Help Understand SR - Steve's Questions

[stovepipe]

Hi everyone,

I recently decided that I wanted to the understand the concepts and applications of SR better. I've been reading a lot about it, but it gets confusing at points. For now I'm only concerned with SR and not GR. Here are some examples that I'm sure many of you would find trivial, and I want to know if I'm understanding them correctly.


-The twin paradox.
Two clocks are synchronized in frame A. Clock 1 stays put in frame A. Clock 2 is accelerated to a relativistic velocity, say v=.8c. Due to time dilation Clock 2 should be running at .6 the speed of Clock 1 for the observer in frame A. Here's my confusion. I read that this dilation is independent of acceleration. Yet I consistently read answers to this problem saying that it is the Clock that was accelerated that runs slower. And when I choose to ignore acceleration I see that from the point of view of an observer that is in frame B, in which Clock 2 is at rest after acceleration, Clock 1 is running slower, and I can no longer make a judgment on which clock is truly slower, because now the slower clock depends on which frame you are in. But examples I read always say that Clock 2 is slower.

-Multiple frames/bodies.
Frame B is traveling at .8c to the right with respect to frame A. Frame C is traveling at .8c to the left with respect to frame A. An obsever in frame B measures and object stationary in frame C to be moving away at 0.9756c. After traveling 1ly according to frame A, an observer in frame B makes a measurement of how far he has traveled. Due to length contraction, he measures that he has only traveled .6 ly. The same happens for frames A and C. According to A, an object in B and an object in C have grown appart by 2 ly. I can't seem to figure out how far the objects have grown appart according to frames B or C. Would it be 2 ly times the contraction factor for .9756c or 1.2 ly or something completely different. I'm leaning towards 1.2 ly assuming the measurement was made based on references in A. Now my confusing twist, again involving acceleration. Suppose both the observers in frames B and C rapidly deccelerate to come to rest in frame A. They make the messurements again and find that they are 2 ly appart not 1.2. If they had been trying to measure the distance during the decceleration would they conclude that the other body was accelerating away from them and exceeding the speed of light since it apparently traveled .8 ly in a very short time.

-Doppler effect.
A human would have a hard time navigating a relativistic rocket. If I'm moving towards a violet (400nm reflected wavelength) object at .8c. The reflected wavelength would be red-shifted to appear 1200nm (invisible infrared light). If I'm moving away from a red (750nm reflected wavelength) object at .8c. The reflected wavelength would be blue shifted to appear 250nm (invisible ultraviolet light). So any objects that you are moving towards or away from that reflects only visible light, would now appear invisible. I guess if you were navigating by stars it wouldn't be such a big problem since the infrared or ultraviolet light that they emit would now be visible. One more question. Silicon is opaque to visible light but transparent to infrared light due to its crystal structure. If a crystal of silicon was moving towards the source of a blue light at .8c, would an observer in the lights source's frame see the silicon as transparent to blue light?

Thanks everyone,
-Steve

 

[RLS.Jr]

Relative speed is the key component. The faster clock (clock 2) measures time slower than the slower one (clock 1). Acceleration is important only as it relates to relative speeds. Clock 1 would always appear running faster than Clock 2...according to this theory.

 

[diazona]

Welcome to the forums, Steve!

-The twin paradox.
Two clocks are synchronized in frame A. Clock 1 stays put in frame A. Clock 2 is accelerated to a relativistic velocity, say v=.8c. Due to time dilation Clock 2 should be running at .6 the speed of Clock 1 for the observer in frame A. Here's my confusion. I read that this dilation is independent of acceleration. Yet I consistently read answers to this problem saying that it is the Clock that was accelerated that runs slower. And when I choose to ignore acceleration I see that from the point of view of an observer that is in frame B, in which Clock 2 is at rest after acceleration, Clock 1 is running slower, and I can no longer make a judgment on which clock is truly slower, because now the slower clock depends on which frame you are in. But examples I read always say that Clock 2 is slower.

I'm not sure where you read that the time dilation is independent of acceleration, but I wonder about that. If it's true in some sense, it certainly doesn't apply here. It is definitely the twin who's accelerated that has the slower clock.

Note that the paradox only appears after the second twin, a.k.a. clock 2, comes back to rest with respect to clock 1. While the two clocks are on the first leg of their journey, traveling apart at .8c, you are correct that you can't make a judgment about which clock is truly slower, because it does depend on which frame you're int.

