Is Special Relativity Correct for Observer A?

[Anama Skout]

So let's say you're sitting on the ground (observer A) and there's a car or something that moves at .99c (observer B). (Of course, this is all hypothetical) Now, my interpretation goes that when A is seeing B, the information he gets is distorted, namely due to time dilation, length contraction, ... And so if he wants to get the real length of B he should use $$\ell' = \frac{\ell}{\gamma}\Longrightarrow\ell=\gamma\ell'$$ to know the real length of B. Is this correct? ($\gamma$ is of course the Lorentz factor)

 

[HallsofIvy]

If, when you say "real length", you mean the length as measured by an observer who is at rest in the train's frame of reference, then, yes, that is correct.

 

[harrylin]

So let's say you're sitting on the ground (observer A) and there's a car or something that moves at .99c (observer B). (Of course, this is all hypothetical) Now, my interpretation goes that when A is seeing B, the information he gets is distorted, namely due to time dilation, length contraction, ... And so if he wants to get the real length of B he should use $$\ell' = \frac{\ell}{\gamma}\Longrightarrow\ell=\gamma\ell'$$ to know the real length of B. Is this correct? ($\gamma$ is of course the Lorentz factor)

That is only correct if he believes that his measurement system is "in reality" distorted, and that by incredible chance the ruler of B happens to be undistorted. According to relativity we cannot make such claims, and consequently we cannot know the "real" length of the car B. But we can know the length of the car as measured in rest - thus, you calculated the "rest length" or "proper length" of the car. Without indication of speed (or temperature), "the length" of a car means the proper length (at standard temperature).

 

[Anama Skout]

If, when you say "real length", you mean the length as measured by an observer who is at rest in the train's frame of reference, then, yes, that is correct.

Is it then true to say that the length of the car never changes, but it only appears to us to have changed (and using $\ell=\ell'\gamma$ we can find out the original length)?

 

[harrylin]

Is it then true to say that the length of the car never changes, but it only appears to us to have changed (and using $\ell=\ell'\gamma$ we can find out the original length)?

It depends somewhat on how you define "length changes". People who think that relativistic effects are "only appearances" (but how could that be possible?) are astounded when they hear that Einstein's prediction of a very real difference has proven to be correct:
"if the clock at A is moved with the velocity v along the line AB to B, then on its arrival at B the two clocks no longer synchronize, but the clock moved from A to B lags behind the other which has remained at B by ½tv²/c² [...]. It is at once apparent that this result still holds good if the clock moves from A to B in any polygonal line, and also when the points A and B coincide."
- §4 of http://www.fourmilab.ch/etexts/einstein/specrel/www/

 

[Nugatory]

Is it then true to say that the length of the car never changes, but it only appears to us to have changed (and using $\ell=\ell'\gamma$ we can find out the original length)?

It might be best not to think as if there is one "real length" and the other lengths are somehow less real. The length of the car, which is by definition the distance between the positions of its ends at the same time, is different in different frames because of relativity of simultaneity.

The rest length of the car is defined to be its length in a frame in which the car is not moving. It is a particularly useful concept (if we know it, we can calculate many interesting things about the car and what's going on inside), so when someone says "length" without any further qualification, they just about always mean the rest length. But that doesn't make it any more real.

You could say "Is it then true that the rest length of the car never changes, although the length is different in other frames (and using $\ell=\ell'\gamma$ we can calculate the rest length from the length in a given frame and vice versa)?" and you'd be on pretty solid ground.

 

[15characters]

Is it then true to say that the length of the car never changes, but it only appears to us to have changed (and using $\ell=\ell'\gamma$ we can find out the original length)?

You should learn the spacetime interval formula. You can use it to find the Proper Distance or the Proper time for any event. Proper Distance and Proper time are invariant for observers because the interval does not change.

Timelike (Time - Distance = Interval)

For any timelike event the Proper Distance is = to the Spacetime Interval = Time - Distance . The person with the lowest time will be the one who records 0 distance between events (i.e.) they did not move, the events came to them, and they timed the gap on a wrist watch.

Spacelike (Distance - Time = Interval)

For any space like event the Proper Time is = to the Spacetime Interval = Distance - Time. The person with the lowest Distance will be the one who records 0 Time between the events (i.e. the events where simultaneous)

The measurement of the length of a ruler is a spacelike event. The question of what is happening on the Sun now at this moment relates to spacelike events.
By person I mean frame of reference, and also all the numbers are squared.

 

[15characters]

For example in a 100m race all you need to analyse it is the spacetime interval formula (Interval = Time - Distance)

The interval for any two events is the same for all observers in ANY frame of reference

Spacelike events (Interval = Distance - Time)

The measurement of the length of the track is a spacelike event.

For Earth the track is measured with a ruler with synchronised clocks at either end and the measurement are simultaeneous. Not even light can be in both places at the same time. For Earth the Spacetime Interval will be 100m - 0seconds = 100 interval

The spacetime interval for all observers in any frame of reference will be equal to 100m.

Distance - time = 100 is the same for all observers in all frames.

The distance could be increased to 100 million but then the time between the measurements in their frame of reference would also increase by 100 million such than (100million + 100 - 100million = 100) . So that the spacetime interval is fixed at 100.

If any observer does not move per the Earth's reference frame they will increase the distance between the the measurements. One such individual will of course be the runner - who will claim the Earth's measurement is too long, because although the track is compressed in his view, the Earth's ruler was not synchronised so the track moved between measurements - as Earth's clock's run very slow in his opinion, the track moved a long way for even a small gap in clock synchronisation.

