I'm getting interested in special relativity and almost all people and explanations about the subject I found on the internet seem to think that when A and B would be in relative motion, they would both observe the other's clock tick slower than their own. However what I found about this is following, in § 4. (Physical Meaning of the Equations Obtained in Respect to Moving Rigid Bodies and Moving Clocks) of Einstein's On the electrodynamics of moving bodies (1905):       From this there ensues the following peculiar consequence. If at the points A and B of K there are stationary clocks which, viewed in the stationary system, are synchronous; and if the clock at A is moved with the velocity $v$ along the line AB to B, then on its arrival at B the two clocks no longer synchronize, but the clock moved from A to B lags behind the other which has remained at B by $\frac{1}{2}tv^2/c^2$ (up to magnitudes of fourth and higher order), $t$ being the time occupied in the journey from A to B.   This sounds like complete opposite to what everyone seems to think.  Now this time discrepancy doesn't have anything to do with Doppler effects or isn't just some observers illusion. This is supposed to be physical fact, after stopping the traveler has physically aged less than the rest of the world.    The question: Why should the stationary clock seem slower to the traveller?    The quote i provided explicitly says that his own clock will have experienced less time. Please note that A and B can be arbitrarily close and there can be an infinite amount of such points for which the quote applies. Also the train need not stop to witness this effect   Therefore it seems to me certain that the traveller would observe the stationary clocks move faster.
Suppose we have two twins travelling away from each other, each twin moving at some speed $v$:    Twin $A$ observes twin $B$’s time to be dilated so his clock runs faster than twin $B$’s clock. But twin $B$ observes twin $A$’s time to be dilated so his clock runs faster than twin $A$’s clock. Each twin thinks their clock is running faster. How can this be? Isn’t this a paradox?
Why would spacetime curvature cause gravity? Do electrons have shape? According to :     The electron has no known substructure. Hence, it is defined or assumed to be a point particle with a point charge and no spatial extent.   Does point particle mean the particle should not have a shape, surface area or volume?  But when I searched Google for the "electron shape" I got many results (like  and ) that says electrons are round in shape. This one never ceases to confuse me. When to use 'such as' and when to use 'like' while giving examples? Is there any clear rule?        Metros like Mumbai, Delhi and Karachi are unsafe after dark.    Metros such as Mumbai, Delhi and Karachi are unsafe after dark.       (Before anyone points out, Delhi and Karachi are unsafe round-the-clock, but it's just an example.) Shell script to send an alert mail if disk usage exceeds 90% What is a use case where one would use an SObject as a Map key? $\|u\|\leq \|u+av\| \Longrightarrow \langle u,v\rangle=0$ Prove that $\langle u,v\rangle=0\Longleftrightarrow \|u\|\leq \|u+av\|$.  So far I can get the $\Longrightarrow$ very easily, but I need some help with the $\Longleftarrow$ implication, any hints would be greatly appreciated.  Please no answers, I just want a small nudge to get me in the right direction, I've been stumped on this for a little bit.  Here's what I have so far:  ($\Longleftarrow$)  Assume $\|u\| \leq \|u+av\|$  $\|u+av\|^2=\langle u+av,u+av\rangle=\langle u,u\rangle+ \langle u,av\rangle+\langle av,u\rangle+\langle v,v\rangle=\|u\|^2+\bar{a} \langle u,v\rangle+a\langle v,u\rangle+\|v\|^2=\|u\|^2+ \bar{a}\bar{\langle v,u\rangle}+a\langle v,u\rangle+\|v\|^2$  Thoughts/ideas? Is "tri-quarterly" a real English word meaning 3 times a year? Is "tri-quarterly" a real English word meaning 3 times a year? Are there any other words that mean 3 times a year? I'm an Indian student currently in Singapore as a Research Associate with a TEP valid for five months and have just been selected for an internship in Switzerland this summer.  Is it possible for me to obtain a Schengen visa from the Swiss Embassy in Singapore? Let $X,Y,Z$ be subspaces of $V$ so that $X$ is a subspace of $Y$. Prove that $Y\cap (X+Z)=X+(Y\cap Z)$ Let $X,Y,Z$ be subspaces of $V$ so that $X$ is a subspace of $Y$. Prove that $Y\cap (X+Z)=X+(Y\cap Z)$  I know that I need to prove that $Y\cap (X+Z)\subseteq X+(Y\cap Z)$ and $X+(Y\cap Z)\subseteq Y\cap (X+Z)$ but I´m having a hard time.  