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Monday, March 27, 2023

When Discussing the Twin Paradox: Learn This First

This text is meant for anybody who needs to begin 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 inclined 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 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 show you how to to border no matter questions you continue to have after studying it, in a method that may show you how 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 selection of various methods of analyzing the usual twin paradox situation, described within the article, every of which provides some perception. I’ll briefly checklist 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 steered methods of analyzing the usual twin paradox that we frequently discover talked about in threads right here, and which seem in varied sources, similar to:

  • 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 function of this text is to not give an in depth research of all of the attainable methods of wanting on the twin paradox. Our function right here is extra common. The assorted analyses, usually talking, will be categorized by how effectively they reply three common questions:

  • (Q1) If the twins have aged in another way once they come again collectively, there should have been some distinction or asymmetry between them through the journey. However doesn’t relativity says that every one frames are equally legitimate? How does the tactic of research cope with this?
  • (Q2) How a lot extra evaluation or inference should be completed, past what’s given in the issue assertion, in an effort 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 instances. 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 form 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 identical factor, however generalized to instances the place it isn’t possible to attract a easy diagram of the situation and one has to depend on equations as a substitute. However the fundamental level is identical as within the Spacetime Diagram evaluation: you’ve gotten two twins who take totally different paths by 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 aren’t 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 firstly.

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

  • (A1) The asymmetry between the twins is easy: it’s the totally different lengths of their paths by spacetime. These path lengths are invariants; they don’t rely upon which body you undertake. So each twins will agree on them. The 2 twins, in the event that they use totally 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 completed 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 seen as guidelines of thumb for recognizing when the trail lengths of the twins by 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 once they flip round, the inertial twin’s path will probably be longer, so Acceleration works right here as an asymmetry to elucidate the distinction in getting old. 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 can at all times work however which requires extra work over and above the Spacetime Geometry evaluation–see A2 under). As we’ll see below A3 under, all of them break down in some unspecified time in the future. Solely the Spacetime Geometry evaluation by no means does.

  • (A2) As a way to apply the Spacetime Geometry evaluation, you must know the paths of the twins by spacetime. But when the situation is effectively specified in any respect, it should embody specs which are ample to calculate these paths–if it doesn’t, you’ll be able to’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 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 would contain tedious computation for extra difficult situations), and would in all probability should be completed 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 strategy to know that’s to know the twins’ paths by spacetime so you’ll be able to in flip calculate the paths of the sunshine alerts 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 when 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 below A1 above, if the situation is effectively sufficient specified in any respect, it should comprise sufficient info to calculate the paths of the twins by 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 decide the one which works one of the best for you.

    For another evaluation, you first would want to test 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 instances (like the usual situation with the Acceleration evaluation, mentioned above), you would want to do all of the work you’ll 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 utilize (for instance, in case your chosen methodology insists on utilizing inertial frames), or might need ambiguities in even outline a body (for instance, there isn’t any distinctive strategy to outline a “relaxation body” for the touring twin in the usual situation, no matter the way you specify their correct acceleration).

Briefly: in case you are a very particular case, similar to the usual situation described within the Usenet article, there are in all probability a number of analyses that may “resolve” the paradox in a technique or one other. However for any evaluation apart from the Spacetime Diagram/Spacetime Geometry evaluation, ultimately you’ll encounter a case that that evaluation can’t resolve. And even earlier than that, you’ll seemingly 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 attainable, is Spacetime Geometry.

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