Timelike events (Interval = Time - Distance)

The race itself is timelike, there is a start and a finish separated by a certain amount of time. The lowest possible amount of time for the race is called the Proper Time. It is measured by a clock present at both start and finish (assuming no acceleration).

The runner measures the Proper Time. For the runner the two events are not separated by any distance. Rather he was standing still, and the start and finish line came speeding past him. Separated by a certain amount of time. The Proper Time.

A runner's wristwatch will measure the Proper Time. The runner did not move between the events (ignoring accleration) in the runner's frame of reference he is stationary and the track moved past him.
No observer can measure a lower time than the runner's Proper Time for the race.

Lets say the runner's watch says 8 seconds. Then 8 seconds will be the Spacetime Interval for the runner and for all observers in all frames of reference.

The interval

For the runner distance is 0 so the interval equals the time:

8 seconds - 0 metres = 8 interval

For the spectators, there was a distance between the start and finish. The events did not occur in the same place. For them the Distance between the start and finish was 100m. Therefore they must add 100m distance to the runners measurement of 0 metres along with an appropriate amount of time that the interval will still be 8.

8 seconds (+ 100 seconds - 100 metres) = 8 interval

 

[15characters]

(Spectators on the left runner on the right)

Measurement of track length.

For the spectators the track measurements are taken at a single point in time and two flares are fired simultaneously to prove this.

For the runner, the track is speeding towards him. He sees the spectators first they measuring one end then they fire a flare. A short period passes, during which the tiny track continues to race towards him, then he sees the spectators measuring the other end and then they fire a red flare. For him the the entire track moved between measurements.

The spectators have Proper Distance which is lowest because the track does not move in their frame of reference.

The timing of the race itself. As seen here, for the spectators standing at the start line, the runner moved through time and distance. Whereas for the runner (right diagram) he only moved through time, he did not move through space. Therefore runner has the Proper time which is always the lowest time.

 

[pervect]

Is it then true to say that the length of the car never changes, but it only appears to us to have changed (and using $\ell=\ell'\gamma$ we can find out the original length)?

I would recommend splitting the single concept of "length" into two different concepts. There is the traditional notion of length, which turns out in special relativity NOT to be a property solely of the object, but rather to be a property that depends on both the object, and the observer.

Then there is a different concept of length, called the proper length, which is a property only of the object and independent of the observer.

To my way of thinking, the "proper length" is the sort of length that is the most qualified to be "real", due to it's observer independence. Different people have diffrent notions of what "real" means, so your mileage may vary.

Specifying that proper length is observer-independent and that "length" is observer dependent gets to what I consider to be the heart of the issue, without the philosophical baggage of attempting to define what the word "real" 'really' means.

 

[15characters]

I would recommend splitting the single concept of "length" into two different concepts. There is the traditional notion of length, which turns out in special relativity NOT to be a property solely of the object, but rather to be a property that depends on both the object, and the observer.

Then there is a different concept of length, called the proper length, which is a property only of the object and independent of the observer.

To my way of thinking, the "proper length" is the sort of length that is the most qualified to be "real", due to it's observer independence. Different people have diffrent notions of what "real" means, so your mileage may vary.

Specifying that proper length is observer-independent and that "length" is observer dependent gets to what I consider to be the heart of the issue, without the philosophical baggage of attempting to define what the word "real" 'really' means.

They're all real.

If a rocket ship twice as long as the diameter of the solar system flew past Earth at 0.99xxxxc then it really would be contracted because at the time that its front end was over the moon, the back end would be over somewhere very close, like Earth; which would be impossible if it was not really contracted in our frame of reference.

Of course we would know the proper length was much longer but that would be irrelevant unless it slowed down. We could even prove it is short by briefly fitting a short barn over it etc...

 

[Nugatory]

Now, my interpretation goes that when A is seeing B...

There is one subtlety here that no one has mentioned yet: The word "see" is used in two different ways in relativity problems, and the answer to your question will be different according to what you mean by that word "see".

I mentioned above that the length of B's car in any frame is (more or less by definition) the distance between the position of the nose of the car and the position of the tail of the car at the same time. So we can imagine A measuring the length of B's car by the following procedure: shine a laser beam across the road in front of the car; the beam hits a detector on the other side of the road; when the nose of B's car reaches that point on the road the beam will be broken and the detector will see the loss of signal; start a timer at that moment; eventually the tail of B's car will pass and the beam will be able to cross the road again; we stop the timer as soon as that happens and the detector sees the beam again. Thus, the time registered by the timer is the time that the car blocked the laser beam. Divide that time by by the speed of the car according to A, and you'll have the length of the car in the frame in which is A is at rest. This will be the result calculated by the length contraction formula (and if you think about it, you'll see that it is equivalent to measuring the distance between the two ends of the car at the same time). Usually when we talk about what A "sees", we really mean what he measures, using this or some equivalent procedure.

But if by "see" you mean literally what A actually sees, the image that is formed on the retina of his eyes or that we'd we get if we were to take a photograph as B's car zoomed by... Then we'll generally get a different result. Suppose we painted a bunch of marks on the road, each one meter apart. You'd think we could measure the length of B's car by seeing which marks the nose and tail line up with as the car passes; we just have to watch carefully as B's car passes, or snap a photograph that we can study at our leisure to see exactly which marks line up. The problem with this is that our eyes and photographic film are recording the light that hits them at the same time - but because the distance between the nose and tail of the car is different, the light from the tail of the car started at a different time than the light from the nose of the car, so we aren't seeing the position of the nose and the tail at the same time, and we aren't getting a true measurement of the length.

This might be a good time to put in a plug for http://gamelab.mit.edu/games/a-slower-speed-of-light/