-Multiple frames/bodies.
Frame B is traveling at .8c to the right with respect to frame A. Frame C is traveling at .8c to the left with respect to frame A. An obsever in frame B measures and object stationary in frame C to be moving away at 0.9756c. After traveling 1ly according to frame A, an observer in frame B makes a measurement of how far he has traveled in frame A. Due to length contraction, he measures that he has only traveled .6 ly in frame A. The same happens for frames A and C. According to A, an object in B and an object in C have grown appart by 2 ly. I can't seem to figure out how far the objects have grown appart according to frames B or C. Would it be 2 ly times the contraction factor for .9756c or 1.2 ly or something completely different. I'm leaning towards 1.2 ly assuming the measurement was made based on references in A. Now my confusing twist, again involving acceleration. Suppose both the observers in frames B and C rapidly deccelerate to come to rest in frame A. They make the messurements again and find that they are 2 ly appart not 1.2. If they had been trying to measure the distance during the decceleration would they conclude that the other body was accelerating away from them and exceeding the speed of light since it apparently traveled .8 ly in a very short time.

I inserted some key points in bold there Anyway, first of all, consider the situation after object B has traveled 1 ly as measured in frame A. Objects B and C are then 2 ly apart in frame A. Now, to figure out the distance between B and C in frame B, you need to multiply the distance in frame A by the transformation factor between frame A and frame B, which is .6, since the relative speed between frame A and frame B is .8c.

As for the issue of speed during the deceleration: that's a really interesting point. But it's often said that, although special relativity prevents an object from moving through space at a speed faster than light, it doesn't prevent transformations of spacetime itself from causing objects to separate faster than light. Another way to think about this is that two objects can never pass each other at a speed faster than c, but if you're comparing objects that are spatially separated, (seemingly) weird things can happen. I think that idea applies here.

-Doppler effect.
A human would have a hard time navigating a relativistic rocket. If I'm moving towards a violet (400nm reflected wavelength) object at .8c. The reflected wavelength would be red-shifted to appear 1200nm (invisible infrared light). If I'm moving away from a red (750nm reflected wavelength) object at .8c. The reflected wavelength would be blue shifted to appear 250nm (invisible ultraviolet light). So any objects that you are moving towards or away from that reflects only visible light, would now appear invisible. I guess if you were navigating by stars it wouldn't be such a big problem since the infrared or ultraviolet light that they emit would now be visible. One more question. Silicon is opaque to visible light but transparent to infrared light due to its crystal structure. If a crystal of silicon was moving towards the source of a blue light at .8c, would an observer in the lights source's frame see the silicon as transparent to blue light?

When you say "silicon is opaque to visible light but transparent to infrared light," that's talking about the rest frame of the silicon. After all, all measurements of silicon's transparency have been made in its own rest frame, or very nearly so. Now, in your situation, I'm assuming that you mean the source of light is blue in its own rest frame? In that case, an observer moving away from the source would see the light from the source as infrared. The silicon counts as an observer, so the light passes through it. From the perspective of an observer at rest with respect to the source, the silicon's crystal structure would appear to be compressed in the direction of motion so that instead of being transparent to infrared light, it would be transparent to blue light.

 

[Fredrik]

I'll just quote myself:

Check out #3 and #142 (page 9) in this thread.

The spacetime diagram that I linked to in #3 should be useful.

 

[Fredrik]

-Multiple frames/bodies.
...
I can't seem to figure out how far the objects have grown appart according to frames B or C. Would it be 2 ly times the contraction factor for .9756c or 1.2 ly or something completely different.

Something completey different. A measurement of length determines the the difference between the spatial coordiantes of two events that are simultaneous in the co-moving inertial frame. So if B is supposed to make his measurement at the event (1.25,1) (expressed in A's coordinates), the first thing you need to figure out is where B's simultaneity line through that event intersects C's world line. Then you need to figure out the distance in B's frame between those two events.

It's very useful to know that if an observer is moving with velocity v, the slope of his world line in a spacetime diagram is 1/v and the slope of his simultaneity lines is v.

Suppose both the observers in frames B and C rapidly deccelerate to come to rest in frame A. They make the messurements again and find that they are 2 ly appart not 1.2. If they had been trying to measure the distance during the decceleration would they conclude that the other body was accelerating away from them and exceeding the speed of light since it apparently traveled .8 ly in a very short time.

