One principle in general relativity is that the wordlines of massless particles are null geodesics. It also seem to be a commonly stated fact (for instance see eq. (3.62) in Section 3.4 of Carroll's GR text) that one can parameterize the worldline of a massless particle as $x(\lambda)$ such that $$\frac{dx^{\mu}}{d\lambda} = p^{\mu}, \tag{1}$$ where $p^{\mu}$ is the four-momentum of the massless particle. To clarify what is meant by four-momentum, the energy of a particle with four-momentum $p$ as measured by an observer moving with four-velocity $U$ is $-p_\mu U^\mu$.  Now I understand why one could pick a parameterization that satisfies (1) at a particular point. But why must such a parameterization satisfy (1) globally?  The only argument I have is that such a parameterization always exists for the world-line of a particle with mass, and taking the limit to light speed suggests that such a parameterization should also hold for the world-lines of massless particles.  My question is: what are some other arguments for why the parameterization should exist?
I am following Carroll's GR book. He explain that it is convention to parameterize geodesics of photons by a parameter $\lambda$ such that  $$p^\mu ~=~ \frac{d x^{\mu}}{d \lambda}.\tag{3.62}$$  But this is the definition of 4-velocity for a massive particle in the case of $\lambda=\tau$ equal to proper time. My question then, is $$u^\mu = p^\mu$$ for all massless particles? I ask because it's well-known that proper time $\tau$ freezes for massless particles.
Best way to start investing, for a young person just starting their career? Ok, I have to admit, I don't really delve too deep into the semantics of the spatial operators list below, I'm more of a user, I write software to drive the SQL most of the time and don't really think about it too much.  However, I have a situation where a spatial operation I'm doing is going dog slow, on a table that has great indexes on and has never caused a problem before now.  As a result, I'm trying to figure out which of the following is my best bet to use, that will give me the best performance, however, sometimes I find my searches don't return anything when I know they should, and sometimes return stuff when they shouldn't.  SO I'm reaching out, to ask if some one can give me the actual definition, such as criteria to match, fully within enclosing geometry, or within outer ring etc etc of each of the following operations:  Equals(Geom1, Geom2):int Disjoint(Geom1, Geom2):int Touches(Geom1, Geom2):int Within(Geom1, Geom2):int Overlaps(Geom1, Geom2):int Crosses(Geom1, Geom2):int Intersects(Geom1, Geom2):int Contains(Geom1, Geom2):int Relate(Geom1, Geom2):int   If you can give a concrete example of when they will and will not match too, that would be great.  For reference, I'm searching on a table of Linestrings, using a rectangular polygon, rotated to be pointing in the direction of travel (It's part of a traffic managment app)  For further reference, here is a sample of the SQL the app generates to perform a search:  SELECT recordID,AsBinary(geometry) AS geometry,Distance(GeomFromText('POINT(-1.84101 54.85078)',4326), geometry) AS distanceFromGps FROM linegrid WHERE Intersects(GeomFromText('POLYGON ((-1.8413149820810311 54.850782468607292, -1.8409507853094111 54.850952257034713, -1.8408279009723911 54.850864894077496, -1.8411920982612455 54.850695105650068, -1.8413149820810311 54.850782468607292))'), geometry)   You can see in this example I'm using intersects trying the others as mentioned doesn't quite give the results I expect, hence why I'm asking the question.  Please note, I'm not particularly bothered about actual faster/other ways of doing this at the moment, BUT if you do happen to spot a better way do feel free to shout up :-)  For now all I'm mostly trying to understand is the differences between each of the operations.  As for doing it faster/better I intend to open a different question for that at a later date.  For more reference, the Spatial engine this is working against is Spatilite 2.3.0, and I can't upgrade it due to the devices it's running on not having a more up to date build for it (Windows CE)    I'm going to choose mintix'es answer here as that provided a super easy to understand grid and description of all the various operations, which answered my question.  I'm still faced with getting performance on the routines up, but I do now have a few strategies to try on that one, I may still seek advice here from the group with regards to that.  @Vince deserves some credit too, as that white paper is definitely the definitive answer, but it's also quite challenging to read (I had to read through it several times) , for anyone wanting the real I am on these operations The Clementini paper is the way to go, if you need an easy to understand overview while developing solutions however, the FME link should be open in another browser tab. Adapt BibTeX style to give all initials I'm using a modification of the plainyr BibTeX style. Although my bib-file contains all initials for some authors (e.g. B.D. McKay), I end up with just the first initial in my document (e.g. B. McKay). I thought about editing the BibTeX style myself, but after seeing the code, I decided against it. Maybe there are some people who are more knowledgeable in this area of expertise.  I want my final document to include the initials for all the names, and not just those of the first name.  This is the relevant code of the BibTeX style, or at least, I think it is the relevant code.  FUNCTION {format.names} { 's :=   #1 'nameptr :=   s num.names$ 'numnames :=   numnames 'namesleft :=     { namesleft #0 &gt; }     { s nameptr "{f.~}{vv~}{ll}{, jj}" format.name$ 't :=       nameptr #1 &gt;         { namesleft #1 &gt;             { ", " * t * }             { numnames #2 &gt;                 { "," * }                 'skip$               if$               t "others" =                 { " et~al." * }                 { " and " * t * }               if$             }           if$         }         't       if$       nameptr #1 + 'nameptr :=       namesleft #1 - 'namesleft :=     }   while$ } LOAD DATA INFILE easily convert YYYYMMDD to YYYY-MM-DD? Present tense in Flashback story for something that is still true now? I'm wondering whether I should use the present tense for a descriptive statement but in a past context. For example:     (1) I went to the zoo and saw a giraffe last week.       (2) A giraffe is a mammal with long neck.       (3) It is the first time I see that animal with my own eyes.   Statement (2) is a description of a giraffe in general, not the specific one in the zoo, so is it correct to use is?  Statement (3) is an event that is still true now, so should I use is and see?  I'm not native speaker so forgive me if the question is very obvious to others. How to find required information in man The man command brings up a nice manual for many programs, but how can I use it more effectively? For example man gcc brings:  NAME        gcc - GNU project C and C++ compiler  SYNOPSIS        gcc [-c|-S|-E] [-std=standard]            [-g] [-pg] [-Olevel]            [-Wwarn...] [-Wpedantic]            [-Idir...] [-Ldir...]            [-Dmacro[=defn]...] [-Umacro]            [-foption...] [-mmachine-option...]            [-o outfile] [@file] infile...         Only the most useful options are listed here; see below for the        remainder.  g++ accepts mostly the same options as gcc. .... many text   But what if I need only some part of it? What is the best way to jump to the section where -Idir is described, for example? Question about arithmetic–geometric mean We have two sequences: $$a_{n+1}=\sqrt{a_nb_n}$$ $$b_{n+1}=\frac{a_n+b_n}{2}$$ I need to prove that those are making Cantor's Lemma.(At the end I shold get that: $\lim_{n\to \infty}a_n=\lim_{n\to \infty}b_n$ by Cantor's Lemma)  Any ideas how?  Thank you. When you hit up a site, such as cnn.com, does the TCP/IP packet contain the MAC address of your network card?  Could cnn.com theoretically record my MAC address? How to use maths symbols in text mode? I try to add a center dot with \circ but it fails to compile. How to use maths symbol in text mode? The error message is   ovak, slovenian, spanish, swedish, turkish, turkmen, ukrainian, uppersorbian,  welsh, loaded. (/usr/local/texlive/2012/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/local/texlive/2012/texmf-dist/tex/latex/base/size10.clo)) (./untitled-1.aux) ! Missing $ inserted. &lt;inserted text&gt;              $ Real analysis derivate problem. How can I prove this? Let $f$ be differentiable in $[a,b]$ and $f(a)=0$. If $\exists M \in R$ such that $\vert f'(x) \vert \leq M \vert f(x)\vert \ $,  $\forall x \in [a,b]$, then $f(x)=0, \ \forall x \in [a,b]$.  $\\$ I have an idea, but I can´t see it clear. I have arrived to the inequality $\vert f(x)\vert \leq  K\vert f(y)\vert $ for $y \in (a,x)$, supposing $k&gt;0$ and repeating the process $\vert f(x)\vert \leq  K^n\vert f(z)\vert$ for $a&lt;z&lt;y&lt;x$. If $K&lt;1$, then if $n \rightarrow \infty \ $, $f(x)=0$. That´s a sort of intuitive idea. Coloring a black and white PDF graphic How to break out of jQuery each Loop My machine is currently running Windows 7. I have 2 partitions: OS (C:) and DATA (D:). I intend to do a native Ubuntu 14.04 install. Assuming I format my Windows OS partition, will I have full access to my D partition in Ubuntu ? Search engines and browser accept-language I'm building a web portal where language content will generally depend on the "accept-language" sent by the browser. The same content-URI will thus serve different content to different users depending on their browser setting.  