Metric Definition and 1000 Threads

Homework Statement Assume some metric space (K,d) obeys Lindelof, take (X,d) a metric subspace of (K,d) and show it too must obey Lindelof. The Attempt at a Solution I'm assuming since I know that (K,d) obeys Lindelof then there is some open cover that has a countable subcover say {Ji | i is a...

I am reading a paper and the authors read the tt component of the metric from the line element ds^2=f(r)[g(r) dt^2+h(r) dr^2] as g_{tt}=g(r) instead of (what I expect to be) g_{tt}=f(r) g(r) Could somebody please explain to me why? Thank you.

Homework Statement Regard Q, the set of all rational numbers, as a metric space, with d(p, q) = |p − q|. Let E be the set of all p ∈ Q such that 2 < p2 < 3. Show that E is closed and bounded in Q, but that E is not compact. Is E open in Q? Homework Equations Definition of interior...

Homework Statement [PLAIN]http://img833.imageshack.us/img833/6932/metric2.jpg The Attempt at a Solution I've shown d_{X\times Y} is a metric by using the fact that d_X and d_Y are metrics. What is a simpler description of d_N with d_X the discrete metric? Is it just: d_N(x,y) =...

Homework Statement Let X and Y be metric spaces, f a function from X to Y: a) If X is a union of open sets Ui on each of which f is continuous prove that f is continuous on X. b) If X is a finite union of closed sets F1, F2, ... , Fn on each of which f is continuous, prove that f is continuous...

My understanding of the Einstein Summation convention is that you sum over the repeated indices. But when I look at the metric tensor for a flat space I know that g^{λ}_{λ} = 1 But the summation convention makes me think that it should equal the trace of the matrix g_{μσ}. So it should...

Why is it that a metric space (X,d) always has two clopen subsets; namely {0}, and X itself? Rudin calls it trivial, and so do about 15 other resources I've perused. What confuses me is that if we define some metric space to be the circle in ℝ2: x2+y2 ≤ r2, then points on the boundary of...

" Let X be a metric space and p be a point in X and be a positive real number. Use the definition of openness to show that the set U(subset of X) given by: U = {x∈X|d(x,p)>r} is open. " I have tried: U is open if every point of U be an interior point of U. x is an interior point of U if there...

This is really a continuation from another thread but will start here from scratch. Consider the case of a static thin spherical mass shell - outer radius rb, inner radius ra, and (rb-ra)/ra<< 1, and with gravitational radius rs<< r(shell). According to majority opinion at least, in GR the...

Hey All, I have been working on some Metric Space problems for roughly 20hrs now and I cannot seem to grasp some of these concepts so I was hoping someone here could clear a few things up for me. My first problem is detailed below... I have the following metric... d(x,y) = d(x,y)/(1 +...

Hello, well I just read a paper by Atish Dabholkar and Ashoke Sen, titled "Quantum Black Holes", pp. 4-5 as shown below and I tried to find d\xi^{2}\frac{2GM}{\xi}=d\rho^{2} like this which is different from the eq. in the paper. So, could somebody please help me to find my...

Hello, Can anyone give me the answer of the following derivative? \frac{\partial{g}}{\partial{g^{\mu \nu}}} Thank you in advance !

Hi, I'm reading this wikipedia article on the metric connection, and it says that the Levi-Civita connection is a metric connection without torsion. If the metric connection is defined so that the covariant derivative of the metric is 0, how can there be torsion? Doesn't this condition force the...

Homework Statement Prove that \rho_{0}(x,y)=max_{1 \leq k \leq n}|x_{k} - y_{k}|=lim_{p\rightarrow\infty}(\sum^{n}_{k=1}|x_{k}-y_{k}|^{p})^{\frac{1}{p}} Homework Equations The Attempt at a Solution My approach was to define a_{m}=max_{1 \leq k \leq n}|x_{k} - y_{k}| and...

I'm having my first differential geometry course and I can't get the concept.

Homework Statement Given (R+, d), R-Real # d= | ln(x/y) | Show that this metric space is complete Homework Equations The Attempt at a Solution Firstly, I know that to show it is complete I need to have that all Cauchy sequences in that space converge... So I'm not 100%...

