This text is meant for anybody who desires to start out a thread right here at Physics Boards on the dual paradox. There are already many, many threads right here on this matter, they usually are likely to cowl the identical floor time and again, so it appears helpful to place an outline of that floor into an article that everybody can learn earlier than beginning yet one more thread. If you’re in that class, and what’s right here solutions your query, nice! However even when it doesn’t, hopefully, this text will provide help to to border no matter questions you continue to have after studying it, in a manner that may provide help to to get higher responses with out repeating issues which have already been stated right here many instances earlier than.
Having stated that, the very first thing we are going to do is shamelessly borrow an already current article on the dual paradox, the one that’s a part of the Usenet Physics FAQ. Earlier than you go any additional, please learn the all pages. It’s not lengthy.
A fast abstract of the above article is that there are a variety of various methods of analyzing the usual twin paradox state of affairs, described within the article, every of which supplies some perception. I’ll briefly listing the methods described within the Usenet article right here:
- The Doppler Shift Evaluation.
- The Spacetime Diagram Evaluation.
- The Equivalence Precept Evaluation.
There are additionally different urged methods of analyzing the usual twin paradox that we frequently discover talked about in threads right here, and which seem in numerous sources, similar to:
- Altering Inertial Frames.
- The Touring Twin’s Relaxation Body.
- Spacetime Geometry.
We’re not going to enter nice element about any of those right here because the objective of this text is to not give an in depth research of all of the doable methods of trying on the twin paradox. Our objective right here is extra common. The assorted analyses, typically talking, might be labeled by how effectively they reply three common questions:
- (Q1) If the twins have aged in another way once they come again collectively, there will need to have been some distinction or asymmetry between them through the journey. However doesn’t relativity says that each one frames are equally legitimate? How does the strategy of research take care of this?
- (Q2) How a lot extra evaluation or inference should be finished, past what’s given in the issue assertion, with a purpose to utterly analyze the state of affairs utilizing the given technique?
- (Q3) Given an evaluation of the usual twin paradox state of affairs, how effectively will that very same technique of research generalize to different situations? For instance, will it work if each twins speed up? Will it work if gravity is current (i.e., in curved spacetime)?
And now we get to the primary level of this text: from the standpoint of those questions, there may be solely one technique of research that can provide a passable response in all circumstances. That’s the Spacetime Geometry evaluation, which is a generalization of the Spacetime Diagram evaluation described within the Usenet article. That article states that the Spacetime Diagram evaluation is a kind of “Common Interlingua” that permits you to take a world view and put every of the analyses in its correct perspective. The Spacetime Geometry evaluation is similar factor, however generalized to circumstances the place it isn’t possible to attract a easy diagram of the state of affairs and one has to depend on equations as an alternative. However the primary level is similar as within the Spacetime Diagram evaluation: you will have two twins who take totally different paths by way of spacetime, and people paths have totally different lengths, and the lengths of the paths are the quantities that every twin ages through the journey. So the totally different ages of the twins once they meet once more are not any extra mysterious than the truth that, if two twins take street journeys between, say, New York and Los Angeles by totally different routes with totally different lengths, their odometers will learn totally different mileages once they meet on the finish even when they had been the identical at first.
Let’s check out how the Spacetime Geometry evaluation responds to our three questions above, and distinction it with a few of the different analyses:
(A1) The asymmetry between the twins is straightforward: it’s the totally different lengths of their paths by way of spacetime. These path lengths are invariants; they don’t rely on which body you undertake. So each twins will agree on them. The 2 twins, in the event that they use totally different frames, would possibly differ within the particulars of how they calculate these invariants, but when their frames are legitimate, they are going to get the identical ultimate solutions. (The calculation finished within the Spacetime Diagram Evaluation web page of the Usenet article is an instance of calculating the paths and path lengths of the twins, utilizing the stay-at-home twin’s relaxation body.)
Different analyses might be seen as guidelines of thumb for recognizing when the trail lengths of the twins by way of spacetime will differ. For instance, if, as in the usual state of affairs, spacetime is flat and one twin stays inertial the entire time whereas the opposite has nonzero correct acceleration once they flip round, the inertial twin’s path shall be longer, so Acceleration works right here as an asymmetry to clarify the distinction in ageing. However that rule of thumb solely works in flat spacetime, and solely when one twin is inertial and the opposite isn’t; it doesn’t generalize. Nor do the principles of thumb concerned in any of the opposite analyses (aside from the Doppler Shift evaluation, which is able to at all times work however which requires extra work over and above the Spacetime Geometry evaluation–see A2 under). As we’ll see underneath A3 under, all of them break down sooner or later. Solely the Spacetime Geometry evaluation by no means does.
(A2) With a view to apply the Spacetime Geometry evaluation, it’s a must to know the paths of the twins by way of spacetime. But when the state of affairs is effectively specified in any respect, it’s going to embody specs which can be adequate to calculate these paths–if it doesn’t, you may’t resolve it by any technique of research (except you might be fortunate sufficient to hit a particular case the place a rule of thumb like Acceleration works–however even then, with out sufficient info to calculate the paths, you received’t be capable of give a numerical reply, only a qualitative judgment of which twin ages extra). And after you have the paths, calculating their lengths is simple (although it’d contain tedious computation for extra sophisticated situations), and would most likely have to be finished anyway it doesn’t matter what technique of research you might be utilizing.
For instance, to even use the Doppler Shift Evaluation, that you must know what the Doppler shifts are–and the one strategy to know that’s to know the twins’ paths by way of spacetime so you may in flip calculate the paths of the sunshine indicators that they ship to one another. (Be aware that the Usenet article glosses over this by simply providing you with the outcomes of that calculation–however in case you didn’t already know these outcomes, you would need to calculate them.)
(A3) As has already been famous, the Spacetime Geometry evaluation is the one one which generalizes to all situations. As we noticed underneath A1 above, if the state of affairs is effectively sufficient specified in any respect, it should include sufficient info to calculate the paths of the twins by way of spacetime. And that’s all you want for this evaluation. Plus, as we noticed above, this evaluation works in any body, since it’s calculating invariants, so that you don’t have to fret about whether or not the body you might be utilizing is the “proper” one. You simply choose the one which works the very best for you.
For some other evaluation, you first would want to test to ensure it really works in any respect for the state of affairs (since all of them have limitations in what sorts of situations they work in). Even when it did, except the state of affairs was one of many easiest particular circumstances (like the usual state of affairs with the Acceleration evaluation, mentioned above), you would want to do all of the work you’d do for the Spacetime Geometry evaluation, plus extra work to evaluate no matter your chosen evaluation tells you to evaluate (like Doppler Shift, as above). You may also be restricted in what frames you should use (for instance, in case your chosen technique insists on utilizing inertial frames), or may need ambiguities in tips on how to even outline a body (for instance, there isn’t any distinctive strategy to outline a “relaxation body” for the touring twin in the usual state of affairs, no matter the way you specify their correct acceleration).
In brief: if you’re taking a look at a very particular case, similar to the usual state of affairs described within the Usenet article, there are most likely a number of analyses that may “resolve” the paradox in a technique or one other. However for any evaluation aside from the Spacetime Diagram/Spacetime Geometry evaluation, eventually you’ll encounter a case that that evaluation can’t resolve. And even earlier than that, you’ll doubtless find yourself doing extra and more durable work than you wanted to. The one absolutely common strategy to resolve all such situations, and do it as effectively as doable, is Spacetime Geometry.