Recently I have been thinking about inertial frames in GR. The definition of an inertial frame in SR and in classical mechanics is one in which the laws of physics "hold good". Or in other words, where the equations take on their textbook forms. One of the things I did not realize about GR is that the laws of physics take the same form in all reference frames, including those that would be considered non-inertial in SR and classical mechanics. Since all reference frames use the same sets of equations that is no longer a distinguishing feature of different frames. So I believe that in GR it does not make sense to apply the terms "inertial" and "non-inertial" to reference frames. Of course, "inertial" and "non-inertial" can be applied to objects as before, but not to reference frames. An inertial object would simply be one where a small attached inertial guidance system would read 0. So, I think I probably owe 2inquisitive an apology. I don't remember all of the details of our recent rather vitriolic argument about inertial and non-inertial vs. flat and curved. I think I made several good points, but I am afraid that I have been mistaken on this issue. Sorry about my unnecessary caustic remarks in general and particularly the ones where I was wrong about this. -Dale
Thanks, Dale, but you owe me no apology. Yes, we have differences of opinion, but this place would be pretty boring if everyone just patted each other on the back in agreement. I know I have learned much from you, and hopefully I may occasionally bring up something of interest to you and others. I really enjoy reading new papers, of new discoveries, and of the puzzles still unexplained.
It seems to me a much deeper issue is being discussed here, one a colleague and I have been debating (in friendly fashion) for some time. It seems Dale is actually talking about the principle of general covariance and its implications (and the associated debate which has endured for nearly 80 years) rather than specifically inertial reference frames. SR defines inertial reference frames more rigidly than described above, for instance there are a class of objects (including the space-time interval, the energy-momentum 4 vector, etc) which maintain Lorentz invariance in inertial frames in minkowski space (ie in SR). The structure of lorentz and minkowski space specifically places inertial reference frames in a different class to accelerated reference frames in SR. On the other hand general covariance more closely addresses the form of the laws of physics under coordinate transforms. It is suggested by some that *any* physical theory can be written in such a way as to meet a condition of general covariance, whereas Einstein originally set the requirement of general covariance as the conceptual basis for GR. It should be remembered that even in curved space-time, those reference frames which are at least locally inertial maintain the special properties of SR (recall that the metric of curved space approaches the minkowski metric in the flat limit), so the traces of various lorentz invariant objects remain invariant in GR's inertial reference frames (chief of interest being vanishing freefalling FoRs).
What is the essence of this debate? I thought that the equivalence principle (same as the principle of general covariance?) was both fundamental and essential to GR. So according to your understanding are there inertial and non-inertial frames in GR and if so how are they defined? I just don't see how you can make the distinction when the form of all of the laws are the same in all frames and space is allowed to curve. I mean, consider a universe with two spatial dimensions on the surface of a sphere. If you set up a typical lattitude/longitude coordinate system and gave it an "eastward" boost then you have a traditional frame at the equator and a traditional rotating frame at the poles, but it is the same coordinate system. It seems to me therefore that it doesn't make sense to talk about inertial or non-inertial reference frames when one frame is both simultaneously and it seems irrelevant anyway since the laws of physics have the same form at the poles as at the equator. Sorry if I am unclear, this is a relatively new line of thought for me. -Dale
The statement that the laws of physics have the same form (everywhere) is a really cuddly idea. But, it was a postulate (everybody knows what that means, right?) which was unproven, is unproven, and CANNOT be proven. How are you going to PROVE it? We can, and have been, for a 101 years, been playing an interesting science game of "what if". But nobody can claim an invincible position of saying that whatever happens in any frame must happen in every frame. Unless you have been in every frame and are a guaranteed not-liar. Dale, I am not even trying to confront you personally. My present forum goals go in an opposite direction. But I believe that we should draw a line between what can be proved and known for certain, and what we enjoy to believe.
