I had something stuck in my mental craw for a while. I have a decent understanding of the basics of Einstein's theory of relativity, especially the "special" version. One of the consequences of it is that time slows down and distances get shorter for objects moving at very high speeds. At first this sounds totally insane (because it is), but once you accept that the speed of light is constant for all observers, always, then it has to follow. The thought experiments that demonstrate this are not that difficult to follow if you're actually willing to sit down and think them through.
A further consequence is that time travel into the future is theoretically possible. If we had a craft that could move close to the speed of light, somebody could get in, zoom out into space for, say, 16 years (their time), and then return to Earth. Since time was moving "normally" here (not slowed down) more than 16 years would have elapsed for us -- let's say, 20 years, for the sake of example -- and thus our rocketperson would have effectively traveled 4 years into the future. Totally trippy, but totally true. (In fact, I believe this is how Ender time travels at the end of Ender's Game, one of the few science fiction/fantasy books I read and enjoyed -- even if the author is kind of a bigot).
So that's all well and good, but here's the part that started bothering me. Speed is a relative concept. In the scenario above, we think of our rocketperson as moving at a very high speed, and that's true, they are, relative to Earth. We could just as easily think of the rocketperson as being still and the Earth moving at a very high speed with respect to them. But if that's the case, then why doesn't time slow down on Earth? How does science know that the rocketperson is supposed to go into the future relative to people on the Earth, rather than vice versa? How does it know?!
This had been bothering me off and on for some time, but I started thinking about it a lot while listening to an episode of Startalk. I was actually going to email the show to see if Neil deGrasse Tyson would talk about it, but before I did this, I thought "maybe Google has an answer for me. Maybe somebody else has thought of this." Somebody has.
In fact, this conundrum has a name, The Twin Paradox, and it is "arguably the most famous thought experiment in relativity theory." Huh, how 'bout that? (Traditionally the problem has been framed by considering two identical twins, one who stays on Earth and one who rockets off into space at a high speed before returning years later. The traveler will be younger than the homebody upon returning.) As I've said before, I have mixed emotions when I think of something interesting and then Google it to find out that people way smarter than me have already thought of the same thing many times over. On the one hand, it sucks because it's not original; on the other it's cool because I came up with the same thing on my own as really smart people. It's a weird sorta of simultaneous rejection and validation.
The article linked above does a really good job of resolving the paradox, but I'm still not completely satisfied. I now understand (I think) how science knows the difference between the traveler and the homebody in this particular example. The travel returns to the reference frame of the other, not vice versa. One way to think of it (I think) is to add a third person who can observe the clocks of both twins from a stationary position in the universe. The clock of the twin on Earth never changes with respect to this third person, the clock of the twin in the spacecraft does when he or she makes the return trip to Earth. If the twin in space never turned around and the Earth somehow shot off into space and caught up with the traveler so that it's clock was moving at the same rate as the traveler, then (I think) the twin on Earth would be younger than the twin in the spacecraft.
But there are still two things that bother me about this problem: (1) What would happen if instead of traveling out into space away from Earth, the traveling twin just orbited Earth at a very high speed before returning -- would he or she be younger then? (2) Why is it the traveling twin specifically who gets younger? I understand there is a difference between the two, but why does this difference manifest itself one way and not the other way?
For (1), I really don't know, but my guess is "no". For (2), I think I will just have to content myself with the answer "because that's how the math works". And by the way, although that's not the most satisfying answer in the world, it's also not a terrible one. There have been many, many times throughout my life in which I tried to conceptualize a bit of mathematics and couldn't really do it, so I just memorized the parts I needed and moved on, and then at random moment in the future, something would click, and it was like "Ah... So that's why that's like that! That's where that formula comes from!" As an example, I went through an entire year of high school calculus (and scored a 4 on the AP Exam, I might add) without really understanding conceptually what the derivative of a function is. This is kinda amazing and kinda impressive in a weird way. I knew how to calculate derivatives, but I didn't really get what they are or why they're important until I got to college and did something I had never done before -- read a calculus textbook.
This brings me to another topic, one I've thought about a lot before and probably even blogged about here before: I wonder if a "concepts driven" approach to math education is the correct one. Back when I was a TA and taught community college, the emphasis was always on the concepts -- teach the concepts, the formulas have no meaning without the concepts, math isn't just rote memorization and manipulating symbols it's about learning concepts. But when I think back over my own experience in learning math, it's almost the exact opposite, especially before I got to college. I memorized almost all the formulas first and then the concepts came, usually at a much later date.
Admittedly, I'm not very well-versed in the research of math education. It could be I'm atypical, and weighing my own experience too heavily as part of a larger group. I'm open to being proven wrong about this, but for now I'm sticking up for memorization and rote learning in mathematics.
