This is more like a conceptual question. Let's say we have two point masses $m_1$ and $m_2$ separated by the distance $r$. According to the Newton's law of universal gravitation, the objects will be accelerated with respect to each other by a force: $$\vec{F}=-G\tfrac{m_1m_2}{r^2}\hat{r}.$$  I was trying to visualize this problem from the General Relativity point of view - in terms of spacetime curvature and the objects following the geodesics.  So my question is: How would General Relativity explain this attraction between the two objects? I know that their mass will warpe the spacetime around them, but how does this curvature of spacetime explain the fact that they are getting closer together - esspecialy when $r$ is very large? Assuming that the object are perfect spheres and the mass distribution is uniform, isn't the spacetime curvature created by each object going to be isotropic?  I started by imagining that the objects will advance through time at 1 sec/sec in their proper frame, but because the spacetime is warped around them - the motion through time implies motion through space. How far off am I?  To put it more technically - How does the spacetime curvature in this particular case lead to geodesics that will eventually make the objects approach each other?
The Newtonian description of gravity can be formulated in terms of a potential function $\phi$ whose partial derivatives give the acceleration:  $$\frac{d^2\vec{x}}{dt^2}=\vec{g}=-\vec{\nabla}\phi(x)=\left(\frac{\partial\phi}{\partial x}\hat{x}+\frac{\partial\phi}{\partial y}\hat{y}+\frac{\partial\phi}{\partial z}\hat{z}\right)$$  However, in general relativity, we describe gravity by means of the metric. This description is radically different from the Newtonian one, and I don't see how we can recover the latter from the former. Could someone explain how we can obtain the Newtonian potential from general relativity, starting from the metric $g_{\mu\nu}$?
Give a combinatorial proof: $ n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k} $ Find a combinatorial argument for the following binomial identity:   $$n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k} .$$   Algebraic proofs can be found at  , and a related identity at  . Why does  get a pass, but I can't ask my own book recommendation question? Can I ask my own book recommendation question in a way that is acceptable to the community, like the C++ book list question? I'm looking for one word which describes what a student who missed classes needs to do Would it be possible to have a "community accepted" feature? What does an unprocessed RAW file look like? Access is denied because the NTFS volume is already exclusively opened Plural or singular verb for collective term Why is it string.join(list) instead of list.join(string)? How to force NSLocalizedString to use a specific language Why is my query suddenly slower than it was yesterday? [Salutations]  (check one)  [ ] Well trained professional, [ ] Casual reader, [ ] Hapless wanderer,   I have a (check all that apply)  [ ] query [ ] stored procedure [ ] database thing maybe     that was running fine (if applicable)   [ ] yesterday [ ] in recent memory [ ] at some point    but is suddenly slower now.  I've already checked to make sure it's not being blocked, and that it's not the victim of some long running maintenance task, report, or other out of band process.  What is the problem, what should I do, and what information can I provide to get some help?  [*Insert appropriate closing remarks*] How to install Opera Browser? I am trying to install opera browser from the terminal.  I downloaded the 'debian' installer because it didn't show up in software center.  It completely destroyed my software center - it is frozen and I have no idea how to fix (I have a different thread for that though).  Can I use this thing called synaptic package manager to install Opera?  or am I stuck with Firefox?    How do I install Opera Browser? Short story with competitors complaining to government about a new very popular synthetic food "None is" or "none are" Let $P$ be a prime. Show that $\exists$ $x \in \mathbb{N}$ such that $f(x) = p$ then $\exists$ $y \in \mathbb{N}$ such that $g(y) = p$ What is given?  $$\text{Let P be a prime}$$ $$\text{Let} \space f(x)= 3x+1$$ $$\text{Let} \space g(x)= 6x+1$$  Show that: If there exists $x \in \mathbb{N}$ such that $f(x) = P $ , then there exists a $y \in \mathbb{N}$ , such that $g(y) = P$  What have I tried?   Given the question says 'show that', that leads me to believe I need to do a proof of sorts. I'm guessing it might be a proof by either strong induction, or just induction (or maybe just a direct proof?).  I don't even know how to set this up If $f(x) =$ $3x + 1 = $ prime...for any integer $x$, then there exists an integer $y$ such that $g(y) = 6y + 1 = $ prime?   Any hints or starting points would be greatly appreciated.   $$\text{Thanks in advance!}$$ How do I find a point a given distance from another point along a line? In music, when two or more pitches are played together at the same time, they form a chord. If each pitch has a corresponding wave frequency (a pure, or fundamental, tone), the pitches played together make a , which is obtained by simple addition. This wave is no longer a pure sinusoidal wave.  For example, when you play a low note and a high note on a piano, the resulting sound has a wave that is the mathematical sum of the waves of each note. The same is true for light: when you shine a 500nm wavelength (green light) and a 700nm wavelength (red light) at the same spot on a white surface, the reflection will be a superposition waveform that is the sum of green and red.  My question is about our perception of these combinations. When we hear a chord on a piano, we’re able to discern the pitches that comprise that chord. We’re able to “pick out” that there are two (or three, etc) notes in the chord, and some of us who are musically inclined are even able to sing back each note, and even name it. It could be said that we’re able to decompose a Fourier Series of sound.  But it seems we cannot do this with light. When you shine green and red light together, the reflection appears to be yellow, a “pure hue” of 600nm, rather than an overlay of red and green. We can’t “pick out” the individual colors that were combined. Why is this?  Why can’t we see two hues of light in the same way we’re able to hear two pitches of sound? Is this a characteristic of human psychology? Animal physiology? Or is this due to a fundamental characteristic of electromagnetism? Isn't every subset of a compact space compact? Let $\mathfrak{C}$ be an open cover of a topological compact space $X$, and let $\mathfrak{B}\subseteq \mathfrak{C}$ be its finite subcover. Then every subset of $X$ also has the same cover and subcover! Shouldn't every such subset also be compact then?  Motivation: I read somewhere that every closed subset of a compact space is compact. I wonder why it shouldn't be the case for every subset of $X$, rather than just the closed ones. How do I get the linksys WUSB6300 wireless adapter to work on linux I noticed that if I transfer a few large files between two hard drives, it's pretty speedy, at around at least 30&nbsp;MB per second, but if I transfer thousands of files less than 5&nbsp;KB, it is pretty damn too slow.. around 1 to 2&nbsp;MB per second.   Is there a way to speed up the copy/paste process with thousands of small files on Windows 7? I have some non-Gaussian distributed variable and I need to check if there are significant differences between the values of this variable in 5 different groups.  I have performed Kruskal-Wallis one-way analysis of variance (which came up significant) and after that I had to check which groups were significantly different. Since the groups are kind of sorted (the values of the variable in the first group are supposed to be lower than the values of the variable in the second group which are supposed to be lower than the values of the variable in the third group and so on) I only performed 4 tests:  Group 1 vs Group 2 Group 2 vs Group 3 Group 3 vs Group 4 Group 4 vs Group 5   I have performed this analysis with two different methods. I started by using Dunn's Multiple Comparison Test but nothing came up significant. On the other hand if I use Mann-Whitney test and correct for the number of tests (4) using Bonferroni, 3 tests come up significant.  What does it mean? Which results should I trust?