-known and not at all in contradiction with relativity.

The relativistic effect is that the frame invariant proper time is only equal to the frame variant coordinate time for an inertial frame where the clock is at rest.

DaleSpam, could you please give an explanation to resolve the contradiction?

There is no contradiction.
1) your "clocks" are not clocks
2) proper time is invariant
3) coordinate time is not invariant
4) your "clocks" don't violate any relativistic effects, the velocity of your "clock" is different in different frames, transforming according to the Lorentz transform exactly as it should, as shown above.

 

[nitsuj]

if you say that the display is not the time in Special Relativity because the ball moving speed or the arm rotating speed has been changed after Lorentz Transformation which makes the displayed time on general physical clocks incorrect in Special Relativity,
then the time in Special Relativity becomes mysterious and Special Relativity is no longer a theory of physics.

What's the physical significance of this understanding you have anyways? An observation has no consequence.

Misunderstood or not. The "contradiction" is conceptual, not physical. So your musing is no longer about physics.

SR has a postulate that "builds in" all mechanical physics as it applies to motion.

A ruler in comparative motion is not a "proper" ruler, same goes for the clock.

those two statements are all that need to be said for the above.

 

[Dale]

Here is how to completely work this problem and show that there is no contradiction. Consider a frame where the x velocity of the "xinhangshen clock" is 0 and the y velocity is k (in units where c=1). In this frame the worldline of the "xinhangshen clock" is given by $(t,x,y,z)=(t,0,kt,0)$. The display on the "xinhangshen clock" would read $t_{xinhangshen}=t=y/k$. A physical clock traveling with the "xinhangshen clock" would read $t_{physical}=\sqrt{t^2-(kt)^2}=t \sqrt{1-k^2} = t_{xinhangshen} \sqrt{1-k^2}$. So we see immediately that the proposed "xinhangshen clock" does NOT represent a general physical clock and does NOT keep proper time as a standard physical clock.

Now, if we transform to a primed reference frame moving at velocity v in the x direction wrt the unprimed frame. Then we find that the worldline of the clock is $(t',x',y',z')=\left(\frac{t}{\sqrt{1-v^2}},\frac{vt}{\sqrt{1-v^2}}, k t, 0\right) = (t',t'v, t' k \sqrt{1-v^2},0)$. Thus, at a time t' the clock reads a time $t_{xinhangshen}=y/k=y'/k=t'\sqrt{1-v^2}$. So even though the "xinhangshen clock" does not keep proper time, it still time dilates as expected. Furthermore, the physical clock would read $t_{physical}=\sqrt{t'^2-(t'v)^2-(t'k \sqrt{1-v^2})^2}=t' \sqrt{1-k^2}\sqrt{1-v^2} = t_{xinhangshen} \sqrt{1-k^2}$, so the error between the "xinhangshen clock" and the physical clock is the same in both frames.

 

[JM]

Clocks and Light

I am pretty confused in the following situation:

Two identical clocks moving at a constant speed v from each other in x-direction. If each clock is made up of a ball moving at a constant speed of 1 on a ruler in y-direction, then the position of the ball of a clock is the time of the clock. According to special relativity, y' = y no matter at what speed the two inertial reference frames move away from each other. Thus, the two clocks will always have the same time in both reference frames if they start from the same time at the same position, which contradicts the time conversion formula in the Lorentz Transformation.

Can anybody give me an explanation how to resolve the contradiction?

You assert two identical clocks in relative motion. Einsteins Special Relativity also is based on identical clocks in relative motion. His transforms, and the formula, result from his Light Postulate, which states that the speed of light is independent of the motion of the source. Its the light that causes the time differences given by the time dilation formula. A good example of how this works is given by Feynman in ' Six not-so-easy Pieces' pages 59-63.

 

[ghwellsjr]

What's the physical significance of this understanding you have anyways? An observation has no consequence.

I thought all of physics was based on observations. What do you mean when you say they have no consequence?

