Okay, this may sound quite stupid, but I am currently learning special relativity in my physics class (2301 OSU). So since both frames assume that their frame is stationary and that time in all other frames is dilated. So if Twin A stays on earth and Twin B travels at constant velocity to another planet, both can say that the other person is aging slower. But since the twin that’s traveling goes through some sort of acceleration to return back to earth, they do age slower (due to effects of general relativity according to my friend). But what if that twin could return to compare clocks without having to accelerate (I know its not possible but its just a what if situation), how would their clocks compare?
Take the following gedankenexperiment in which two astronauts meet each other again and again in a perfectly symmetrical setting - a hyperspherical (3-manifold) universe in which the 3 dimensions are curved into the 4. dimension so that they can travel without acceleration in straight opposite directions and yet meet each other time after time.  On the one hand this situation is perfectly symmetrical - even in terms of homotopy and winding number. On the other hand the Lorentz invariance should break down according to , so that one frame is preferred - but which one?  So the question is: Who will be older? And why?  And even if there is one prefered inertial frame - the frame of the other astronaut should be identical with respect to all relevant parameters so that both get older at the same rate. Which again seems to be a violation of SRT in which the other twin seems to be getting older faster/slower...  How should one find out what the preferred frame is when everything is symmetrical - even in terms of GRT?  And if we are back to a situation with a preferred frame: what is the difference to the classical Galilean transform? Don't we get all the problems back that seemed to be solved by RT - e.g. the speed limit of light, because if there was a preferred frame you should be allowed to classically add velocities and therefore also get speeds bigger than c ?!? (I know  is only a local theory but I don't understand why the global preferred frame should not 'override' the local one).  Could anyone please enlighten me (please in a not too technical way because otherwise I wouldn't understand!)  EDIT Because there are still things that are unclear to me I posted a follow-up question:
What is the scientific name of this evergreen plant? This plant is found in India. It is an evergreen plant. Would love to know the scientific name. Can you help me with my software licensing issue? This is a  about Licensing.    Questions on licensing are off-topic on Server Fault.  If your question has been closed as a duplicate of this question, then this is because we want to help you understand why licence questions are off topic rather than just telling you "it just is". In all likelihood, this question will not address your question directly, it was not meant to.    I have a question regarding software licensing. Can the Server Fault community please help with the following:   How many licenses do I need? Is this licensing configuration valid? What CALs do I need to be properly licensed? Can I run this product in a virtual environment? Can I downgrade this product to an earlier version? Am I entitled to feature X with license Y? Pullback of a Volume Form Under a Diffeomorphism. I have an exercise here, which I have no idea how to do.     Problem: Let $ U $ and $ V $ be open sets in $ \mathbb{R}^{n} $ and $ f: U \to V $ an orientation-preserving diffeomorphism. Then show that   $$   {f^{*}}(\text{vol}_{V}) = \sqrt{\det \! \left( \left[   \left\langle {\partial_{i} f}(\bullet),{\partial_{j} f}(\bullet) \right\rangle   \right]_{i,j = 1}^{n} \right)} \cdot   \text{vol}_{U}. $$   Notation:         $ \text{vol}_{U} $ and $ \text{vol}_{V} $ denote the volume forms on $ U $ and $ V $ respectively.   $ f^{*}: {\Lambda^{n}}(T^{*} V) \to {\Lambda^{n}}(T^{*} U) $ denotes the pullback operation on differential $ n $-forms corresponding to $ f $. What is the meaning of equation of a circle or equation of a cube? We have an equation of a Circle $$x^2 + y^2 = 1$$ and equation of a Cube $${||x-y|+|x+y|-2z|+||x-y|+|x+y|+2z|=1}$$  When I substitute x,y in first equation or x,y,z in the second equation then the equations are said to be satisfied when LHS and RHS are equal.  I have a quick question. If the first equation or second equation get satisfied, because LHS becomes equal to RHS, then those values of x and y of first equation are said to be lying on the circumference of the circle and not inside or outside of the circle and x,y,z of the second equation are said to be lying on the surface of the cube and not on the inside and not on the circle of the cube. Correct? I have a Minecraft 1.8.1 vanilla server and would like to reward certain users. I would like to /give some users a 99999/99999 durability renamed Diamond Pickaxe (with all enchantments on it). How do I use commands to change/repair the durability of items? If possible, using only one command and use /give. There are 'n' candies and 't' boxes. Find the number of ways to place the candies in the boxes for each of the conditions (given in the problem). There are 'n' candies and 't' boxes. Find the number of ways to place the candies in the boxes for each of the conditions (all the candies must be spread out) : (a) candies and boxes are different; (b) candies of the same diiferent boxes should not be empty cartons: (c) candies equally new, boxes are different; Edit :(d) candies are different, the boxes are the same, there should be no empty boxes; (e) candies of different boxes alike.