Metric Definition and 1000 Threads

The Friedmann–Lemaître–Robertson–Walker metric is a solution of the field equations of GR. It tells us the local behavior of spacetime, that is, g(x) at a given spacetime point x If the matter density is high enough, the curvature is positive. It is said then that the universe is closed...

Given a metric tensor gmn, how to calculate the inverse of it, gmn. For example, the metric g_{\mu \nu }= \left[ \begin{array}{cccc} f & 0 & 0 & -w \\ 0 & -e^m & 0 &0 \\0 & 0 & -e^m &0\\0 & 0 & 0 & -l \end{array} \right] From basic understanding, I would think of divided it, that is...

Homework Statement let d be a metric on an infinite set m. Prove that there is an open set u in m such that both u amd its complements are infinite. Homework Equations If d is not a discrete metric, and M is an infinite set (uncountble), then we can always form an denumerable subset...

Homework Statement I am wondering if anyone understands why this metric is defined the way it is because i can't seem to make sennse of it. I get that way we use the underlying metric space to define the borel sigma field and then the set of all borel measures, but the actual definition...

In the context of Friedmann's time, 1922, how did he know to make the metric scale factor, a, a function of time when Hubble's redshifts were not yet published? I understand that he took the assumption that the universe is homogenous and isotropic, but does that naturally imply that the universe...

Hi all, Given a metric space (X,d), one can take its completion by doing the following: 1) Take all Cauchy sequences of (X,d) 2) Define a pseudo-metric on these sequences by defining the distance between two sequences to be the limit of the termwise distance of the terms 3) Make this a...

I was told the extended real \hat{R}=R\cup\{-\infty,\infty\} is homeomorphic to [0,1], I was wondering if the mapping h: [0,1]\rightarrow\hat{R}, h(x)=\cot^{-1}(\pi x), 0<x<1, h(0)=\infty, h(1)=-\infty is a valid homeomorphism, so that a metric may be defined by the metric on [0,1]? Thank...

I'm well aware and understand that homeomorphism do not need to preserve metric completeness, I'm just trying to work out a simple counterexample. I have tried searching around just for kicks, but only seem to find more complex ones. I'm wondering if the one I have works for it for sure? On...

I am trying to understand how the Schwartzschild metric works and coming to the conclusion that if I drop an object that it will not fall to the Earth but just stay there when I open my hand. Therefore, I'm confident I have made a mistake, but I don't see where it is. Here is the metric...

How to calculate Riemannian Metric by distance function?? Dear Folks: Here is the problem: in |z|<1, we difine a distance between any two points z1 and z2 by d(z1 , z2) = ln((z1 - b)(z2 - a)/((z1 - a)(z2 - b))) ,where a is the intersection of line z1z2 and the circle which is nearer to...

So imagine your on Earth at a latitude of 30 to 45° N, between two rotating Kerr Metric Blackholes with detached event horizons (dual singularities) allowing you to be shielded from the crushing force of the black holes. Which way do the rotating black holes need to rotate for the past and...

For the following two-dimensional metric ds^2 = a^2(d\theta^2 + \sin^2{\theta}d\phi^2) using the Euler-Lagrange equations reveal the following equations of motion \ddot{\phi} + 2\frac{\cos{\theta}}{\sin{\theta}}\dot{\theta}\dot{\phi} = 0 \ddot{\theta} -...

Another "christoffel symbols from the metric" question Homework Statement Find the Christoffel symbols from the metric: ds^2 = -A(r)dt^2 + B(r)dr^2 Homework Equations \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{x^a}} \right) = \frac{\partial L}{\partial x^a} The...

So far I have learned a bit about topological spaces, there has been several occasions regarding metric spaces where I had to invoke completeness (or at least uncountability) of real numbers R, which is itself a property of the usual metric space of R. For example, to show that any countable...

Disregard. I done figured it out.Homework Statement Find equations of motion for the metric: ds^2 = dr^2 + r^2 d\phi^2Homework Equations L = g_{ab} \dot{x}^a \dot{x}^b \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{x}^a} \right) = \frac{\partial L}{\partial x^a} The Attempt at a...

(X,\rho) is a pseudometric space Define: x~y if and only if ρ(x,y)=0 (It is shown that x~y is an equivalence relation) Ques: If X^{*} is a set of equivalence classes under this relation, then \rho(x,y) depends only on the equivalence classes of x and y and \rho induces a metric on...

Homework Statement Let X be a metric space and A a subset of X. Prove that the following are equivalent: i. A is dense in X ii. The only closed set containing A is X iii. The only open set disjoint from A is the empty set Homework Equations N/A The Attempt at a Solution I can...

Suppose that (X,d) is a metric Show \tilde{d}(x,y) = \frac{d(x,y)}{\sqrt{1+d(x,y)}} is also a metric I've proven the positivity and symmetry of it. Left to prove something like this Given a\leqb+c Show \frac{a}{\sqrt{1+a}}\leq\frac{b}{\sqrt{1+b}}+\frac{c}{\sqrt{1+c}} I try to...

Got it, thank you

I have very little knowledge in general relativity, though I do have a decent understanding of the theory of special relativity. In special relativity, points in space-time can be represented in Minkowski space (or a hyperbolic space) so that the metric tensor (that is derived in order to...

Hello, I have a problem to understand what people say by "induced metric". In many papers, it is written that for brane models, if we consider the metric on the bulk as g_{\mu\nu} hence the one in the brane is h_{\mu\nu}=g_{\mu\nu}-n_\mu n_{\nu} where n_{\mu} is the normalized spacelike...

I left this following question from my excercises last, hoping that solving the others will give me an insight onto how to proceed. But I still don't have a plan on how to start it: Consider the sequence s_n = Ʃ (k=0 to n) (t_k * p^k) in Q(rationals) with the p-adic metric (p is prime); where...

I left this following question from my excercises last, hoping that solving the others will give me an insight onto how to proceed. But I still don't have a plan on how to start it: Consider the sequence s_n = Ʃ (k=0 to n) (t_k * p^k) in Q(rationals) with the p-adic metric (p is prime); where...

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...

Homework Statement A spaceship is moving without power in a circular orbit about an object with mass M. The radius of the orbit is R = 7GM/c^2 (1) Find the relation between the rate of change of angular position of the spaceship and the proper time and radius of the orbit. Homework...

Homework Statement Show that an open (closed) subset of a metric space E is connected if and only if it is not the disjoint union of two nonempty open (closed) subsets of E. Homework Equations The definition of connectedness that we are using is as follows: A metric space E is...

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...

So I have the hyperboloid in \mathbb{R}^3 given by x^2 + y^2 - z^2 = -1 . I define the scalar product of 2 tangent vectors (a1, b1, c1) and (a2, b2, c2) at a point as a1a2 + b1b2 - c1c2. I want to show that this defines a Riemannian metric on the hyperboloid. I know that a Riemannian...

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 Let a,b \in \mathbb{R} and we define I = [a,b] ([] means closed set). Let \mathcal{C}_{\mathbb{K}}(I) be the space of all continuous functions I \to \mathbb{K} with the norm f \mapsto ||f||_I = \displaystyle \sup _{x \in I} |f(x)|. Let U be the set of all continuous functions...

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...