Misunderstood or not. The "contradiction" is conceptual, not physical. So your musing is no longer about physics.

SR has a postulate that "builds in" all mechanical physics as it applies to motion.

A ruler in comparative motion is not a "proper" ruler, same goes for the clock.

I thought every ruler measures proper length and every clock measures proper time. What do you mean by these statements?

those two statements are all that need to be said for the above.

 

[Samshorn]

I thought ... every clock measures proper time.

What do you mean by that? The literal readings on an actual clock need not correspond to the proper time along the worldline of the clock. (Actual clocks run fast or slow.) Do you just mean that an ideal clock, i.e., a device constructed in such a way as to read the proper time along its worldline, will read the proper time along its worldline? True enough, but circular. Or do you mean that the temporal state of a clock will progress in proportion to proper time, even though this may not correspond to the literal readings on the clock? If so, then it doesn't need to be a "clock", you could just as well refer to any physical system, and then you would need to say what is meant by "temporal state", which you can't define with reference to elapsed proper time or it is circular.

Telling people that "every clock measures proper time" is not good, because it could only be literally true if we simply defined 'proper time' to be whatever any clock reads, which of course would be utterly incoherent. That's what beginners tend to think you must mean, which totally sends them down the wrong track.

There is a non-circular way of correctly saying what you are probably trying to say, but it's quite a bit more subtle and complicated than just saying "every clock reads proper time".

 

[ghwellsjr]

Telling people that "every clock measures proper time" is not good, because it could only be literally true if we simply defined 'proper time' to be whatever any clock reads, which of course would be utterly incoherent. That's what beginners tend to think you must mean, which totally sends them down the wrong track.

You don't understand--I am a beginner and most of what I've learned is from experts like these:

As Ibix pointed out, every clock measures its own proper time.

In SR and GR there is a well-defined mathematical procedure to calculate proper time for moving objects along trajectoreis through spacetime (as measured by a co-moving clock).

I prefer to define "proper time" mathematically, as a property of a timelike curve in spacetime, and then take one of the axioms that define SR to be "A clock measures the proper time of the curve in spacetime that represents its motion".

Proper time is measured by a single clock and can be used only for events that occur locally, right next to the clock.

The proper time, in both SR and GR, is the time actually measured by a single clock.

So I think you will have to agree, I'm not just sending beginners down the wrong track, I've been sent down there with them.

There is a non-circular way of correctly saying what you are probably trying to say, but it's quite a bit more subtle and complicated than just saying "every clock reads proper time".

So then why don't you say it in a way that all of us beginners can understand?

 

[Dale]

True enough, but circular. ...

There is a non-circular way of correctly saying what you are probably trying to say, but it's quite a bit more subtle and complicated than just saying "every clock reads proper time".

I would be interested to hear it because personally I think the statement you are objecting to is fine.

 

[WannabeNewton]

I agree. This is unequivocally a pointless exercise in semantics. See the following passage from Wald: http://postimg.org/image/aq9amdtkh/

Also see here (from MTW): http://postimg.org/image/s16qrlpnj/

 

[atyy]

I agree. This is unequivocally a pointless exercise in semantics. See the following passage from Wald: http://postimg.org/image/aq9amdtkh/

I think Samshorn has a point that a clock is defined as a device that reads proper time. The theory says such a device can be made since proper time is a coordinate-independent quantity along a worldline.

An example of a "clock" that does not read proper time is a pendulum.

In our modern age, we have the luxury of defining atomic vibrations as clocks, then it is indeed derived, not defined, that those read proper time.

