How would you go about teaching relativity to someone who refuses to take anything on faith or argument from authority? That is, to someone to whom you must actually prove experimentally each postulate or aspect of the theory? This hypothetical student is interested in learning. They just don't want to have to believe anything without fully understanding how it was derived from experiment. They are willing to take the reality of the experiments themselves on faith. If the experiment was actually documented in some way that is enough to prove that the experiment actually took place. You aren't dealing with someone for instance who doubts the moon landings because he wasn't actually there himself. I'm thinking the best method is simply to thoroughly investigate and understand every experiment conducted and based on the experimental evidence derive what you can through deductive logic. So in that vein, what are the most convincing experiments conducted, particularly regarding the most non-intuitive aspects of the theory? Does anyone have any links or book references on such experiments? Is relativity ever taught this way? From the perspective of a strict experimentalist deriving all theory strictly from the experiments that demonstrate it? Are there any experiments that are simple enough to replicate oneself as a demonstration? Building a cyclotron or a relativistic spaceship are not options. The most useful experiments I think would be ones that demonstrate/prove the physical reality of space-time and/or Minkowski 4-space and the massless nature of light despite the fact that its path is affected by the proximity of massive objects. Remember you're dealing with a skeptic. You'd have to start with observational/experimental evidence of some kind. They aren't just going to accept for instance the assertion that massive objects change the path of light because massive objects curve space into a fourth temporal or spatio-temporal dimension. First you would have to actually prove the existence of and then the curvature of space-time. And even then you'd have to show evidence for the force that causes objects and/or light to follow those paths. For instance without gravity a ball does not roll down a hill even though the hill is there. Some kind of an external force (or acceleration) is needed. Assumptions: 1) The student genuinely wants to learn. 2) The teacher genuinely wants to teach and is not annoyed/angered by the lack of faith. 3) Everything you teach, every step, must be proven as rigorously as a mathematical proof. The chain of logic has to be tight. The evidence has to be unequivocal. I do realize that teaching such a student in such a manner would take longer and be more difficult than teaching one who just accepts whatever he is told without question. OTOH, perhaps that student would have a deeper understanding of the theory in the end because of his familiarity with the physical experiments from which the theory was derived. I guess it's sort of like teaching Calculus by starting with the problems Newton was trying to solve. That has always seemed to me to be a rigorous and effective way to teach, despite the fact that it takes longer. Also for teaching in this manner, would it be better to start with the math and then do the theory or start with the theory and then do the math? Or should it matter? I think the equations themselves are pretty easy to prove. All you have to show in that case is that the equations work, that they make working useful predictions. The quantitative aspects are no problem. I'd be mostly concerned about proving the qualitative aspects of the theories.
I would start by emphasizing the huge number of experiments which support special relativity and contradict Newtonian physics: http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html and http://relativity.livingreviews.org/Articles/lrr-2005-5/index.html There is also one other approach that I particularly like: Robertson, Rev. of Mod. Phys. 21, pg 378 (1949) In this paper they outline a few basic assumptions, like the laws of physics should be isotropic and so forth, and derive the most general theory possible from those assumptions. Then, he uses the results of 3 experiments to fix the undetermined constants in the general theory. The result is SR to within a fraction of a percent.
I wouldn't explain the content of any theory to this person until I've been able to make him/her drop those naive expectations. (I'll just say "him" or "he" from now on). If you think you can teach him what a theory is by teaching the content of a theory, I wouldn't have a problem with that, but it seems like he has already made up his mind about what a theory is, and is wrong about it. You need to get those ideas out of his head before you do anything else. Postulates aren't proved. They are guessed. This is true regardless of whether the term "postulates" refers to the statements that define the theory or to the loosely stated ideas that can help you guess an appropriate definition of the theory. "Einstein's postulates" for SR are loosely stated ideas that helped him and others guess an appropriate definition of the theory. The theory is defined by the mathematics of Minkowski spacetime and statements about how to measure time and length. There's a lot of guessing involved when a theory is found. In principle, you could start by guessing a definition of a theory, and then find that it makes excellent predictions. In reality, this is probably too difficult. Einstein started with the observations that 1) the principle of relativity is an essential ingredient of pre-relativistic classical mechanics, and 2) the only solutions of Maxwell's equations that describe waves say that the speed of the waves is c, regardless of such things as the velocity of the source that produced them. He then guessed that there might exist a theory that includes both, and figured out some of the things that such a theory would predict. Two years later Minkowski figured out how to properly define the purely mathematical part of the theory. You could mention 1 and 2, and tell him about experiments that prove that the predictions of pre-relativistic classical mechanics and classical electrodynamics are