Is Relativity Hard?

By Sean Carroll | February 15, 2011 9:15 am

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.

  • Gizelle Janine Vozzo

    I tend to tie quantum mechanics with relativity, just because it seems natural that one really leads you to the other, rather, that was my experence with quantum mecanics and then relativity. It’s nice some one notes that connection just this once.

  • Blunt Instrument

    I once tried to understand quantum mechanics, but I couldn’t find this damn “Hilbert space” where it all seems to happen.

  • Chris

    I would rank the hardness by the math involved. General relativity (you need new symbols and techniques just to get the basics), quantum mechanics (mostly calculus), and special relativity (mostly algebra). I am simplifying but that’s where most people start off.

  • Eugene

    My shortened version :

    GR : you change the equations of motion of “stuff”.

    QM : you change the notion of what you think is “stuff”.

  • Matthew Dodds

    A series of epiphanies, yes that is a great description. I remember the first time I calculated and completely understood a set of Christofell symbols for a given metric. After that, I never quite understood why I thought it was hard.

  • Solomon

    Blunt Instrument @2: I think that’s the point at which you should just say “Fock it” and move on to quantum field theory…

  • Chris J.

    I actually found quantum mechanics near impenetrable and finally after self studying for a while (Using Griffith’s and Shankar’s books) gave up altogether. Just this past December I bought a copy of Schutz’s A First Course Course In General Relativity and have been able to get through the first quarter of it with very few troubles. In fact, it’s the most enjoyment I have ever had using a textbook! So yes, I agree that on some level GR is much easier to understand than QM – if only because I find the math a lot more “elegant” and the assumptions that lead up to it are much more open to intuitive reasoning.

    In fact, my next purchase is going to be a copy of Spacetime and Geometry: An Introduction to General Relativity (I was planning on going: Schutz, Carroll and then Wald. Hopefully), but I was actually wondering if there was any plans for a second edition on the horizon? I’m going to order a copy either way, I just didn’t to shell out the cash and then not long after see that an updated version is coming out! Sorry, I hope that’s not too off topic!

  • Ron

    What’s really hard is understanding why there’s an advertisement for some new-age snake medicine on this page. Quantum Pendant, indeed! (It’s not always there: it shuffled out on a reload.)

  • Vladimir Kalitvianski

    Relativity is not that hard if in classical mechanics some typical examples of “relativity” are considered first. A tree from a distance looks smaller and needs calculations to get the right (proper) size, for example. In relativity such calculations involve time intervals too but the principle is the same: one obtains different raw experimental data in different RFs and needs recalculations to get the “proper” data.

  • Low Math, Meekly Interacting

    Yeah, I thought the deal with GR isn’t so much conceptual, it’s just that solving realistic metrics is incredibly difficult. Of course, calculating the complete electronic structure of ammonia, or the mass of a proton straight from the equations of QCD, is also incredibly difficult, so they say.

    As for the conceptual stuff, I agree. If the fact that all observers, no matter where, what, how, measure the same speed of light (in a vacuum, of course) sinks in, a lot that’s important about relativity (so I’m told) sinks in too. Throw in the equivalence principle, and you’re pretty much the rest of the way, at a pop-sci level of understanding.

    All discussion of QM is strange and metaphorical. One could as justifiably say “weird shit happens in Hilbert Space, and out pops the world you at least think you’re seeing” as “God throws dice where you can’t see them!” It’s all pretty much meaningless. There are these equations, with the most profound symmetries, built at times from the most abstract and unintuitive (to me, anyway) mathematical objects, and they seem to describe nature as precisely as humans could ever hope to measure it.

    But is that really any more hard to get?

  • cormac

    Another difference is that, while surprising, relativity fits firmly within the tradition of ‘classical’ physics, whereas qt clearly doesn’t

  • Cusp

    Relativity is only “hard” because we don’t teach classical mechanics properly.

  • Vladimir Kalitvianski

    In my opinion, classical and quantum mechanics are equally easy to understand if we point out that in any case we study complex objects needing many bits of information. One point on a photo film is not sufficient in both cases. Only many-many points give an idea what the “object” is.

  • spyder

    “Hardness” is not a property that inheres in a theory itself

    You mean “hardness” isn’t a quantitative measure of a particular quality of: metal, rock, elements, etc? I am so easily confused.

  • Jason Dick

    Well, I would say it depends upon whether you’re talking about General Relativity of Special Relativity. Special Relativity, of course, is quite easy. But General Relativity, not so much. I’d rank it as roughly on par with Quantum Mechanics, with no definitive way of saying which is harder.

