Contradiction in Taylor-Wheeler's example of spacetime invariance

[GregAshmore]

In section 1.2 of Taylor and Wheeler's Spacetime Physics, a rocket moves past a laboratory (on Earth). Attached to the rocket is a pin. From that pin a spark is emitted at two locations in the lab, separated by 2 meters. The observer in the rocket measures the elapsed time between the sparks, as does the observer in the lab.

The following data is given in Table 1-3:

Lab: elapsed time = 33.69 nanoseconds, distance = 2.000 meters
Rocket: elapsed time = 33.02 nanoseconds, distance = 0 meters.

It is then shown that the spacetime interval, (time interval)^2 - (distance)^2, is the same in both frames.

Notice that the time interval measured in the rocket frame is less than the time interval measured in the lab frame. That makes sense if one considers the rocket to be moving.

However, the spacetime invariance is calculated with a distance of zero in the rocket frame. This means that the spacetime invariance is calculated with the rocket considered to be at rest. That is the meaning of proper time, as I understand it--the time measured by an observer at rest.

But in that case, the elapsed time measured in the stationary rocket should be greater than the elapsed time measured in the moving lab.

The only way I see to resolve this contradiction is to declare that the motion of the rocket is in some sense absolute with respect to the earth.

Consider how the problem would read if the pin were placed in the lab, and the receivers on the rocket. In that case, the distance in the lab would be zero, and the distance in rocket frame would be 2 meters. If the time readings in each frame are the same as before, then the spacetime interval is not the same in the two frames. Of course, the spacetime intervals will agree if the time measurements are also exchanged. But what physical connection can there be between the elapsed time in each frame and the location of the pin and receivers?

Please understand that I am not arguing against time dilation as seen in the measurements of muons and other particles. I am questioning the ability to consider any frame of reference to be at rest. Indeed, if the muon is considered to be at rest, then the measured time in the lab should be less than the half life of the particle, not greater.

 

[bcrowell]

In both of these examples (the rocket and the muons), you simply seem to have the time dilation effect the wrong way around. A clock runs fastest in a frame in which it's at rest.

 

[chronon]

The observer in the lab will need to syncronise the time measurements at the two positions - effectively two clocks will be required, which the observer in the lab claims are syncronised. However, the observer in the rocket will claim that these two clocks are not syncronised (but that each one is affected by time dilation)

 

[Janus]

Here's what you have to consider. The pin is at a single location in the ship frame. The receivers are at two different locations in the lab frame.

To measure the time elapsed in the the ship frame, you just need a single clock located at the pin. In the lab frame, the most straight forward way is to put a clock at each receiver and synchronize them. The time elasped is then just the difference in the readings on the clocks when the pin causes a spark at each.(You could use a single clock, but then you would have to account for signal propagation delay from receivers and clock)

In the lab frame, the ship takes 33.69 ns to pass between the receivers. Meaning that the difference in the two clock readings when the pin passes them will be 33.69 ns.
The ship clock will be dilated and show 33.02 ns for the same time interval.

Now we examine events from the Ship frame. In this frame, the distance between the two receivers is not 2m but contracted to 1.96m, At the relative speed between lab and ship, means that 33.02 seconds pass on the ship's clock between the passing of the interval. This also means that 32.36 ns will pass on the lab clocks.
However, this does not mean that the difference in the readings of the two clocks in the lab at the time of the pin passing will be 32.36ns. This is because, in the ship frame the two lab clocks are not synchronized and the clock at the second receiver will be 1.33 ns ahead of the first clock. Thus if clock 1 read 0 at the moment the pin passed, according to the ship frame, then clock 2 will read 1.33 ns at that same instant. When the pin passes clock 2 it will read 33.69 ns (32.36+1.33) just like it does in the Lab frame.

Both frames agree as to how much time passes on the ship's clock and what the time difference readings are between the lab clocks. They just don't agree as why.

If you reverse the situation and place the pin in the lab and the receivers in the ship, you will also reverse the measurements; Both frames will agree that 33.02 ns elapses on the lab clock and that the time reading difference between the two ship clocks will be 33.69 ns.

It is perfectly symmetrical, and you cannot say which frame is absolutely moving.

 

[GregAshmore]

In both of these examples (the rocket and the muons), you simply seem to have the time dilation effect the wrong way around. A clock runs fastest in a frame in which it's at rest.

No, I have copied the numbers correctly.