 

[Enigman]

Greetings,
Okay, crash course of SR
All laws of physics hold true in inertial frames- no observer can tell if (s)he is moving.
There is NO correct time, time flows differently for different inertial frames; ie. All observers disagree on the matter of time and length and as far their frame is concerned they are ALL correct.
Now proper time is time measured by a non accelerated clock which pases through both events-this is the closest thing SR has to "correct" time - but pretty irrelavent to this post.
Proper length (not "correct" length -as all observers believe they are correct...) is length measured in the frame where object being measured has zero relative velocity.
Okay coming back to the post :
let the clocks start with 2#u# relative velocity in opposite directions from origin along x axis

For the observer at origin both clocks would show the same time as both have same magnitude of velocity wrt to O viz. #u#. But then if there were to be a similarly synced clock at O it would show a different time as the other clocks (note: magnitude of oscillation would be same ie. y=y' time taken to reach max displ. would differ ie. Time period)

Let's take another observer who is at rest wrt to one of the clock let's say A
To that observer the origin is moving away with velocity #u#
- and hence the time period of clock at O would greater than that clock A. This time period will increase by the same factor that O thinks A -clock has increased by. As for the other clock at let's say B, will have still greater time period as it moves at2#u#

B will have the same oppinion about A.
The calculations have aldready been done by Janus

Appologies; if there are any mistakes I've started SR only in the last weekend (that too by a book which calls Gallileo the father Of modern physics and talks about Einstein in present tense...)

Regards

 

[ghwellsjr]

I think Samshorn has a point that a clock is defined as a device that reads proper time. The theory says such a device can be made since proper time is a coordinate-independent quantity along a worldline.

An example of a "clock" that does not read proper time is a pendulum.

In our modern age, we have the luxury of defining atomic vibrations as clocks, then it is indeed derived, not defined, that those read proper time.

I don't understand your post. First you agree with the idea "that a clock is defined as a device that reads proper time" and then you say that it is "not defined" that clocks "read proper time".

Also, Special Relativity cannot account for gravity and therefore it cannot account for a pendulum clock.

 

[ghwellsjr]

Greetings,
Okay, crash course of SR

You've done quite well for only having started on SR since last weekend. However, there are a few "mistakes".

All laws of physics hold true in inertial frames- no observer can tell if (s)he is moving.
There is NO correct time, time flows differently for different inertial frames; ie. All observers disagree on the matter of time and length and as far their frame is concerned they are ALL correct.
Now proper time is time measured by a non accelerated clock which pases through both events-this is the closest thing SR has to "correct" time - but pretty irrelavent to this post.

This description is related to the spacetime interval of a timelike pair of events and is one example of Proper Time but it completely misses the point of Proper Time which is that an accelerated clock which passes through the same two events will accumulate a different Proper Time. This is the whole point of the so-called Twin Paradox--two clocks start at the same event with the same Proper Time and then take different paths through spacetime (at least one accelerates) and finally end up at the second event with different Proper Times on them.

Proper length (not "correct" length -as all observers believe they are correct...) is length measured in the frame where object being measured has zero relative velocity.

This also is too restrictive. In this situation, the Proper Length is equal to the Coordinate Length but even when the object is moving, its Proper Length can be measured by a ruler that is comoving with it even though both of them are not equal to the Coordinate Length.

Okay coming back to the post :
let the clocks start with 2#u# relative velocity in opposite directions from origin along x axis

For the observer at origin both clocks would show the same time as both have same magnitude of velocity wrt to O viz. #u#. But then if there were to be a similarly synced clock at O it would show a different time as the other clocks (note: magnitude of oscillation would be same ie. y=y' time taken to reach max displ. would differ ie. Time period)

Let's take another observer who is at rest wrt to one of the clock let's say A
To that observer the origin is moving away with velocity #u#
- and hence the time period of clock at O would greater than that clock A. This time period will increase by the same factor that O thinks A -clock has increased by. As for the other clock at let's say B, will have still greater time period as it moves at2#u#

A will not see or measure the speed of B to be 2#u# but something less as determined by the relativistic velocity addition formula (or by applying the Lorentz Transformation process to the different scenarios).

B will have the same oppinion about A.
The calculations have aldready been done by Janus

If you're going to mention something like this, it would be nice if you would provide a link or reference.

