Metric Definition and 1000 Threads

Homework Statement I was working on a problem and think I might have run across an issue. Is the usual metric defined on R - {0}? (Where R is the real numbers) Reworded, can I say that I have a space R - {0} with the usual metric on it? Thank you.

Why must we have g_{\mu\nu}=g_{\nu\mu}? What are the physical consequences if this did not hold?

1. Let M be a compact metric space. If r>0, show that there is a finite set S in M such that every point of M is within r units of some point in S. 2. Relevant theorems & Definitions: -Every compact set is closed and bounded. -A subset S of a metric space M is sequentially compact...

Hey, For a problem I am working on I need to know if the energy-momentum tensor of the Reissner- Nordstrom metric obeys the weak energy condition. Since I need the result for the following calculations, I just want to be sure that it is correct. Does anyone of you know the correct answer and...

Hey guys! I am going crazy... most books don't cover this and instead assume that the manifold is Riemannian or pseudo-Riemannian and has a metric tensor defined on it. I want a "generalized" hodge star. I have an orientable smooth manifold, that's IT. I have heard that there is a way to...

If we use the mapping (r,phi)---->(x,y)= (2tan(r/2)cos(phi), 2tan(r/2)sin(phi)) Which metric do we have to impose on R^2 in order that the mapping preserves distance. Any help would be greatly appreciated. Thank you very much

I've never actually seen a proof that a space is compact just from the definition. In metric spaces it was usually after the notion of closed and bounded or sequential compactness was introduced. For example is there a way to prove [a,b] is compact (with the usual topology on the real numbers)...

hey, does anyone of you know the reissner-nordström metric, if there is a magnetic field instead of the coulomb field of a charged gravitating body? i just need the formula, but i could not find it in the internet, maybe someone here knows it? sorry for my english, but i am german...

How does one redefine the metric of a given set, such as the reals? I thought it would be an interesting concept to have a metric defined like so: d_X:X^n \times X^n \to \Re (x_1, x_2, x_3, \cdots, x_n), (y_1, y_2, y_3, \cdots, y_n) \mapsto \sum^{n}_{i=1}{\frac{x_i+y_i}{2}} Does it have to be...

How many cubic meters are in 5.43 x 10^6 cm^3? There are .001 meters in the centimeter. m= meter cm= centimeter This is what I did: 5.43 x 10^6 cm^3 (.01m/1 cm)^3 And then did the calculations, which gave me: .0000043 x 10^6 m^3 or 4.3 m^3. Not sure if I set up the...

I’m sorry, but I find dark matter and dark energy problematic. It’s hard to think of a Universe made up of about 95 % of stuff we have no idea about, except that maybe dark matter and dark energy have some properties. So I’m thinking maybe there’s something wrong with the data, but I can’t...

Hi, I'm new to Physics Forum and wasn't really sure where to post this since its not strictly speaking a homwork question. So if it happens to be in the wrong place I apologise. I was looking through some lecture notes from when I did my Physics degree years ago and come across a problem...

Homework Statement Prove that the sequence space l^2 (the set of all square-summable sequences) is complete in the usual l^2 distance. Homework Equations No equations.. just the definition of completeness and l^2. The Attempt at a Solution I have a sample proof from class to show...

What are the symmetries determined by FRW spacetime? I guess they include Lorentz symmetry, rotationally and translationally symmetries, but not time symmetry. Is this right? Thanks

I'm not very fund of the subject, but what i know is that the Schwarzschild metric and other known solutions for Einstein's action have some constant representing mass. However, I encountered some solutions where no such constant existed. Can someone explain to me what does this mean exactly? In...

So I am an engineering graduate trying to teach myself some general relativity. I have tried to solve the Einstein Field equations for a wormhole metric and some others. After pages and pages of calculating Christoffel Symbols, Riemann Tensors, Ricci Tensors and Scalars, and so on, I end...

hi, in my textbook there ist a problem, i cannot solve: let T be a linear bijective map from R^4 to R^4, which preserves the light cone. show: T* ds^2 = (constant)^2 * ds^2, where ds^2 ist the minkowsky metric and T* ist the pullback of the metric. can someone show how to do it. michiherlin

1. What are the value of physics constant in Kerr metric, including G, M, c, a, r, or others? I expect to simplify Gamma 2. why g_compts[1,4] has element and not [4,1]? 3. Some book assume G = c = 1, what is the meaning of this setting? 4. Different material have different metric, are...

