Sometimes the stress-energy tensor $T_1^{\mu\nu}$ is defined as $$T_1^{\mu\nu}=\frac{\partial \mathcal{L}}{\partial(\partial_\mu\phi)}(\partial^\nu\phi)-\eta^{\mu\nu}\mathcal{L}$$ which for translational symmetries of spacetime satisfies $\partial_\mu T_1^{\mu\nu}=0.$ It is not always symmetric. In general relativity, stress-energy tensor is defined as $$T_2^{\mu\nu}=\frac{\delta S}{\delta g_{\mu\nu}}$$ where $S=\int d^4x\sqrt{-g}\mathcal{L}(\phi,\partial_\mu\phi)$ is the action of the theory. It is always symmetric because the metric tensor $g_{\mu\nu}$ is always symmetric.   Can we say that the second definition is more general? If so, can we derive the $T_1^{\mu\nu}$ starting from the second expression $T_2^{\mu\nu}$ in the flat spacetime limit?   Please keep the discussion elementary.
What is the correspondence between the conserved canonical energy-momentum tensor, which is $$ T^{\mu\nu}_{can} := \sum_{i=1}^N\frac{\delta\mathcal{L}_{Matter}}{\delta(\partial_\mu f_i)}\partial^\nu f_i - \eta^{\mu\nu}\mathcal{L}$$ (the four conserved Noether currents corresponding to four possible spacetime translations)  where $\{f_i\}_{i=1}^N$ are the $N$ matter fields in the theory, and we assume $f_i\mapsto\alpha^\nu\partial_\nu f_i$ for translations,  and stress-energy tensor from the Einstein-Hilbert action, which is: $$ T_{\mu\nu}\equiv-\frac{2}{\sqrt{-g}}\frac{\delta\mathcal{L}_{Matter}}{\delta g^{\mu\nu}} $$   In particular, how do you get that the two are equal (are they?) for Minkowski space, for which there is no variation in the metric?
Is it possible to have a rate of change of acceleration? I know this may seem a weird question, but it always bothers me. My physics book (Resnick,Halliday,Walker), and also various sites never say anything beyond acceleration.   But when a moving body is being acted by a variable force , its acceleration will definitely change: it will either increase or decrease. Then there will be rate of change of acceleration with respect to time. So, why don't books mention this? What is the cause for not measuring $\frac{d\vec{a}}{dt}$ ? If it exists, what is the use of it? Upon the risk of being down voted as no MWE is present, but at the moment just trying to get some pointers. I updated my MiKTeX installation to the newest available version at the moment (MiKTeX-pdfTeX 4.1 (MiKTeX 20.12)) and now I get an error while building my documentation: Appendix D. (../html/examples/group/latex/refman_doc.tex (../html/examples/group/latex//gro up__group1.tex) [243] ! I can't write on file `../html/examples/group/latex//group__group2.aux'. \@include ...mmediate \openout \@partaux &quot;#1.aux&quot;                                                   \immediate \write \@partau... l.3 \include{group__group2}  Please type another output file name ! Emergency stop. \@include ...mmediate \openout \@partaux &quot;#1.aux&quot;                                                   \immediate \write \@partau... l.3 \include{group__group2}  *** (job aborted, file error in nonstop mode)  some small background:  Started normally with pdflatex doxygen_manual.tex in here is the statement: \subinputfrom{../html/examples/group/latex/}{refman_doc} and in this file again: \include{group__group2} as far as I know the disk is not full and the protections of the files and directories are OK as well  It looks like that now an attempt is made to write to the directory ../html/examples/group/latex/ whilst in the past the file was written to the current directory. I already tried -shell-escape and -enable-write18 (as well as --shell-escape and --enable-write18) but to no avail.  Any known changes in respect to writing the aux file? Any hints  edit I asked another question () with the details and MWE but that was closed with the reason that it was a duplicate of this one ... So I'm adding the information now here: In the question  asked about the problems I encountered due to an update of my MiKTeX distribution, but didn't take the time (at that moment) to create a MWE, so it didn't attract much response. I have the directory structure:  main  manual.tex   group  group__group2.tex refman_doc.tex    manual.tex \documentclass{book} \usepackage{import} \usepackage{hyperref}  \begin{document} \subinputfrom{../group}{refman_doc} \end{document}  