In general, you are absolutely correct that postulates are unproven. If a theory is based on some postulates and if experimental evidence agrees with the predictions of the theory then it is reasonable to accept the postulates. But the evidence never proves the postulates. It is often possible to come up with a different theory that uses different postulates to explain the same evidence (e.g. LET and SR), and it is just as reasonable to accept the other set of postulates as any choice between the two is purely aestetic or philosophical (e.g. Occham's razor). I understand this concept as do most other scientists and engineers; this is fundamental to science and is not particularly a flaw of relativity. Personally, I don't think that the line you want to draw is particularly useful. Solipsism is the only thing that you can put on the "know for certian" side and then everything else is on the "enjoy to believe" side. Solipsism, while being quite certain is also quite useless. I think "reasonable to believe" and "unreasonable to believe" is a more practical line. The line you want to draw is generally much more interesting to philosophers than to scientists, and even less interesting to engineers like myself than to scientists. At this point I am simply trying to understand the theory itself and not worrying very much about experimental evidence. I am sorry if my outlook is unsatisfying to you, but luckily there is room in this world for philosophers and engineers and everything in between. -Dale
Let's get something straight about postulates. Yes, scientific postulates aren't "proveable". Only mathematical theorems can be proven. But that doesn't mean that scientific postulates aren't testable. In the case of the postulates of relativity, not only have the postulates been tested but they have been verified. The only thing that is lost taking certain statements in a scientific theory to be postulates is that they cannot be derived from the theory, not that they can't be verified.
The debate is over whether it adds any content to Einstein's theory. Most physicists agree that mathematically general covariance is a general principle which applies equally to all physical laws. Einstein thought it was a special condition for a relativistic theory, and some still agree with him. The principle of general covariance is that all physical laws should be written as equations which are covariant under general coordinate transforms. A covariant equation maintains its form under a transform. You can write any physical law this way if you have the appropriate tools (including newton's theory). This is quite different from the equivalence principle which is specific (and as you suggest fundamental) to GR. Sure. An object in an inertial reference frame feels no external resultant force. (that's the easiest way) The crucial distinction to make is between the form of the laws and their content. Covariance addresses the form of the laws, invariance addresses content. As we've seen above, covariance is a pre-requisite of the field equations, invariance is something new. Einstein's theory maintains invariance of certain absolute objects (eg physical constants, invariant objects, etc). Some of these objects maintain invariance only in inertial reference frames, eg the lengths of the space-time interval (proper time) and the energy momentum 4-vector (rest mass). The equivalence principle itself refers to inertial reference frames - infinitesimal freefalling reference frames are inertial, which means that transforms from such frames to rest frames maintains not only covariance (which you've focused on) but also invariance (which you haven't focused on yet). Does that help? MC
I disagree emphatically. Any of the tests that you may have used to verify the postulates of SR can equally be taken as tests to verify those of LET. The only sense in which the postulates have been experimentally verified is that it is now very reasonable to accept them. But it is just as reasonable to accept any alternative set of postulates from which you can derive the Lorentz transform. You can certainly use Occham's razor to pick SR over LET or any other such theory, but that is essentially just a personal preference for simpler explanations. -Dale
So the debate is whether or not covariance applies to other theories than GR, but there is no debate about how it applies specifically to GR. I already mentioned this in the first post, but this is a definition of an inertial object not an inertial reference frame. Also, the reading of an attached accelerometer (the external resultant force "felt" by an object) must be a frame invariant quantity regardless of if the frame is inertial or not. Otherwise you have paradoxes. True, I haven't focused on invariance here. I understand invariance for SR, but is the SR and GR version of invariance somehow different? Covariance is still something new to me and it is new to me in the context of GR. I suppose that you could use covariance and tensors in SR to do something like transform between inertial and rotating reference frames in flat spacetime without changing the form of the laws. Would that be an example of covariance in SR? Perhaps that would be a good exercise for me to try in order to understand covariance. In any case, if that approach were used then my usual definition of inertial frames would not apply to SR either. As far as invariance goes, all frames must agree on the results of any experimental measurement or you will get paradoxes. So my understanding is that things which can be measured are frame invariant in general, not just between inertial frames. The examples you mentioned above are proper time and rest mass. The proper time at least is simple to measure, just use a clock, so that must be generally frame invariant and not just for inertial frames. -Dale
I don't see that you disagree at all. The relativity postulate of SR is also an implicit postulate of LR, and the speed of light postulate is a theorem of LR. Both have been observed to hold experimentally. The only thing that experiments can tell you is whether a prediction is true, not if it should be taken as a postulate. True, the postulates of SR aren't the postulates of LR. So what? All of the predictions are the same, and that's the only thing that counts.
