<--- continued from above -->
Now, let's take the pen beside you that is 5 1/2" long in *your* reference frame and the pen on the spacecraft going past you at 86% of c which is 5 1/2" long in *its* reference frame (that of the spacecraft). Each is measured to be 2 3/4" long in the other reference frame.
Now introduce a *second* spaceship that goes past you at 86% of the speed of light, but in the opposite direction as the first. And yes, it also has a pen that is moving along with it and is measured to be 5 1/2" long in the frame of the second spacecraft. So far, not much changes. You see the pen on the second spacecraft as being 2 3/4" long. The second spacecraft sees your pen as 2 3/4" long. Not much different than with the first spacecraft.
But, what does the first spacecraft see concerning the second? This is where things start getting more strange. We first have to ask how fast the second spacecraft is moving in the frame of the first: how fast do they see each other passing by? Now, you might suspect it is 86% + 86% =172% of c. But this isn't possible! NOTHING MOVES FASTER THAN c in any frame. The actual speed they see the other moving past is (.86+.86)/(1+.86*.86) = .9887 c!
Now, in the frame of the first ship, the pen on the *second* ship will be measured to be 5.5*sqrt(1-(.9887)^2 ) = .82". Similarly, the pen on the first will be measured to be .82" from the second ship.
So, which of these lengths is 'correct'? ALL of them are. They can all be used *in the appropriate frame* to do any calculation required to do physics, chemistry, etc. Now, it makes sense in this case to work with the length of the pen in the frame where the pen is at rest (so, from the respective spacecraft where the pens are). But there is NOTHING that is required about that. And, in fact, it is often easier to do calculations from some frames rather than others, depending on the problem. They are all equivalent!
Next, we can go one step up on the strangeness factor.Let's look at clocks. Suppose you have an atomic clock calibrated to give accurate time. On each of the spacecraft, there are also similarly calibrated atomic clocks. These are all clocks that are as accurate as we can get them and are all calibrated to measure via the definition of a second that is typically used. So, let's suppose, for the sake of this example, that you time some process as taking exactly 10 seconds in your frame.
Now, the first spacecraft going past you watches and times this same process with its atomic clock. it measures that same process as taking 20 seconds. Your atomic clock appears to be going slow according to the atomic clock on the spacecraft! The same is true for the second spacecraft going past in the other direction at 86% of c. Both spacecraft see that process as taking 20 seconds and not 10 seconds. This is known as time dilation.
But, now, let's have some process happen on the first spacecraft. A careful measurement is done on that spacecraft and that process is measured to take 10 seconds in the frame of the first spacecraft. What do you see? You might think you would measure it to be 5 seconds, but what you would really measure is that this process would take 20 seconds! Again, the situation is symmetric. Each of you sees the clocks of the other as being slower.
Now, finally, we ask the second spacecraft what it sees of the process that takes 10 seconds on the first. Now, recall that the two spacecraft see each other going past at 98.87% of c. It turns out that the 10 seconds on the first spacecraft will be measured to take 66.8 seconds from the second spacecraft! So, the process that took 10 seconds as measured by the first spacecraft will take 20 seconds as measured by your atomic clocks and will take 66.8 seconds as measured by the atomic clocks on the second spacecraft.
In this, we have had three frames of reference: yours, and two from the two spacecraft. The two spacecraft are going at 86% of the speed of light past you in your frame. Because they are in different frames, they measure different lengths and different time durations. Length and time are relative to the frame of reference. They are NOT absolute. Furthermore, there is not one frame that 'gets it right'. All the frames give equivalent, but different descriptions of what is going on.
Now, let's take the pen beside you that is 5 1/2" long in *your* reference frame and the pen on the spacecraft going past you at 86% of c which is 5 1/2" long in *its* reference frame (that of the spacecraft). Each is measured to be 2 3/4" long in the other reference frame.
Now introduce a *second* spaceship that goes past you at 86% of the speed of light, but in the opposite direction as the first. And yes, it also has a pen that is moving along with it and is measured to be 5 1/2" long in the frame of the second spacecraft. So far, not much changes. You see the pen on the second spacecraft as being 2 3/4" long. The second spacecraft sees your pen as 2 3/4" long. Not much different than with the first spacecraft.
But, what does the first spacecraft see concerning the second? This is where things start getting more strange. We first have to ask how fast the second spacecraft is moving in the frame of the first: how fast do they see each other passing by? Now, you might suspect it is 86% + 86% =172% of c. But this isn't possible! NOTHING MOVES FASTER THAN c in any frame. The actual speed they see the other moving past is (.86+.86)/(1+.86*.86) = .9887 c!
Now, in the frame of the first ship, the pen on the *second* ship will be measured to be 5.5*sqrt(1-(.9887)^2 ) = .82". Similarly, the pen on the first will be measured to be .82" from the second ship.
So, which of these lengths is 'correct'? ALL of them are. They can all be used *in the appropriate frame* to do any calculation required to do physics, chemistry, etc. Now, it makes sense in this case to work with the length of the pen in the frame where the pen is at rest (so, from the respective spacecraft where the pens are). But there is NOTHING that is required about that. And, in fact, it is often easier to do calculations from some frames rather than others, depending on the problem. They are all equivalent!
Next, we can go one step up on the strangeness factor.Let's look at clocks. Suppose you have an atomic clock calibrated to give accurate time. On each of the spacecraft, there are also similarly calibrated atomic clocks. These are all clocks that are as accurate as we can get them and are all calibrated to measure via the definition of a second that is typically used. So, let's suppose, for the sake of this example, that you time some process as taking exactly 10 seconds in your frame.
Now, the first spacecraft going past you watches and times this same process with its atomic clock. it measures that same process as taking 20 seconds. Your atomic clock appears to be going slow according to the atomic clock on the spacecraft! The same is true for the second spacecraft going past in the other direction at 86% of c. Both spacecraft see that process as taking 20 seconds and not 10 seconds. This is known as time dilation.
But, now, let's have some process happen on the first spacecraft. A careful measurement is done on that spacecraft and that process is measured to take 10 seconds in the frame of the first spacecraft. What do you see? You might think you would measure it to be 5 seconds, but what you would really measure is that this process would take 20 seconds! Again, the situation is symmetric. Each of you sees the clocks of the other as being slower.
Now, finally, we ask the second spacecraft what it sees of the process that takes 10 seconds on the first. Now, recall that the two spacecraft see each other going past at 98.87% of c. It turns out that the 10 seconds on the first spacecraft will be measured to take 66.8 seconds from the second spacecraft! So, the process that took 10 seconds as measured by the first spacecraft will take 20 seconds as measured by your atomic clocks and will take 66.8 seconds as measured by the atomic clocks on the second spacecraft.
In this, we have had three frames of reference: yours, and two from the two spacecraft. The two spacecraft are going at 86% of the speed of light past you in your frame. Because they are in different frames, they measure different lengths and different time durations. Length and time are relative to the frame of reference. They are NOT absolute. Furthermore, there is not one frame that 'gets it right'. All the frames give equivalent, but different descriptions of what is going on.