Consider the spacetime metric ds^2=-(1+r)dt^2+\frac{dr^2}{(1+r)} + r^2 ( d \theta^2 + \sin^2{\theta} d \phi^2) where \theta, \phi are polar coordinates on the sphere and r \geq 0. Consider an observer whose worldline is r=0. He has two identical clocks, A and B. He keeps clock A with...

Hi, I'm a physics undergrad working through Carroll at the moment. In the section on the Kerr black hole, he states that K= \partial_t is a Killing vector because the coefficients of the metric are independent of t. He then states in eq. 6.83 that K^\mu is normalized by: K^\mu K_\mu = -...

I am a bit perplexed by the consequences of the fact that all covariant first derivatives of the metric tensor are zero. I think I can follow some of the proofs, as presented for example in John Lee "Riemannian Manifolds - An Introduction to Curvature". But intuitively it "seems wrong" to me...

Hi guys I have a quick question on the Schwarzschild Metric: Since the metric is a solution to the EFEs does it intrinsically have the curvature of the gravitational field embedded in the metric? If so is it represented by the time and spatial components of the metric? If not could you please...

I cannot work out the following functional derivative: \frac{\delta}{\delta g_{\mu\nu}} \int d^4 x f^a_{\phantom{a}b} \nabla_a h^b Where f is a tensor density f= \sqrt{\det g} \tilde{f} ( \tilde{f} is an ordinary tensor) and should be consider as independent of g. In my opinion this is not...

General four-dimensional (symmetric) metric tensor has 10 algebraic independent components. But transformation of coordinates allows choose four components of metric tensor almost arbitrarily. My question is how much freedom is in choose this components? Do exist for most general metric...

Homework Statement Let X be the integers with metric p(m,n)=1, except that p(n,n)=0. Show X is closed and bounded but not compact. Homework Equations I already check the metric requirement. The Attempt at a Solution I still haven't got any clue yet. Can anyone help me out?

I'm following a slightly confusing set of notes in which I can't tell what exactly the timelike geodesic equations for the Schwarzschild metric are (seems to have about 3 different equations for them). How are these derived, or alternatively, does anyone have a link to a site in which they...

Hi, I am in the process of trying to teach myself GR Maths, at the A101 level, and have been working through the idea of tensors as scalars, vectors and matrices, i.e. rank-0, 1 and 2 tensors. Think I have also acquired some idea of the concept of contravariance and covariance, which then seems...

I'm having trouble thinking of an example of one. I'm aware that all metrics induced by a norm are translation-invariant, but I can't think of any example of a metric that isn't induced by a norm. I guess I'm stuck focusing on one aspect and I can't think of a simple example.

Hi! I'm trying to derive the hyperbolic distance formula for the upper-half plane model. It is given here: http://en.wikipedia.org/wiki/Poincar%C3%A9_half-plane_model" I have the first formula, (ds)2= ... But I can't figure out how they got the distance formula below it. I understand...

Homework Statement Prove that f:(M,d) -> (N,p) is uniformly continuous if and only if p(f(xn), f(yn)) -> 0 for any pair of sequences (xn) and (yn) in M satisfying d(xn, yn) -> 0. Homework Equations The Attempt at a Solution First, let f:(M,d)->(N,p) be uniformly continuous...

What is the rank of the matrix of a reimannienne metric ?

A metric consistent with interval: \mathrm{d}s^2=-\mathrm{d}\tau^2+\frac{4\tau^2}{(1-\rho^2)^2}\left(\mathrm{d}\rho^2+\rho^2\mathrm{d}\theta^2+\rho^2\sin(\theta)^2\mathrm{d}\varphi^2\right) get zero for riemann's tensor, therefor must be isomorphic with minkowski tensor. But I don't find thus...

Given a metric of the form ds^2 = A(r) dt^2 + B(r) dr^2 + C(r)^2 (d\theta^2+sin^2\theta d\phi^2 + sin^2\theta sin^2\phi d\psi^2) I want to find the ADM mass of this black hole. Can anyone help me with the formula, or method to follow?