Can somebody clarify how the formula for variation of the auxilliary worldsheet metric is obtained due to reparametrization of the worldsheet in string theory??

Hello, I was wondering if if has any sense of talking about angles on an arbitrary http://en.wikipedia.org/wiki/Metric_space" (where only a distance function with some properties is defined) At first sight it seems to not has any sense, only some metric spaces has angles, namely does that...

I have started with the space-time metric in a weak gravitational field (with the assumption of low velocity): ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2) Where \phi<<1 is the gravitational potential. Using the standard form for the Christoffel symbols have found: \Gamma^0_{00}=\phi_{,0}...

Homework Statement Prove that lim_{n} p_{n}= p iff the sequence of real numbers {d{p,p_{n}}} satisfies lim_{n}d(p,p_{n})=0 Homework Equations The Attempt at a Solution I think I can get the first implication. If lim_{n} p_{n}= p, then we know that d(p,p_{n}) = d(p_{n},p) <...

This is kind of a weird question. I like to think about how I would explain things to other people, and I realized that I don't know a great way to explain in general how terms defined in the context of metric spaces are generalized to the context of topological spaces. It's not at all difficult...

Hi, The typical representation of the Dirac gamma matrices are designed for the +--- metric. For example /gamma^0 = [1 & 0 \\ 0 & -1] , /gamma^i = [0 & /sigma^i \\ - /sigma^i & 0] this corresponds to the metric +--- Does anyone know a representation of the gamma matrices for -+++...

I haven't learned much of advanced mathematics. It seems that we can use metric tensors to lower or raise index of christoffel symbols. But isn't christoffel symbols made of metric tensors and derivatives of metric tensors? How can we contract indices of a derivative directly with metric tensors...

Homework Statement Let \left (X,d \right) be a metric space, and let \left\{ x_n \right\} and \left\{ y_n \right\} be sequences that converge to x and y. Let \left\{ z_n \right\} be a secuence defined as z_n = d(x_n, y_n). Show that \left\{ z_n \right\} is convergent with the limit d(x,y)...

I'm trying to verify that if (M,d) is a metric space, then (N,e) is a metric space where e(a,b) = d(a,b) / (1 + d(a,b)). Everything was easy to verify except the triangle inequality. All I need is to show that: a <= b + c implies a / (1 + a) <= (b / (1 + b)) + (c / (1 + c) Any help...

Hi guys, two problems, first one I understand for the most part, the second one, I do not know how to set up and solve. Homework Statement Let X = R^{n} for x = (a_{1},...,a_{n}) and y = (b_{1},...,b_{n}), define d_{\infty}(x,y) = max {|a_{1}-b_{1}|,...,|a_{n}-b_{n}|}. Prove that this is a...

Suppose (X,d) is a metric space and A, a subset of X, is closed and nonempty. For x in X, define d(x,A) = infa in A{d(x,a)} Show that d(x,A) < infinity. I really don't have much of an idea on how to show it must be finite. An obvious thought comes to mind, namely that a metric is...

Homework Statement Consider a submarine cruising 32 ft below the free surface of seawater whose density is 64 lbm / (ft^3). What is the increase in the pressure in psi exerted on the submarine when it dives to a depth of 172 ft below the free surface? Assume that the acceleration due to...

I have never been formally trained in GR, and have a question regarding the basics of how to calculate the energy density from a metric. This question arises from thought experiments involving a field with a negative energy density. This is important only because I expect the energy density...

I've encountered the term Hausdorff space in an introductory book about Topology. I was thinking how a topological space can be non-Hausdorff because I believe every metric space must be Hausdorff and metric spaces are the only topological spaces that I'm familiar with. my argument is, take two...

I've worked through a common-sense argumenthttp://www.mathpages.com/rr/s8-09/8-09.htm" showing the time-time component of the Schwarzschild metric g_{tt} = \left (\frac{\partial \tau}{\partial r} \right )^2\approx 1-\frac{2 G M}{r c^2} On the other hand, I've not worked through any...

I'm currently reading Ross's Elementary Analysis, which presents the definition of continuity as such: (not verbatim) Let x be a point in the domain of f. If every sequence (xn) in the domain of f that converges to x has the property that: lim f(xn) = f(x) then we say that f is...