Appologies; if there are any mistakes I've started SR only in the last weekend (that too by a book which calls Gallileo the father Of modern physics and talks about Einstein in present tense...)

Regards

 

[Nugatory]

I think Samshorn has a point that a clock is defined as a device that reads proper time. The theory says such a device can be made since proper time is a coordinate-independent quantity along a worldline.

An example of a "clock" that does not read proper time is a pendulum.

In our modern age, we have the luxury of defining atomic vibrations as clocks, then it is indeed derived, not defined, that those read proper time.

I've seen this discussion before, and to me it always comes down to people (myself included) not being completely clear about the distinction between coordinate time along the worldline of an observer at rest at the spatial origin of a coordinate system and proper time along the same worldline. The distinction between the two usually isn't very useful; we generally try to choose coordinate systems in which the value of the time coordinate for an inertial observer following a given worldline is the same as proper time; or equivalently $g_{tt}$ expressed in that coordinate system is equal to 1 along that worldline.

The readings of an ideal clock give us both coordinate time in that coordinate system and proper time; as WbN points out above they're equal so discussing which the clock is measuring is sterile.

On the other hand, a non-ideal clock still provides a perfectly good time coordinate; it labels each point on that worldline with a unique value and with appropriate choice of simultaneity convention will supply a time coordinate for points off that worldline as well. All that's going on is that the imperfections of the clock are encoded in the value of $g_{tt}$ along its inertial worldline - when the metric tensor is expressed in coordinates in which the clock is providing the t coordinate. This can still be a perfectly flat spacetime; the non-unity metric components are compensating for the less than ideally simple choice of coordinates..

 

[PeterDonis]

I think Samshorn has a point that a clock is defined as a device that reads proper time.

In which case the statement "a clock reads proper time" is true. In most discussions of the topic, such as the discussion ghwellsjr was having with nitsuj in this thread, simply stating that is sufficient.

If you really want to get into the nitty-gritty of *how* clocks read proper time, then concerns like those Samshorn raised might be relevant; but Nugatory gave good responses to those concerns, which indicate why, most of the time, just saying "a clock reads proper time along its worldline" is sufficient.

 

[PeterDonis]

a non-ideal clock still provides a perfectly good time coordinate; it labels each point on that worldline with a unique value and with appropriate choice of simultaneity convention will supply a time coordinate for points off that worldline as well. All that's going on is that the imperfections of the clock are encoded in the value of $g_{tt}$ along its inertial worldline - when the metric tensor is expressed in coordinates in which the clock is providing the t coordinate. This can still be a perfectly flat spacetime; the non-unity metric components are compensating for the less than ideally simple choice of coordinates..

Would another way of stating this be to say that the time an ideal clock reads can be used as an affine parameter along its worldline, whereas the time a non-ideal clock reads can't? (The latter can still be used as a time *coordinate*, but since the scaling of the clock varies along its worldline, its reading can't be used as an affine parameter, for which I believe the scaling has to be constant.)

 

[Dale]

The readings of an ideal clock give us both coordinate time in that coordinate system and proper time; as WbN points out above they're equal so discussing which the clock is measuring is sterile.

In my opinion, the distinction in such a case is that the proper time is only defined along the worldline of the clock whereas the coordinate time is defined over the whole coordinate chart. They are equal where both are defined, but since they are defined for different regions of the manifold they are not the same.

 

[Enigman]

Thanks for the corrections. I'll have to admit I haven't quite come to terms with the twin paradox yet - I've read Feynman who simply dismissed me as a cocktail party phillosopher saying symmetry breaks down at acceleration and only inertial frames are relative. How does the acceleration affect the time isn't mentioned in his lecture. (I issued the lectures from library after returning that ancient book of intro to "modern" physics.)
So you are essentially saying that proper time can be measured even in non inertial frames? I am going to look a bit into that for now that and twin paradox.
And I am kicking myself for that line about 2u.
(Also as a clarification I started reading SR from 9th grade
from children's biographies of einstein, documentries and such like. It's only last weekend I read an official text on SR for the first time. Am in btech 1st yr now.)
Regards
P.S. Calculations of Janus are somewhere in the beginning of the post. Sorry about the confusion.

