Metric Definition and 1000 Threads

Hi Suppose (X,d) is a bounded metric space. Can we extend (X,d) into (X',d') such that (X',d') is compact and d and d' agree on X? ( The reason for asking the question: To prove a theorem in Euclidean space, I found it convenient to first extend the bounded set in question to a compact one (...

In general 4d space time, the Riemann tensor has 20 independent components. However, in a more symmetric metric, does the number of independent components reduce? Specifically, for the Schwartzchild metric, how many IC does the corresponding Riemann tensor have? (I think it is 4, but I...

Hi all, I am trying to argue that an ellipse is a good approximation for some discontinuity in a material. The ellipse and the interpolated curve I get from the photo look very much alike, but I need an actual number to show how much the two curves differ from each other. I thought of...

Please explain me how to derive the Timelike geodesics in Schwarzschild Metric. Thank you.

Hi, I have been reading about metric spaces and came across an elementary property that I am having difficulty proving. A quick search on these forums and google has also failed. Given a metric space with distance function d, and an increasing, concave function f:\mathbb{R} \rightarrow...

Today while working I found that an M6 x 1.0 Thread bold will easily screw into a hole tapped 1/4"-28. At first when I found this, I thought one of my boxes of bolts was labeled wrong, but I tried another set and it work. Has anyone else come across this similarity before? -CR

Was trying to learn differential geometry, had to take time off of it to develop some knowledge of topology, namely compactness and Hausdorff's condition. I'm using Sutherland's book on topology and came across something I didn't understand concerning metric spaces, Sutherland speaks of the so...

The sequential characterisation of continuity says that f is continuous at x_0 if and only if for every sequence (x_n)_{n\in\mathbb{N}} in X, f(x_n)\to f(x_0) as x_n \to x_0. f is continuous on X if this is the case for all x_0 \in X. I think I've done all the parts of this question up to the...

in many research of kaluza klein theory to unified electromagnetic & gravity fields these research begin with 5D metric how can i drive this metric? please any research drive this metric show me. note russian research's by english is very good if you know it show me . please it is very important

Homework Statement Homework Equations The Attempt at a Solution I've done the first 3 parts. I've come to the bit on Cauchy sequences at the end. How do I show x_n = n is/isn't a Cauchy sequence in the 2 metrics? (x_n) is a Cauchy sequence in a metric space (X,d) if for any...

i remember the French came up with metric meter by measuring the distance between equator and north pole and then divided by an integer to come up meter. it that still the defintion for meter? also, it seems now that a second is defined by the integer number of oscillation of atomic clock...

Homework Statement Consider a metric space (X,d) with subsets A and B of X, where A and B have non-zero intersection. Show that diam(A\bigcupB) \leq diam(A) + diam(B) Homework Equations The Attempt at a Solution A hint would be very much appreciated. :smile:Let x\inA, y\inB, z\inA\bigcupB...

Given X=R∞ and its element be squences let d1(x,y)=sup|xi-yi| let d∞(x,y)=Ʃ|xi-yi| then there exists some some x(k) which convergences to x by d1 but not by d∞ ,for example let x be the constant squence 0, i.e xn=0 ,and let x(k)n=(1/k2)/(1+1/k2)n then d1(x(k),x)=1/k2 and...

Hello, I'm having problems solving this problem I got in class. I want to learn the concept and how to approach the solution. Here it is: Consider the metric ds=dx^2+dy^2-dudv+2H(x,y,u)du^2 What form must the function H have for this metric to represent a plane gravitational wave...

Given X=R∞ and its element be squences let d1(x,y)=sup|xi-yi| let d∞(x,y)=Ʃ|xi-yi| then there exists some some x(k) which convergences to x by d1 but not by d∞ ,for example let x be the constant squence 0, i.e xn=0 ,and let x(k)n=(1/k2)/(1+1/k2)n then d1(x(k),x)=1/k2 and...