This is tricky stuff. There's a synchronization convention you can use to construct a coordinate system which we can think of as B's point of view during the deceleration, but things get weird pretty quickly. What event on C's world line is simultaneous with the event on B's world line that you're considering depends on B's motion both in the past and in the future. So you really can't measure the distance during the acceleration. You can only gather data, and later put the pieces together and claim to have performed a measurement. I think the result would be what you suspected, i.e. that we would end up describing C as moving away faster than c. That part isn't so weird actually.

 

[stovepipe]

You guys have been very helpful. Thank you! I especially liked the graphic, it helped a ton. I do have a quick question though.

It's very useful to know that if an observer is moving with velocity v, the slope of his world line in a spacetime diagram is 1/v and the slope of his simultaneity lines is v.

When looking at the diagram it appears that the the slopes of the simultaneity lines (red and blue) for the rocket are .3 which isn't v. I might be confused about what you are calling simultaneity lines.

 

[Fredrik]

When looking at the diagram it appears that the the slopes of the simultaneity lines (red and blue) for the rocket are .3 which isn't v. I might be confused about what you are calling simultaneity lines.

Or maybe about what I'm calling "slope". Yes, the red and blue lines are what I call simultaneity lines. I call them that because all the events on such a line are assigned the same time coordinate by the inertial frame that's co-moving with B at the event where the simultaneity line intersects B's world line. Take the upper blue line for example. It goes from (7.2,0) to (20,16). (I'm writing the time coordinate first, and 16 is 20*0.8, i.e the distance in A's frame from Earth to the turnaround event). So its slope is (20-7.2)/16=0.8.

 

[stovepipe]

Oh, I just made a stupid mistake, haha. I used 12 (B's age) instead of 20 (the time on the axis).

But this diagram still really helped my understanding. The time dilation is only a function of velocity (i.e. when the ship is moving with a constant velocity, A's clock appears to be ticking at .6 of B's clock to observers in frame B, while B's clock appears to be ticking at .6 of A's clock to observers in frame A.) But the actual change in aging comes from the change in simultaneity due to the acceleration. At the beginning of the flight, the acceleration doesn't change the apparent simultaneity because both observers are close to each other and same with the final deceleration, but when the ship turns around the distance between the observers allows it to change.

Three things that I'm curious about now are:

1. What happens during the turn around? I assume their are four main effects for the observer in the rocket. His simultaneity line sweeps from the one position to the other. A clock at rest in frame A at the turn around point would appear to speed up then slow down again. The distance from the Earth to the rocket would appear to expand then contract again. Lastly, light from the Earth would go from red-shifted to blue-shifted.

2. Someone mentioned a using grid of clocks to tell the time in frame A in the forum you linked to. I'm interested in the visual effects of that scenario to the observer on the rocket, but for simplicity only 2 clocks in the grid. When on Earth, the ship clock and the Earth clock read the same. A clock at the turn around reads 16 years behind, since it is 16 ly away, but ticks at the same rate.
The clocks read (0,0,-16) (ship,earth,turn) to the observer on the ship.
After an instantaneous acceleration to .8c, the clock on Earth reads the same, and the clock at the turn around would read 3.2 years behind, since the point that the simultaneity line crosses increased by 12.8 years. But due to length contraction it is now only 9.6 ly away, so it actually reads 3.2 years ahead.
So the clocks read (0,0,3.2) at this point to the observer on the ship. (Is this right? am I seeing the turn around clock's future?)
Just before the rocket turns around, the clocks read (12,-2.4,20). The time on Earth's clock increased by 7.2 but is now 9.6 ly away. The time on clock at the turn increased by 7.2 years but read 12.8 in it's own frame just after the ship accelerated.
After coming to a stop, the clocks read (12,4,20)
After accelerating back towards Earth, the clocks read (12,23.2,20) (Again, I seem to be seeing the future, this time Earth's)
Just before landing, the clocks read (24,40,17.6)
After coming to a stop on Earth, the clocks read (24,40,24)
So correct me if I'm wrong in the above, since this says that if you accelerate towards a distant object, you can see it's future. It also says that a clock you are moving towards will appear tick faster than one that is passing you. One that you're moving away from will appear to tick more slowly, and if you're going fast enough it will even appear to tick backwards, as is the case with this example.

