I am really confused about the use of tensors in special relativity. I'm not sure if I understood what is a covariant and a contravariant tensor and vector. I am looking for a book of relativity that teach this.
I am looking for a textbook that treats the subject of Special Relativity from a geometric point of view, i.e. a textbook that introduces the theory right from the start in terms of 4-vectors and Minkowski tensors, instead of the more traditional "beginners" approach. Would anyone have a recommendation for such a textbook ?  I already have decent knowledge of the physics and maths of both SR and GR ( including vector and tensor calculus ), but would like to take a step back and expand and broaden my intuition of the geometry underlying SR, as described by 4-vectors and tensors. What I do not need is another "and here is the formula for time dilation..." type of text, of which there are thousands out there, but something much more geometric and in-depth.  Thanks in advance.
Error: You are missing the following 32-bit libraries, and Steam may not run: libXtst.so.6 [...] I'm aware that many people are having problems on launching Steam (downloaded from Steam not the repos) on Ubuntu 16.04 64-bit with this error message:  svetlana@svetlana-desktop:~$ steam Running Steam on ubuntu 16.04 64-bit STEAM_RUNTIME is enabled automatically Error: You are missing the following 32-bit libraries, and Steam may not run: libXtst.so.6 libXrandr.so.2 libgobject-2.0.so.0 libglib-2.0.so.0 libgtk-x11-2.0.so.0 libpulse.so.0 libgdk_pixbuf-2.0.so.0 Installing breakpad exception handler for appid(steam)/version(1463778979) Installing breakpad exception handler for appid(steam)/version(1463778979) Installing breakpad exception handler for appid(steam)/version(1463778979) Installing breakpad exception handler for appid(steam)/version(1463778979) [2016-05-22 16:33:15] Startup - updater built May 20 2016 18:57:58 [2016-05-22 16:33:15] Opted in to client beta 'publicbeta' via beta file You are in the 'publicbeta' client beta. Looks like steam didn't shutdown cleanly, scheduling immediate update check [2016-05-22 16:33:15] Checking for update on startup [2016-05-22 16:33:15] Checking for available updates... [2016-05-22 16:33:15] Download skipped by HTTP 304 Not Modified [2016-05-22 16:33:15] Nothing to do [2016-05-22 16:33:15] Verifying installation... [2016-05-22 16:33:15] Performing checksum verification of executable files [2016-05-22 16:33:15] Verification complete [2016-05-22 16:33:16] Shutdown   My question has anyone found a solution yet? It'd be nice to see reputation updates in the app, as we get on both the desktop and mobile sites.  Mock-up of how this would look:        Other possible icon states:        While I understand reputation is shown in the feed, it's a bit unwieldy, and has no notification.   With this feature implemented, the feed could arguably lose both reputation and inbox items, and be left with the interesting questions that fill it up. This would make for a much more consistent UX across app and desktop. What is the most efficient way to write the old-school:  StringBuilder sb = new StringBuilder(); if (strings.Count &gt; 0) {     foreach (string s in strings)     {         sb.Append(s + ", ");     }     sb.Remove(sb.Length - 2, 2); } return sb.ToString();   ...in LINQ? Which packages should be loaded after hyperref instead of before? How is the number / tetris code in Fez intended to be solved? In the same spirit as :   I was able to crack the language code using tradition substitution cipher breaking techniques.  But I ended up cheating on the number/tetris symbols by looking on the Internet.  The number code is logical enough now I know it -- but I can't see how I would have arrived at it without help.  With regard to tetris shapes:      What would cause you to go down the path of associating Tetris shapes with controller inputs? What should be created first in a video game? I'm Egyptian and flying to Cairo from Tbilisi, via Istanbul. The first flight Tbilisi-Istanbul is on Atlasjet, then from Istanbul to Cairo I am flying on Egyptair. For that, I have to recheck in at Istanbul to my Egyptair flight  Will I need to pass passport control at Istanbul airport to collect my bags and check in on Egyptair? Will I need a transit visa for that? Or it can be all done through neutral zone? I have installed a package using snap and I need to modify one of the files but when I try to change its ownership or permissions, I always get the following message:  sudo chmod +x ./my_file.js chmod: changing permissions of '/snap/my_app/my_file.js': Read-only file system   How can I modify files that are installed via snap? Google Sheets Customize View Using Create Custom Menu and Hiding/Showing Specifc Columns Trying to determine a way that I can use a custom menu (.createMenu) with views that will allow me to show / hide specific columns.  I need the capability to pick specific multiple columns and not ranges.  In the example below how can you pick a custom view from the menu (say "Customer Info") that will only show columns A, B, C, and H?  Maybe another view could show "Financial". Convergence of a sequence implies convergence of the norm of the sequence If $\langle X_n \rangle$ is a sequence in $\mathbb R^n$ then $\lim\limits_{n\to \infty}X_n=A$. Show that $\lim\limits_{n\to \infty} \| X_n \| = \| A \|$ .   