Relativity is a postulate of LR, but the speed of light is considered frame variant. Rod contraction and clock dilation are added as ad hoc postulates in order to explain the measured invariance despite the actual variance. But even if there were no alternative to a given theory experimental results would not "prove" any of the theory's postulates. If a postulate logically implies a result there is still always the possibility that some correct but unknown alternative postulate also implies the same result and the known postulate is wrong even though it leads to the right conclusion. I agree here. The predictions are really the only thing that counts which is also why I think that the distinction about "proveable" is not very useful. As long as the predictions follow logically from the postulates and as long as experiments agree with the predictions then there is no real reason to not believe the postulate. -Dale
???? I'm not sure what you're talking about. Observers in different reference frames potentially disagree on most measurements: distance and time intervals, energy, momentum, stress, fluxes of these quantities, etc. The paradoxes which arise (eg twin paradox, pole and barn paradox, bell's spaceship paradox, etc) embody different measurements by different observers and are usually resolved precisely by identifying the asymmetry between what different observers see. Frame invariant quantities are special objects in relativistic theories, eg proper time is invariant, local time intervals are generally not. Rest mass is invariant, relativistic mass or mass-energy is not. The trace of Tij is, but the individual energy and momentum components are not.
I am addressing your above comment that: "Some of these objects maintain invariance only in inertial reference frames, eg the lengths of the space-time interval (proper time)". This is incorrect, since it is experimentally measurable. All experimentally measurable values must be generally frame invariant not just invariant for inertial frames or you would get a true, unresolvable, paradox. For example, an ideal clock starts at one event moves along some path through spacetime and stops at a second event. Using an inertial reference frame some observer predicts that the clock will record proper time A and using a non-inertial reference frame some observer predicts that the clock will record proper time B. If A≠B then you have an unresolvable paradox, the two predictions cannot both be right since one clock cannot possibly give two different readings for a single measurement. The two observers, of course, can disagree on how much time "really" elapsed while the clock made its measurement since they can disagree about which direction in spacetime is the time direction. But all frames, inertial and non-inertial, must agree about any proper time, spacetime interval, or other Lorentz invariant, and any experimental measurement must be a Lorentz invariant. My time may disagree with your time, but you and I both agree on your time between any two events on your worldline and on my time between any two events on my worldline. -Dale
You say it yourself right here: The measrued speed of light does not vary among inertial frames in LR. The measured speed is the only one that matters. Thus, the speed of light postulate of SR is a theorem of LR. Take a postulate p and a prediction q. Then take a simple theory of the form "If p, then q". You are correct when you state that experimental verifications of q do not constitute a verification of p. However this is inapplicable to the SR/LR issue on two grounds. 1.) The postulates of SR have been verified by experiments specifically designed to test them directly. 2.) If you take p to be the postulates of SR and q to be the Lorentz transforms, then SR is not of the form "If p, then q". It is of the form "p if and only if q". That's because you can do the reverse derivation. That is, if you start from the Lorentz transforms you cannot help but recover the postulates of SR. This doesn't happen in every theory. Take the Bohr model of hydrogenlike atoms for instance. From this model you can derive the exact same energy spectrum that follows from the Schrodinger equation. Yet the postulates of Bohr and Schrodinger are not consistent with eath other. This is an instance of an "If p, then q" theory. If you start from the energy spectrum you cannot recover the postulates of either Bohr or Schrodinger. This is an example of the type of theory you are talking about.
