This text is meant for anybody who desires to begin a thread right here at Physics Boards on the dual paradox. There are already many, many threads right here on this subject, they usually are inclined to cowl the identical floor again 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 enable you to to border no matter questions you continue to have after studying it, in a method that can enable you 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’ll 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 situation, described within the article, every of which provides some perception. I’ll briefly record 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 regularly discover talked about in threads right here, and which seem in numerous sources, resembling:

• Altering Inertial Frames.
• Acceleration.
• 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 examine of all of the doable methods of wanting on the twin paradox. Our objective right here is extra normal. The assorted analyses, usually talking, will be categorized by how effectively they reply three normal questions:

• (Q1) If the twins have aged otherwise after they come again collectively, there will need to have been some distinction or asymmetry between them in the course of the journey. However doesn’t relativity says that every one frames are equally legitimate? How does the tactic of research take care of this?
• (Q2) How a lot further evaluation or inference should be executed, past what’s given in the issue assertion, with a view to utterly analyze the situation utilizing the given methodology?
• (Q3) Given an evaluation of the usual twin paradox situation, how effectively will that very same methodology 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’s solely one methodology of research that may give 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 allows 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’s not possible to attract a easy diagram of the situation and one has to depend on equations as a substitute. However the fundamental level is similar as within the Spacetime Diagram evaluation: you have got two twins who take completely different paths via spacetime, and people paths have completely different lengths, and the lengths of the paths are the quantities that every twin ages in the course of the journey. So the completely different ages of the twins after 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 completely different routes with completely different lengths, their odometers will learn completely different mileages after they meet on the finish even when they have been the identical in the beginning.

Let’s check out how the Spacetime Geometry evaluation responds to our three questions above, and distinction it with a number of the different analyses:

• (A1) The asymmetry between the twins is easy: it’s the completely different lengths of their paths via 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 completely different frames, may differ within the particulars of how they calculate these invariants, but when their frames are legitimate, they may get the identical remaining solutions. (The calculation executed 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 will be considered as guidelines of thumb for recognizing when the trail lengths of the twins via spacetime will differ. For instance, if, as in the usual situation, spacetime is flat and one twin stays inertial the entire time whereas the opposite has nonzero correct acceleration after they flip round, the inertial twin’s path will likely be longer, so Acceleration works right here as an asymmetry to clarify the distinction in getting older. 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 foundations of thumb concerned in any of the opposite analyses (aside from the Doppler Shift evaluation, which is able to all the time work however which requires further work over and above the Spacetime Geometry evaluation–see A2 beneath). As we’ll see underneath A3 beneath, all of them break down sooner or later. Solely the Spacetime Geometry evaluation by no means does.

• (A2) As a way to apply the Spacetime Geometry evaluation, it’s important to know the paths of the twins via spacetime. But when the situation is effectively specified in any respect, it can embody specs which are enough to calculate these paths–if it doesn’t, you possibly can’t resolve it by any methodology of research (except you’re 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 gained’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 must be executed anyway it doesn’t matter what methodology of research you’re utilizing.

  For instance, to even use the Doppler Shift Evaluation, you might want to know what the Doppler shifts are–and the one approach to know that’s to know the twins’ paths via spacetime so you possibly can in flip calculate the paths of the sunshine indicators that they ship to one another. (Notice that the Usenet article glosses over this by simply supplying 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 situation is effectively sufficient specified in any respect, it should include sufficient info to calculate the paths of the twins via 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’re 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 examine to ensure it really works in any respect for the situation (since all of them have limitations in what sorts of situations they work in). Even when it did, except the situation was one of many easiest particular circumstances (like the usual situation with the Acceleration evaluation, mentioned above), you would want to do all of the work you’d do for the Spacetime Geometry evaluation, plus further work to evaluate no matter your chosen evaluation tells you to evaluate (like Doppler Shift, as above). You may additionally be restricted in what frames you should utilize (for instance, in case your chosen methodology insists on utilizing inertial frames), or may need ambiguities in find out how to even outline a body (for instance, there is no such thing as a distinctive approach to outline a “relaxation body” for the touring twin in the usual situation, no matter the way you specify their correct acceleration).

Briefly: if you’re a very particular case, resembling the usual situation described within the Usenet article, there are most likely a number of analyses that can “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 probably find yourself doing extra and more durable work than you wanted to. The one absolutely normal approach to resolve all such situations, and do it as effectively as doable, is Spacetime Geometry.