The time measured in the lab is 33.69 ns, the time measured in the rocket is 33.02 ns. The lab clock is therefore running faster than the rocket clock. In the statement of the example, the rocket is said to be moving; hence the smaller time interval.

With the muon, the lab time is 200 micro-seconds. The muon time is 4.5 micro-seconds. That makes sense when one considers the muon to be moving. But if the muon time is proper time, then the muon is at rest, and the lab is moving. Then, the lab time should be less than 4.5 microseconds.

 

[GregAshmore]

The observer in the lab will need to syncronise the time measurements at the two positions - effectively two clocks will be required, which the observer in the lab claims are syncronised. However, the observer in the rocket will claim that these two clocks are not syncronised (but that each one is affected by time dilation)

I'm not sure how this relates to my question.

 

[GregAshmore]

Here's what you have to consider. The pin is at a single location in the ship frame. The receivers are at two different locations in the lab frame.

To measure the time elapsed in the the ship frame, you just need a single clock located at the pin. In the lab frame, the most straight forward way is to put a clock at each receiver and synchronize them. The time elasped is then just the difference in the readings on the clocks when the pin causes a spark at each.(You could use a single clock, but then you would have to account for signal propagation delay from receivers and clock)

In the lab frame, the ship takes 33.69 ns to pass between the receivers. Meaning that the difference in the two clock readings when the pin passes them will be 33.69 ns.
The ship clock will be dilated and show 33.02 ns for the same time interval.

Now we examine events from the Ship frame. In this frame, the distance between the two receivers is not 2m but contracted to 1.96m, At the relative speed between lab and ship, means that 33.02 seconds pass on the ship's clock between the passing of the interval. This also means that 32.36 ns will pass on the lab clocks.
However, this does not mean that the difference in the readings of the two clocks in the lab at the time of the pin passing will be 32.36ns. This is because, in the ship frame the two lab clocks are not synchronized and the clock at the second receiver will be 1.33 ns ahead of the first clock. Thus if clock 1 read 0 at the moment the pin passed, according to the ship frame, then clock 2 will read 1.33 ns at that same instant. When the pin passes clock 2 it will read 33.69 ns (32.36+1.33) just like it does in the Lab frame.

Both frames agree as to how much time passes on the ship's clock and what the time difference readings are between the lab clocks. They just don't agree as why.

If you reverse the situation and place the pin in the lab and the receivers in the ship, you will also reverse the measurements; Both frames will agree that 33.02 ns elapses on the lab clock and that the time reading difference between the two ship clocks will be 33.69 ns.

It is perfectly symmetrical, and you cannot say which frame is absolutely moving.

I appreciate the detailed accounting, but I'm not sure it answers my questions.

Perhaps my problem is that I don't understand the meaning of proper time.

The definition given by Taylor is this:

"We carry our wristwatch at constant velocity from one event to the other one. It is not enough just to pass through the two physical locations--we must pass through the actual events; we must be at each event precisely when it occurs. Then the space separation between the two events is zero for us--they both occur at our location."

The language of this paragraph is torturous. If the two events occur at our location, then it is peculiar to talk about moving at constant velocity between the two physical locations.

After thinking about this for a while, and studying an x-ct diagram in Born, I think the idea of proper time can be stated this way:

The proper time between two events which are spatially separated in frame A is measured by a clock which is at rest in frame B, where frame B moves at just the constant velocity relative to frame A which puts the two events at the same space location in frame B.

That said, I see that my problem is with the symmetry of relativity. How can both frames see the clocks in the other frame running slow?

Suppose that the rocket has 100 pins on it, spaced 1 meter apart. The lab has 100 receivers, spaced 1 meter apart. The pins and receivers are installed while the rocket is at rest in the lab. The spacing is verified by putting all 100 pins in contact with the 100 receivers simultaneously (at rest, of course).

The rocket is brought up to half light speed and streaks through the lab. How will the clocks read as they record the spark events?

Will there be an instant when all 100 pins spark across to the receivers simultaneously?

According to Taylor (and Born), the rocket does not physically contract at the high speed. If so, then there must be an instant when all pins are physically aligned with the receivers, and all pins spark simultaneously. And yet, according to relativity, the sparks cannot be simultaneous in both frames.

If you wish, you may point me in the right direction to figure this out for myself, rather than giving the answer. (That's what I would do if one of my kids asked me such a question.)