very accurate. But if you do, you should explain that this is mainly of historical interest. This is not the way to justify SR. These are just the observations that helped Einstein state a couple of ideas that helped him (and Minkowski) guess that the theory we now call special relativity might be useful. Once we have guessed a theory, we can use it to predict results of experiments. Then we can do experiments to see how accurate those predictions are. Once we've done that, it's completely irrelevant how we came up with the theory in the first place. This is why it's OK to just guess a theory or an idea that eventually leads to a theory, and it's also why it doesn't make sense to try to "experimentally prove" statements that are part of a definition of a theory, or statements that express ideas that can be developed into a theory. This also indicates a fundamental misunderstanding about what a theory is. This is not how things are done. Theories are guessed. General relativity is defined by the assumption that matter curves spacetime. (To be more precise: It's defined by the assumption that the relationship between the geometry of spacetime and the properties of the matter it contains is given by Einstein's equation). GR says that spacetime is a manifold with a metric that satisfies Einstein's equation. Motion is represented by curves in spacetime. The metric tells us which curves in spacetime to consider "straight lines". The acceleration of an object at an event p is then defined as a measure of how much the curve that represents that object's motion is different from the "straight line" through p. Now you can define the force acting on the object through an equation like F=ma. Start with special relativity and spacetime diagrams. This way you can explain the essential features of the theory almost completely without math. You should however do the derivation of the gamma factor based on the pythagorean theorem. Don't bother with general relativity until he understands special relativity pretty well. The mathematics of SR is mainly about Lorentz transformations. I think these should be taught using matrices. I firmly believe that I could teach a person matrices and SR in less time than it would take me to teach him/her SR. This is how I would introduce Lorentz transformations in 1+1 dimensions. In 3+1 dimensions, I think I would just use the matrix definition of a Poincaré transformation: ##x\mapsto T(\Lambda,a)(x)=\Lambda x+a##, where ##a## and ##x## are 4×1 matrices, and ##\Lambda## is a 4×4 matrix that satisfies ##\Lambda^T\eta\Lambda=\eta##.
Our best current theory is already probably wrong, yet it describes available data well enough. Since our wrong best current theory and the true theory (assuming it exists) both describe available data well enough, there is not a unique theory that can be derived from the data.
Thanks for the links. I figured proving special relativity itself would be relatively trivial. I'm more concerned about experimental evidence for the existence of space-time or Minkowski 4-space as a physical reality as opposed to a useful mathematical model or construct which may or may not exist in the actual world. I am assuming that there is strong and direct evidence for every aspect of both SR and GR at this point. I know there is for SR. I'm just unfamiliar with GR experiments which don't themselves rely on assumptions which can be questioned. Eclipse experiments for instance are often mentioned as evidence for various aspects of GR, but all I really see it as proving is that light is bent when passing close to a massive object (gravitational lensing). Even if you can prove that light or photons don't have mass it still seems that space-time distortions which somehow 'attract' the massless light are not the only possible explanation. For instance it could just mean that light behaves as if it had mass in the presence of a sufficiently strong gravitational field. It certainly isn't inconceivable. It's not like light behaves like anything else in most other ways either. Quanta, wave-particle duality, c invariance, some properties of matter with mass and some without... It think it's fair to say that light is strange and difficult to accurately predict under previously untested conditions. Using some expected property of light to either validate or invalidate any theory based on how we think it should behave seems a bit shaky to me. The eclipse experiments are nice quantitative evidence for the utility of the equations as engineering devices, but as a proof for the theory attached to the mathematics I find it kind of weak. OTOH if we could demonstrate that some massless object that isn't light is affected by gravity and we were 100% certain that the object really was massless then it would seem to invalidate the gravity equation. If m1=0 then F=0. So it would definitely cause some problems. In any case, if someone knows about experimental evidence for some of the more difficult to prove aspects of the GR I would really appreciate any links or references. Direct proofs that even a (reasonable) skeptic couldn't challenge would be ideal. It seems like some of these experiments may be a bit obscure. Difficult to track down. That's why I'm asking. I'm sure they must exist. Otherwise people wouldn't be so certain about the truth of the theories. I would very much like to read and become familiar with the experiments. Thank you. That sounds amazing. I'll try to find that paper.
I don't understand this at all. AFAIK, there is no such thing as a "existence"-ometer or a "physical relaity"-ometer. Suppose that you have a mathematical model (aka theory), and suppose further that you have a set of 1000 different experiments all of which agree with the theory to very high precision. What more are you looking for? Obviously not. There are always an infinite number of theories that you could produce to explain the same data, simply by introducing additional free parameters. This is the essence of the scientific method. If you are not willing to accept experimental evidence as validation of a theory then you are not doing science. Regarding GR, I would agree with Frederik that you should avoid GR until the student has SR well in hand.