 

[Nugatory]

Would another way of stating this be to say that the time an ideal clock reads can be used as an affine parameter along its worldline, whereas the time a non-ideal clock reads can't? (The latter can still be used as a time *coordinate*, but since the scaling of the clock varies along its worldline, its reading can't be used as an affine parameter, for which I believe the scaling has to be constant.)

I think it's still an affine parameter, just nowhere near as convenient as proper time. The metric coefficient captures the scaling when you remember to express it in the appropriate coordinate system.
(This is a pragmatists's answer, not a mathematician's. If a mathematician says I'm wrong, they're right, but it doesn't stop me from evaluating my line integrals just as I always did).

 

[Nugatory]

So you are essentially saying that proper time can be measured even in non inertial frames? I am going to look a bit into that for now that and twin paradox.

Yes. Indeed all of special relativity works just fine in non-inertial frames as long as the spacetime is flat (If not flat, then there are significant gravitational forces at work and you have to use the methods of general relativity).

There are several reasons why people often haven't realized this:
1) Non-inertial frames require appreciably more complicated math which tends to obscure the basic concepts; so most basic texts use only inertial frames in their examples. It's easy to jump to the conclusion that the inertial frame is a necessary as well as a sufficient condition for applying SR.
2) There aren't that many situations in which considering an SR problem from a non-inertial frame contributes any new insight; so again you don't seem it done very often. (The Rindler solution is one of the more important exceptions, but it's not generally considered an introductory-level problem).
2) Just about every explanation of GR uses the equivalence principle between acceleration and gravitation to introduce GR. It's easy to think that if problems involving gravity require GR, and if there's an equivalence principle between gravity and acceleration, then problems involving acceleration must also require GR. This syllogism is bogus, but awfully tempting.

 

[atyy]

I don't understand your post. First you agree with the idea "that a clock is defined as a device that reads proper time" and then you say that it is "not defined" that clocks "read proper time".

Also, Special Relativity cannot account for gravity and therefore it cannot account for a pendulum clock.

If we define a clock using atomic vibrations, then we because we have equations for how the atom interacts with gravity, we can derive that the clock reads proper time. So given an atomic clock, that a clock reads proper time is derived, rather than put in by hand at the start. This is still not exactly right, since real atomic clocks are not just isolated atoms.

The more traditional way and very proper way of doing things is to define an "ideal clock" as a device that reads proper time. Here atoms are not specified at all in the definition of a clock. Given such an abstract definition, the theory must at least give some assurance that an ideal clock can be built. Typically one says that proper time is coordinate invariant, so it could be the output of a device traveling on the worldline. The argument is also given that since acceleration is absolute in relativity, acceleration can be sensed and corrected for.

Either point of view leads to the same experiemental predictions, so it's just a matter of taste.

Also, Special Relativity cannot account for gravity and therefore it cannot account for a pendulum clock.

Yes, that would be a better example for GR in which clocks still read proper time. An example for SR would be a non-ideal clock like a wristwatch that's been run over by a truck.

@Peter Donis and @Nugatory - yes, I agree with your points.

Thanks for the corrections. I'll have to admit I haven't quite come to terms with the twin paradox yet - I've read Feynman who simply dismissed me as a cocktail party phillosopher saying symmetry breaks down at acceleration and only inertial frames are relative. How does the acceleration affect the time isn't mentioned in his lecture. (I issued the lectures from library after returning that ancient book of intro to "modern" physics.)
So you are essentially saying that proper time can be measured even in non inertial frames? I am going to look a bit into that for now that and twin paradox.
And I am kicking myself for that line about 2u.
(Also as a clarification I started reading SR from 9th grade
from children's biographies of einstein, documentries and such like. It's only last weekend I read an official text on SR for the first time. Am in btech 1st yr now.)
Regards
P.S. Calculations of Janus are somewhere in the beginning of the post. Sorry about the confusion.

