Trying to Understand Relativity (part 7)

I had planned to start off this post by correcting an error I made in the last relativity post.  I had said that a trip across a light year at 75% the speed of light would take 18 months, when actually it would be 16 months.  I was going to go into a lengthy explanation of how to figure this, but a lot has changed since then, and pursuing this mistake would be a waste of time.  You see, I've come to the conclusion that I've gotten completely off the track with this, and I need to approach the problem from a fresh angle.  I'm pretty sure that everything I've said up this point, and every piece of progress that I've thought I made is completely wrong.  So not only have I been making your brains hurt, I've been doing so with gross errors in logic and my ignorance of the concepts I've been trying to deal with.  For this I apologize.

But I'm getting ahead of myself here.  I've really been trying to put some genuine effort into figuring this out lately, and I want to give you some kind of idea where I'm at with this.  The best place to start would be with a comment I left on the last relativity post:

I was doing some actual reading on this today, and I came across an interesting explanation. Imagine you have a clock that works by bouncing a beam of light between two mirrors stacked on top of each other and facing each other. The interval between each bounce is one second. If you started to move the clock to the right, then it would look like the beam of light was traveling at an angle to a stationary observer because the position of one mirror would always be slightly to the right of where the other had been when the beam hits it. So from the stationary observer's perspective the beam of light would be traveling a farther distance between the mirrors. Since the speed of light is constant then the time it takes the light to reach each mirror is actually extended and therefore time becomes distorted. One second becomes 1.3 then 2 then 6 and so on, as the clock is pushed faster to the right. The exact amount of distortion can actually be calculated from the angle that the beam is skewed. But, if you were to run along side the clock, keeping pace with it, the beam would be straight up and down again and bouncing at perfect one second intervals. It's not just a matter of appearances. The distance the beam is traveling is actually relative to the observer.

So see, THIS I understand. It's the best explanation of relativity I've found. For the first time it gives me a perfectly clear idea of not only how but why relativity works. I just have to figure out how this relates to my scenario. Or maybe I need to come up with a new scenario. I don't know.

I was thrilled when I first discovered this illustration.  Finally, an explanation of relativity I understood.  I spend pretty much the entire shift at work that night bouncing that beam of light around between those two mirrors in my head as I went about the business of filling the machines and mopping the floor.  It actually made sense.  I just had to figure out how to apply this to Bob and Ann and I was home free.  

Unfortunately, as always, it turns out that none of this was going to be easy.  A few days later I had some quiet time to really think and dig into the situation.  It wasn't long before I came face to face with another huge problem.  You see, although the above illustration does a great job of explaining how an observer in one frame of reference can see time moving differently in another frame of reference that's in motion relative to his own, it doesn't help me understand how that time distortion affects the motion itself between the two frames.  That's the issue that I think is really at the heart of my thought experiment.  A message I sent to my friend, secondscout, who assured me he has the answers to all this, will demonstrate what I mean: 

Hey, since you're throwing your hat into the ring on this, I figured out something you can help me with. I found a good thought experiment the other day that explained the time dilation effect. I described it in the comments of my last post on my blog. It might be one you're already familiar with. Anyway, I think I have a good idea how and why the clock would look like it's running slower on a ship speeding away from you at near light speed. I'm having a problem now with the speed and distance the ship itself would appear to be going. Would it look like it's going it's actual speed, or would it look like it's traveling slower to the same degree as the time is slowed on board? Either way, I'm coming up with a problem.

I'll explain with a simpler version of my experiment that's closer to the classic twin paradox. Let's say Bob and his mother live in the same house. Bob hops aboard his rocket and takes off at 75% the speed of light. Now, to his mother watching at the window time appears to run slower aboard the ship. For every minute that passes aboard the ship, she sees 90 seconds pass on her clock at home. Now, if Bob looks back at his house, he sees the same thing. From his point of view, it looks like he's standing still and the house is zipping away from him at 75% the speed of light with a slower clock. From a relative standpoint it doesn't matter who's in a ship and who's in a house, it just matters that there's a gap widening between them and the speed that it's widening. It only makes sense to say he's going 75% the speed of light away from the house. His speed has to be calculated from a reference point.

Okay, so after Bob has traveled for a year, he stops. At 75% the speed of light, he's now 9 light months from the house. So when he looks back at the house, I would think it would appear that only three months have passed there since he left. At this point, I'd almost be willing to chalk all this business of slowing clocks and time dilation up to the most elaborate optical illusion of all time, caused simply by the delay in the light's travel time, and go take a nap, if it wasn't for one thing: The Mother.