Suppose we're in a general normed space, and we're considering a sequence \{x_n\} which is bounded in norm: \|x_n\| \leq M for some M > 0. Do we know that \{x_n\} has a convergent subsequence? Why or why not? I know this is true in \mathbb R^n, but is it true in an arbitrary normed space? In...

Let us consider the General Relativity metric: {ds}^{2}{=}{g}_{00}{dt}^{2}{-}{g}_{11}{{dx}_{1}}^{2}{-}{g}_{22}{{dx}_{2}}^{2}{-}{{g}_{33}}{{dx}_{3}}^{2} ---------------- (1) Using the substitutions: {dT}{=}\sqrt{{g}_{00}}{dt} {dX}_{1}{=}\sqrt{{g}_{11}}{dx}_{1}...

I'm currently working through General Relativity and I'm wondering how you would express the variation of a general metric tensor, or similarly, how you would write the total differential of a metric tensor (analogous to how you would write the total derivative for a function)? Also, on a...

Question: have some sense that in a space time with metric ds^2 = g_{tt}dt^2+ g_{xx}dx^2+ g_{yy}dy^2+g_{zz}dz^2, the coordinates x,y,z \in ]-\infty, \infty[ , but t \in [0, \infty[ ?

Homework Statement Having a little trouble on number 24 of Chapter 3 in Rudin's Principles of Mathematical Analysis. How do I prove that the completion of a metric space is complete? Homework Equations X is the original metric space, X^* is the completion, or the set of...

Dear readers, Let X be the product space of a countable family \{X_n:n\in\mathbb{N}\} of separable metric spaces. If X is endowed with the product topology, we know that it is again separable. Are there other topologies for X such that is separable? Is there a natural metric on X such that X...

Hi, I'm trying to attack a problem where the Riemannian metric depends explicitly on time, and is therefore a time-dependent assignment of an inner product to the tangent space of each point on the manifold. Specifically, in coordinates I encounter a term which looks like...

any help would be appreciated. not sure where to start.

When I was studying general relativity, I learned that to change a vector into a covector (or vice versa), one used the metric tensor. When I started quantum mechanics, I learned that the difference between a vector in Hilbert space and its dual is that each element of one is the complex...

Hi there, I came across the following problem and I hope somebody can help me: I have some complete metric space (X,d) (non-compact) and its product with the reals (R\times X, D) with the metric D just being D((t,x),(s,y))=|s-t|+d(x,y) for x,y\in X; s,t\in R. Then I have some sequences...

From the definition of an open set as a set containing at least one neighborhood of each of its points, and a closed set being a set containing all its limit points, how can we show that the complement of an open set is a closed set (and vice versa)? Usually this is taken as a definition, but...

Hi, Is a metric signature of (-,-,-,-) unphysical? Thanks,

This is an interesting hypothesis that doesn't seem to have been discussed yet. What are its flaws? Mark Hadley at the University of Warwick argues that galactic rotation causes gravitational frame-dragging sufficient to put a local asymmetric twist into spacetime and explain observed CP...

Do we need a metric field on a manifold so as to specify a coordinate system on it?

I am learning about General Relativity. The planetary orbits can be calculated with more precision especially Mercury. I am stuck on how to get from the Schwarzschild Metric: a four variable Differential Equation to a radius(r,theta,phi,t) and velocity(r,theta,phi,t) of a single planet...

Homework Statement T is a compact metric space with metric d. f:T->T is continuous and for every x in T f(x)=x. Need to show g:T->R is continous, g(x)=d(f(x),x). Homework Equations The Attempt at a Solution f is continuous for all a in T if given any epsilon>0 there is a delta>0...

Hi, All: Let X be a metric space and let A be a compact subset of X, B a closed subset of X. I am trying to show this implies that d(A,B)=0. Please critique my proof: First, we define d(A,B) as inf{d(a,b): a in A, b in B}. We then show that compactness of A forces the existence of a in A...

Hi This might be a stupid question, so I hope you are patient with me. When I look for the definition of the energy-momentum tensor in terms of the Lagrangian density, I find two different (?) definitions: {T^\mu}_\nu = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\partial_\nu...