Okay, gotta go. We're going to a friend's annual pool party. Last year I jumped into the pool with my phone and wallet in my pocket. The wallet dried out fine, the phone, not so much.
Until next time...
A further consequence is that time travel into the future is theoretically possible. If we had a craft that could move close to the speed of light, somebody could get in, zoom out into space for, say, 16 years (their time), and then return to Earth. Since time was moving "normally" here (not slowed down) more than 16 years would have elapsed for us -- let's say, 20 years, for the sake of example -- and thus our rocketperson would have effectively traveled 4 years into the future. Totally trippy, but totally true. (In fact, I believe this is how Ender time travels at the end of Ender's Game, one of the few science fiction/fantasy books I read and enjoyed -- even if the author is kind of a bigot).
So that's all well and good, but here's the part that started bothering me. Speed is a relative concept. In the scenario above, we think of our rocketperson as moving at a very high speed, and that's true, they are, relative to Earth. We could just as easily think of the rocketperson as being still and the Earth moving at a very high speed with respect to them. But if that's the case, then why doesn't time slow down on Earth? How does science know that the rocketperson is supposed to go into the future relative to people on the Earth, rather than vice versa? How does it know?!
This had been bothering me off and on for some time, but I started thinking about it a lot while listening to an episode of Startalk. I was actually going to email the show to see if Neil deGrasse Tyson would talk about it, but before I did this, I thought "maybe Google has an answer for me. Maybe somebody else has thought of this." Somebody has.
In fact, this conundrum has a name, The Twin Paradox, and it is "arguably the most famous thought experiment in relativity theory." Huh, how 'bout that? (Traditionally the problem has been framed by considering two identical twins, one who stays on Earth and one who rockets off into space at a high speed before returning years later. The traveler will be younger than the homebody upon returning.) As I've said before, I have mixed emotions when I think of something interesting and then Google it to find out that people way smarter than me have already thought of the same thing many times over. On the one hand, it sucks because it's not original; on the other it's cool because I came up with the same thing on my own as really smart people. It's a weird sorta of simultaneous rejection and validation.
The article linked above does a really good job of resolving the paradox, but I'm still not completely satisfied. I now understand (I think) how science knows the difference between the traveler and the homebody in this particular example. The travel returns to the reference frame of the other, not vice versa. One way to think of it (I think) is to add a third person who can observe the clocks of both twins from a stationary position in the universe. The clock of the twin on Earth never changes with respect to this third person, the clock of the twin in the spacecraft does when he or she makes the return trip to Earth. If the twin in space never turned around and the Earth somehow shot off into space and caught up with the traveler so that it's clock was moving at the same rate as the traveler, then (I think) the twin on Earth would be younger than the twin in the spacecraft.
But there are still two things that bother me about this problem: (1) What would happen if instead of traveling out into space away from Earth, the traveling twin just orbited Earth at a very high speed before returning -- would he or she be younger then? (2) Why is it the traveling twin specifically who gets younger? I understand there is a difference between the two, but why does this difference manifest itself one way and not the other way?
For (1), I really don't know, but my guess is "no". For (2), I think I will just have to content myself with the answer "because that's how the math works". And by the way, although that's not the most satisfying answer in the world, it's also not a terrible one. There have been many, many times throughout my life in which I tried to conceptualize a bit of mathematics and couldn't really do it, so I just memorized the parts I needed and moved on, and then at random moment in the future, something would click, and it was like "Ah... So that's why that's like that! That's where that formula comes from!" As an example, I went through an entire year of high school calculus (and scored a 4 on the AP Exam, I might add) without really understanding conceptually what the derivative of a function is. This is kinda amazing and kinda impressive in a weird way. I knew how to calculate derivatives, but I didn't really get what they are or why they're important until I got to college and did something I had never done before -- read a calculus textbook.
This brings me to another topic, one I've thought about a lot before and probably even blogged about here before: I wonder if a "concepts driven" approach to math education is the correct one. Back when I was a TA and taught community college, the emphasis was always on the concepts -- teach the concepts, the formulas have no meaning without the concepts, math isn't just rote memorization and manipulating symbols it's about learning concepts. But when I think back over my own experience in learning math, it's almost the exact opposite, especially before I got to college. I memorized almost all the formulas first and then the concepts came, usually at a much later date.
Admittedly, I'm not very well-versed in the research of math education. It could be I'm atypical, and weighing my own experience too heavily as part of a larger group. I'm open to being proven wrong about this, but for now I'm sticking up for memorization and rote learning in mathematics.
Okay, gotta go. We're going to a friend's annual pool party. Last year I jumped into the pool with my phone and wallet in my pocket. The wallet dried out fine, the phone, not so much.
Until next time...
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