(Edit : candies are different, boxes are equal) Specify the display type that matches the placement, if possible. My Answers : (a) Each layout is encoded with a word of $n$ letters from the alphabet of 't' letters $\implies$ possible $n^t$ variants. (b) Write, $n$ candies in the form of balls ina line, we need to put $(t-1)$ partition in $(n-1)$ place, but we can't put two partitions on one place, so we get : $^{t-1}C_{n-1}$. (c) First we need to choose a candy in a box (in the first box $n$ ways, in the second box $(n-1)$, $\cdots$ in the $t-th$ : $(n-t+1)$ ways $\implies$ total $n!/(n-t)!$ methods, and then we distribute the remaining $(n-t)$ candies into $t$ boxes, this is encoded with owrds from the alphabets (for each candy $t$ variants) $\implies$ $t^{n-t}$. The Answer is : $\frac {n!}{(n-t)!}*t^(n-t)$ Are my answers (a), (b) &amp; (c) are correct? For (d) &amp; (e) I don't know how to proceed? Please help me. I am trying to find examples of maps between topological space which are local homeomorphism but not covering maps. Especially, how twisted has to be such a counterexample : can it be a local diffeomorphism between connected manifolds which is not a covering map ?   I found here() a nice proposition which state that a local homeo from a compact space to a connected Hausdorff space is a covering map.   I am interested in all type of counterexamples, from non-Hausdorff spaces to surfaces, to get a better picture of the differences between covering maps and local homeomorphisms.  Thank you ! I just received a decision letter from a journal editor (after three rounds of major revision). The letter shows that the reviewers "have asked for some other minor revisions". After getting the minor revision letter, I immediately completed the revision in accordance with the feedback from the reviewers (the suggested changes just involve adding a few sentences) and immediately resubmitted the revision to ScholarOne. A week has passed since I submitted the revision, but the status in ScholarOne remains unchanged. Is this normal? Can I assume that this basically means "acceptance with minor revision"? Thanks for your insight and sharing. Grub rescue after installing Ubuntu on a external HDD from USB Field level permissions in Drupal 7 What is the best way to create a piece of content with multiple fields... allowing users to edit the content as a whole yet blocking the ability to edit.  The setup I have is a piece of content that has 3 fields: Body, PDF, SWF... and a "Selector" (List Text) that's tied via dependancy. So the user chooses BODY, PDF or SWF and that's what available to edit and shown on the page when live.  What I want to do is add a PHP field... that's either:     greyed out (unable to edit) if your not admin...   doesn't show up in the Drop Down list... Can't select it if you can't see it Shows up on the drop down, but the field doesn't show up in the editor without permissions.   I hope I'm making sense... I can provide an example if needed. JSON URL from StackExchange API returning jibberish? Additional needed documents for UK trip if someone will shoulder all expenses Someone (a relative) from the UK (already a naturalized British Citizen) will be financing my trip, as well as taking care of my accommodation. What are the documents I need to submit aside from bank statements? Does the sponsor need to fill-out the Sponsorship Undertaking Form? ()  something to consider from my question: -my relative is a naturalized citizen in the UK. Is proof of naturalization needed? I'm absolutely new to Linux. I would like to know how to install Ubuntu alongside the pre-installed Windows 8+ OS.  Should I do it with Wubi, or through the Live USB/DVD?  What steps do I need to take to correctly install Ubuntu? I have seen many folks doing this  concluding that shadow from point A on the Moon can travel to point B on the Moon faster than light.  What I fail to understand here (and I am sure I am wrong), is nothing can travel faster than light. I also read about Relativity where Einstein stated that every event happening for you depends on how fast the light gets from where the event is happening to you.  This is my particular problem with this Moon shadow experiment. You cannot make a shadow travel faster than light, because in order for the "disappearance" of light to get to the Moon, it has to travel at light speed.  Here is one way to understand my point. Consider we shine a flashlight on Moon, like the guy in the linked video did, we move the finger across the face of the flashlight, before we moved the finger it was casting the shadow at Point A on Moon, after we finish moving our finger, the shadow is at Point B on the Moon.  