Brad DeLong, in the course of something completely different, suggests that the theory of relativity really isn’t all that hard. At least, if your standard of comparison is quantum mechanics.
He’s completely right, of course. While relativity has a reputation for being intimidatingly difficult, it’s a peculiar kind of difficulty. Coming at the subject without any preparation, you hear all kinds of crazy things about time dilating and space stretching, and it seems all very recondite and baffling. But anyone who studies the subject appreciates that it’s a series of epiphanies: once you get it, you can’t help but wonder what was supposed to be so all-fired difficult about this stuff. Applications can still be very complicated, of course (just as they are in classical mechanics or electrodynamics or whatever), but the basic pillars of the theory are models of clarity.
Quantum mechanics is not like that. The most on-point Feynman quote is this one, from The Character of Physical Law:
There was a time when the newspapers said that only twelve men understood the theory of relativity. I do not believe there ever was such a time. There might have been a time when only one man did, because he was the only guy who caught on, before he wrote his paper. But after people read the paper a lot of people understood the theory of relativity in some way or other, certainly more than twelve. On the other hand, I think I can safely say that nobody understands quantum mechanics.
“Hardness” is not a property that inheres in a theory itself; it’s a statement about the relationship between the theory and the human beings trying to understand it. Quantum mechanics and relativity both seem hard because they feature phenomena that are outside the everyday understanding we grow up with. But for relativity, it’s really just a matter of re-arranging the concepts we already have. Space and time merge into spacetime; clocks behave a bit differently; a rigid background becomes able to move and breathe. Deep, certainly; inscrutable, no.
In the case of quantum mechanics, the sticky step is the measurement process. Unlike in other theories, in quantum mechanics “what we measure” is not the same as “what exists.” This is the source of all the problems (not that recognizing this makes them go away). Our brains have a very tough time separating what we see from what is real; so we keep on talking about the position of the electron, even though quantum mechanics keeps trying to tell us that there’s no such thing.
Everyone who has commented on it says that special relativity is easy. So here is an easy challenge problem:
A transponder is launched along the x axis at a speed v=1/2 starting at x=1 Meter and time 0 in the rest frame. At time t=1 Meter a meter stick passes the origin at the same speed v=1/2 and is synchronized so it reads 0 at x=0. (Yes we also have to synchronize moving meter sticks!) What is the reading on the moving meter stick when the transponder intersects it? The meter stick may be considered of indefinite length. What is the clock reading on the transponder?
Time on clock = 0.866 and length = 1.414m
But it’s this kind of question that makes learning general relativity harder – As I mentioned, I think it is better to start off with the concept of 4-vectors, coordinate transformations and tensors.
Just to clarify
>> concept of 4-vectors, coordinate transformations and tensors.
in a physics-first context.
Pingback: Which is Easier to Understand? | Follow Me Here…
I calculated that the time on the clock and the reading on the meter stick are both Sqrt[3].
The intersection occurs at {x,t}={2,2} in the rest frame. The dilation factor is Sqrt[1-(1/2)^2] = Sqrt[3]/2 and since t=2 that gives Sqrt[3] for the transponder clock reading.
There is an x-t symmetry in the problem and that gives the same result for the reading on the meter stick.
Inertial clocks are fairly intuitive and easy to visualize. Inertial tape-measures are quite unintuitive. It’s not too difficult to solve by equations, or a spacetime diagram, but much more difficult to visualize – say as an animation.
Let me recommend the book “Spacetime Physics” by Taylor and Wheeler. Taylor tested the book out on real students like me 🙂 Students are comfortable with x^2 + y^2 + z^2 = R^2. Things get odd by tossing in time with a different sign. Taylor and Wheeler handle these issues with clarity.
Why quantum mechanics is different from classical mechanics remains an open question. I think quantum mechanics is the result of a collision of the 2 biggest ideas ever in physics: calculus and spacetime. Do the calculus of spacetime _correctly_, and the questions are answered. It is our tools of tensor calculus that have created the fog because they treat all dimensions as the same. Time is not space, and the behavior of differential time is different from differential space.
First, freeze changes in space, then changes in time. Looks like you have a movie, which you do. Each frame has the space stuff frozen. Watch it fast or slow if you like, but one frame follows the other. Movies are the stuff of classical physics.
