Metric Definition and 1000 Threads

Given a manifold as algebraic variety, say sphere, how do we obtain possible metrics? how do we classify them? If spcaetime manifold is n-sphere, Einstein's vacuum (for now) equation would be some special metric among many other possible metrics? i'm curious what role Einstein's equation...

Hi, If you have a spherically symmetric spacetime metric in a set of spherical coordinates t,r,theta,phi: [P,Q,0,0;Q,R,0,0;0,0,S,0;0,0,0,Ssin^(theta)]. Here P,Q,R,S are functions of t and r. Now, if I want to choose cooridnates to get the metric in the generic diagonal form (that is by...

Hi, I'm trying to determine the exact transformation that brings a spherically symmetric spacetime metric in spherical coordinates to the Sylvester normal form (that is, with just 1 or -1 on its main diagonal, with all other elements equal to zero.) Assuming that the metric has Lorentzian...

Does anyone know what the metric tensor looks like for a 2 dimensional sphere (surface of the sphere)? I know that it's coordinate dependent, so suppose you have two coordinates: with one being like "latitude", 0 at the bottom pole, and 2R at the northern pole, and the other being like...

Hello, I am trying to understand what the differences would be in replacing the symmetry equation: g_mn = g_nm with the Hermitian version: g_mn = (g_nm)* In essence, what would happen if we allowed the metric to contain complex elements but be hermitian? I am not talking about...

If already known the form of Action and unkown the Metric, how to derive the geodesic equation?

Hello, In Godel's paper: an example of a new type of cosmological solutions of einstein's field equations of gravitation, he passes from his original metric to cylindrical coordinates by giving some transformation formulas. Can someone tell me how is this transformation obtained, or at least...

Alright, first things first, I'm a grade 12 student residing in Ontario, Canada and I'm relatively new to these forums and to the world of physics. I'm doing my grade 12 ISU and I've taught myself how to work around spherical coordinate systems, however, the schwarzschild metric confuses me...

[FONT="Georgia"][FONT="Tahoma"]Homework Statement Assume that we have a metric like: ds^{2}=f dr^{2}+ g d\theta^{2}+ h d\varphi^{2} where r,\theta , \varphi are spherical coordinates. f,g and h are some functions of r and theta but not phi. Homework Equations How can I calculate...

Hi all, I have a GR exam on tuesday and getting a bit confused as to how to find the metric for an observer in free fall a distance two schwarzchild radii from a black hole. I know this is a bit of a basic question but I am just wondering if I am correct to substitute r=2rs and dt=d(tau)...

The Robertson-Walker metric applies a time-dependent scale factor to model the expansion of the universe. The scale factor is only applied to the spatial coordinates (in the frame of the "comoving observer"). That is not covariant and it is hard to see how c could remain constant if the same...

Homework Statement I'm stuck on how to start this. The Hammin metric is define: http://s1038.photobucket.com/albums/a467/kanye_brown/?action=view&current=hamming_metric.jpg and I'm asked to: http://i1038.photobucket.com/albums/a467/kanye_brown/analysis_1.jpg?t=1306280360 a) prove...

Homework Statement Let X and Y be subsets of R^2, both non-empty. If X is open, the the sum X+Y is open. This is either supposed to be proved or disproved. Homework Equations The Attempt at a Solution This strikes me as false since we are only given the X is open...

Homework Statement Let A be a non-empty set and let d be the discrete metric on X. Describe what the open subsets of X, wrt distance look like. Homework Equations The Attempt at a Solution I think that the closed sets are the subsets of A that are the complement of a union of...

A friend of mine had the following funny question: Imagine I have a metric ansatz with two unknown functions. The Einstein equations give both real and complex solutions for the unknown functions. Question: Is there a decent interpretation of these complex solutions in GR? We know about...

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...

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...

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...