 

[Janus]

Suppose that the rocket has 100 pins on it, spaced 1 meter apart. The lab has 100 receivers, spaced 1 meter apart. The pins and receivers are installed while the rocket is at rest in the lab. The spacing is verified by putting all 100 pins in contact with the 100 receivers simultaneously (at rest, of course).

The rocket is brought up to half light speed and streaks through the lab. How will the clocks read as they record the spark events?

Will there be an instant when all 100 pins spark across to the receivers simultaneously?

No

According to Taylor (and Born), the rocket does not physically contract at the high speed. If so, then there must be an instant when all pins are physically aligned with the receivers, and all pins spark simultaneously. And yet, according to relativity, the sparks cannot be simultaneous in both frames.

I think you are misinterpreting what they mean here. Both the lab frame and the rocket frame will measure the other as contracted. (From the Lab frame the pin on the rocket will be 0.866 m apart, and from the Rocket frame the receivers will be 0.866 m apart.)

One of the things to keep in mind with Relativity is that Time dilation, Length contraction, and the Relativity of Simultaneity all work together in concert.

 

[GregAshmore]

No I think you are misinterpreting what they mean here. Both the lab frame and the rocket frame will measure the other as contracted. (From the Lab frame the pin on the rocket will be 0.866 m apart, and from the Rocket frame the receivers will be 0.866 m apart.)

One of the things to keep in mind with Relativity is that Time dilation, Length contraction, and the Relativity of Simultaneity all work together in concert.

Taylor-Wheeler, in Box 3-4:

Whether a free-float clock is at rest or in motion in the frame of the observer is controlled by the observer. You want the clock to be at rest? Move along with it...Therefore the time between its ticks as measured in your frame is determined by your actions. How can your change of motion affect the inner mechanism of a distant clock? It cannot and does not...We conclude that free-float motion does not affect the structure or operation of clocks (or rods).

Born, in VI-5:

The view in the preceding paragraphs does away with the controversy as to whether the contraction is "real" or only "apparent". If we cut a cucumber, the slices will be larger the more obliquely we cut them. It is meaningless to call the various oblique slices "apparent" and call, say, the smallest which we get by slicing perpendicularly to the axis the "real" size.

It is clear from these quotes that the structure of a physical body does not change simply because some other body moves in relation to it.

I build two identical bodies, each with two pins. I set the distance between the bodies so that there is always at least a small gap between the pins, and arrange that a spark will pass from one body to the other whenever two pins are at this smallest approach. I introduce relative motion between the bodies. Neither body actually changes; only its view of the other body changes. Therefore, as the bodies pass there must be an instant at which the two pins are both at the minimum distance. At that one instant, there will be two sparks.

It may be argued that neither party will measure the sparks to be simultaneous. For the moment I will not dispute that. I will simply say that both sets of measurements are wrong.

The disagreement boils down to this, I think. One can argue that reality is established by measurement; whatever we measure is what is real. Or, one can argue that our view of reality is limited, or even distorted, by the limitations of the tools which we use to measure it.

I take the second view. It seems to me that my view is supported by GR, which (according to Born and Wald) allows for real velocities of real objects greater than the speed of light.

 

[George Jones]

I take the second view. It seems to me that my view is supported by GR, which (according to Born and Wald) allows for real velocities of real objects greater than the speed of light.

Speed depends on how distance is defined.

In special relativity (flat spacetime), there is a standard definition of (spatial) distance that can be applied both locally and globally. In other words, this definition of distance applies to nearby objects, and to objects that are far away. Speed is change in distance divided by elapsed time, so this standard definition of distance can be used to calculated speeds of objects that are near and far. Speeds of objects, near and far, calculated in this way always have the speed of light as their speed limit.

The situation in general relativity (curved spacetime) is far different. Because of spacetime curvature, the definition of (spatial) distance used in the flat spacetime of special relativity can only be applied locally, just as the Earth looks flat only locally. This leads to speeds of nearby objects that limited by the the speed of light, but it say nothing about the behaviour of objects that are far away.

Even though the special relativity definition of distance cannot be applied globally in curved spacetime, there is a standard cosmological definition of distance that is used in the Hubble relationships. Strangely, this cosmological definition of distance can be applied to the flat spacetime of special relativity (Milne universe), and when this is done, it produces a definition of distance (for special relativity) that is different than the standard definition of distance in special relativity!

