Metric Definition and 1000 Threads

Homework Statement show that (the range of) a sequence of points in a metric space is in general not a closed set. Show that it may be a closed set. 2. The attempt at a solution I don't know where to start. For example, if we are given a sequence of real numbers and the distance...

Why Yab=Xab-kHab+k2HacHcb-... and not Yab=Xab-kHab+(1/2)k2Haccb-...? Y is the curved space-time metric X is the planespace-time metric

Can someone help me understand the notion of complete metric space? I've read the definition (the one involving metrics that go to 0), but I can't really picture what it is. Does anyone have any examples that could help me understand this?

Can an alternative topology of spacetime be defined upon a "mertic" of alternating forms? A less stringent question: Can a topology be defined, without first premising a metric, but premising an alternating structure instead?

I am studying topology. There I learn that the open sets given by the metric can be used to define a topology, e.g. the usual metric topology on R^n given by the Euclidean metric. Now I try to understand the topology of (flat) spacetime. Is there a metric? The Lorentz 'metric' gives...

1. Is its possible diagonalization of metric matrix (g_{uv}) in general relativity? 2. If we include imaginary numbers, can this help?

I have here a quote from Hartle's Gravity, page 321: "The fraction of rest energy that can be released in making a transition from an unbound orbit far from an extremal black hole to the most bound innermost stable circular orbit is (1-1/\sqrt{3})\approx 42\%". My question is about...

I have posted this here for two reasons, one I think you all will really appreciate this as a teaching tool to hopefully increase your students understanding (and hopefully grades) & second because you all might find/make more concept maps on more advanced topics or with focus shifted in a...

If torsion = anti-symmetric part of the connection coefficients, and \Gamma_{\alpha\beta\gamma}=\frac{1}{2}(g_{\alpha\beta,\gamma}+g_{\alpha\gamma,\beta}-g_{\beta\gamma,\alpha}) then doesn't the metric have to have an antisymmetric component? The first two terms on RHS are together...

Homework Statement There's something about the Riemannian metric that I don't understand. I don't understand the compatibility with the metric. If X,Y,Z are vector fields, the Riemannian connection (I'll call it D) satisfies DX(g(Y,Z)) = g(DXY,Z) + g(Y, DXZ), where g is the Riemannian metric...

I realize this is a "simple" mathematical exercise, in theory, but I'm having a lot of trouble finding some algorithmic way to do it. The problem is this: I want to expand the contravariant metric tensor components g^{\mu\nu} in terms of the covariant metric tensor g_{\mu\nu}. The first order...

Okay so I have: Eqn1) Tij=\rhouiuj-phij = \rhouiuj-p(gij-uiuj) Where Tij is the energy-momentum tensor, being approximated as a fluid with \rho as the energy density and p as the pressure in the medium. My problem: Eqn2) Trace(T) = Tii = gijTij = \rho-3p My attempt: Tr(T) = Tii...

My first question is the following. Does the radial component of the schwarzchild metric account for just the radius of the body in study or is it the distance between the body and the observer, where the body is treated as a singularity (Point mass particle)? My second question is about how...

Hi, Everyone: I am trying to understand the meaning of a statement that two embedded manifolds intersect normally*. The statement is made in a context in which any choice or existence of a metric is not made explicit, nor--from what I can tell-- implicitly either. If...

We know that every discrete metric space with at least 2 points is totally disconnected. Yet I read this: A MS that is countable with more than 2 pts is disconnected. Is it that I'm misreading this statement. It sounds like if it has 2 or less points it is connected? more means greater than.

Hello! When using the Schwarzschild exterior metric in the klein-gordon equation one can perform the standard tortoise(E-F) coordinate transform to yield a wave equation which has a well defined potential that is independent of the energy term. My understanding is that the motivation for this...

I am trying to proove that the following relation: A_{\nu} \partial_{\mu} \partial^{\nu} A^{\mu} = A_{\nu} \partial^{\mu} \partial^{\nu} A_{\mu} The only way I found is by setting: A_{\nu} \partial_{\mu} \partial^{\nu} A^{\mu} = A_{\nu} \partial_{\mu} \partial^{\nu} g^{\mu \sigma}...

Can someone explain the difference between the two? 2 topo spaces are isometric if they have the same metric properties and homeometric if they have the same topological properties. If a space is homeo it is iso, but not vice verse. Which begs the conclusion that every topological...

hello everyone, following the book of Landau&Lifsitz I managed to understand the Schwarzschild solution. At the end, it finds this formula for the mass of the spherical body generating the gravitational field: M=\frac{4\pi}{c^2} \int^a_0 \epsilon(r) r^2 dr in which \epsilon(r) is...

