I have seen a few posts here on stack exchange that suggest that energy is not conserved in general relativity on a large scale. I am a pretty confused with this. I don’t understand how such a conclusion is reached. $\nabla_\mu T^{\mu\nu}$ $=0$ Doesn’t the above equation signify that energy momentum is conserved in GR. Please note there are other questions already on such issues. But they seem to suggest that energy is not conserved in GR. See for eg:  But I am asking that when we have $\nabla_\mu T^{\mu\nu}$ $=0$ why will energy not be conserved in the first place itself Edit after the comment: I am asking why $\nabla_\mu T^{\mu\nu}$ $=0$ does not imply energy conservation
For example, the radiation dominated cosmology, the energy density of radiation is proportional to $a^{-4}$ and the volume is proportional to $a^3$, where $a$ is the scale factor. So the total energy of radiation is propotional to $a^{-1}$. So where is the loss of energy of radiation? Is it because the gravitational field has the energy? Does $\nabla_aT^{ab}_{\rm matter}=0$ represent the conservation of energy and momentum of matter field in GR?
Is the C# "explicit implementation" of the interface present in Java? In C#, if you have two base interfaces with the same method (say, F()) you can use explicit implementation to perform different impl. for F(). This alloes you to differently treat the object, corresponding to the current point of view: as IMyInterface1 or IMyInterface2. Is this possible in Java? How can I install a .bin file? Simple as the title states: Can you use only Java commands to take a screenshot and save it?  Or, do I need to use an OS specific program to take the screenshot and then grab it off the clipboard? Recently, the titles to some questions have been displaying as "undefined":    "undefined" is replaced with the question's true title upon refreshing the page. Is this a bug? Custom Layout in android I would like to convert a large LaTeX project (i.e. spanning multiple files) into plain text. The purpose is generation of statistics, so representing mathematics is not an issue. In fact, all mathematics is ideally ignored.  I have found  but could not get it to run. It seems unfinished, but is exactly what I am looking for otherwise. What are the most important things I need to do to encourage Google Sitelinks? R: loading a table with number as the column heading all were wrongly added a X, how to solve it? How do I change my username? The overly simplified explanation I'm giving myself right now is  causes the opposite of what gravity does, that's why the universe is expanding. Now where gravity is a force, why dark energy is "energy"? Why it's not called "Dark Force" instead? I think I must be missing something here. Is $2^{|\mathbb{N}|} = |\mathbb{R}|$? I run a job every minute to reindex my site's content.   Today, the search engine died, and when I logged in there were hundreds of orphan processes that had been started by cron.  Is there another way using some kind of existing software that will let me execute a job every minute, but that won't launch another instance if that job doesn't return (i.e. because the search engine process has failed)? Inversion in "only [adverb] have they" I have seen this construction quite often:      Online ads have been around since the dawn of the Web, but only in   recent years have they become the rapturous life dream of Silicon   Valley.   What is the rule there?. When your sentence doesn't start with pronoun + verb, invert them as verb + pronoun?. I know it sounds awkward but is it possible (grammatically correct) to use something similar to:      Online ads have been around since the dawn of the Web, but only in   recent years they have become...   And in any case, does this only work with have (or has)? Maybe it works fine with 'had' but I can't think of an example right now. Bias and variance in leave-one-out vs K-fold cross validation How do  methods compare in terms of model variance and bias?   My question is partly motivated by this thread: . The answer there suggests that models learned with leave-one-out cross-validation have higher variance than those learned with regular $K$-fold cross-validation, making leave-one-out CV a worse choice.  However, my intuition tells me that in leave-one-out CV one should see relatively lower variance between models than in the $K$-fold CV, since we are only shifting one data point across folds and therefore the training sets between folds overlap substantially.  Or going in the other direction, if $K$ is low in the $K$-fold CV, the training sets would be quite different across folds, and the resulting models are more likely to be different (hence higher variance).  If the above argument is right, why would models learned with leave-one-out CV have higher variance? Two different solutions to integral Given the very simple integral  \begin{equation} \int -\frac{1}{2x} dx \end{equation}  The obvious solution is  \begin{equation} \int -\frac{1}{2x} dx = -\frac{1}{2} \int \frac{1}{x} dx = -\frac{1}{2} \ln{|x|} + C \end{equation}  However, by the following integration rule \begin{equation} \int \frac{1}{ax + b} dx = \frac{1}{a} \ln{|ax + b|} + C \end{equation}  the following solution is obtained \begin{equation} \int -\frac{1}{2x} dx = -\frac{1}{2}\ln{|-2x|} + C \end{equation}  Why are these solutions different? Which is correct?   The second solution can be simplified \begin{equation} -\frac{1}{2}\ln{|-2x|} + C = -\frac{1}{2}\ln{|-2|} -\frac{1}{2}\ln{|x|} + C= -\ln{\frac{1}{\sqrt{2}}} - \frac{1}{2}\ln{|x|} + C \end{equation} but they still differ. (This is on NT based Windows like 2000, XP, Vista, 7)  On the command-line, the percentage characters are ignored:  wget "http://www.justitie.nl/images/Handleiding%20voor%20verwerkers%20persoonsgegevens_tcm34-3940.pdf"   So it correctly downloads this file (each %20 becomes a space):  "http://www.justitie.nl/images/Handleiding voor verwerkers persoonsgegevens_tcm34-3940.pdf"   But inside a batchfile, all the %20 are being interpreted as expanding parameter 2 (which is empty) resulting in this file to get downloaded (each %20 becomes 0):  "http://www.justitie.nl/images/Handleiding0voor0verwerkers0persoonsgegevens_tcm34-3940.pdf"   Is there a way to circumvent the percentage parameter expansion?  --jeroen How to concatenate multiple lines of output to one line? List of lists changes reflected across sublists unexpectedly I needed to create a list of lists in Python, so I typed the following: my_list = [[1] * 4] * 3  The list looked like this: [[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1]]    Then I changed one of the innermost values: my_list[0][0] = 5  Now my list looks like this: [[5, 1, 1, 1], [5, 1, 1, 1], [5, 1, 1, 1]]    which is not what I wanted or expected. Can someone please explain what's going on, and how to get around it? Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$. Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G   has subgroup of order $\operatorname{lcm}(n,m)$.   This is a statement that my lecturer mentioned in my (beginners') Abstract Algebra class. I'm not sure I understand why it's true.  What I have so far: Use the abelian group structure theorem on $\langle H, K\rangle$ (finite group generated by $H$ and $K$). Then $\langle H,K\rangle=C_{a_1}C_{a_2}\dotsm$ and $n, m |\langle H,K\rangle$. Which means it's also true that $\operatorname{lcm}(n,m)|\langle H,K\rangle$. Can I leverage this to say there's a subgroup of order $\operatorname{lcm}(n,m)$? This is NOT an installation problem question. I've had both op sys working fine side-by-side for a couple of weeks now. I'm using Ubunutu 12.04 64-bit.  This morning when I tried to switch from Win8 to Ubuntu, it just rebooted into Win8. No grub screen whatsoever. This is the 3rd machine I've had dual-booting &amp; have never seen this, but then Win8 is a 1st for me &amp; has thrown some pretty awesome blocks in the path to get to here. Atrocious op sys...  I'm assuming the ASUS updater did this yesterday. (ASUS R503U)  So, do I just put the Ubuntu disk in &amp; reinstall grub?