A different definition of distance gives a different concept of speed, since speed is distance over time. This alternative definition of speed, even within the context of special relativity, produces speeds of material objects that are greater than the standard speed of light! In other words, this definition of speed produces, in both cosmology and in special relativity, speeds that are greater than the standard speed of light.

If v is standard speed in special relativity, and V is cosmological "speed" applied to special relativity, then some corresponding values (as fractions of the numerical value of the standard speed of light) are:

  v                   V
0.200                0.203
0.400                0.424
0.600                0.693
0.800                1.10
0.990                2.65

Even though there can be different definitions of spatial distance, there is no ambiguity with respect to the prediction of experimental measurements. One just has to keep in mind what definition is being used.

 

[JesseM]

In section 1.2 of Taylor and Wheeler's Spacetime Physics, a rocket moves past a laboratory (on Earth). Attached to the rocket is a pin. From that pin a spark is emitted at two locations in the lab, separated by 2 meters. The observer in the rocket measures the elapsed time between the sparks, as does the observer in the lab.

The following data is given in Table 1-3:

Lab: elapsed time = 33.69 nanoseconds, distance = 2.000 meters
Rocket: elapsed time = 33.02 nanoseconds, distance = 0 meters.

It is then shown that the spacetime interval, (time interval)^2 - (distance)^2, is the same in both frames.

Notice that the time interval measured in the rocket frame is less than the time interval measured in the lab frame. That makes sense if one considers the rocket to be moving.

However, the spacetime invariance is calculated with a distance of zero in the rocket frame. This means that the spacetime invariance is calculated with the rocket considered to be at rest. That is the meaning of proper time, as I understand it--the time measured by an observer at rest.

You're talking as though it makes sense to "consider the rocket to be moving" or "consider it to be at rest" in some absolute, frame-independent sense. In relativity such an idea is physically meaningless! Velocity is a relative quantity, all you can say is that in the rocket's rest frame the rocket is at rest, and in the lab's rest frame the rocket is moving. Proper time isn't "time measured by an observer at rest" in any absolute sense, since "at rest" has no absolute meaning at all in relativity. Rather, the proper time between two events that occur on the worldline of an inertial object (like the pin) is the time between them in the frame that considers the object to be at rest. More generally, the proper time between two events on the worldline of an arbitrary object (whether it's moving inertially or not) is the time between those events as measured by a physical clock moving along with the object. In the case of the pin, the proper time between sparks is equal to the coordinate time between sparks in the frame where the pin is at rest, which is also equal to the time that would be measured between sparks by a miniature clock riding on the pin.

Consider how the problem would read if the pin were placed in the lab, and the receivers on the rocket. In that case, the distance in the lab would be zero, and the distance in rocket frame would be 2 meters. If the time readings in each frame are the same as before, then the spacetime interval is not the same in the two frames.

But the times couldn't possibly be the same as before, not if the pin was at rest in the lab frame. In this case it would be guaranteed that the time between sparks measured in the rocket's frame would be larger than the time between them measured in the lab's frame, this can be proven directly from the Lorentz transformation which relates the coordinates of an event in one frame to the coordinates of the same event in another.

 

[JesseM]

That said, I see that my problem is with the symmetry of relativity. How can both frames see the clocks in the other frame running slow?

Suppose that the rocket has 100 pins on it, spaced 1 meter apart. The lab has 100 receivers, spaced 1 meter apart. The pins and receivers are installed while the rocket is at rest in the lab. The spacing is verified by putting all 100 pins in contact with the 100 receivers simultaneously (at rest, of course).

The rocket is brought up to half light speed and streaks through the lab. How will the clocks read as they record the spark events?

Will there be an instant when all 100 pins spark across to the receivers simultaneously?

"Simultaneously" has no absolute meaning in relativity either, two events at different locations in space which happen simultaneously in one frame are defined to happen non-simultaneously in another (the relativity of simultaneity which Taylor and Wheeler discuss on pp. 62-63). You might find it helpful to take a look at the illustrations I did for this thread showing two rulers moving at relativistic speeds relative to one another, each with clocks placed at each ruler-marking that are synchronized in the ruler's rest frame. You can see from the diagrams how length contraction, time dilation and the relativity of simultaneity all work together to make it possible for the situation to be completely symmetrical, with each frame saying that the other ruler is contracted and that the clocks on it are slowed-down and out-of-sync, without there being any contradictions in their predictions about local events like what times a given pair of clocks will read at the moment they pass next to one another.