In LET the measured speed is different from the actual speed because of distortions in the rods and clocks used to do the measurements. I personally agree with you that the measured speed is the only one that matters and that it is pretty useless to make any distinction between a measurable speed and a completely unmeasurable "real" speed, but it is nonetheless without question a distinction made by LET. A spiteful demon could have doctored the results. It is not an "if and only if" proposition because you can always take the LET approach and allow fundamentally unmeasurable but real quantities. Then you can derive "if q then you will measure p, but in reality r because your measurements are wrong". Or the universe could be a fundamentally illogical place. You can also always take the solipsist's position that nothing really exists besides yourself and all of your experiments and measurements are just hallucinations, or that they are facets of some VR simulation ala "Matrix". I know that the above caviats and unobservables are all completely useless propositions, but nonetheless that is why postulates cannot possibly be proven through experiment. -Dale
Then the distinction has to be made evident when formalizing the theorems. For instance one could use c' to indicate the "real" speed of light relative to the "moving" observer, and c to indicate the measured speed. Now Lorentz didn't do this in his 1904 paper, but only because he wasn't concerned with the "real" speed c'. He was only concerned with the measured speed. Then we might as well give up on science. It is "if and only if" because you can recover the postulates of SR from the Lorentz transforms. The derivation can proceed both ways, and that is precisely what "if and only if" means. If we want to admit the existence of "unmeasurable but real quantities" then we would have to make a formal distinction between them. This distinction could be made in either SR or LR. All you would have to do is define a symbol c' to stand for the "real" speed of light and let it be related to the measured value c by c+v or c-v, or any appropriate quantity.
Exactly. If we insist on having things "proved and known for certain" then we have to give up on science. This is by no means a criticism of science but rather an indication of the uselessness of insististing on having things proved for certain. -Dale
No! If you admit the existence of a spiteful demon who can fix the results of experiments, then science is impossible. Furthermore logic itself is impossible. If the demon is capable of snookering us into believing that 2 meters is really 1 meter, then he can snooker us into believing that "I am" is consistent with "I am not". Even worse he can have us believe that "I am" implies "I am not". If you insist on admitting the "spiteful demon" postulate, then I say we have nothing more to talk about, as we will disagree on both what is reasonable and what is evident. But if you do acknowledge the same logic, mathematics, and physics as the rest of us, then it is clear that the Lorentz transforms bear an "if and only if" relationship to the postulates of SR. I'll happily agree that the postulates of SR must refer to "measured" as opposed to "real" quantities to make that statement true, because only measured quantities mean anything.
This is not the case. Even if the spiteful demon is capable of deceiving me about everything else, the fact that he is deceiving me implies that I am. This is the philosophical bedrock of solipsism and solipsism is, IMO, the only position that can be "proved and known for certain". It is also a completely useless and boring philosophical position. Let me be clear: I do not admit the spiteful demon postulate because there is no reason to admit it. However, I recognize that if you insist on dealing only with what can "be proved and known for certain" then you must admit the possibility of the spiteful demon. Since the "known for certain" position requires the admission of an unreasonable postulate, my opinion is that the position itself is unreasonable. I think that you and I agree on what is reasonable and evident. My only point here is that demanding that things "be proved and known for certain" is not useful because it demands that you admit the possibility of unreasonable postulates. Science is great, and it gives us plenty of reasons to believe the things that we do and learn about the things we don't know. But I am sure that you will agree with me that it is fundamentally incapable of giving certain knowledge. All scientific principles are essentially tentative and subject to contradiction by future experimental results. If you want certain knowledge then you need to talk to a preacher or philosopher, not a scientist. -Dale
