Why is the metric used in physics a squared norm? The length metrics used in physics are usually a squared norm, like the following: $$ds^2 = g_{\mu \nu} \, dx^{\mu} \, dx^{\nu}. \tag{1}$$  What other kinds of continuous metrics could we define?  Why not the following multi-linear candidates? \begin{align} ds &amp;= g_{\mu} \, dx^{\mu}, \tag{2} \\[1ex] ds^3 &amp;= g_{\mu \nu \lambda} \, dx^{\mu} \, dx^{\nu} \, dx^{\lambda}, \tag{3} \\[1ex] \vdots \\[1ex] ds^n &amp;= g_{\mu_1 \mu_2 \mu_3 \dots \mu_n} \, dx^{\mu_1} \, dx^{\mu_2} \, dx^{\mu_3} \dots dx^{\mu_n}. \tag{4} \end{align} What are the basic rules that "forces" physics to use the "squared" metric (1)?
Can I use a higher rank tensor field as the metric field? What a metric should necessarily do is it should give me a way to associate a frame invariant number with a given pair of spacetime events. Now, if I use a higher rank tensor field (say, for example, a tensor field of rank 3: $g_{\mu \nu \rho}$) then also I can certainly produce a frame invariant scalar out of a given displacement $\vec{A}$ in this trivial way: $ I := g_{\mu \nu \rho} A^{\mu}A^{\nu}A^{\rho}$. Since the components of the higher rank tensor and that of the displacement vector are going to transform in a covariant and a contravariant manner respectively, the cooked up quantity is certainly a scalar.   Another crucial property, which I think has more direct physical content than the previous one, is that between two inertial frames, at least one such transformation should exist that leaves at least one metric invariant. i.e., There should exist at least one combination of transformation matrix $\displaystyle\frac{\partial{x^{\alpha}}}{\partial{x^{\mu '}}}$ and metric $g_{\alpha \beta \gamma}$ that satisfies the following equation:      $\displaystyle\frac{\partial{x^{\alpha}}}{\partial{x^{\mu '}}}\displaystyle\frac{\partial{x^{\beta}}}{\partial{x^{\nu '}}}\displaystyle\frac{\partial{x^{\gamma}}}{\partial{x^{\rho '}}} g_{\alpha \beta \gamma} - \delta_{\mu '}^{\alpha}\delta_{\nu '}^{\beta}\delta_{\rho '}^{\gamma} g_{\alpha \beta \gamma} =0$    The last thing I can think of that can put a restriction on the choice of a tensor as a metric is the existence of a possibility of finding a metric compatible symmetric connection field. Following the usual procedure of finding the expression for a metric compatible symmetric connection field, I reached following condition (unlike the case of the usual two rank metric where we get a full-fledged expression) for the connection in the terms of the metric:      $g_{\nu \rho k_1} \Gamma^{k_1}_{\mu \lambda} - g_{\lambda \mu k_2} \Gamma^{k_2}_{\rho \nu} = \displaystyle\frac{1}{2} (\partial_{\nu}g_{\rho \lambda \mu} + \partial_{\rho}g_{\lambda \mu \nu}  - \partial_{\lambda}g_{\mu \nu \rho}  - \partial_{\mu}g_{\nu \rho \lambda}) $   My question is that if it is possible to satisfy the two highlighted conditions then can we use such a 3 rank (or even higher rank tensors with similarly produced conditions) tensor fields as metric fields?   PS: This is NOT a proposal for a new home-production theory of gravity (or that of anything for that matter) but rather it is just that I am trying to understand why a two rank tensor is used in General Relativity as the metric. Thank you.
Determinant of a specially structured matrix ($a$'s on the diagonal, all other entries equal to $b$) I have a program which I am trying to run, however when I run it; it just complains that it can't find a particular file.  However I have no idea which folder it is trying to find this particular file in. I have a copy of the required file, I just need to know which folder to copy it too.  Is there any way to show in real time which files are being accessed or which files are trying and failing to be accessed?  I am using Ext4 filesystem if that helps.  Thanks Reverse String order Can I travel to Schengen countries that were not on my itinerary when I applied? How can I find local IP addresses (i.e. 192.168.x.x or 10.0.x.x) in Python platform independently and using only the standard library? How does system testing differ from System Integration Testing (SIT)? I was asked this question during an interview. My thoughts are SIT is performed on a complete systems and System Testing can be done on a single module as well. There is a socket related function call in my code, that function is from another module thus out of my control, the problem is that it blocks for hours occasionally, which is totally unacceptable, How can I limit the function execution time from my code? I guess the solution must utilize another thread. How to prepare data for ListVectorPlot[]? Neural network - meaning of weights Firefox 33 doesn't see Java Plugin I am trying to enable Java (not JavaScript) in my Firefox v. 33 so that I can use applets within the browser.  Currently, I just get a dark gray box across the whole screen when I try to load an applet, without any error message or prompt to install plugin.  As per , I tried going into the Firefox plugin page to activate it, however, it wasn't there listed.  