Can you lend me a hand please? I would really appreciate it I have trouble on this question: "Characterize the groups G for which the map $$\iota: G \to G$$ sending $$x \mapsto x^{-1}$$ for all $x \in G$, is an automorphism of $G$." From  post I have seen that $G$ must be abelian. Can anyone propose another characteristic of $G$ that I can derive from the given conditions besides abelian? Thank you so much!! sum of the residue at all poles of the rational function p/q is 0. We are given two polynomials p, q with deg(q)>1+deg(p). Show that the sum of the residue at all poles of the rational function p/q is 0.   How do I go about solving this? With regards to  post about  and the part about retaining reputation earned from deleted posts that have a score of 3 or better and were visible on the site for 60 days, does this also apply to self-deleted posts and will it continue to do so? I culled a few of my own answers over the weekend that were less than stellar, and my reputation dropped 1.8k, but the rep is back today after the changes were announced. I thought it would be useful to clarify that this also applies to self-deleted stuff. How to change text entry in Ubuntu 18.04 I was using Ubuntu 16.04. There it was so easy to change the text entry option. Settings > Text entry.  But when I switch to Ubuntu 18.04 I can't find such option. So how can I write in another language? How to change text entry i.e. by super+shift or something like that? How do I convert a SMP server world into a Single-player compatable world? I had a Minecraft world on Minefold.  I archived my Minecraft world just before Minefold closed about a year ago.  I'm trying now to open the world in my local Minecraft installation on OS X, but it doesn't show up in the worlds list.  To import it, I simply unzipped the archive from Minefold into my ~/Library/Application Support/minecraft/saves folder.  I'm running Minecraft 1.8, and I don't have any mods installed.  There's no error; the world just doesn't show up in my list of worlds after I click [Singleplayer] (or [Multiplayer]).  Now, the big clue is that the directory structures are different between my locally-created worlds and the world created on Minefold, but I can't figure out how to adapt the files to make them match:  This is a screenshot of my saves folder. I've circled in blue "local" and "minefold", which contain data files for the local and Minefold worlds, respectively.    I bracketed similar filenames in both worlds, but I'm flummoxed in that the Minefold version has three (3) sets of data files, whereas the local world has only one (1) set of files (even though I have visited Nether in the local world).  How do I load my old Minefold world locally? In the , the key $k = (a, b)$ should have the following property, i.e. GCD of $a$ and the modulus should be 1:     $\gcd (a, m) = 1$, where $m$ is the set of possible moduli   My question is, why it is required that $a$ should have the above property? What is the significance of this? And why is $b$ excluded from this condition? We are given two independent standard normal random variables $X$ and $Y$. We need to find out the M.G.F of $XY$.  I tried as follows : \begin{align} M_{XY}(t)&amp;=E\left(e^{(XY)t}\right)\\&amp;=\int_{- \infty}^{\infty}\int_{- \infty}^{\infty}e^{(XY)t}f_X(x)f_y(y)dxdy \\ &amp;=\dfrac{1}{2\pi}\int_{- \infty}^{\infty}\int_{- \infty}^{\infty}e^{(xy)t}e^{\frac{-x^2}{2}}e^{\frac{-y^2}{2}}dxdy \\ &amp;= \dfrac{1}{2\pi}\int_{- \infty}^{\infty}\int_{- \infty}^{\infty}e^{-\dfrac{(x-ty)^2}{2}}e^{\dfrac{-t^2y^2}{2}}e^{\frac{-y^2}{2}}dxdy \\ &amp;= \dfrac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty}e^{\dfrac{-t^2y^2}{2}}e^{\frac{-y^2}{2}}dy \\ &amp;= \dfrac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty}e^{-y^{2}(\frac{1}{2}+t^2)}dy \\ &amp;= \dfrac{1}{\sqrt{1+2t^2}} \end{align} Is this correct ? Schrödinger Equation for Two Electrons? 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I am able to show that $f'$ never vanishes by showing that $\vert f'-a_1\vert &lt;\vert a_1\vert$, but that doesn't imply that $f$ is injective. For example, $e^z$ is not inejective but its derivative never vanishes. What is required to prove that $f$ is injective? How do you show $f$ is injective on any set?