group__group2.tex \hypertarget{group__group2}{}{The Second Group}  refman_doc.tex {Module Documentation} \include{group__group2}  When running in the directory main the command pdflatex manual.tex I get the error: ... (../group/../group/refman_doc.tex [1{C:/Users/User/AppData/Local/MiKTeX/2.9/pdf tex/config/pdftex.map}] ! I can't write on file `../group//group__group2.aux'. \@include ...mmediate \openout \@partaux &quot;#1.aux&quot;                                                   \immediate \write \@partau... l.2 \include{group__group2}  Please type another output file name:  but in a non interactive mode (the usual way to run. i.e. with \batchmode) one doesn't get an opportunity to specify the file and the generation crashes. As written in the original question adding options like -shell-escape and / or -enable-write18 didn't help.  Is there a solution for this problem?  edit 2 I did some further test. I replaced the \subinputfrom with an \input statement, although this worked for the limited example above it didn't work for my larger codebase as here (again just a limited view) we have in the refman_doc.tex {Module Documentation} \input{group__group1} \include{group__group2}  with the contents of group_group1.tex similar to group__group2.tex and the file is also parallel to group__group2.tex. This results in the error: (manual.out) (manual.out) (../group/refman_doc.tex  ! LaTeX Error: File `group__group1.tex' not found.  Type X to quit or &lt;RETURN&gt; to proceed, or enter new name. (Default extension: tex)  Enter file name: ! Emergency stop. &lt;read *&gt;  l.2 \input{group__group1}  !  ==&gt; Fatal error occurred, no output PDF file produced! Exercise from Atiyah-Macdonald, Chapter 1, 2.iv) Let $A$ be a ring and let $A[x]$ be the ring of polynomials in an indeterminate $x,$ with coefficients in $A.$ Let $f=a_0 + a_1x+\cdots+a_nx^n \in A[x].$ $f$ is said to be primitive if $(a_0,a_1,\ldots,a_n)=(1).$     Prove that if $f,g\in A[x],$ then $f$ and $g$ are primitive $\Rightarrow$ $fg$ is primitive.   I know Gauss lemma holds true for UFDs, but in this case it isn't required $A$ to be UFD: is it possible to drop the assumption, or is some hypothesis missing here? I want to prove:  $$\int_{0}^{2\pi}\frac{1}{a^{2}\cos^2t+b^{2}\sin^{2}t}dt=\frac{2\pi}{ab}$$  by the deformation theorem of complex variable.  Then I consider a parameterization $\gamma:[0,2\pi]\rightarrow A$, traveled in the opposite direction of the clock hand, of the ellipse (in $\mathbb{C}$):  $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$  I thought of searching for a function $f:A\rightarrow \mathbb{C}$ (analytic in $A$) and a curve $\lambda:[0,2\pi]\rightarrow A$ (homotopic to $\gamma$) and use the deformation theorem:  $$\int_{\gamma}f = \int_{\lambda}f$$  But I can not find a function $f(z)$ that meets what I need, can you help me to find the f?   P.D: Necessarily I have to use the theorem of deformation Convert LaTeX to HTML Is there a converter that will convert LaTeX to HTML? I have tried several services but the LaTeX diagrams and some of the maths came out weird.  I actually converted the Latex to PDF without a problem. Then I used  to convert from PDF to HTML. The square root signs did not appear properly and some of the Cartesian coordinate plane diagrams had nothing on them except the grid.   \documentclass[12pt,a4paper]{report} \usepackage{amssymb} \usepackage{amsthm} \usepackage{amsmath} \usepackage{pstricks,pst-plot} \usepackage{pst-eucl} \addtolength{\hoffset}{0.2cm} \addtolength{\textwidth}{1.1cm} \addtolength{\textheight}{1.0cm} \addtolength{\voffset}{-0.7cm} \linespread{1.3} \begin{document}  If $x$ is non-positive then $x \leq -\sqrt{\frac{1}{2}}$     $\hspace{30 mm} $     (6)\\  (4) and (6) tell us that\\  (4)  $-1 \leq x \leq 0$  \\  (6)  If $x$ is non-positive then $x \leq -\sqrt{\frac{1}{2}}$\\  Therefore $-1 \leq x \leq -\sqrt{\frac{1}{2}}$\\  \vspace{50 mm}  \begin{align*} \psgrid[unit=2cm](0,0)(-2,-2)(2,2) \pnode(-1.8,.87){b}\uput[50](b){b} \pnode(-1.8,1.8){a}\uput[50](a){a} \psset{linecolor=green} \psline[linewidth=1pt](-1.8, 0)(-1.8,4) \psset{linecolor=green} \psline[linewidth=1pt](-1.8, 0)(-1.8,-4) \psset{linecolor=blue} \psellipticarc(0,0)(2,2){0}{180} \psset{linecolor=red} \psline[linewidth=1pt,linestyle=dashed,dash=2pt](-2,2)(-1.41,1.41) \psset{linecolor=red} \psline[linewidth=1pt](-1.41,1.41) \psset{linecolor=red} \psline[linewidth=1pt]{-&gt;}(4,-4) \psset{linecolor=red} \psline[linewidth=1pt]{-&gt;}(-2,2)(-4,4) \end{align*}  \end{document}   PDF output:    PDF->HTML output: I'm not sure how to solve this problem. Can you guys provide some input/hints?      