How to obtain Kerr Metric via Spinors (Newman-Penrose Formalism)? I am a bit confused with Ray d'Inverno's Book. Why perform the coordinates transformation: 2r-1 -> r-1 + r*-1 I am bit confused of it. And I am a bit confused, too, of how to write out null tetrad...

Homework Statement Consider a coordinate transformation from (t,x) to (u,v) given by t=u\sinh vx=u\cosh v Suppose (t,x) are coordinates in a 2-dimensional spacetime with metric ds^2=-dt^2+dx^2 Sketch, on the (t,x) plane, the curves u=constant and the curves v=constant.Homework Equations None...

Can someone please explain to me how exactly you write down a metric, say the FLRW metric in matrix form. Say we have the given metric here. ds^2 = dt^2 - R(t)^2 * [dw^2 + s^2 * (dθ^2 + sin^2(θ)dΦ^2)] Thank you.

It's really a question about convention. Does such a metric have to be linear on each fiber?

Could someone tell me what would be the partial derivative... \frac{ d g_{ab} }{ d g_{cd}} ?? Such expressions occur when one does variations of gravitational action... Note: For some reason, the d needed for derivative is not appearing in the post...although it was visible in the preview...

Hey all, My question pertains to interior metrics, for example the Schwarzschild interior metric given in post #5 of https://www.physicsforums.com/showthread.php?t=323684 The radial derivative of the first term, the dt^2 coefficient, matches the radial derivative of the Schwarzschild...

To show that some T(x,y) = something is a metric on a set for which it is compact, we have to prove that it respects the 3 axioms of distance. right?

Hello, I'm currently studying general relatively and am trying to plot orbits of planets around the sun using the schwarzchild metric. I've worked out the geodesic equations, working with c=1 to simplify things, and written a MATLAB script to plot trajectories, but I'm struggling to work out...

We know the background metric has the description; g_{\mu \nu}\sim\eta_{\mu \nu}+h_{\mu \nu} I would like to know what the physical meaning is of the difference then? h_{\mu \nu} - \eta_{\mu \nu} When I've read field theories describing gravitons, they are usually denoted as...

What exactly is the physical meaning of the fact that the covariant derivative of the metric tensor vanishes?

Hi. This is example 7.2 from Hartle's "Gravity" if you happen to have it lying around. Metric of a sphere at the north pole The line element of a sphere (with radius a) is dS^{2}=a^{2}(d\theta^{2}+sin^{2}\theta d\phi ^{2}) (In (\theta , \phi ) coordinates). At the north pole \theta = 0 and at...

I would like to ask, how these identities are true \partial_{\mu}(-g)=(-g)g^{\alpha\beta}\partial_{\mu}g_{\alpha\beta} and \partial_{\mu}g^{\alpha\beta}=-g^{\alpha\lambda}g^{\beta\rho}\partial_{\mu}g_{\lambda\rho} Sorry I meant" derivative of metric tensor and its determinant", I was able to...

Hi. I'm trying problem 1 on p138 of this http://arxiv.org/PS_cache/gr-qc/pdf/9707/9707012v1.pdf Now when I try and get the Euler Lagrange equation for \phi I get (the Kerr metric in BL coordinates can be found at the bottom of p77) \frac{\partial L}{\partial \phi} = \frac{d}{d \tau}...

Homework Statement Let (X,d) be a metric space and let {x_n} be a sequence in X converging to a. Show that d(b, x_n ) ->d(b,a) Homework Equations The Attempt at a Solution For every eps > 0 there is an N such that d(x_n,a) < eps for all n>= N But where do I go from here...

I wish I could calculate the contraction: gabgab I wish someone could show me how to get n! Unfortunately, I find it difficult, for I am not familiar with Tensor Algebra ... My wrong way to calculate it: gabgab= gabgba (since gab is symmetric) = δaa = 1Why is it wrong?

If (xn) and (yn) are two Cauchy sequences in a metric space (X, d), and we define d((xn), (yn)) = lim d(xn, yn). Is this a metric on the set of all Cauchy sequences? I'm thinking yes since all 3 properties work.

I have the following question on metric spaces Let (X,d) be a metric space and x1,x2,...,xn ∈ X. Show that d(x1, xn) ≤ d(x1, x2) + · · · + d(xn−1, xn2 ), and d(x1, x3) ≥ |d(x1, x2) − d(x2, x3)|. So the first part is simply a statement of the triangle inequality. However, the metric...

1. For integers m and n, let d(m,n)=0 if m=n and d(m,n) = 1/5^k otherwise, where k is the highest power of 5 that divides m-n. Show that d is indeed a metric. 2. The attempt at a solution Here is what I have come up with: PROOF: Clearly by definition d(m,n) = 0 iff m=n and d(m,n)>0...

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...