    The reason I’d rank GR up there is the concept of general covariance, which leads to all sorts of weird behavior that, to a lot of people, doesn’t seem to make any sense. Examples include things which apparently move at faster than the speed of light (as is the case with the usual definition of recession velocity and most of the visible galaxies), or an apparent failure of conservation of energy (which happens for anything that has any sort of pressure in an expanding universe).

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  • Low Math, Meekly Interacting

    If you surf around on the web, you can find fun little movies of gravitational waves, what you might see if you fell into a black hole, etc. These graphical depictions may not have the horsepower to provide total realism, but they are “realistic” in that they’re not metaphorical depictions.

    There’s a fun little app. I have called “Atom in a Box”. This lets you fiddle with quantum numbers, and shows you the probability density of various excited states of an electron in a hydrogen atom. “Superposition” is especially cool, giving you a look of sorts at hybrid orbitals.

    However, if I understand things correctly, these pictures I see are pure fantasy. True, they accurately describe the odds, but the “real” picture at any given time is rather boring: A point. All those beautiful lobed shapes don’t “exist” anywhere. That’s not “really” what’s happening to the electron when I’m not looking at it. In fact, the whole idea that I can “not look” and thus see what the electron is “really” up to when I’m not observing it is an utter contradiction, right?

    Sure, it’s difficult at first to get your head around the fact that while quasar X has only been receding from us for some number of billions of years, it’s really some-or-other-many billions of light years away right now. But, by God, it’s a big ancient glob of stars and whatever’s left of that young galaxy we can see in our telescope is right where we say it is in spacetime. If you can get all the light cones straight in your head, it’ll make sense eventually.

    What the heck am I legitimately supposed to say about a picture of a d orbital? It’s in there somewhere? It’s smeared out? It’s discrete, yet everywhere at once? All of those things might be legitimate so long as you preface each statement with “It’s as if…” We can describe what electrons can be thought of as doing whenever we’re not looking, but taking the square of the absolute value of a (complex) amplitude to even call the odds does seem more than a tad mysterious when it gets down to what’s “actually” going on. I get that the numbers work out, but what, exactly, is being worked out?

  • ZoneSeek

    The premise of Greg Egan’s Incandescence was that relativity’s not that hard, so a neolithic-level insect species figures it out because of their asteroid’s unusual dynamics.

  • Neil

    GRT hard, SRT not.

  • Cusp

    >> GRT hard, SRT not.

    Not really – replace n by g and , by ; and carry on.

  • Albert Zwiestein

    LM, WI: “What the heck am I legitimately supposed to say about a picture of a d orbital? It’s in there somewhere? It’s smeared out? It’s discrete, yet everywhere at once? All of those things might be legitimate so long as you preface each statement with “It’s as if…” We can describe what electrons can be thought of as doing whenever we’re not looking, but taking the square of the absolute value of a (complex) amplitude to even call the odds does seem more than a tad mysterious when it gets down to what’s “actually” going on. I get that the numbers work out, but what, exactly, is being worked out?”

    Tout le monde interprets the wavefunction as a probability distribution.

    But there is another interpretation that was favored by Schroedinger: that the electron, after it is bound in the atom, is decomposed into myriad virtually infinitessimal particles which are physically distributed throughout the atom, and the wavefunction is then interpreted as the actual physical distribution of the electron’s mass/charge.

    Schroedinger said he much preferred changes in excitation to be viewed as deterministic changes in the vibrational properties of the envelope, rather than as a point-like electron jumping from orbit to orbit acausally, and the probability smoke and mirrors.

    A.O. Barut and others have made attempts along these lines, but with limited success. Some would say that the approach is destined to fail. Others say that this highly intuitive approach, with its pictorial/conceptual potential and lack of black-box magic, has never been adequately explored.

    Bottom line: A new physics that unifies GR and QM in a way that makes both of them equally easy to understand, at least conceptually, is quite possible. But before physicists can make their own “jump” into the new paradigm, they will need to admit that some of their most cherished assumptions are only limited approximations, such as: differentiability, reversibility, strict reductionism, non-dissipative systems, and all the other Platonic fantasies that are holding us back.

    Albert Zwiestein

  • Elliot Tarabour

    “Hard” is the operative word. I think we need to separate “conceptually hard” from “computationally hard.