Hi everybody, I'm having difficulties sizing a compressor for natural gas. First,my assumption is to take natural gas as mixture of ideal gases.The compression must be between 9 and 32 bar and it starts at 45 °C. I know that volumetric flow rate must be 85000 Sm3 per day ,where I assume 15°C and...

Homework Statement X, Y are metric spaces and f: X \rightarrow Y Prove that f is discontinuous at a point x \in X if and only if there is a positive integer n such that diam f(G) \geq 1/n for every open set G that contains x Homework Equations diameter of a set = sup{d(x,y): x,y...

Hello, when measuring length of geodesic shortest paths, or more in general, when measuring the length of a parametric curve in the space, what we usually do is to sum the length of infinitesimal arcs of that curve, assuming an euclidean norm. Why this choice? I have not found in literature any...

Homework Statement Let (X, d) be a metric space and let A,B\subsetX be two disjoint closed sets. Show that X is normal by using the function f(x)=d(x,A)/[d(X,A)+d(x,B)] The Attempt at a Solution I'm somewhat stuck on this. I'm guessing the proof is pretty short, but any help would be...

Hello, let's supppose I am given a unit-quaternion q expressed as an element of \mathcal{C}\ell_{0,2}(\mathbb{R}) as follows: \mathit{q} = a + b \mathbf{e_1} + c \mathbf{e_2} + d \mathbf{e_{12}} I now rearrange the terms in the following way: \mathit{q} = (a + d \mathbf{e_{12}}) +...

HI I've got this question I don't know how to do; Let X be a metric space, and let Q,J be non-empty subsets of X. prove that inf{dist(x,J):x is a member of Q}= inf{dist(Q,y):y is a member of J}. I know that the dist(x,J):= inf{d(x,y)|y is a member of J}, I thought maybe if I tried to...

Homework Statement Prove that if a metric space (X, d) has an \epsilon-net for some positive number \epsilon, then (X, d) is bounded. Homework Equations The Attempt at a Solution I think that (X, d) might be not bounded. For example, let X be a subspace of real line with usual...

hi friends can someone help me and explain for me what is the difference between a standard and metric?

Homework Statement Suppose M is a metric space and A \subseteq M. Then A is totally bounded if and only if, for every \epsilon >0, there is a finite \epsilon-dense subset of A. Homework Equations The Attempt at a Solution I have already done the \Rightarrow but need to verify...

can we define a metric on a submanifold of as follows: M is a manifold equipped with a Riemannian metric g, we denote (M, g), and S is a submanifold of M. an application i of S to M is an immersion, i * is the linear tangent; then a metric on S G is given by G = goi*.

Is component(maximal connected set) of a metric space open or closed or both(clospen)?or even can be half-open(not open and not closed)? I know it is a silly question as (3,5] is a component in R,right? However some theorem i encountered stated that component must be closed or must be open. I...

Homework Statement Let E,E' be metric spaces, f:E\rightarrow E' a function, and suppose that S_1,S_2 are closed subsets of E such that E = S_1 \cup S_2. Show that if the restrictions of f to S_1,S_2 are continuous, then f is continuous. Also, if the restriction that S_1,S_2 are closed is...

Hi, I have been wondering if there is a Lorentz-invariant quantity that satisfies the definition of a metric for space-time. The space-time interval s2 = t2 - r2 [where r is the vector (x,y,z)] does not satisfy the requirement for a metric m that m(t1,r1, t2,r2) = 0 if and only if (t1,r1)...

Homework Statement One needs to show that a connected metric space having more than one point is uncountable. The Attempt at a Solution First of all, if (X, d) is a connected metric space, it can't be finite, so assume it's countably infinite. Let x be a fixed point in X. For any x1 in...

hello! just a quick question, does the covariant derivative of the metric give zero even when the indices(one of the indices) of the metric are(is) raised? also another question not entirely related, does the covariant deriv. of exp(2 phi) where phi is the field, also give zero or not...

I'm trying to find Schwarzschild solution for 3-dimensional space-time (i.e. time\otimes space^2). The problem is, I can't take the 4-dimensional solution \[ds^2=\left(1-\frac{r_g}{r}\right) dt^2-\left(1-\frac{r_g}{r}\right)^{-1} dr^2-r^2\left(d\theta^2+sin^2\theta d\phi^2\right)\] and...

let M be a manifold and g a metric over M . is it true that every subbundle from M must have the same metric g ?