My thought process, $\| X_i \|= (\sum_{i=1}^n x_i^2)^{(1/2)}$ therefore be the multiplicative property of limits $x_i^2 = a_i^2$. Therefore the limit of norm of the components is equivalent to the limit of the corresponding limit of each component of the sequence. Best book of topology for beginner? I am a graduate student of math right now but I was not able to get a topology subject in my undergrad... I just would like to know if you guys know the best one.. So basically , we all know the direction outside the magnet its from NORTH to SOUTH. but what i don't understand is that what is the direction inside is it the same ?? Number of zero digits in factorials Here is a riddle someone has been asked in a job interview: How many zero digits are there in $100!$?  Well, I found the first $24$ quite fast by counting how many times five divides $100!$ ($5$ divides $20$ times and $25$ divides it $4$ times).   However, there are more zero digits in the middle of the number (these can be found by hand, by typing factorial(100) in sage).  My question is whether there is a smart way to determine the number of zero digits in $100!$, and more generally in $n!$.  By the way, this will not affect the job interview as it was finished some time ago. Does it make any difference for SEO benefits to buy dropped and already available domains other than backorder domains that are expiring soon?  I heard somewhere that whois gets deleted after being dropped. But I didn't find any confirmation if it's true or not.  Anybody can tell me if there are benefits of one against the other, if yes what? I am on Ubuntu and I have forgotten the password for my Gnome Keyring (yes, stupid me, I know). Any way I can get it back? I have already looked at , but it says it's not possible if my home directory is encrypted and unfortunately my home directory is encrypted. brute force password guessing on SSH server isn't going to work? I'm running an SSH server on my personal computer. The log had many people(or bots?) repeatedly trying to log in to my server (that is before I changed the default port), which made me a little freaked out. I was worried about their dictionary attacks or brute force password guessing.  But then if you type wrong password (or wrong username), the server waits for about 2 seconds before it asks for password again. So an attacker can try one password per 2 seconds. So I concluded that these attacks aren't that great a threat (as long as your password does not consist of dictionary words.) Am I correct? Add a "voted to close" activity event somewhere on the user profile page Often when I vote to close, I want to go back and check the comments on the question, to see if it's been improved, to comment further, or to answer the question.  It would be helpful to add a bit of a trail so I can find those questions again. I'm suggesting adding an event to the Activity page on the user profile, because this is the logical spot for the given action.  I realize the Activity page is public, but I don't see why this content couldn't be added only for the currently logged-in user.  An alternative would be to add a new tab on the user profile page that only lists current-user content, and is not shown publicly. I'm sure there are other events, data, or statistics that could be shown here as well, so I'm certain a new tab like this wouldn't be underutilized.    One thing that could be done instead of this is to always add a comment stating why I voted to close. This works well enough, but only some of the time -- for example, if there is already a vote to close as a duplicate and a comment with a link to that duplicate, I'm not going to add a comment myself if I have nothing to add aside from a close vote.  Another way to do it would be to favourite the question. This only works well to a point since questions that get voted to close don't necessarily end up being downvoted (i.e., for my "closed favourites," I couldn't just look at the last page of my favourites).    Related: Jon Skeet  that votes to close on your own questions be shown in the Recent Activity area, which is not what I'm requesting here. Colors and legend in groupplots/barplot In order to compare three items I use the groupplot command. For each item I have n different criteria to compare.    Is it possible to set the color for each criteria itself? Meaning the first bar in every group goes gray, the second pink etc.. The legend than should contain the n compared criteria.  And... how do I manage to let the whole tikz-picture have a width of 0.9\textwith  \documentclass{standalone} \usepackage{pgfplots,pgfplotstable} \usetikzlibrary{pgfplots.groupplots}  \pgfplotstableread{ nr  fii faa foo a   10  15  12 b   12  18  14 c   13  19  15 d   9   13  10 e   7   11  8 f   10  15  12 }\datatable  \begin{document} \begin{tikzpicture} \begin{groupplot}     [group style={columns=3},         ybar,         ymin=0,         ymax=20,         xtick=\empty,         tickpos=both]  \nextgroupplot[xlabel=fii] \addplot [fill=blue] table [x expr=\coordindex, y=fii] {\datatable}; \nextgroupplot[xlabel=faa] \addplot [fill=green] table [x expr=\coordindex, y=faa] {\datatable}; \nextgroupplot[xlabel=foo] \addplot [fill=red] table [x expr=\coordindex, y=foo] {\datatable};  \end{groupplot} \end{tikzpicture} \end{document} Is there such a thing as a Ubuntu Certified Engineer? From Wikipedia :     Suppose you're on a game show, and   you're given the choice of three   doors: Behind one door is a car;   behind the others, goats. You pick a   door, say No. 1, and the host, who   knows what's behind the doors, opens   another door, say No. 3, which has a   goat. He then says to you, "Do you   want to pick door No. 2?" Is it to   your advantage to switch your choice?   The answer is, of course, yes - but it's incredibly un-inituitive. What misunderstanding do most people have about probability that leads to us scratching our heads -- or better put; what general rule can we take away from this puzzle to better train our intuition in the future?