I'm very curious to know how this will affect search indexing. Does Google index using all languages, and is it handled well? A  not only grants me a Sentinel Raven, but also the handy ability to see through its eyes.  Unlike the telepathic connection with a regular familiar, I don't have to use an action to do so.    If I am blinded, but my Sentinel Raven is not (and within telepathic range), can I use my raven's sight to cast spells that require me to see my target? Would this negate any disadvantage resulting from me being blinded? Or would I still suffer disadvantage, for example, because I have to cast from my raven's perspective and not my own point of view? Help with completing the proof: Any unbounded sequence is either infinitely large or has a convergent subsequential limit Given $\{x_n\}$ is an unbounded sequence, prove:   $$ \lim_{n\to\infty}x_n = \infty\ \  \text{or}\ \ \lim_{n_k \to \infty} = A $$   In other words any unbounded sequence either infinitely large or has a convergent subsequence.     Everything before udpate section is wrong.  Part 1. Proving $|x_n| &gt; M \implies \lim_{n\to\infty} x_n = \infty$  $\Box$ Start with the definition of an unbounded sequence: $$ \forall M &gt; 0 \in \Bbb R \ \exists N \in \Bbb N: \forall n &gt; N\implies |x_n| &gt; M $$  Suppose that: $$ \lim_{n\to\infty}x_n \ne \infty $$ That means: $$ \forall n\in\Bbb N\ \exists C &gt;0 \in\Bbb R : |x_n| &lt; C $$  Now taking $M^\prime = \max\{M, C\}$ we have that: $$ \begin{cases} \begin{align} \exists N \in \Bbb N : &amp;\forall n&gt;N \implies &amp;|x_n| &gt; M^\prime \\ &amp;\forall n \in \Bbb N: &amp;|x_n| &lt; M^\prime \end{align} \end{cases} $$  So we have arrived to a contradiction which means the assumption is wrong and: $$ \lim_{n\to\infty}x_n = \infty $$ $\Box$     Part 2. It's not clear to me how to prove that if unboundedness does not imply infinity limit then it must imply the existence of a convergent subsequence.   Two questions is my mind:   Is part 1 correct?  How do I proceed with the second part?     I'm using the following definition. $$ \lim_{n\to\infty}x_n = +\infty \stackrel{\text{def}}{\iff} \forall \epsilon &gt; 0 \exists N \in \Bbb N :\forall n &gt; N \implies x_n &gt; \epsilon \\ \lim_{n\to\infty}x_n = -\infty \stackrel{\text{def}}{\iff} \forall \epsilon &gt; 0 \exists N \in \Bbb N :\forall n &gt; N \implies x_n &lt; -\epsilon \\ $$ A sequence is considered infinitely large when: $$ \lim_{n\to\infty}x_n = \infty\ \ \text{when}\ \ \lim_{n\to\infty}x_n = + \infty \ \ \text{or} \ \ \lim_{n\to\infty}x_n = - \infty $$    Update.   Looks like i've messed up a lot of things. I will try once again. As shown in comments and answers the statement holds only in case: $$ \lim_{n\to\infty} |x_n| = \infty \iff \forall \epsilon &gt; 0 \exists N \in \Bbb N: \forall n &gt; N \implies |x_n| &gt; \epsilon $$  Consider the following: $$ \lnot P = \lnot\left(\lim_{n\to\infty}|x_n| = \infty \right) \iff \exists \epsilon &gt; 0\ \forall N_1 \in \Bbb N : \exists n &gt; N_1 \land |x_n| &lt; \epsilon $$  But on the other hand it is given that $x_n$ is unbounded: $$ Q = \forall M &gt; 0\ \exists N_2 \in \Bbb N : |x_{N_2}| &gt; M $$  Construct a negative expression for boundedness: $$ \lnot P = \exists M &gt; 0\ \forall N_2 \in \Bbb N : |x_{N_2}| &lt; M $$  If $S = \lnot P \implies \lnot Q$ then $S = P \lor \lnot Q$  Let $\epsilon = M$, choose $N = \max\{N_1, N_2\}$ then both statements are true and: $$ \exists \epsilon &gt; 0\ \forall N = \max\{N_1, N_2\}: \exists n &gt; N \land |x_n| &lt; \epsilon $$  Now either $P$ is true, which would mean $\lim |x_n| = \infty$, or $\lnot Q$ is true which would mean the sequence is bounded and hence contains a convergent subsequence. Nignx - worker_connections are not enough How to make a bibliography (for total beginners)? In the sentence     What’s funny is Cat’s dry humour.   Wouldn’t it be better to put a comma after “funny”:     What’s funny, is Cat’s dry humour.   This question entails a (at the time of writing) heated discussion in the . Why does my document 'error' when using '%' I am designing gamecards in LaTeX  Everything is going pretty ok so far but when i try to insert the %-sign for a question about percentages my file keeps giving errors.  I tried putting the percentage between dollarsigns because i know just % will make it a 'remark' and that the dollarsign makes it 'mathematical data'.   Whats the reason for the error?   P.s. the error says: file ended while scanning use of \kaart. ...    Codesample:   {\kaart{Bekijk statistiek kaart F:   Schat het percentage mensen dat als lievelingseten spaghetti opgaven, je mag er slechts $ 2% $ langs zitten. } { $ 11% $}}