3. If I wanted to compare times for a second non-inertial observer, could I just draw his world line on the same graph and compare where the green lines cross?

Thanks again!
-Steve

 

[v2kkim]

For first item, moving 2 systems are equivalent so we can not tell which one is slower, but accelerating system is different, so I think we should not forget about acceleration. If you got confused oversimplifying the fact by leaving accel.. behind then it is your own problem.

 

[Fredrik]

The time dilation is only a function of velocity (i.e. when the ship is moving with a constant velocity, A's clock appears to be ticking at .6 of B's clock to observers in frame B, while B's clock appears to be ticking at .6 of A's clock to observers in frame A.) But the actual change in aging comes from the change in simultaneity due to the acceleration. At the beginning of the flight, the acceleration doesn't change the apparent simultaneity because both observers are close to each other and same with the final deceleration, but when the ship turns around the distance between the observers allows it to change.

Yes, you got that right. I just want to add a comment about a physical observer's "point of view" and how it relates to what you said, in particular the statement "the time dilation is only a function of velocity".

If we choose to define "the observer's point of view" to be the co-moving inertial frame, then what you said above becomes the "correct" description of this scenario. It's important to be aware that the correct description depends on a choice we've made, and that other choices are possible. The co-moving inertial frame is the simplest coordinate system that covers all of spacetime and also agrees with measurements that the observer performs in his immediate vicinity, but that doesn't mean that we have to think of it as the observer's point of view.

Why does the time dilation in this scenario only depend on the speed? To understand this, we need to know that time dilation (or at least the "standard" type of time dilation) is, by definition, a disagreement between two inertial frames about the difference between the time coordinates of two events that have the same position coordinates in one of the frames. Note that acceleration doesn't even enter into it. The standard type of time dilation only depends on speed, by definition, and the choice we made ensures that all the coordinate systems we'll be talking about are inertial frames, which ensures that we can describe what's happening in terms of two separate effects: a) standard time dilation, and b) the changing slope of the simultaneity lines as the rocket turns around.

So the short version of the answer is "by choice and by definition". This is probably why you've been seeing conflicting answers to that question. I don't mean that other people use different definitions. I mean that they just don't understand these things, because very few people have really taken the time to think these things through.

Let's return to "the observer's point of view". If we drop the requirement that a coordinate system should cover all of spacetime, then there's at least one more simple way to construct a coordinate system that agrees with local measurements. The idea is to construct a coordinate system in exactly the same way that we construct the global inertial frames that we associate with physical observers that always have been, and always will be, moving with constant velocity. This post has some of the details.

Since this construction is exactly the same construction that led to global inertial frames in the first place (the only difference is that the observer we're starting with isn't assumed to be eternally inertial), one could argue that it's more natural to use this coordinate system to represent the observer's point of view. If we do, the description will be different. See this article for more.

1. What happens during the turn around? I assume their are four main effects for the observer in the rocket. His simultaneity line sweeps from the one position to the other. A clock at rest in frame A at the turn around point would appear to speed up then slow down again. The distance from the Earth to the rocket would appear to expand then contract again. Lastly, light from the Earth would go from red-shifted to blue-shifted.

Sounds about right, assuming that we define the observer's point of view at each moment to be the co-moving inertial frame. Note however that there are some subtleties involved in describing the ticking rate of the clock on Earth. Does it just speed up to the normal rate or does it go insanely fast? You should elaborate on this if you explain it to someone other than me.

I don't have time for the rest right now, but maybe the above can help you figure it out on your own.

 

[stovepipe]

Note however that there are some subtleties involved in describing the ticking rate of the clock on Earth. Does it just speed up to the normal rate or does it go insanely fast?

I actually specifically left out what would happen with a clock on Earth, because I didn't know. That's why I described a clock adjacent to the ship in Earth's reference frame, which I would assume would only speed up to normal rate.

I don't have time for the rest right now, but maybe the above can help you figure it out on your own.

That's quite alright, you've definitely been a big help. I think I got my #2 wrong. After working out a different example, I think that I might have a better idea of how to approach it. So you should skip over that. I'm still confused about what to do with multiple non-inertial observers, which I'd like to know. And I started a new thread about simultaneity, when I realized that it confused me. Basically, I realized that while all inertial observers can agree on what the others saw, they don't agree on what happened. So if I were to try to extend that to a non-inertial observer, I'm just even more confused.