  • Bee

    Relativity is “hard” because there’s like one million books full of confusing stories about spaceships and lasers and somebody observing somebody’s something, which is all completely irrelevant decoration. As a teenager I read a whole stack of these books and failed to make much sense out of them because one starts asking all sorts of questions about the construction of clocks and what it means to actually ‘see’ something etc. Then, hallelujah, somebody handed me a book in which it said the Poincaré-group is the symmetry group of Minkowski-space.

    Yes, I know, you’ve written one of these math-less books. I’m just saying I believe that pop sci shouldn’t shy away from mathematical definitions and equations to accompany the blabla. We all know the math is clearer and sometimes I think lack of it doesn’t only give the average reader a completely wrong impression about theoretical physics, it actually makes matters more complicated.

  • The Numbers

    Yes, but with a Complete Set of Commuting Observables I can completely describe a state. What QM tells us is that it is nonsensical to talk of position and momentum as absolutes, instead we must talk of states. QFT tells us that we can only talk about input and output particles and the states of those particles. These things are not hard.

  • AJKamper

    As someone who qualitatively understands a heck of a lot, but hasn’t bothered to learn the math and isn’t likely to, I can’t say that I’d be in the faintest way pleased to learn that the Minkowski-group is the symmetry space of the Poincare group. I topped out at Calc II in HS, and the underlying foundations that would have to be learned for me to make heads or tails of most mathematical equations wouldn’t exactly be satisfying.

    As for what’s “harder,” I find special relativity totally nonintuitive. I mean, I still “get it,” in that once I conceive of c being the same in all frames, I can intellectually see why length shrinks and time dilates. But I haven’t really internalized it; every time I envision it I have to think through the individual steps each time.

    QM, on the other hand, I feel like I get pretty well, because I decided long ago that “wave/particle duality” is simply our failure to properly perceive the underlying probabilistic nature of reality. Discovering that there are wave equations for everything that “entangle” or whatever only reinforced that. It’s quite possible I’m getting something horribly wrong, but it’s not difficult for me to conceive of “things” as bounded by no more than their likelihood of being in that spot at that time.

    But if I had to work with the MATH of it? Good gracious, no please.

  • Doug

    @Zwiestein: Schro’s interpretation of the wave function is a lovely interpretation for electrons, but for any multiple particle states you need to live in a Hilbert space equipped with tensor products, and cannot put the distribution over configuration space.

    It seems like relics of Schro’s style have survived in the popular press too. An image of the electron as smeared *in space* is better than thinking of it as a misbehaving point particle, but it’s not quite enough.

  • AnotherSean

    General Relativity is hard for the same reason that it took Enstein so long to develop it. People try and understand it without knowing the mathematics. I remember my undergraduate quantum class, I had no problem with it. I could do all the calculations, no sweat. But I understood nothing. I look back on it, and part of the problem was the insistence on thinking the wave function “collapsed”. This was encouraged by the text book and the professor, and as you can tell, I remain quite bitter about the experience!

  • David George

    There is an electron, it is not a point particle, it has a position, but when you measure it you bump it out of position. The problem is that once you eggheads learn the current jargon and the math you figure you understand it, stop thinking about what is really going on, and forget what Feynman said (above) and (elsewhere): “No one has found any machinery. . . . We have no ideas about a more basic mechanism. . . .”

  • Chris Duston

    When I was a physics undergrad at the University of Massachusetts, I was interviewed by the school newspaper. They asked me what my hardest final was going to be that semester and I said “Quantum Mechanics.” The interviewer asked me why, and my reply was “Because it’s quantum mechanics”. It was published (my first publication?), and the result was my professor (William Gerace, retired now) put a question on the final, “Why does Chris think quantum mechanics is hard?” I think my solution to the problem was similarly circular.

  • ChuckWhite

    Thanks for that. Now I’m comfortable knowing what I don’t (can’t?) know … {grin}

  • TimG

    It’s certainly true that the nature of measurement in quantum mechanics is hard to understand, perhaps so much so that no one really *does* understand it. But I find there are other things that people *think* of as quantum weirdness that really aren’t much different than what one sees in classical wave mechanics. Superposition, for instance. Even the uncertainty principle has an analog in classical wave packets, where one must superpose waves of increasingly many wavelengths to reduce the width of the packet in space (as one can see from the Fourier transform). Of course the connection between wavelength and momentum is from QM.

