Metric Definition and 1000 Threads

What is the formula to evaluate a multi-integral by a change of coordinates using the squareroot of the metric instead of the determinate of the Jacobian? Thanks.

Hi, I have posted this question in the PF relativity forum because I am trying to understand the derivation of the Lense-Thirring (L-T) metric. Various sources suggest that L-T produced a solution of Einstein’s field equations of general relativity in 1918, just a few years after Schwarzschild...

Homework Statement Show that if (x_{n}) is a sequence in a metric space (E,d) which converges to some x\inE, then (f(x_{n})) is a convergent sequence in the reals (for its usual metric). Homework Equations Since (x_{n}) converges to x, for all ε>0, there exists N such that for all...

Barbour writes: the metric tensor g. Being symmetric (g_uv = g_vu) it has ten independent components, corresponding to the four values the indices u and v can each take: 0 (for the time direction) and 1; 2; 3 for the three spatial directions. Of the ten components, four merely...

I was attempting to find a counterexample to the problem below. I think I may have, but was ultimately left with more questions than answers. Consider the space, L, of all bounded sequences with the metric \rho_1 \displaystyle \rho_1(x,y)=\sum\limits_{t=1}^{\infty}2^{-t}|x_t-y_t| Show that a...

Homework Statement The problem is I am wanting to know if the expression on the right hand side is dimensionless. Homework Equations ds^2 = (1 - \frac{2GM}{c^2 r})c^2 dt^2 The Attempt at a Solution Since the Schwarzschild radius is r = \frac{2GM}{c^2} would I be right in saying that...

H is a contravariant transformation matrix, M is a covariant transformation matrix, G is the metric tensor and G-1 is its inverse. Consider an oblique coordinates system with angle between the axes = α I have G = 1/sin2α{(1 -cosα),(-cosα 1)} <- 2 x 2 matrix I compute H = G*M where M =...

In this Wiki link for the derivation of the Schwarzschild metric, in the section "simplifying the components", g_22 and g_33 are derived. The problem is that upon deriving them, they first set those local measurements of the components for the metric upon a 2_sphere (on the left side) equal to...

Homework Statement Let M be a metric space with metric d, and let d_{1} be the metric defined below. Show that the two metric spaces (M,d), (M,d_{1}) have the same open sets. Homework Equations d_1:\frac{d(x,y)}{1+d(x,y)} The Attempt at a Solution I tried to show that the neighborhoods...

Would someone here be able to write down for me an example of a metric on a manifold with both macroscopic dimensions, and microscopic "curled up" dimensions with some radius R ? Number of dimensions and coordinates used don't matter. Not going anywhere with this, I am just curious as to how...

Hello, everybody. I have some doubts I hope you can answer: I have read that the n+1-dimensional Anti-de Sitter (from now on AdS_{n+1}) line element is given, in some coordinates, by: ds^{2}=\frac{r^{2}}{L^{2}}[-dt^{2}+\sum\limits_{i=1}^{n-1}(dx^{i})^{2}]+\frac{L^{2}}{r^{2}}dr^{2} This can be...

Hello, So, given two points, x and x', in a Lorentzian manifold (although I think it's the same for a Riemannian one). If in x the determinant of the metric is g and in the point x' is g'. How are g and g' related?By any means can g=g'? In what conditions? I'm sorry if this is a dumb...

This is a fairly trivial question I think. I'm only asking it here because after some googling I was unable to find its answer. I was at one point led to believe that the form of the Lorentz-transformation matrix is dependent on the convention used for the Minkowski metric. Specifically it...

In deriving the Schwarzschild metric, the first assumption is that the transformation of r^2 (dθ^2 + sin^2 θ dψ) remains unchanged due to the spherical symmetry. What does that mean exactly? What is the logic behind it? Please apply any math involved in algebraic form. Thanks.

hi friends, Suppose we have a vector bundle E equipped with a hermitian metric h, and in a subbundle of E noted SE . I would like tocalculate explicitly the induced metric Sh defined on SE. How to proceed?

I have a very limited knowledge of tensor calculus, and I've never had proper exposure to general relativity, but I hope that the people reading this forum are able to help out. So I'm doing some stat. mech. and a part of a system's free energy is \mathcal{F} = \int V(\rho)\nabla^2\rho dx I'd...

Ok, so I'm trying to plot the unit circle using the chebyvhev metric, which should give me a square. I am trying this in MATLAB, using the 'pdist' and 'cmdscale' functions. My uber-complex code is the following: clc;clf;clear all; boundaryPlot=1.5; % Euclidean unit circle for i=1:360...

Why is the unit vector for time in Minkowski space i.e. the fourth dimension unit vector always opposite in sign to the three other unit vectors? The standard signature for Minkowski spacetime is either (-,+,+,+) or (+,-,-,-). Is there some particular reason or advantage for making time...

Now let's say I have the metric for some curved two surface ds^2=G(u,v)du^2+P(u,v)dv^2 ( the G and P functions being the 00 and 11 components, assuming the metric is diagonal) Now my question is, since the metric defines the scalar product of two vectors, let's say (1,0) and (0,1), for...

Hello, We know that NUT spacetime is just like a massless rotating black hole, that this consideration introduces a new concept "magnetic mass", and I know just a little about its metric form and the parameters appear in it. While I was searching for NUT spacetime and its metric, I mostly...

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...

Completion of Metric Space Proof from "Intro. to Func. Analysis w/ Applications" Homework Statement I have started studying Functional Analysis following "Introduction to Functional Analysis with Applications". In chapter 1-6 there is the following proof For any metric space X, there is...

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...