Here's my synopsis of Chapter 5. I'm still hoping that someone will help me out:
1. Everything travels through spacetime at speed c, which is to say at the speed of light. However, light travels only in space, not in time. Everything else does some of its traveling in time. Therefore, we don't get from place to place as fast as light does. In fact, we move so slowly (that is, we use so much of our share of c on traveling in time) that our experience of spacetime is quite different from light's experience -- or what light's experience would be if it were sentient.
2. We can think of distances between two points in space (Fayetteville and Shreveport) or two points in time (5:00 and 6:00). We can also think about two points in spacetime (waking up in bed and breakfast at the kitchen table). If the distance needs to have a direction, then we can think of it as an arrow, or a vector (studiously ignoring mosquitoes as disease vectors). Depending on what's important, we can describe distances in spacetime by as many as three numbers: length, height, width, and passage of time.
3. The spacetime distance between two points is always the same, mathematically speaking; it can be distributed differently, though. So if we consider the spacetime between my great-grandfather at Ellis Island and me, there's the distance in miles between us of 1314.68 or so; then he was at sea level so I'm higher; I don't know how width could be measured; and then there's the time difference of a century. Presumably, there could have been a great burst of light at Ellis Island at just that moment that the ship arrived from Antwerp and the spacetime between that and me now would be identical to the spacetime between Robert Allen Haden and me -- but it would be taken up entirely by space and not by time.
I have no idea what #3 would mean. "So much," as the authors say, "for vectors."
4. Billiard balls colliding will take off with the same amount of energy with which they collided, but in opposite directions -- except for friction, which spoils this story, so we leave it out. All the energy used in any undertaking is conserved: it doesn't get used up or anything. It's still there, just as the water we're now drinking is the same water the dinosaurs drank. Energy can be stored and used and moved around, but it doesn't diminish.
5. This hooks up with the idea of all the movement through spacetime adding up to c. But it's better because it makes a nicer equation with y.
What am I missing?
You are missing the sound effects of my poor little brain exploding.
ReplyDeletelol. Where's Darin? He could help, I bet.
ReplyDeleteI'm sure he could. He's uber smart. I haven't heard much from him lately though. He's been sick and busy I think. May have to find a new mind to pick!
ReplyDeleteSorry for the absence. As Rosie said something got ahold of me a couple weeks ago and still hasnt completely let go. I'm hoping these antibiotics will finally do the trick. Anyway...
ReplyDeleteIn #1, did you mean to say "However, -light- travels only in space, not in time." or is that a bad guess? We know light takes time to travel, we are just now seeing that it happens to travel at the fastest speed that anything can possibly travel (cosmic speed limit). From one of the previous chapters, if light (or anything) could travel faster than c, then we would lose causality.
I can see where some confusion might have been from the author stating that light uses up all of its speed quota. That is not to say that light only uses speed and does not use time. It just means that there is a max amount of speed that can possibly be used by anything, and light uses that max amount.
#2 is pretty tricky, and I'm not sure how well the authors really explained it. I had to look at some other sources. A vector in this context is basically just a distance with a direction attached. While the distance from point-A to point-B is the exact same as the distance between point-B to point-A, the same cannot be said for a vector from point-A to point-B and a vector from point-B to point-A. The direction is reversed, so the vector is different.
The way we measure distance in space (x,y,z) is to take the difference of the coordinates of the two points we want to measure. To measure distance in spacetime, we use the same spacial difference but we have to add in a difference in time as well.
To take this back to the previous post about the observer in bed... There is no difference in the distance in space he moves because he stays at the same location. The difference is wholly in time, and so on the spacetime graph you only see the arrow going up the time axis. If he had moved during that time interval, you would see the arrow going both up the time axis, and across the space axis. To the person looking out their window from a plane, however, the observer in bed has moved in space and so the time arrow must be adjusted to compensate.
#3 -- I didnt understand the authors to be saying that the spacetime distance between two points is always the same, but rather that the spacetime interval for a particular event has to work out to be the same for any observer.
For the example of your grandfather, I think it might be more accurate to use spherical coordinates but I dont want to break out my old math books to plug in any real numbers. You could find the spacial distance between each location described in latitude, longitude, and altitude, and then the difference in time of 100 years. The spacetime interval squared would then be the difference between the spacial distance squared and the product of the time difference squared and the speed of light squared.
I dont completely follow the last part of your example... but I think what the authors were trying to say is that we wanted to find a formula that would give us a single spacetime number that could be agreed upon whether from the perspective of you or from the perspective of someone flying past Earth at the speed of light (or any speed).
One of my main complaints about this chapter was that they whizzed through the mathematical conversions too quickly. I wouldnt have complained about that except for the fact that they felt the need earlier in the book to explain that x squared means x multiplied by x. I just have to work through the equations on paper for myself to really see and understand the conversions.
That does help, thanks.
ReplyDeletePerhaps the authors don't realize that the math in this chapter is more complex than x squared = x times x. Perhaps for them it's about equally complex.
Okay, I now feel much better about the vectors and can move on. Thanks.
I think perhaps I was hindered by the fact that I use the term "vector" a lot with different meandings from this one, and I didn't fully get this usage of the term.
Again, thanks -- big help!
Um, that would be "meanings."
ReplyDeleteWith all of their discussion about the speed of light, I am now convinced that I have no idea what is meant by the term "light year." Thought I knew that. Now, I don't.
ReplyDeleteMy understanding is that one light year is the distance light can travel in a year. So an object 30,000 light years away from us is one for which the light would leave there and reach us 30,000 years later. We could convert that into miles, but the number would be inconveniently big.
ReplyDeleteYes, but if light is still traveling that speed away from me if I'm traveling really fast, then that speed of light only works if everything is still. Since we've learned in this book that speed is totally relative, that messes up the concept of the speed of light in my mind.
ReplyDeleteAnd now that I think about it some more, that's the whole point. The speed of light is NOT relative, it always stays that fast regardless of how fast whatever else is moving. So the whole idea is to harness that property to whatever end we want to use it.
ReplyDelete