  • jackd

    AJKamper @25 – I’m in a similar situation to you, probably even worse with respect to math. But much as I enjoy reading popularized accounts of relativity and QM, I constantly remind myself that it’s ridiculous to say I _understand_ them in any significant way. Being able to work the appropriate mathematics may not be sufficient to understanding physics, but I would bet a large sum that it’s necessary.

  • rob

    all i know is that:

    In Hilbert Space no one can hear you scream.

  • The Cosmist

    If anyone reading this is interested in learning GR by self-study, I highly recommend this course: There are video lectures and problems, and the professor is *excellent*.

    In general, I’m finding GR to be challenging but much more satisfying than QM. GR was created in Einstein’s head without experimental evidence, so it is by definition more comprehensible to human minds than QM. No one would have imagined something as bizarre as QM unless they were forced to by strange experimental results. QM is something from an alien Lovecraftian universe, whereas GR are laws of nature for a universe that at least *might* have been created by a rational God!

  • Cusp

    From Bee >> which is all completely irrelevant decoration.

    I could not agree more!! But I also feel that the flip side is true – Slightly more advanced textbooks are to “maths-first” with physics added right at the end (I have an ex-student who did GR in maths and didn’t understand a single bit of physics at the end of it).

    So – I made a decision to use Hartle’s book for my GR course (but mix it up a bit with most of the maths concepts) with a physics-first (not decoration first though) approach. I *think* students finish up with a workable knowledge of GR.

  • TimG

    @rob: That’s because an unbounded scream with definite momentum is not normalizable. Try a rigged Hilbert space instead.

  • Low Math, Meekly Interacting

    While I’m sure knowing all about the Poincaré group is truly necessary to grasp the significance of Lorenz invariance and the beauty of Very Important Things like Noether’s Theorem, can we glean much of any of that from good pictorial descriptions of translations in Minkowski Space? I’m afraid if I have to learn group theory I may be permanently out of luck. The Road to Reality (the book) has taught me that life’s too short, I’m afraid.

  • here

    Some lectures by Leonard Susskind are up on youtube. They cover various subjects and don’t skimp on the math, but present the minimum necessary to see what’s going on. They can provide a nice jumping off point from pop. science coverage for folks with some maths background.

  • Curious Wavefunction

    The other thing of course is that the mathematics of special relativity is far simpler than that of quantum mechanics or general relativity.

  • Cusp

    >> The other thing of course is that the mathematics of special relativity is far simpler than that of quantum mechanics or general relativity.

    This is a curious statement – I think part of the problem is that SE is taught with one sort of maths (a little algebra and calculus) where as GR is another (4-vectors and tensors etc). If we taught SR properly (with 4-vectors and tensors), deal with SR in non-Minkowski coordinates (still flat space-time) so things like Christoffel symbols are non-zero and we start using covariant derivatives early, then the transition to GR is easier.

    As Bee said – a lot of SR teaching is window dressing.

  • el_dhulqarnain

    @Chris J. post #7.

    It’s disappointing to hear that you gave up on learning QM, especially as you’ve shown remarkable tenacity in pursuing GR.

    A glance at your reading material might suggest why you found QM difficult to self-learn. Griffiths has a truly excellent E & M book. Truly magnificent, a gem of pedagogy. But I don’t think his QM book is anywhere near that.

    As an introductory text, I’d strongly recommend you pick up ‘A Modern Approach to Quantum Mechanics’ by John S. Townsend. It begins quite simply with Stern-Gerlach two-state systems, illustrating all (most) of the nuances of the new mechanics. I can’t recommend it enough. You’ll see at once that the mathematics and the physics illuminate each other.

    Unfortunately, I don’t have a particular graduate-level QM text to recommend for self-learning. Le Bellac’s book ‘Quantum Physics’ comes to mind, as well as Gottfried and Yan’s ‘Quantum Mechanics: Fundamentals’; I recommend these because I find books like Shankar’s are somewhat outdated. There have been many recent developments in quantum mechanics, and I find Le Bellac’s and Yan’s books capture these developments much better.

    The recent paper ‘Graduate Quantum Mechanics Reform’ , Carr L.D. , McKagan, S.B., American Journal of Physics, v. 77, p. 308 (2009) does a good job of chronicling the various ‘epochs’ of quantum mechanics and recommends what a graduate syllabus should cover. You’ll find that books like Shankar’s are a bit light on the modern stuff. Just saying.