I do have Java 7 installed and it is in the path:  &gt;java -version java version "1.7.0_71" Java(TM) SE Runtime Environment (build 1.7.0_71-b14) Java HotSpot(TM) 64-Bit Server VM (build 24.71-b01, mixed mode)   What do I need to do to get Firefox to see my Java installation and recognize it as a plugin so that I can enable it and view applets inside the browser? For example: $$7x \equiv 1 \pmod{31} $$ In this example, the modular inverse of $7$ with respect to $31$ is $9$. How can we find out that $9$? What are the steps that I need to do? Update If I have a general modulo equation: $$5x + 1 \equiv 2 \pmod{6}$$ What is the fastest way to solve it? My initial thought was: $$5x + 1 \equiv 2 \pmod{6}$$ $$\Leftrightarrow 5x + 1 - 1\equiv 2 - 1 \pmod{6}$$ $$\Leftrightarrow 5x \equiv 1 \pmod{6}$$ Then solve for the inverse of $5$ modulo 6. Is it a right approach? Thanks. For you that have tested this, should we use Ubuntu and install different Desktop Environments (KDE/XFCE...) or we should install different distributions (Kubuntu/Xubuntu...). Are there any notable issues in performance / CPU consumption?  PS:  I know how to install them (already did), I wanted to know if anyone has any problems regarding the performance and what is better to install the desktop or to use the ditribution What methods can I use to create balance and consistency between a group of differing logos? A regular challenge that I face is balancing two (or more) logos. Whether they need to be equal or one needs to be given priority. Some logos however might be squares, others rectangles, circles, etc...   As a quick sample I snagged a nice high resolution photo from  and pretend its going to be an advertisement that needs to show some footwear brand logos on it. In this case I used Adidas and Saucony.  This is certainly not balanced even though they're height is the same:    Likewise this is not balanced even though they're width is the same:    So then the question becomes when are they in balance, is this balance:    Maybe a little more adjustment, is this balance:    And in these examples I'm only working with two Black Logos. Factor in color and it can get even more difficult to balance. Another issue being when there's tiny text as part of the logo.  So the question is, other than estimating / eye-balling it, what tips, methods or even tools are there to balance different sized logos? I heard the following dialog in a British English movie:     Words go round quick. All the little tongues go clack, clack, clack.   What does this mean, and why was the word "quick" used instead of "quickly"? How do I start Nautilus as root? I've got a problem with Nautilus in Ubuntu 12.04 LTS.  When I run the command gksu nautilus /, the password ask box appears and I type my super-user password, then the password box disappears and Nautilus is not opened.  When I try to open Nautilus as a normal user in with the command nautilus, the folder is opened.  When I try in terminal   $ sudo nautilus   I got the following error  $ sudo nautilus /  ** (nautilus:8523): WARNING **: Command line `dbus-launch --autolaunch=2c8ce9b7da2257c2609b749700000007 --binary-syntax --close-stderr' exited with non-zero exit status 1: Autolaunch error: X11 initialization failed.\n Could not parse arguments: Cannot open display:   How can I fix this error? Floor function in complex plane My question is very simple. How would one take the floor of a complex number and what is the floor of i? I understand that the  is given by  $$F^{\mu\nu}\mapsto\begin{pmatrix}0 &amp; -E_{x} &amp; -E_{y} &amp; -E_{z}\\ E_{x} &amp; 0 &amp; -B_{z} &amp; B_{y}\\ E_{y} &amp; B_{z} &amp; 0 &amp; -B_{x}\\ E_{z} &amp; -B_{y} &amp; B_{x} &amp; 0 \end{pmatrix}$$  where $\mu$, $\nu$ can take the values {0,1,2,3} or {$t,x,y,z$}.  So, for example  $$F^{01}=F^{tx}=-E_x$$  My question is, what would the following expression be?  $$F^{t\rho}=?$$ or $$F^{z\rho}=?$$  where $\rho=\sqrt{x^{2}+y^{2}}$ is the radial coordinate in ?  And more generally, how can we construct the Electromagnetic Tensor in cylindrical coordinates? Where $\mu$, $\nu$ now take the values {$t,\rho,\varphi,z$}. Add space between headings and body in the home of the iPad app It's a pain in the ass for me to quit the app, draw a red circle in another app and return to the app, so...    If you notice the three questions in the bottom, you will notice that the emboldened headings (New Answer, Comment) are stick to the next character. Imperfect.    App Version: 1.4.0.249 Device: iPad 2 (GSM) OS Version: Version 9.1 (Build 13B143) Why is Sauron called "the Necromancer"? During the time period of The Hobbit, a character is mentioned called the Necromancer.  Radagast even (in the movie) specifically states that the Necromancer can "summon the spirits of the dead."  (It's been a while since I read the book so I don't know if this exact phrasing is used there, but given the name it seems reasonable.) This character is later revealed to be Sauron.  Since when can Sauron summon the spirits of the dead?  He can extend life ad infinitum, like Gollum's, for example, and he can twist people into half-alive monsters, like the Ringwraiths, and he himself seems impossible to kill so long as the One Ring remains, but I can't recall ever actually seeing him "summon" something that was once fully dead.  In fact, the only character I remember doing that was actually Aragorn, though he merely commanded them and didn't raise them himself.  Is this just an unfounded rumor about the power in Dol Guldur, some hushed-whisper nightmare that isn't actually an accurate description, or is there a reason he is called the Necromancer?  It seems like if Sauron could actually resurrect the dead, what little chance Middle Earth had of defeating his hordes would be long, long gone... What would happen if every person in the world gathered at a minimum possible area and jumped all at once? Would that have any effect on Earth?