Let $A=(x_1,y_1)$, $B=(x_2,y_2)$ and $C=(x_3,y_3)$ be three points in $\mathbb{R}^{2}$.      Show that the area of $\Delta ABC$ is given by   $\frac{1}{2}\left| \det(M) \right|$, where   $$M = \begin{pmatrix} 1 &amp; 1  &amp; 1\\  x_1 &amp; x_2 &amp; x_3\\  y_1 &amp; y_2 &amp; y_3 \end{pmatrix}$$ How do I split a string by a multi-character delimiter in C#? What is wrong with this query? (Unknown error parsing query) Does using an extender actually change the aperture of the lens? I just bought a Canon 2X extender and I always believed that it would only result in 2 stops' loss of light and not change the aperture.    However, when I tested this extender on a Canon 135mm f/2 lens, the maximum aperture I could set went down to F/4.  So it kind of confused me, because if I step down the aperture to f/5.6, is it actually setting the aperture to f/5.6 or f/2.8 (i.e., 1 stop below f/2)? Twitter Bootstrap uses these break points for responsive design:  Large Display: 1200px Default: 980px Tablet: 768px Mobile: 480px  A client said they are using 1280px for their Large Display breakpoint.  Is there an industry standard for these or are they just made up as we go right now? According to general relativity, inertial mass and gravitational mass are the same, and all accelerated reference frames (such as a uniformly rotating reference frame with its proper time dilation) are physically equivalent to a gravitational field of the same strength.  Refer: Einstein's thought experiment of a Physicist in an Accelerated Box.  I find that the 'How' part of this question is somewhat easy to understand Mathematically but hard to visualize physically and on the 'Why' part I am totally blank. "pros and cons", "black and white", "ups and downs". Always in a fixed sequence, is there a word or phrase for these? Is there a word or phrase for two nouns or adjectives joined by a conjunction (usually "and") in a fixed sequence?        alive and well           fast and furious   hat and gloves   pen and pencil   law and order   wit and wisdom   salt and pepper   bacon and eggs   horse and carriage   Adam and Eve   ladies and gentlemen   loud and clear   heads or tails   yes or no   dead or alive Favorite questions in the Android app I haven't been able to find my favorite questions in the official Stack Exchange Android app.  Looking at my accounts page gives nothing on them. How to perform a true GIS clip of polygons layer using a polygon layer in R? Can inductor voltage and capacitor current change abruptly? I understand that inductor current and capacitor voltage cannot change abruptly, but can inductor voltage and capacitor current change abruptly?  I have a feeling the answer is no but I cannot explain why. Calculating $\sum_{n=1}^\infty\frac{1}{(n-1)!(n+1)}$ I want to calculate the sum:$$\sum_{n=1}^\infty\frac{1}{(n-1)!(n+1)}=$$  $$\sum_{n=1}^\infty\frac{n}{(n+1)!}=$$ $$\sum_{n=1}^\infty\frac{n+1-1}{(n+1)!}=$$ $$\sum_{n=1}^\infty\frac{n+1}{(n+1)!}-\sum_{n=1}^\infty\frac{1}{(n+1)!}=$$ $$\sum_{n=1}^\infty\frac{1}{n!}-\sum_{n=1}^\infty\frac{1}{(n+1)!}.$$  I know that $$\sum_{n=0}^\infty\frac{1}{n!}=e$$  so $$\sum_{n=1}^\infty\frac{1}{n!}=\sum_{n=0}^\infty\frac{1}{n!}-1=e-1$$  But, what can I do for $$\sum_{n=1}^\infty\frac{1}{(n+1)!}$$ ?  Am I allowed to start a sum for $n=-1$ ? How can I bring to a something similar to $$\sum_{n=0}^\infty\frac{1}{n!}$$? 64 bit enum in C++? Is there a way to have a 64 bit enum in C++?  Whilst refactoring some code I came across bunch of #defines which would be better as an enum, but being greater than 32 bit causes the compiler to error.  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Change default XML comment snippet in Visual Studio When I hit /// in Visual Studio, is it possible to change the resulting snippet from this:  /// &lt;summary&gt; ///  /// &lt;/summary&gt;   to this?:  /// &lt;summary&gt;&lt;/summary&gt;