See, it doesn't work out the same from her point of view. She can't look across the 9 light months and see him as he was 3 months after he left the house, because he wasn't stopped on the 9 light month spot 3 months after he left. He didn't stop there until a year after he left. So, supposedly she shouldn't see him reach that point and stop until 21 months after he's left the house.

But how can this be? They're both traveling at 75% the speed of light relative to each other. When Bob stops, they both stop. When they both look back, shouldn't they both see the same distance and difference between each other? Yet, it seems for the mother that she has to see Bob's speed slowed to the same degree as the time distortion. Meanwhile, Bob has to see the house move away from him at it's actual speed. At one year, he is 9 light months from the house. How could he truly say he was traveling at 75% the speed of light otherwise?

Now, you could try to look at from the opposite point of view. You could say the house was speeding away from Bob, and the mother looked back after a year across the 9 light month distance and saw Bob as he was three months after they parted. But the problem is that Bob stopping the ship is an event that happened at a specific position in time and space. It was that action that stopped the widening of the gap. It seems like everything would have to add up in agreement with exactly when and where it happened.

So, I hope this makes sense? I'm totally stumped. If you can help me out of this one, then I think I can figure the rest out. I think it's all a matter of flipping it around for his approach to House A. I think I was actually closer to the answer in my first couple of posts. If time on the ship appears to slow as it speeds away, shouldn't it appear to go faster as the ship approaches? I don't know. Maybe you can answer that one too.

I'm expecting a reply on this before too long.  I might copy it into the next post, with secondscout's permission, of course.  In the meantime, if anyone sees what I'm talking about and has any ideas, as always your contributions are quite welcome.  As you can see, it's two steps back and maybe half a step forward.  Now I can't even get Bob to leave his house without running into problems, let alone get him over to Ann.  Don't worry, we'll get these two crazy kids together yet.

EDIT: At the risk of making this the longest post ever, I'm going to go ahead with the edit I mentioned below.  Although the concept of relative motion is fundamental to the understanding of relativity, I haven't really delved too deeply into the matter in these posts.  Perhaps in the past I didn't fully appreciate what an important element it was.  Perhaps I took for granted that more people were familiar with the concept.  Either way, I think it's time to spare a few moments for a basic explanation of the concept.

In the comments below, Chanel was confused about how the house could be moving at 75% the speed of light in the scenario I laid out above.  I explained that it was a matter of relative motion.  We put some cars on a highway and I explained how their speeds calculated relative to one another.  If you're going 50 MPH and the car in front of you is traveling at 40 MPH, then from your perspective the car ahead of you is backing up towards you at 10 MPH.

Now, all this might seem needless confusing and a complete waste of time, but the fact of the matter is that the speed of nearly everything with one crucial exception is calculated from a reference point.  Generally we use the Earth itself as a reference point, and since most of our experience is confined to the surface of the Earth, this works out nicely and we never give it another thought.  When we're driving at 50 MPH in our car we never consider that this is 50 MPH relative to the Earth.  We just figure that's the speed we're moving.  But now let's say we stop at red light.  At that point, we figure we're standing still, and someone standing at the side of the road would agree.  They would look at our car and say it's stopped.  However, the Earth itself rotates at about 1,000 MPH give or take.  So, to someone out in space, they would say you're spinning at 1,000 MPH on the surface of the Earth.  The guy standing by the roadside thinks you're stopped because he's spinning at 1,000 MPH along with you.  Now, suppose there was a guy further out in space.  He would say, no, you're spinning at 1,000 miles an hour AND orbiting the sun at 67, 062 MPH.  Standing still indeed!  Now, suppose there was someone even further out in space.  He would say, no, again.  He would say you're spinning at 1,000 MPH, orbiting the sun at 67,062 MPH AND orbiting the galaxy at 447,000 MPH.  Finally, if you asked someone in another galaxy how fast you were moving, they would say you're spinning at 1,000 MPH, orbiting the sun at 67,062 MPH, orbiting the galaxy at 447,000 MPH and all the while you'd also be speeding away from them at 2,250,000 MPH (depending on which galaxy you talk to.)  So you see, the question of how fast you're going depends entirely on the reference point you're figuring your speed from.  In reference to the Earth, you're standing still.  In reference to the sun, you're spinning at 1,000 MPH and orbiting it at 67,062 MPH.  In reference to the center of the galaxy....you get the point.