In the twin paradox, a clock is defined to be an ideal clock, one which reads proper time. A clock that reads proper time is not "directly" affected by acceleration, in contrast to a pendulum which is "directly" affected by acceleration. Proper time can be read in non-inertial frames, because it is a property of one's trajectory in spacetime. It is the spacetime analogue of distance or the number of rotations a wheel makes when traveling from San Francisco to Los Angeles - that number doesn't depend on whether you use latitute and longitude to describe the path you took in space. The number of rotations the wheel makes depends on the spatial route one took. Similarly, proper time is simply "spacetime distance", and the proper time of the two twins is different because they took different spacetime paths even though they started and ended at the same event.

The nonintuitive thing is that the formula for spatial distance is d²=x²+y² (straight line in space), whereas proper time is T² = -t²+x²+y² (straight line in spacetime), with a minus sign instead of a plus. For curved paths, you use the same formula but cut the line into little pieces which are essentially straight, and add up the results from all the pieces.

 

[PeterDonis]

I think it's still an affine parameter, just nowhere near as convenient as proper time.

It's certainly not as convenient, but the reason I question whether a non-ideal clock's time can be an affine parameter is that affine parameters are supposed to be linearly related to each other. An ideal clock's time, which is just proper time, is an affine parameter; but the point of a non-ideal clock is that its period is not constant, so the time it keeps would not be a linear function of proper time.

If a mathematician says I'm wrong, they're right, but it doesn't stop me from evaluating my line integrals just as I always did).

Yes, I wasn't questioning the fact that, no matter how poorly a non-ideal clock keeps time, you can still set up coordinates and define a metric using its reading as the time coordinate, and use those to do integrals--including the integral that gives proper time. (The practical problem here is that, if you don't know the exact relationship between the non-ideal clock's time and an ideal clock's time, you don't know what the actual metric coefficient $g_{tt}$ should be. But in principle you can always put an ideal clock next to the non-ideal one to find that out.)

 

[nitsuj]

I don't understand your post. First you agree with the idea "that a clock is defined as a device that reads proper time" and then you say that it is "not defined" that clocks "read proper time".

The observation is not part of the physics being observed. So musing over what the clock that has length dependent displays reads while in comparative motion isn't going to yield anything regarding the physical processes of the clock itself. The clock isn't different because it has been observed

A ruler at rest with you is a proper length. Same goes for the clock, I am unsure how else to word it. It is such a blatant point, but was raised to make the distinction between these measuring devices in motion are not the same as when at rest.

for the quoted part remember clocks are not perfect. I'm gunna assume you agree that a clock doesn't "read" proper time at all, it displays it. The variance between the two could be idealized away, even my mechanical watch that loses minutes over days is accurate enough for my scheduling . Sometimes extremely accurate measures of proper time are needed(CERN Neutrino mearement, probably gravity wave detection ect), sometimes it's just a fun distinction to make.

 

[PeterDonis]

A ruler at rest with you is a proper length. Same goes for the clock, I am unsure how else to word it.

Try these wordings:

A clock (an ideal clock, if we need to be precise about that) measures proper time along its worldline.

A ruler (an ideal ruler, if we need to be precise about that) measures proper length in its instantaneous rest frame.

It is such a blatant point, but was raised to make the distinction between these measuring devices in motion are not the same as when at rest.

But the distinction here is in you, not the devices. The clock and ruler don't know that they are "in motion", because motion is relative anyway; they're in motion relative to you, but not relative to themselves.

In other words, the distinction you are making is in the observation, not in the thing observed; but you said that the observation is not part of the thing observed. So it would seem appropriate to choose wording that makes it clear that the distinction is in the observation; but your wording seems to me to obfuscate that issue.