Metric Definition and 1000 Threads

So the axiom of choice is confusing to me, apperently there is a distinction between the exsistence of an element and the actual selection of an element? I'm confused as to how much the axiom of choice is needed in elementary metric space theorems? As an example, is the Axiom of Choice needed...

Hi! Given the schwarzschild metric ds^2=-e^{2\phi}dt^2+\frac{dr^2}{1-\frac{b}{r}} I can make this coordinate transformation \hat e_t'=e^{-\phi}\hat e_t \\ \hat e_r'=(1-b/r)^{1/2}\hat e_r and I will get a flat metric. Is this correct? Another thing I'm a lot confused about: if I am at...

Hey all, I've never taken a formal class on tensor analysis, but I've been trying to learn a few things about it. I was looking at the metric tensor in curvilinear coordinates. This Wikipedia article claims that you can formulate a dot product in curvilinear coordinates through the following...

The (0,0) and (r,r) components are: g_{00}= -e^{2\phi},g_{rr}=e^{2\Lambda}. From the first component, combined with the fact that the dot product of the four velocity vector with itself is -1, one can find in the MCRF, U^0=e^{-\phi}. What does this mean? In the MCRF, the rate of the two clocks...

I'm beginning self-study of real analysis based on 'Introductory Real Analysis' by Kolmogorov and Fomin. This is from section 5.2: 'Continuous mappings and homeomorphisms. Isometric Spaces', on page 45, Problem 1. This is my first post to these forums, but I'll try to get the latex right...

I have a simple question about the notation. I want to be more correct with notation, I don't understand exactly what the notation is saying. In regards to a Metric space A metric space is an ordered pair (M,d) where M is a set and d is a metric on M, i.e., a function {{\bf{d: M \times...

In Elements of the Theory of Functions and Functional Analysis (Kolmogorov and Fomin) the definitions are as follows: An open sphere S(x_0,r) in a metric space R (with metric function \rho(x,y)) is the set of all points x\in R satisfying \rho(x,x_0)<r. The fixed point x_0 is called the...

I need to transform cartesian coordinates to spherical ones for Minkowski metric. Taking: (x0, x1, x2, x3) = (t, r, α, β) And than write down all Christoffel symbols for it. I really have no clue, but from other examples I've seen i should use chain rule in first and symmetry of...

Hi all, In flat space-time the metric is ds^2=-dt^2+dr^2+r^2\Omega^2 The Schwarzschild metric is ds^2=-(1-\frac{2MG}{r})dt^2+\frac{dr^2}{(1-\frac{2MG}{r})}+r^2d\Omega^2 Very far from the planet, assuming it is symmetrical and non-spinning, the Schwarzschild metric reduces to the...

Hi all, I am wondering if it is possible to derive the definition of a Christoffel symbols using the Covariant Derivative of the Metric Tensor. If yes, can I get a step-by-step solution? Thanks! Joe W.

Officially, the US adopted metric units as the legal standard in 1866, but never seriously attempted to implement a plan to phase out "customary" units. As a result, the US is the only industrialized nation which still uses non-metric units widely in commerce and law...

Guys, I read that integrating the ds gives the arc length along the curved manifold. So in this case, I have a unit sphere and its metric is ds^2=dθ^2+sin(θ)^2*dψ^2. So how to integrate it? What is the solution for S? Note, it also is known as ds^2=dΩ^2 Thanks!

Hi! I'm trying to give a few examples of symmetric manifolds. In the article "Introduction to Symmetric Spaces and Their Compactification" Lizhen Ji mentions the Poincaré disk as a symmetric space in the following way: D = \{z \in \mathbb C | |z| < 1\} with metric ds^2 =...

This is the course description: I want to take this class because the professor comes highly recommended, but I'm a little worried that I won't be entirely prepared for it. Normally this class requires Real Analysis as a prerequisite, and even though the professor explicitly states that...

Hello friends, I would ask if anyone knows the métrqiue on the quotient space? ie, if one has a metric on a vector space, how can we calculate the metric on the quotient space of E?

