Metric Definition and 1000 Threads

Hello, the following is a post that was in progress and I am continuing it here after I received a message saying that most of the members had moved from mathhelpforum here. Me: I have a problem where I am asked to show that for a complete metric space X, the the natural Isometry F:X --> X* is...

Hi, i was thinking about the metric tensor transformation law: g_{cd}(x) = \frac{{dx'}^a}{{dx}^c} \frac{{dx'}^b}{{dx}^d} g'_{ab}(x') and, in view of this definition, the differences between Poincare transformations and reparametrization-like transformation (f.e. various conformal...

(1) (X,d) is a COMPACT metric space. (2) f:X->X is a function such that d(f(x),f(y))=d(x,y) for all x and y in (X,d) Prove f is onto. Things I know: (2) => f is one-one. (2) => f is uniformly continuous. I tried to proceed by assuming the existence of y in X such that y has no...

This is the problem I'm trying to slove: Consider the sequence s_n = Sumation (from k=0 to n) p^k (i.e. s_n=p^0+p^1+p^2...+p^n) in Q(rationals) with the p-adic metric (p is prime). Show that s_n converges to a rational number.[/B] Now, I do get some intuition on showing that the...

I've been doing some amateur simulations of particle trajectories in a few well-known metrics (using GNU Octave and Maxima), and in the case of the Schwarzschild and Gullstrand-Painleve metrics I have the ability to check my results using freely available equations for conservation of energy and...

Homework Statement Find (X,d) a metric space, and a countable collection of open sets U\subsetX for i \in Z^{+} for which \bigcap^{∞}_{i=1} U_i is not open Homework Equations A set is U subset of X is closed w.r.t X if its complement X\U ={ x\inX, x\notinU} The Attempt at a Solution Well...

I am using the schwartzchild metric given as ds^2 = (1 - \frac{2M}{r})dt^2 - (1 - \frac{2M}{r})^{-1} dr^2 , where I assume the angular coordinates are constant for simplicity. So if a beam of light travels from radius r0 to smaller radius r1, hits a mirror, and travels back to r0, I am...

Hello everyone. I want to study topology ahead of time (it begins next semester only) and I have two options: I could go straight for general topology (among the books I searched I found Munkres to be the one I felt most comfortable with) or go for a thorough study of metric spaces (in which...

Homework Statement Is empty set a metric space? Homework Equations None. The Attempt at a Solution It seems so because all the metric properties are vacuously satisfied. Mabe the question had better be put like this: Does mathematicians tend to think empty set as a metric space...

Are there different kinds of metric in GR? For instance. I read in http://www.astronomy.ohio-state.edu/~dhw/A682/notes3.pdf there the FRW Metric is about: "In 1917, Einstein introduced the first modern cosmological model, based on GR, in which the spatial metric is that of a 3-sphere:"...

I'm coming with a good background in Big Bang expansion as the following sci-am article shows (which I've mastered): http://space.mit.edu/~kcooksey/teaching/AY5/MisconceptionsabouttheBigBang_ScientificAmerican.pdf What I'd like to understand is this. Expansion can only be felt in unbound...

So the confusion I'm having here really has to do with parity inversion in spherical (or boyer-linquist) coordinates. I've been looking at the discrete symmetries of the Kerr-Newman metric, and I've noticed that depending on how you define parity-inversion, you can get very different results...

Homework Statement let {xi} be a sequence of distinct elements in a metric space, and suppose that xi→x. Let f be a one-to-one map of the set of xis into itself. prove that f(xi)→x Homework Equations by convergence of xi, i know that for all ε>0, there exists some n0 such that if i≥n0...

Hello guys. I was told to prepare a presentation on perturbed Einstein's field equation by my advisor. I got some of the things I needed to start with in the Weinberg's Cosmology book but it was not enough. Can anyone please tell me a book or anything with information on metric perturbation? Thanks

I am trying to match little square patches in an image. You can imagine that these patches have been "vectorized" in that the values are reordered consistently into a 1D array. At first glance, it seems reasonable to simply do a Euclidean distance style comparison of two of these arrays to get a...

I typed the problem in latex and will add comments below each image. The supremum of |1 - x| seems dependent on the interval [a, b]. For instance, if [a, b] = [-500, 1], then 501 is the supremum of |1 - x|. But if [a, b] = [-1, 500], then 499 is the supremum of [1 - x]. So what should I...

When reading other threads, following question crept into my mind: When given a manifold, why shouldn't I give it distance function by giving it a simple metric function, that is MxM→ℝ with the usual axioms? I could happily measure distances in coordinate-independent way for ever after...

Homework Statement Prove the triangle inequality for the following metric d d\big((x_1, x_2), (y_1, y_2)\big) = \begin{cases} |x_2| + |y_2| + |x_1 - y_1| & \text{if } x_1 \neq y_1 \\ |x_2 - y_2| & \text{if } x_1 = y_1 \end{cases}, where x_1, x_2, y_1, y_2 \in \mathbb{R}...

[topology] "The metric topology is the coarsest that makes the metric continuous" Homework Statement Let (X,d) be a metric space. Show that the topology on X induced by the metric d is the coarsest topology on X such that d: X \times X \to \mathbb R is continuous (for the product topology on X...

So I'm working on a problem (Hartle problem 6, chapter 6) dealing with tachyons. So far, I have determined the four-velocity and the four-momentum (up to a sign) of a tachyon. I have, with the four-velocity being a unit spacelike four-vector, u^{\alpha}=\frac{\pm...

Homework Statement suppose that a metric space A is a union A = B U C of two subsets of finite diameter. Prove A has finite diameter. Homework Equations The Diameter of a metric space M is sup D(a,b) for all a,b in M. The Attempt at a Solution Really, no idea where to begin. I just...

Let C(X) be the set of continuous complex-valued bounded functions with domain X. Let {fn} be a sequence of functions on C. We say, that fn converges to f wrt to the metric of C(X) iff fn converges to f uniformly. By definition of the uniform convergence, for any ε>0 there exists integer N...

A metric space of equivalent Cauchy sequence classes (Z, rho) is defined using a metric of the sequence elements in the space (X,d), where d is from XX to R (real numbers). The metric of the sequence classes is rho = lim d(S, T), where S and T are the elements of the respective sequences. To...

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

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...

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))?

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...