So my argument goes like this:  say we have an object that is nearly at the density required for a black hole.  Then we Lorentz boost into a new frame, the object is Lorentz contracted and therefore the density increases above the black hole limit.  As a mere change of frame can't have physical effects it surely can't create a black hole and therefore a moving black hole needs a larger density.  Is this correct and if it is how does this come out of General Relativity?
I'm a big fan of the podcast Astronomy Cast and a while back I was listening to a Q&amp;A episode they did. A listener sent in a question that I found fascinating and have been wondering about ever since.   From the show :      Arunus Gidgowdusk from Lithuania asks: "If you took a one kilogram mass and accelerated it close to the speed of light would it form into a black hole? Would it stay a black hole if you then decreased the speed?"   Dr. Gay, an astrophysicist and one of the hosts, explained that she'd asked a number of her colleagues and that none of them could provide a satisfactory answer. I asked her more recently on Facebook if anyone had come forward with one and she said they had not.  So I thought maybe this would be a good place to ask.
I started thinking about the number of associative (binary) operations on a set with $n$ elements today.  Looking online I found  which indicates only $113$ of the possible $19,683$ operations on a three-element set satisfy association.  So, $50\%$ of binary operations on a two-element set satisfy association, and less than $0.6\%$ of all binary operations on a three element set satisfy association.  For an $n$-element set, where $n$ denotes a natural number, is the ratio $R_{n}=A_{n}/B_{n}$, of the number of associative binary operations $A_{n}$ to the number of all binary operations $B_{n}$, in general small?  I don't mean the following questions as equivalent, but since they seem more concretely answerable in principle, does   $$ \lim_{n \to +\infty} \frac{A_{n}}{B_{n}} = 0 ? $$       Also, is $F : n \to R_{n}$ a monotonically decreasing function?  Some Background of the Question: In his Linear Algebra Problem Book 1995 on p. 6 Paul Halmos writes "The commonly accepted attitudes toward the commutative law and the associative law are different.  Many real life operations fail to commute; the mathematical community has learned to live with that fact and even to enjoy it.  Violations of the associative law, on the other hand, are usually considered by specialists only."  For all I know, Halmos might only have written that to motivate the study of Linear Algebra and doesn't quite literally mean what he appears to say.  But, if he means what he appears to say, and if $F : n \to R_{n}$ is monotonically decreasing, or $R_{n}$ is generally small, I think there's something amiss with what Halmos says, since non-associative operations seem so common that one may as well enjoy them. Why no tag Badges for Good questions Why not make  but for questions too?  I love that everything is well though behind SO but I can't see no reason as to why great questions can't be prized with "Badges by subject". Why does the inverse of the Hilbert matrix have integer entries? Mention a specific potential adviser and project in the PhD statement of purpose I am writing my statement of purpose for my Ph.D. and I want to work in a specific field with a specific professor in the department that I am going to apply for and I have a very good experience in this field. I sent an email for that professor asking if he/she is accepting Ph.D. students and he/she did not respond. I built my whole statement of purpose based on my previous achievements and future plan for that specific area. I also mentioned that I am willing to work with that professor. My statement of purpose tells that I am very passionate about this field and I am only willing to work with that professor (if there were more than one professor working on the same area, I would mention all of them). if that professor doesn't accept new Ph.D. students, will I risk my admission to that school?   Will the committee response be something like this:  "the applicant really knows what he/she is going to do and there is a specific future plan and he/she also got a good experience in this field, but he/she sounds very strict to his plan and doesn't show any flexibility...REJECTED"? How can I go about determining the number of unique simple paths within an undirected graph? Either for a certain length, or a range of acceptable lengths.  Recall that a simple path is a path with no cycles, so I'm talking about counting the number of paths with no cycle. In a  of Amstat News, the authors (Mark van der Laan and Sherri Rose) stated that "We know that for large enough sample sizes, every study—including ones in which the null hypothesis of no effect is true — will declare a statistically significant effect.".  Well, I for one didn't know that. Is this true? Does it mean that hypothesis testing is worthless for large data sets? Change templates in Xcode How would I change the initial templates created by Xcode when creating a new Cocoa Class.  