Homework Statement Show that \frac{\partial\overline{\mathcal{L}}_G}{{\partial}\mathfrak{g}^{ab}_{,c}}=\Gamma^c_{ab}-\frac{1}{2}\delta^c_a\Gamma^d_{bd}-\frac{1}{2}\delta^c_b\Gamma^d_{ad}.Homework Equations...

Consider a flat 2-dimensional plane. This can be described by standard Cartesian coordinates (x,y). We establish a oblique set of axes labelled p and q. p coincides with x but q is at an angle θ to the x-axis. At any point A has unambiguous co-ordinates (x,y) in the Cartesian system. In...

hi guys, I have a question I would like assistance with: let (v,||.||) be a norm space over ℝ, and let f:v→ℝ be a linear functional. if f is continuous on 0 (by the metric induced by the norm), prove that there is k>0 such that for each u in v, |f(u)| ≤ k*||u||. thanks :)

Homework Statement Let f be a continuous real function on a metric space X. Let Z(f) be the set of all p in X at which f(p) = 0. Prove that Z(f) is closed. Homework Equations Definition of continuity on a metric space. The Attempt at a Solution Proof. We'll show that X/Z(f) = {p...

Homework Statement For x,y \in\mathbb{R} define a metric on \mathbb{R} by d_2(x,y) = |\tan^{-1}(x) - \tan^{-1}(y) | where \tan^{-1} is the principal branch of the inverse tangent, i.e. \tan^{-1} : \mathbb{R} \to (-\pi/2 ,\pi/2). If (x_n)_{n\in\mathbb{N}} is a sequence in \mathbb{R} and...

Homework Statement The Attempt at a Solution I've got through this question up to the last bit. I've got B(0,1) = \{0\} and B(0,2) = \{y\in\mathbb{R} : -1<y<1 \} (i.e. the open interval (-1,1).) How do I show that every subset of \mathbb{R} is open (A \subseteq X is open if it...

Hi! I'm a beginner for a subject "topology". While studying it, I found a confusing concept. It makes me crazy.. I try to explain about it to you. For a set X, I've learned that a metric space is defined as a pair (X,d) where d is a distance function. I've also learned that for a set...

I have an apparently simple question, which is foundamental for a new approach to General Relativity. Is any diagonal metric with constant determinant a solution of Eintein Equations in vacuum? Does someone have the answer?

Hello there. I would like to find the metric tensor produced by the existence of two massive bodies. Does the principle of superposition work for metrics as well? The first idea I got was to add the two metrics for each separate body in order to obtain the resulting one. Is this approach valid...

Are there any down to Earth examples besides the empty set? Edit: No discrete metric shenanigans either.

If (X, d) and (X, r) are metric space, is {X, max(d, r)} necessary a metric space? what about (X, min(d, r))?

Homework Statement Show that \frac{{\partial}g^c{^d}}{{\partial}g_a{_b}}=-\frac{1}{2}(g^a{^c}g^b{^d}+g^b{^c}g^a{^d}) Homework Equations The Attempt at a Solution It seems like it should be simple, but I just do not see how to come up with the above solution. This is what I am coming...

Can someone direct me to the solution to the space-time metric, ds^2 = -dt^2 + dx^2 + dy^2 + dt^2? Thanks.

Hello. In my analysis book, it says that "Any closed bounded subset of E^n is compact" where E is an arbitrary metric space. I looked over the proof and it used that fact that E^n was complete, but it does not say that in the original condition so I was wondering if the book made a mistake in...

Can we define a metric space (\emptyset, d)? The metric is the part that confuses me, since it seems like all of the required properties of d are satisfied since they are "not not satisfied", but I'm not sure. Thank you!

Hi, Let's say I have a 10 dimensional Lie algebra over some field of functions, something along the lines of at least twice differentiable with twice differentiable inverses. The structure constants have inputs from this field. Is it possible to build a metric from these structure...

The Reissner Nordstrom metric considers charge apart from mass in its composition. Both charge and mass appear in the temporal as well as the spatial components of the metric. By considering a large amount of charge against a small amount of mass we can have an estimate the individual...

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...