To me this would seem like an excellent way to justify the natural language aspects of SR theory. 1) Link SR with intuitive classical theory that is not in question. 2) Mop up with the large number of experiments which confirm SR. Ideally one could even perform an experiment which demonstrated any areas in which the student was having doubts. I wonder if one day we will be able to do this at sub-relativistic speeds for every aspect of SR as we have already done with time dilation using highly accurate measuring devices. In any case I don't really see SR as a potential problem for teaching in the way that I mentioned. Are you implying that theory = equation? That the equation is the theory? That the words associated with the mathematics don't matter? The way I see it there are two ways to look at a theory. From the point of view of a mathematician or from the point of view of a philosopher. I don't think either is right or wrong as long as the scientific method is respected and used. Mathematical assertions should be proved. Natural language statements should also be proved. The only way to prove either is through experimental data. In order to get experimental data you actually have to do some experiments. Also, I didn't mean to imply any particular order in the scientific method. Guesses are fine as long as they are confirmed at some point through experiment.
[emph. added] Before doing anything else, it's a good idea for your skeptic to start understanding what scientific proof entails. Science does not 'prove' (except in the classic sense) anything. It sets up experiments that can potentially falsify other things to a high level of confidence. Competing theories survive, to a high level of confidence, this process of falsification. So scientific 'proof' is really quite aligned with its traditional meaning: pound away at something with every test: and if it survives, it is proved. Science doesn't really have truth, it has to make do with high confidence levels. Therefore, science differs little from Conan Doyle's Sherlock Holmes: eliminate what is (demonstrably) false, and the remainder, however improbable, is the answer. Relativity invites natural skepticism, since it defies millennia of common human (and probably animal) horse-sense and instinct; to a bystander: 1. moving objects shrink in their movement-direction, and seem to 'rotate'. ?! 2. moving clocks slow. ?! Who ordered this? But the contrary assertions have been shown to be false.
I would agree with this. I would call the mathematics the "theory" and I would call the words associated with the mathematics an "interpretation" of the theory. You cannot prove an interperation through experimental data, only the math. Often there are multiple interpretations for the same theory, and experiment can never, even in principle, distinguish between the two interpretations.
I define the term "theory" as "a set of statements that can be used to make predictions about results of experiments". (This is the rough version. I could write down a longer and more precise definition, but I don't think we will need it here). No piece of mathematics makes predictions about results of experiments, so the mathematics of Minkowski spacetime is not a theory of physics. The theory is defined by the statements that tell us how to interpret the mathematics as predictions about results of experiments. This is the terminology that A. Neumaier advocated in a long discussion I had with him, so you're not alone. It has the advantage that it's possible to write down a complete definition of each theory, but it has the disadvantage that theories aren't falsifiable (because now it's the interpretations that tell us what the predictions are). The reason I choose another terminology is that lack of falsifiability bothers me more than inability to write down complete definitions.
Nothing. That would be perfect. I suppose I didn't explain myself well. Look at those concepts, at those ideas, from the point of view of a skeptic. He's going to want to see some proof of the ideas themselves. Not just of the mathematics. Unless I am mistaken I believe that we are not just presented with a list of equations when learning relativity. Proving the quantitative relationships between a set of variables does not inherently also prove the explanation for what is causing those relationships. That needs to be proved separately. Wait a second. Is that what you guys are thinking? That once you prove the math you automatically prove the associated natural language theory as well? Proving the math just proves the quantitative relationships between variables. In some cases that's all you need. For the purposes of engineering it's certainly all you need. But from a philosophical perspective that doesn't work so well. The leap from the math to a model of the world where massive objects create 'depressions' in a 4th (temporal) dimension needs its own proof. In this particular case that experimental proof may very well exist. In fact I'm assuming that it does and I'm trying to find it. Even if there is no direct proof, it may be possible to prove starting from the equations. The first thing I would do is examine the equations and try to prove that they are unique. That the same relationships between variables cannot be represented in any other form except through Minkowski's math. Once you've proven that then you just have to show spacetime is the only option. That without it the equations, with their great predictive value, just wouldn't work. I think the scientific method does allow you to choose which theory you believe a particular set of experimental data supports. One may believe for instance that there are simpler or just more plausible hypothesis to explain a given set of data.
I would not. I think that in order for someone to be taught he must take some trust in his teacher. If he does not then I think the teacher is wasting his and the student's time.