Alcubierre metric: ds^2 = \left( v_s(t)^2 f(r_s(t))^2 -1 \right) \; dt^2 - 2v_s(t)f(r_s(t)) \; dx \; dt + dx^2 + dy^2 + dz^2 What formal conditions are required to verify a valid metric solution of the Einstein field equations? How many possible valid metric solutions are there in General...

Hello everyone, Let r(u_i) be a surface with i=1,2. Suppose that its first fundamental form is given as ds^2 = a^2(du_1)^2 + b^2(du_2)^2 which means that if r_1 = ∂r/∂u_1 and r_2= ∂r/∂u_2 are the tangent vectors they satisfy r_1.r_2 = 0 r_1.r_1 = a^2 r_2.r_2 =...

Hello, I believe this is a really stupid question but I can't seem to figure it out. So given a Minkowski spacetime one can choose either the convention (-+++) or (+---). Supposedly it's the same. But given the example of the four momentum: Choosing (+---) in a momentarily comoving...

This short work will help to calculate the varying energy for a non-rotating spherical distribution of mass. The Energy changing in a Schwartzschild Metric It is not obvious how to integrate an energy in the Schwartzschild metric unless you derive it correctly. The way this following metric...

Homework Statement Let S={1/k : k=1, 2, 3, ...} and furnish S with the usual real metric. Answer the following questions about this metric space: (a) Which points are isolated in S? (b) Which sets are open and closed in S? (c) Which sets have a nonempty boundary? (d) Which sets...

I am working on the weak gravitational field by using linearized einstein field equation. What if the metric tensor, hαβ turn out to be a complex numbers? What is the physical meaning of the complex metric tensor? Can I just take it's real part? Or there is no such thing as complex metric...

Homework Statement Show that the following three conditions of a metric space imply that d(x, y)=d(y, x): (1) d(x, y)>=0 for all x, y in R (2) d(x, y)=0 iff x=y (3) d(x, y)=<d(x, z)+d(z, y) for all x, y, z in R (Essentially, we can deduce a reduced-form definition of a metric space...

Need to find the Ricci scalar curvature of this metric: ds2 = e2a(z)(dx2 + dy2) + dz2 − e2b(z)dt2I tried to find the solution, but failed to pass the calculation of Riemann curvature tensor: <The Christoffel connection> Here a'(z) denotes the first derivative of a(z) respect to z...

Homework Statement Need to find the Ricci scalar curvature of this metric: ds2 = e2a(z)(dx2 + dy2) + dz2 − e2b(z)dt2 Homework Equations The Attempt at a Solution I tried to find the solution, but failed to pass the calculation of Riemann curvature tensor: <The Christoffel...

Hey all, making my way through Landau and Lifgarbagez classical theory of fields and i had a specific question on the Einstein equations. Following the palatini approach, we assume that the connection and metric are independent variables and are not related a priori. In the footnote, they say...

I have looked at the definition of the metric tensor, and my sources state that to calculate it, one must first calculate the components of the position vector and compute it's Jacobian. The metric tensor is then the transpose of the Jacobian multiplied by the Jacobian. My problem with this...

Hi there, I've been asking in the Quantum Physics Forum about some foundational stuff and I've got stuck in why we assume in QM that the states space is linear. I have not found any deeper answer than "because it fits with observation". So I am now trying the other way around. I believe that...

Homework Statement Calculate the Riemann curvature for the metric: ds2 = -(1+gx)2dt2+dx2+dy2+dz2 showing spacetime is flat Homework Equations Riemann curvature eqn: \Gammaαβγδ=(∂\Gammaαβδ)/∂xγ)-(∂\Gammaαβγ)/∂xδ)+(\Gammaαγε)(Rεβδ)-(\Gammaαδε)(\Gammaεβγ) The Attempt at a Solution...

Homework Statement Consider a 2D spacetime where space is a circle of radius R and time has the usual description as a line. Thus spacetime can be pictured as a cylinder of radius R with time running vertically. Take the metric of this spacetime to be ds^{2}=-dt^{2}+R^{2}d\phi^{2} in the...

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...

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...

(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...