So I was reading this article because I'm new to this stuff and don't quite understand the ways in which quantum mechanics and GR contradict each other. Why does general relativity require space to be continuous?
In general relativity (GR), spacetime is viewed as a differentiable manifold of dimension $D$ with a metric of Lorentzian signature $(-,+,+,...,+)$.  My question is why differentiable?
I've noticed something curious about the rotation of a rectangular prism. If I take a box with height $\neq$ width $\neq$ depth and flip it into the air around different axes of rotation, some motions seem more stable than others. The 3 axes which best illustrate what I mean are:  (1) Through the centre of mass, parallel to the longest box edge.  (2) Through the centre of mass, parallel to the shortest box edge.  (3) Through the centre of mass, parallel to the remaining box edge.  It's "easy" to get the box to rotate cleanly around (1) and (2), but flipping the box around (3) usually results in extra twisting besides the rotation around (3) that I'm trying to achieve (obviously a "perfect" flip on my part would avoid this twisting, which is why I call it an instability). If you're not quite sure what I'm talking about, grab a box or book with 3 different side lengths and try it out (but careful not to break anything!).  What's special about axis (3)?   Image taken from . How to uninstall broken programs in windows 7 Could anyone tell me how to uninstall programs that their installation had a problem and therefore their uninstall doesn't work?  I removed all their references in the registry and their folder on the hdd but still don't know how to get rid of them completely from Add/Remove Programs list.  Thanks Access denied when creating keystore for Android app Reputation has been changed but not reflected on tooltip. Your summary page shows "N Edits to your posts" for the selected time period but you really have to be looking for it.  Could we add edits to the notification feed? You could put the reason for the edit (entered--or not--when the editor changes it) and just a link to the post. Logical explanation why exponentiation operation is not commutative and associative Considering Peano axioms we'll define addition, multiplication and exponentiation operations.  We can then prove that addition and multiplication operations are commutative and associative. The proof for addition operation is pretty straightforward but for for multiplication it a bit more difficult. For example, to prove associativity, we have to prove distributivity first.  Moreover, for the definition itself it is not obvious that commutativity should take place, because the definition is not symmetric.  $a \cdot 0 := 0$  $a \cdot S(b) := a \cdot b + a$  Here $S(b)$ is the "next" function from Peano axioms  This definition is not symmetric with respect to $a$ and $b$, so commutivity is not obvious.  For exponentiation operation neither commutivity nor associativity take place. And I am very curious about the logical explanation why that happens  NOTE: The explanation I am looking for is somehow similar to a Abel–Ruffini theorem which explains WHY there is no an algebraic solution for a general polynomial equation Given a 4-vector $p^\mu$ the Lorentz group acts on it in the vector representation: $$ \tag{1} p^\mu \longrightarrow (J_V[\Lambda])^\mu_{\,\,\nu} p^\nu\equiv \Lambda^\mu_{\,\,\nu} p^\nu. $$ However, I can always represent a 4-vector $p^\mu$ using left and right handed spinor indices, writing $$ \tag{2} p_{\alpha \dot{\alpha}} \equiv \sigma^\mu_{\alpha \dot{\alpha}} p_\mu.$$ So the question is: in what representation does the Lorentz group act on $p_{\alpha \dot{\alpha}}$?    There are a lot of questions about this and related topics around physics.se, with a lot of excellent answers, so let me clear up more specifically what I am asking for.  I already know that the answer to this question is that the transformation law is  $$ \tag{3} p_{\alpha \dot{\alpha}} \rightarrow (A p A^\dagger)_{\alpha\dot{\alpha}}$$ with $A \in SL(2,\mathbb{C})$ (how is mentioned for example in ). I also understand that $$ \tag{4} \mathfrak{so}(1,3) \cong \mathfrak{sl}(2,\mathbb{C}),$$ (which is explained for example  by Edward Hughes,  by joshphysics,  by Qmechanic).  