My question is about a hypothetical person who is not me. I am looking at this stuff and wondering how I could possibly teach it to someone who needed to be convinced and wasn't willing to take my word for it. Or anyone else's for that matter. In general I am sympathetic with the skeptical viewpoint of requiring evidence for a belief. In this case however I do believe that the special and general theories are the most correct theories available and that they describe the available data better than anything else. Certainly better than the Newtonian laws of motion. In addition I quite like them. If they aren't correct they certainly should be because to me the world is a more interesting place with relativity than without it. Time dilation even gives you a form of time travel. General relativity gives you interesting things like black holes. It's hard for me to imagine that any aspect of special relativity could be wrong in any way and it's extremely well proven in any case. I'm curious to see if GR is as well proven as SR however. I want to thoroughly look into whatever experimental data is available.
I got my hypothetical person into a lot of trouble many years ago when he/she didn't have a clear enough idea about what 'proof', 'evidence', 'belief', 'wrong', etc., meant in the formal sense. One can nit-pick over the finer points of these ideas, but he/she could not get very far until a good rough approximation was established. General relativity gives black holes, so does classical physics; the notion predates Einstein. Certainly, since the value of c has been known for centuries, and the idea of 'escape velocity' is likewise old. One also has to weed out legitimate aspects of relativity from accreted crud ascribed to relativity: relativistic mass-increase for example. Real mass does not increase with relative velocities. This notion comes up repeatedly.
Get him/her to read Einstein's "Relativity: The Special and General Theory" with Feynman as reference. You know Newton's calculus was very non-rigorous. He never proved the most basic things rigorously; just assumed them. So there is only so much rigor a beginner, who is also not Newton (a non-Newtonian beginner :D), can handle.
I think that I would add a couple of qualifiers, like "required". Basically, I would like to distinguish between the parts of a theory which are needed to make experimental predictions (e.g. the Minkowski metric and a description of clocks and rulers) and the parts which are used as background (e.g. the aether or the two postulates) to motivate the theory or as menmonic devices. The former I would call the "theory" and the latter I would call the "interpretation".
OK, then special relativity fits that bill. No, there is no choice allowed. If the data is predicted by the theory then the experiment supports the theory, if the data is not predicted by the theory then the experiment falsifies the theory. Certainly, that is an aesthetic or philosophical preference. It is not possible to prove or disprove such preferences either via math/logic or experiment. However, if you are familiar with Bayesian inference it is possible to quantify the idea of a "simpler" hypothesis in such a way that an experiment which supports both a simple hypothesis and a complicated hypothesis is rationally interperted as evidence favoring the simple hypothesis over the complicated one.
I guess the biggest problem that I would have with teaching someone GR, or more to the point, the interpretation of it is that I have basically accepted the space-time distortion model/interpretation on blind faith. The rubber sheet, bowling ball, marble model has never made sense to me. I cannot imagine a 2D rubber sheet. Only a 3D one. The depression in the rubber sheet created by the bowling ball into which the marbles are supposed to 'fall' would only be present if there were a classical gravitational field behind the sheet. I don't see how you can rely on the very model you are attempting to replace in order to demonstrate your own. Also I think the fourth dimension is supposed to be temporal. Not spatial. The rubber sheet model seems to rely on a spatial fourth dimension. Otherwise why would you end up with a physical depression in the flat sheet representing 3 dimensional space? If the distortion in the rubber sheet were strictly temporal that might explain why time would seem to slow down or even stop in a sufficiently strong gravitational field, but it wouldn't seem to explain the acceleration created by the presence of a massive object. So why do I accept this interpretation/model of gravity as absolutely true despite the fact that it seems about as silly/nonsensical to me as a square circle? Basically because I consider men like Einstein or Feynman or Wheeler or Thorne or Dyson or anyone who might work at a place like the Institute for Advanced Study or anyone who could be accepted into the physics programs at MIT, Caltech, Princeton, Harvard, or Berkeley, to be more intelligent than I in the way that I am more intelligent than, say, a spider monkey. The idea that they could be wrong, and I could be right about something like that seems even more improbable to me than the rubber sheet idea. In fact I actually consider it to be impossible. But if I were teaching the idea to someone I couldn't possibly make that argument. Hence my interest in eventually being able to prove that interpretation or at least demonstrate in any way how it could possibly be true without relying on blind faith. If there are no experiments which directly demonstrate this interpretation of the equations then it seems like some degree of blind faith would be required of the student in order to accept it. Not every student is going to be willing to do that. Of course these are all philosophical concerns about some sort of 'truth'. I assume that the field equations could still be used as kind of a more accurate version of the classical model of gravity. That conceptually you could continue to treat gravity as a field as long as you use Einstein's equations to solve the actual problem. Or maybe that's wrong. Is it necessary to accept the rubber sheet idea in order to use the equations correctly?