When the finger reaches the end of it's moving length, the light waves that have left the flashlight before the finger reached there will have to hit Point B and it will take them a second and a half to get there, and another second and half for us to see that shadow on point B, which makes it 3 seconds + finger moving time for us to actually see the shadow at point B. That means shadow traveled (say) across the diameter of the Moon in 3 seconds.   So. about 3000 Km in ~3 seconds make it 1000 Km per seconds which is very tiny compared to the speed of light. How can these experiments conclude that Shadow could be made to travel faster than light? What is the obvious clue that I am missing here? Harry: "He made seven Horcruxes?"   Dumbledore: "...Not seven Horcruxes: six. The seventh part of his soul, however maimed, resides inside his regenerated body..." (HP6, US Hardback, page 503).    I would think that Dumbledore had already known that there were 6 horcruxes:    the diary  Gaunt's ring  Salazar's locket Helga Hufflepuff's cup Nagini (the whole essence divided thing)   Now those are the five that he was sure of. He knew that when Riddle came to Hogwarts to ask for the DADA position, he was really there so he could hide a horcrux. (But he never got a chance to tell Harry that, right?) Did Dumbledore know of the diadem? Or did he just think Riddle wanted to hide the cup?   Did Dumbledore say there were 6 and not 7 because he didn't know about the diadem but did know about Harry being a horcrux? Or did he say there were 6 to ensure Harry didn't guess he was a horcrux? Sorry if this makes no sense.  I know that in the end there were 7 horcruxes and an 8th part of Voldemort's soul right? How to find files between two dates using "find"? XML to CSV Using XSLT I have the following XML document:  &lt;projects&gt;   &lt;project&gt;    &lt;name&gt;Shockwave&lt;/name&gt;     &lt;language&gt;Ruby&lt;/language&gt;     &lt;owner&gt;Brian May&lt;/owner&gt;     &lt;state&gt;New&lt;/state&gt;     &lt;startDate&gt;31/10/2008 0:00:00&lt;/startDate&gt;    &lt;/project&gt;   &lt;project&gt;    &lt;name&gt;Other&lt;/name&gt;     &lt;language&gt;Erlang&lt;/language&gt;     &lt;owner&gt;Takashi Miike&lt;/owner&gt;     &lt;state&gt; Canceled &lt;/state&gt;     &lt;startDate&gt;07/11/2008 0:00:00&lt;/startDate&gt;    &lt;/project&gt; ...   And I'd like to get this from the transformation (XSLT) result:  Shockwave,Ruby,Brian May,New,31/10/2008 0:00:00 Other,Erlang,Takashi Miike,Cancelled,07/11/2008 0:00:00   Does anyone know the XSLT to achieve this? I'm using .net in case that matters. Let $a_1,\dots ,a_n \in \mathbb{R}^n$ with $a_i = (a_{i1}, \dots , a_{in})$.  Show that when $\mid a_{jj}\mid \gt \sum_{i \neq j} \mid a_{ij} \mid \; \forall \; j\in \{1,\dots ,n\}$, then $a_1,\dots ,a_n$ are linearly independent.  I want to show that the matrix \begin{matrix} a_{11} &amp; a_{21} &amp; \cdots &amp; a_{n1} \\ a_{12} &amp; a_{22} &amp; \cdots &amp; a_{n2} \\ \vdots &amp; &amp; \ddots &amp; \\ a_{1n} &amp; a_{2n} &amp; \cdots &amp; a_{nn} \end{matrix} can be transformed into an upper triangular matrix with all entries in the main diagonal $\neq 0$.  For the cases $n=1$ and $n=2$ this is easy. However, I fail to show this for the general case; the problem being that the tranformations mess up the coefficients such that I can't use the assumption on the $a_{jj}$'s anymore.  I'm pretty sure that this has already been solved somewhere on here, but I don't know how to search for it. Divergence of $\sum\limits_{n=1}^{\infty} \frac{\cos(\log(n))}{n}$ Fastest way to remove non-numeric characters from a VARCHAR in SQL Server I'm writing an import utility that is using phone numbers as a unique key within the import.  I need to check that the phone number does not already exist in my DB. The problem is that   phone numbers in the DB could have things like dashes and parenthesis and possibly other things. I wrote a function to remove these things, the problem is that it is slow and with thousands of records in my DB and thousands of records to import at once, this process can be unacceptably slow. I've already made the phone number column an index.  I tried using the script from this post:   But that didn't speed it up any.  Is there a faster way to remove non-numeric characters? Something that can perform well when 10,000 to 100,000 records have to be compared.  Whatever is done needs to perform fast.  Update Given what people responded with, I think I'm going to have to clean the fields before I run the import utility.   To answer the question of what I'm writing the import utility in, it is a C# app. I'm comparing BIGINT to BIGINT now, with no need to alter DB data and I'm still taking a performance hit with a very small set of data (about 2000 records).   Could comparing BIGINT to BIGINT be slowing things down?  I've optimized the code side of my app as much as I can (removed regexes, removed unneccessary DB calls). Although I can't isolate SQL as the source of the problem anymore, I still feel like it is.