I am referring to the comments and class name created when using Xcode's new class wizard. Arrays, linked lists and other data structures in cmd.exe (batch) script Who is / was the "Lord of the Rings"? Who is the Lord, as referred to in the title? Frodo? Sauron? Someone else? Or is it more of an abstract title, not referring to a single person, like "He who wields the One Ring"? I have a Dell Studio 17. It's a few years old, and I wanted to upgrade the processor. It currently has an Intel Core 2 Duo T5750 @ 2.00GHz  My question is how can I find out which processors are compatible? Assuming this is on the way, but having not seen it previously reported: I can't view myself from the sidebar, or anyone else by tapping them. View Self From Sidebar:   View Asker/Answerer:   View Commenter: Why can't root on one machine change nfs mounted content from another machine? On my NFS server, I have the following export defined:  #NFS exports Database /shared -alldirs -network=192.168.1 -mask=255.255.255.0   On my NFS client:  192.168.1.7:/shared /shared nfs rw 0 0   Obviously, as root on the server, I can do whatever I want.  On the client however, my regular user 'gabe' can make changes to the nfs mount (assuming I have permissions to), but root cannot.    As my regular user:   gabe@client$ cd /shared gabe@client$ ls -l total 8 drwxrwxrwx  4 gabe  wheel   512 Mar 20 19:20 tmp gabe@client$ cd tmp gabe@client$ touch test.txt gabe@client$ rm test.txt   As root:  # cd /shared/tmp # touch test.txt touch: test.txt: Permission denied   Again, this is all on the NFS client side of things, and I suspect perhaps it has something to do with the -maproot option.  This is the first time I'm setting up NFS and I just noticed this peculiarity.  I'm going to do some reading now, to see if I can figure this out, but if anyone has any insight, I would appreciate it. How to make Portable QGIS 3 Error while updating debug log trace flag I am trying to update the debug log trace flag for my user but getting the following error     Having an active trace flag triggers debug logging. You have 259 MB of the maximum 250 MB of debug logs. Before you can edit trace flags, delete some debug logs.   I have removed all existing debug logs but it is still throwing the error. I'm referring to the colored circles with numbers next to them on the top of the page, to the right of my reputation counter. What do they represent? Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$ I've been looking at  $$\int\limits_0^\infty  {\frac{{{x^n}}}{{1 + {x^m}}}dx }$$  It seems that it always evaluates in terms of $\sin X$ and $\pi$, where $X$ is to be determined. For example:  $$\displaystyle \int\limits_0^\infty  {\frac{{{x^1}}}{{1 + {x^3}}}dx = } \frac{\pi }{3}\frac{1}{{\sin \frac{\pi }{3}}} = \frac{{2\pi }}{{3\sqrt 3 }}$$  $$\int\limits_0^\infty  {\frac{{{x^1}}}{{1 + {x^4}}}dx = } \frac{\pi }{4}$$  $$\int\limits_0^\infty  {\frac{{{x^2}}}{{1 + {x^5}}}dx = } \frac{\pi }{5}\frac{1}{{\sin \frac{{2\pi }}{5}}}$$  So I guess there must be a closed form - the use of $\Gamma(x)\Gamma(1-x)$ first comess to my mind because of the $\dfrac{{\pi x}}{{\sin \pi x}}$ appearing. Note that the arguments are always the ratio of the exponents, like $\dfrac{1}{4}$, $\dfrac{1}{3}$ and $\dfrac{2}{5}$. Is there any way of finding it? I'll work on it and update with any ideas.    UPDATE:  The integral reduces to finding   $$\int\limits_{ - \infty }^\infty  {\frac{{{e^{a t}}}}{{{e^t} + 1}}dt} $$  With $a =\dfrac{n+1}{m}$ which converges only if   $$0 &lt; a &lt; 1$$  Using series I find the solution is     $$\sum\limits_{k =  - \infty }^\infty  {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}}} $$ Render looks darker than Look Dev mode I have been texture painting in look dev mode. But when i try to render it, everything looks very dark. I have 3 sun lights in my scene, ambient turned on in Eevee settings. What am i missing? I really like the look of my scene in Look Dev mode, how can i achieve the same look when my scene is rendered too? How are multiple volunteers handled during the Reaping? While skimming through the first few chapters of The Hunger Games to answer another question, I noticed this sentence when Katniss is describing the rules surrounding volunteers after Prim's name is called (at the beginning of the second chapter):     In some districts, in which winning the reaping is such a great honor, people are eager to risk their lives, the volunteering is complicated.   The only reason I can see for volunteering being complicated is if there's more than one volunteer for a particular year, and I don't recall this ever being addressed in the books.  Is there anything in the books that I've missed or any reliable, external sources (interviews or other supplementary material by the author) that explain exactly how this situation is resolved? Is there a way to export each rendered layer as its own image? My scene uses multiple render layers.  I'd like to edit each rendered layer in Photoshop, and composite it in Photoshop.  Is there a way to export each layer as an individual image? Array with labeling columns Is there an intelligent way to label (or name) columns in a matrix? I would like to have something like  \[M=\left[\begin{array}{c|cc} 1 &amp; 2 &amp; 3\\ 4 &amp; 5 &amp; 6 \end{array}\right],N=\left[\begin{array}{c|cc} 7 &amp; 8 &amp; 9\\ 10 &amp; 11 &amp; 12 \end{array}\right]\] \[A \ B \ C \ D \ E \ F\]   where A, B,..., F are the labels (or rather names) of the first, second, ..., sixth columns, respectively. What I want is to label A be exactly below the first column, label B be below the second column, etc., of course, nicely aligned. So far I have forced some extra spaces before the labels manually, and also experimented with the \phantom command.