  • el_dhulqarnain

    @Cusp #40

    There is an interesting book out by Joel Franklin: “Advanced Mechanics and General Relativity”; his approach is interesting, but he uses variational principles from the jump. His viewpoint is pretty much what you describe, starting with the Minkowski metric.

    He then moves on to discuss scalar and tensor fields. Some may find it dry, but I think it’s elegant.

  • David Park

    Special relativity is all about moving objects and running clocks. So where, in any textbook, do you actually see moving objects and running clocks? Most students can go through an entire course and never see them. It’s like studying music theory and reading scores but never listening to any music.

    (I also agree that SR teaching is too full of personalization and frills, not to speak of the meter stick lattices and clocks crashing through each other! )

    Some of this can be overcome by making videos and animations using Java and other tools, but these are usually stand alone items not tightly integrated with the rest of the material.

    To seque slightly to a different topic, technical communication and pedagogy is today in something of a state of chaos.

    I am somewhat an advocate of Mathematica because of its active and dynamic properties. I consider it to be a revolutionary new technical communication medium. It would be possible, in my opinion, to write far superior tutorials if we can:

    1) Get the physical principles and mathematics correct.
    2) Devise presentations that illustrate and teach the material.
    3) Write clear textual explanations.
    4) Supply documented tools for readers to use on their own.

    I know that many of the readers will not like Mathematica because of its commercial nature and expense. Furthermore WRI has not quite turned it into a communication medium such that anyone could have a free Mathematica reader to read and operate the controls of any notebook or application (but not write or edit them). Basically the Adobe Acrobat model.

    But something like this is going to come and learning how to best use such a medium is not necessarily easy.

    And I do like the Bondi k factor and calculus as one tool in learning SR.

    David Park

  • Claire C Smith

    Try understanding applied vector analysis! Whihc is basically calculus, in electrical engineering.

  • matthiasr

    I’m on a train right now, I can force myself to see my worldline (Poincaré-)tilted against that of the trees outside. (I usually don’t, because I have a hard enough time not bumping into stuff in the Galileian world).

    GR is a bit harder because I just can’t really keep all the features and twists of more complex spacetimes in mind at once. But locally it’s just Lorentz anyway.

    I also at some point started to imagine (classical particle-)QM as series of waves rippling outwards and interacting. I also tend to favour the many-worlds-interpretation (the waves just keep on rippling) over the conventional wave function collapse. The latter is a useful tool to make sense of practical (apparent?) measurements though. I think the Hilbert space is not really the place to “grasp” QM, although taking it, ignoring infty – 3 dimensions and thinking of states as vectors helps in making sense of the mathematics of QM.

    I’m not sure about QFT though. Waves with different modes of vibration rippling through fields, maybe?

  • matthiasr

    Of course, in practice it’s always “shut up and calculate”.

  • Sean Peters

    It’s been a looong time since college (BS, physics, 1987)… but to the best of my recollection, it went like this: special relativity – pretty easy. Mindblowing, but not hard. General relativity – harder, still mindblowing, but not too bad. QM – uhh… pretty hard. Finally got it. But the real killer: statistical mechanics. Holy crap. I was at best an indifferent student, and failed a few courses along the way – but my general experience in retaking them was that if I had just buckled down the first time around I would have been fine. But stat mech nearly unmade me – failed it abjectly the first time, and barely managed to pull a C the second… and that was a huge struggle. I felt like I got an acceptable grasp on the topic by the time I was done, but… yeah, hard.

    I read “From Eternity to Here” last year and had to struggle with Vietnam-style flashbacks to stat mech… canonical ensembles… Bose-Einstein condensates… GAAAAHHH!


  • Cusp

    >> Of course, in practice it’s always “shut up and calculate”.

    Not in my course it isn’t.

  • Anonymous_Snowboarder

    @Chris J: not sure the level you are looking for in QM but a good soph. level QM book is Morrison’s isbn 0137479085

  • perry

    harris – nonclassical physics. Best soph level QM book ever.

  • David Park

    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?

  • Cusp

    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.

  • Cusp

    Just to clarify

    >> concept of 4-vectors, coordinate transformations and tensors.

    in a physics-first context.

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  • David Park

    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.

  • The Stand-Up Physicist

    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.

  • Cusp

    >> 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?

  • David Park

    If Cusp can send me an email address I will send you a PDF showing the problem and calculation in more detail.

    David Park

  • Baby Bones

    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.

  • Chris J.

    @ 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!

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    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.


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Cosmic Variance

Random samplings from a universe of ideas.

About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] .


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