Now freeze changes in time, then changes in space. Oops, gonna be a challenge to watch a movie if it doesn’t change in time. What do you do in this situation? Take the same pictures taken before, put them all together, and look at them with a bright light. That is superposition, all possible states. If you make a measurement, you get to see one of them.
The 2 limit definition of spacetime calculus is put to use for the function f=q^2, where q is a quaternion in spacetime. When time goes to zero last, you get the expected f’=2q. If space goes to zero last, the most you can figure out is the norm of the derivative, f’=2 q* q. Fun stuff.
http://bit.ly/qm2limitdef
>> The intersection occurs at {x,t}={2,2} in the rest frame.
OK – I think we have lines crossed here – do you mean the point x=0 on the metre stick? and what do you mean by indefiniate length?
If Cusp can send me an email address I will send you a PDF showing the problem and calculation in more detail.
David Park
djmpark@comcast.net
http://home.comcast.net/~djmpark/
I took special relativity in second year of a four year physics course. I was surprised how easy it was and how even when confronted with a number of paradoxes I could work them out. Lately though, I’ve seen articles on Wikipedia that have posted what I think to be false resolutions to new paradoxes. One goes like this: two objects not in relative motion to each other are connected by a taught string. When viewed by an observer in relative motion with respect to them, they undergo an apparent length contraction. The “paradox” presupposes that the empty space or the background thru which the objects move does not contract, and hence, the taught string should snap from the observer’s perspective. I don’t know about you all here but that sounds absurdly wrong to me. Because any taught string would also go under an apparent length contraction and ‘effectively’ pull the objects closer together. This pull would not be apparent to observers traveling on the objects, and it would not actually be a pull that obeys Newton’s laws; that is, length contractions themselves are not Newtonian motions in themselves that obey the “law” that a motion in motion stays in motion. SO if you still don’t believe me, you must thus acknowledge that some force must be applied to the objects to keep them at a fixed distance as measured by the reference observer and this force would be different for different observers. In fact, the force to keep the objects at the same relative distance and break the taut string would be nearly zero for very slow relative motion and rise to infinity for very fast relative motion. All this rests on some sort of fallacy that that space shouldn’t contract. Special Relativity is not about that though. It’s just about measuring lengths and times and how that relates to electromagnetism.
Another paradox, one I came up with. A hole is cut in a sheet of paper and the cut out circle and paper are then taken to opposite ends of a long course and shot at each other relativistically so as they would meet in such a way that the circle fits in the hole at the midpoint of the course. Imagine both moving relativistically at each other along the x axis and their lengths are also oriented along the x axis. A small displacement has been made in the y axis of one and a small restoring non-relativistic velocity component along the y axis has been acquired by it as well. If the situation were Galilean, the circle and the hole would match up at all points at the mid point of their converging courses, and then pass by each other thanks to the small displacement in the y direction. Observers are evenly placed around the hole; their job in the relativistic case is to determine how many points on the circumference of the hole match up with points on the circumference of the cut-out circle. Corresponding observers are on the circle and observers in the rest frame of the experimental course are placed where the expected coincidence will take place.
In the relativistic case, the question is which observers observe coinciding points, the ones at the z axis diameters, the ones at the leading edge of the circle and trailing edge of the hole, and/or the ones at the tailing edge of the circle and leading edge of the hole?
The observers on the the circle would see a too small oval hole to fit through and the observers on the paper rim would see a too small oval circle for coincidence to take place. The observers on the frame of the course would see two ovals, hole and cut out, coincide perfectly if the velocities were equal and opposite.
Now if I said that my answer is that all observers observer coincidence in all reference frames, how do you think I reconcile that with all the various length contractions? Further, if the observers wrote a message on the cut out paper, when could the observers on the paper read it? The answer is different from the purely Newtonian case.
@ el_dhulqarnain: Thanks for the recommendation, I’ll actually look into Townsend’s QM book – it appears my library has a copy of it. Although, I would really like to to try and get through Schutz as I literally dedicated every waking moment of my life trying to get through a differential geometry course and I’d like to use it before I forget it all!
@ Anonymous_Snowboarder: Thanks a bunch, I’ll check it out! I’m always looking for good textbooks!
Pingback: Relativity | On The Web
I could explain it to you but that would take all of the fun out of watching everyone else struggle with such a beautiful creation.