According to the General Theory of Relativity, the coordinate time distance per spacetime distance traveled by a particle freely falling into a black hole gets closer and closer to $0$ as the particle approaches the event horizon   $$\mathrm{d}s^2 ~=~ -\left(1-\frac{2M}{r}\right)\mathrm{d}t^2 + \left( 1-\frac{2M}{r} \right)^{-1}\mathrm{d}r^2 + \dots.$$  Equivalently, an observer looking from the outside will never see the particle cross the event horizon, even if he/she looked for an arbitrarily long time. However, Hawking radiation implies that black holes don't last forever, but instead shrink and fade away? What happens to everything that is "currently" falling into the black hole during that time?   Edit: What this seems to imply is that as a particle gets close to the event horizon, the rate and intensity of Hawking radiation will increase and increase until most of the mass of the black hole is ejected, in the form of Hawking radiation, out towards the falling particle.
Recently, I read in the journal  that Stephen Hawking wrote a paper claiming that black holes do not exist.  How is this possible? Please explain it to me because I didn't understand what he said.  References:   Article in Nature News:  (Zeeya Merali, January 24, 2014). S. Hawking, Information Preservation and Weather Forecasting for Black Holes, .
Why there are real numbers with infinite digits, but no such natural numbers (or another reason why real numbers are uncountable) This question is me trying to understand (again) why there can be no one-to-one correspondence between the sets of natural and real numbers.  The source of confusion is this: if we abstract completely from the properties and operations on numbers and just think of them as strings of digits, it appears to be possible to list all the real numbers between $0$ and $1$ and all positive integers having the same digits in the same (or reverse) order, for example:  $$0.1 \to 1$$  $$0.01 \to 10$$  $$0.0101 \to 1010$$  $$0.110101 \to 101011$$  I use binary expansion for simplicity.  Since irrational numbers have infinite expansions, we would have to consider some kind of 'infinite natural numbers', like:  $$101110001\dots$$  or  $$\dots101110001$$  I think the main (and only) reason why such numbers don't exist is this requirement:   We need the sets of real and natural numbers to be ordered. We need to be able to do arithmetics on real and natural numbers.   Obviously, if we consider any of two types of 'infinite natural numbers' and the usual absolute value, we find that it's impossible to order them, or do arithmetics with them.  The only way is to change the norm (to $p-$adic norm for example).  While with the real numbers it's still possible to order them, and do arithmetics with them when they have infinite amount of digits in their expansion (or at least in principle).     Is my reasoning correct? Are the two requirements (ordering and arithmetics) enough to explain why natural numbers with infinite amount of digits do not exist, while real numbers do?   This confusion is not restricted to me alone, see  for example Proving an integral equation. Subdomain versus subdirectory I recently moved from having a main site like www.example.com with a subdomain associated site yyy.example.com, to moving everything off the subdomain to a subdirectory like www.example.com/yyy/, with the same hierarchy underneath the subdirectory, and now yyy.example.com is just a redirect page to the subdirectory, so yyy.example.com/abc/page-x is redirected to www.example.com/yyy/abc/page-x and so on.  The effect has been that the traffic increased by several times (the total page views are more than ten times at the present rate) what it was to the yyy subdomain. Previously almost all of the traffic to the yyy subdomain was from the www site. This is from monitoring the old yyy site for a year or more.  I'm curious to know if there's any research or results which would indicate whether this is universal (subdirectories beat subdomains) or "just me". Applying for a UK Tourist Visa from outside country of origin The annual student intake of Hogwarts seems about 100. Since in book 7 Hogwarts is declared mandatory for all children, this means that the total magical population of Britain cannot be much more than 10,000.  How do these people make a living?  All jobs mentioned in the books are non-productive occupations (government, stores, sports &amp;c) - not farmers or engineers. There are exceptions - makers of wands/robes/broomsticks - but where do they get the raw materials from?  Who and how "creates goods" in the wizarding world?  Note that if you say "wizards exchange their gold for Muggle money and buy stuff from Muggles (through intermediaries, if necessary)" then you would have to explain where they get their gold: they would either have to mine it, or produce something of value to the Muggles (in which case the separate financial system of galleons/sickles/knuts does not make sense).  PS. A simple issue of maintaining a steam locomotive for Hogwarts express is pretty much insurmountable for 10k people - without Magic, that is. So, I can imagine that the skilled tradesmen can transfigure dirt into raw materials for their trade - except for food, which "is the first of the five Principal Exceptions to Gamp's Law of Elemental Transfiguration". How do I change a close vote to a duplicate? Voted to close a question based on "lack of minimal understanding".  Later realized the question was a duplicate.  How to I change vote to mark as duplicate instead? Why is Steam only showing me "Buy" instead of "Play" option for a game shared via Family Sharing? I thought the whole point of Family Sharing was so that multiple copies of the game didn't have to be bought. I have no problem downloading the game, but instead of "Play", my only option once it's downloaded is "Buy". Am I missing something? $\forall$ At the beginning or at the end? I have a set of real numbers $x_1, x_2, \ldots, x_n$ and two functions $f:\mathbb{R} \rightarrow \mathbb{R}$ and $g:\mathbb{R} \rightarrow \mathbb{R}$.  What are the differences between the following statements?   $\forall v \in \{1, \ldots, n\} ~f(x_v) = 0 \vee g(x_v) = 0$ $f(x_v) = 0 \vee g(x_v) = 0 ~\forall v \in \{1, \ldots, n\} ~$ This is the proof from the book:     Theorem. There are infinitely many primes of the form $4n+3$.    Lemma. If $a$ and $b$ are integers, both of the form $4n + 1$, then the product $ab$ is also in this form.  Proof of Theorem: Let assume that there are only a finite number of primes of the form $4n + 3$, say  $$p_0, p_1, p_2, \ldots, p_r.$$ Let $$Q = 4p_1p_2p_3\cdots p_r + 3.$$ Then there is at least one prime in the factorization of $Q$ of the form $4n + 3$. Otherwise, all of these primes would be of the form $4n + 1$, and by the Lemma above, this would imply that $Q$ would also be of this form, which is a contradiction. However, none of the prime $p_0, p_1,\ldots, p_n$ divides $Q$. The prime $3$ does not divide $Q$, for if $3|Q$  then $$3|(Q-3) = 4p_1p_2p_3\cdots p_r,$$ which is a contradiction. Likewise, none of the primes $p_j$ can divides $Q$, because $p_j | Q$ implies $p_j | ( Q - 4p_1p_2\cdots p_r ) = 3$, which is absurd. Hence, there are infinitely many primes of the form $4n +3$.   END  From "however, none of the prime ...." to the end, I totally lost!    My questions:   Is the author assuming $Q$ is prime or is not? Why none of the primes $p_0, p_1,\ldots, p_r$ divide $Q$? Based on what argument?     Can anyone share me a better proof?     Thanks. How to disable javascript modal popup ads Why can't I edit attribute table imported through 'text as layer' I imported the coordinates from an Excel table. They show as single points as intended. Now I want to add columns to the attribute table, but I'm not able to edit it. I wanted to import the following columns from Excel as well, but that doesn't seem to work either.  Sorry for the noob-question, but I couldn't find anything relating the theme through google and such.  Ok, with pictures then: I want to use that to add columns with information to these       You see, the small pensil thingy is greyed out, so I can't edit anything in this layer and I can't add any information to my points.  If I try Ale suggestion I get this error: How did Harry breach the Statute of Secrecy? Something always bothered me with Fudge's accusations:     “Disciplinary hearing of the twelfth of August,” said Fudge in a   ringing voice, and Percy began taking notes at once, “into offences   committed under the Decree for the Reasonable Restriction of Underage   Sorcery and the International Statute of Secrecy by Harry James   Potter, resident at number four, Privet Drive, Little Whinging,   Surrey." - OotP Ch. 8 "The Hearing"   Harry cast the Patronus Charm, so he violated the Decree for Reasonable Restriction of Underage Sorcery alright - extinuating circumstances aside - but how did he breach the Statute of Secrecy?  The only witnesses were Dudley and Mrs. Figg, both of whom already knew that Harry is a wizard. Earlier, even Hermione gets it wrong (don't tell her!):     They haven’t even reported that you broke the International Statute of   Secrecy. We thought they would,... - OotP Ch. 4 "#12"   Yes, Harry does use magic in front of a Muggle, Dudley, but even Dudley is not so thick that the fact that Harry's a wizard would come as a revelation to him. Harry cannot breach the Statute of Secrecy to someone already in the know.  Shouldn't Dumbledore point out that glaring error in the indictment sooner rather than later? Or at least mention it, at all?  Is there maybe a source that mentions the Statute of Secrecy containing a clause, that even the mere possibility of magic being witnessed by an uninitiated muggle is sufficient for its breach?? Cause if so, wouldn't even adult wizards have a really tough time doing anything, ever.  Possibly related: Often while playing Pathfinder, my group will run into problems where rules for 3-dimensional play simply don't exist, so far as we can find. Arguments will come up over things like carrying-capacity, movement speed, area-effects, and a variety of other things.  Simply put, are there rules for 3D play for Pathfinder, and if so where can I find them? How small does a land-mass have to be before you live "on" it, rather than "in" it? I'm sure virtually everyone agrees that people live on the Isle of Wight, but in Ireland.  Apparently the usage depends somewhat on physical size, but that can't be the whole story. How exactly do we decide which form to use? And are there any really glaring 'outliers' that don't fit the normal pattern?  LATER - I must just add that the most uncertain case I've found so far is The Falklands. Most people go for in there, but a substantial minority (about 1 in 3) opt for on. When to use LinkedList over ArrayList in Java? Why is using a wild card with a Java import statement bad? It is much more convenient and cleaner to use a single statement like  import java.awt.*;   than to import a bunch of individual classes  import java.awt.Panel; import java.awt.Graphics; import java.awt.Canvas; ...   What is wrong with using a wildcard in the import statement? If I have Optimize iPhone Storage enabled on my iPhone and my phone storage is completely full, I understand that iOS will only store a low-res thumbnail of many of my photos on the device. (The full high-res version of the image will only exist in iCloud and on devices with Download and Keep Originals enabled.)  But when only thumbnails exist on the device, how does that work in relation to other photo apps on iOS? For example, do third-party cloud-based photo apps (like Google Photos, Dropbox's Carousel, or Amazon Prime Photos) have access to the full high-res images from the iCloud Photo Library? Or will they only see the low-res thumbnails in some cases? Very similar to , except for Java.  What is the recommended way of encoding strings for an XML output in Java. The strings might contain characters like "&amp;", "&lt;", etc. Are other players able to see you on their map in Pokemon Go? I am concerned that if I use the game, stalkers and other such people would be able to find me. I have searched for an answer for this question on other forums, but have yet to find the answer. How do I get the value of a textbox using jQuery?