Here's another example that doesn't involve a trip to deep space.  Let's say you're on a bus.  You're sitting in the back, and your friend is sitting up front.  You throw a tennis ball to your friend.  A guy sitting across the aisle clocks of the speed of this tennis ball, using a radar gun or something.  He gets a result of 5 MPH.  From his point of view, that's how fast the ball is moving.  Okay, now let's say you pass a guy standing on the street just as you throw the ball, and let's say the bus is going 50 MPH.  If he also had a radar gun and he had the slightest interest in knowing how fast the tennis ball was moving, he would get a result of 55MPH.  From his point of view, that's how fast the ball is moving, because the frame of reference that the ball is being throw at 5 MPH is itself moving at 50 MPH, so the results are compounded.

You'll remember though, that I said above that there was one crucial exception to all this.  That's the speed that light travels.  Light stands apart from all these interlocking frames of reference.  It travels at the same speed regardless of who is looking at it.  Let's say that instead of throwing a tennis ball, you shined a flash light up at your friend.  The guy across the aisle would clock the beam as traveling at the speed of light.  The guy standing on the street would also clock it as the speed of light, not the speed of light plus 50 MPH like the tennis ball.  The movement of the bus isn't a factor.  The light travels the same speed regardless.  Hell, even the guy in the other galaxy would agree on the speed.

Well, this created a problem.  Light is so much faster than the speeds we usually deal with (186, 282 miles per second) that it isn't real an everyday problem, but it was a problem nonetheless.  How could light cut through all these frames of relative motion and always end up being the same?  Something had to give.  Einstein came along and figured out that that something was space and time itself.  Why is that?  Well, that brings up back to the clock experiment I mentioned way up at the top of this post.

Fig. 1

The clock (fig. 1) operates by bouncing a beam of light between two mirrors.  The beam of light is represented by the yellow lines going up and down.  For the sake of argument, we'll say that the interval between each bounce is one second.  That would make the mirrors 186,282 miles apart.  That would be one huge friggin' clock, but again, for the sake of argument.  Every time the beam hits the mirror, it makes the clock tick one second.  Up, down, tick, tock.   It's a clock; it measures time; yawn...whatever.

Fig. 2

But, let's say you started to watch the clock move to the right.  Fig. 2 is suppose to represent the clock in motion, but you know how my Paint skills are.  The grey clocks are the same clock as it's moving, alright?  So, since the clock is moving, when the beam of light bounces from the bottom mirror to the top one, the top mirror is no longer in the same spot directly above where the bottom mirror was when the beam of light left it.  During the time the beam of light is traveling from bottom mirror to the top, the top has shifted to the right, because the clock is moving to the right.  From where you stand the light traveling in a zig-zag pattern of angles (fig. 2) which means it's traveling a farther distance.  It's not only covering the distance between the mirrors up and down.  It's also covering the distance that the clock has moved.  If we were dealing with tennis balls, there wouldn't be a problem.  We would just add the speed that the clock is moving to it's bouncing speed, and everything would fall nicely into place.  But light is different.  We still measure it traveling at the same speed.  So, since it's traveling a farther distance, but yet going the same speed, that means it takes longer to travel between the mirrors.  Since the clock is still ticking a second for each bounce, you begin to see it take longer and longer to tick a second as the clock moves faster and faster and the beam of light has to travel a farther distance and a more acute angle to keep up with it.  The clock begins to slowwww dowwwwnnn.  Tiiiiicccccckkkkk.....tooooooocccccckkkkk.  Gradually you notice that the time on the clock is falling behind the time on your watch.  Time is moving slower for the moving clock.

Now, suppose your friend was standing there with you.  They don't feel like just standing there watching the clock, so they run to catch up with it.  As they reach a speed where they're keeping a perfect pace with the clock, they look over and see fig.1 again, because relative to them, the clock is once again standing still.  So the years pass, and you stand there watching your friend and you grow old, but they stay young because time has slowed for them relative to you.

This "relative to you" is the crucial point.  If your friend looks back at you, it seems that you are receding from them.  If you're leaning against another one of these fancy clocks, they'll see the zig-zag effect on your clock and they'll think time is slowed for you.  You see, it's all relative.  So, who's growing the long white beard here?  Well, that brings you to the Twin Paradox.  The problem I propose above is a little different.  

Let's say your friend runs for a year.  They run at speed that will put them at a distance where the light will take 9 months to travel the distance between you after a year.  Then they stop and look back.  Now, you're both standing still relative to each other and time is ticking the same.  The light that reaches your friend has taken nine months to travel from you to the spot they're standing on, so they see you as you were nine months in the past.  But you can't look across that same distance and see your friend as they were nine months in the past, because they weren't stopped on that spot nine months ago.  They were still running with the clock.  This is where I'm stuck.

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