So what is missing? Not much really. Two things:   How do I obtain (3) and what is the specific form of $A$, i.e. its relation with the vector representation $\Lambda^\mu_{\,\,\nu}$? Defining the following $$ (\tilde p) \equiv p^\mu, \qquad \Lambda \equiv \Lambda^\mu_{\,\,\nu},$$ $$ \sigma \equiv  \sigma_{\alpha \dot \alpha}, \qquad \hat p \equiv p_{\alpha \dot \alpha},$$ we can rewrite (1) and (2) in matrix form as $$ \tag{5} \hat p \equiv \sigma \tilde p \rightarrow \sigma \Lambda \tilde p  = ( \sigma \Lambda \sigma^{-1}) \hat p,$$ however, this disagrees with (3) which I know to be right, so what is wrong with my reasoning? Why does the transformation law (3) has a form $$\tag{6} A \rightarrow U^{-1} A U,$$ while the usual vector transformation (1) has a form $V \rightarrow \Lambda V$? I suspect this comes from a similar reason to that explained  by Prahar, but I would appreciate a confirmation about this. After upgrading to 12.04 from 10.04, can't login to PHPMYADMIN Graph Proof by induction. Can you prove via induction that there exists a node in a directed graph of n nodes that can be reached in at most two edges from every other node in the graph. Every node in the graph is required to have an edge with every other node that either points out or in (For every pair of nodes X and Y, either X points to Y or Y points to X). Graph Theory: Tree has at least 2 vertices of degree 1 How can I change the local origin of an object, without changing the object location in the world space.  This is the object located at position (0,0,0) in the world space    This is the same object located at position (0,1,0) in the world space, but with its local origin offset so that the object is in the same place    Is there a way to do this in a single step? Chmod to allow read and write permissions for directory I have created directories in root. I am looking for the chmod command to allow all users read and write permissions to a specific directory. I have done chmod 775 for a file but I need this for a directory. This includes permissions on all files and sub directories. How to subtract days from a plain Date? Multi-core vs Multi-CPU I am requesting a new server from our Server team. Which is better for SQL Server performance: 2 CPUs with 8 cores per CPU or 1 CPU with 16 cores? Getting Started to Xetex for Devangari? I have a few satellites and stations orbiting my planets, but I'd like to add onto them. I've seen numerous videos on using docking clamps to attach my new ship to the existing station, but can I do that on a part-by-part basis? For example, I take an extra fuel tank up there, can I attach just the tank to my station, then send my vessel back to Kerbin? I'm not talking about refueling, but rather transferring individual parts from my ship to the station in orbit.  I'd prefer to not have to use an unmanned probe and jettison everything but the desired part in space after docking if possible, but if that's the only way I guess I don't have a choice. List all files in all subfolders I am looking for a list of all English words that are their own antonyms. Off the top of my head, I can only think of "either", "fast", "to dust" and "to lease", but there must be dozens more. Can you provide a link to a comprehensive list, if such exists?  Also, what are such words most commonly called? I like the term "Janus word" I once heard, but is that widely understood?  Edited by popular demand to make it absolutely clear that this is not a community wiki poll. Are antiparticles just particle-shaped holes? If particles are simply regions of space where certain quantum fields have non-zero divergence, are anti-particles simply the corresponding regions of opposite divergence?    This seems like the intuitive answer, especially when considering the process of annihilation.  I have heard before that anti-particles are analogous to particles moving through time in reverse, which seems to indicate that an annihilation event is just a point of symmetry in that (arguably, single) particle's history.  This analogy breaks down, however, when it comes to gravitation, since there seems to be no evidence of a negative-mass particle.    So what then is the meaning of an anti-particle, when their symmetry is preserved across certain fields, but not other? I'm trying to factorise $$ x^3z - x^3y - y^3z + yz^3 + xy^3 - xz^3 $$ into four linear factors.  By plugging it into WolframAlpha I've learned that it's $$-(x-y)(x-z)(y-z)(x+y+z)$$  My question is: what are the steps involved in factorising the expression? Is there a method I don't know about that I'd have access to with my limited maths?  Really appreciate any help!