Metric Definition and 1000 Threads

I have started with the space-time metric in a weak gravitational field (with the assumption of low velocity): ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2) Where \phi<<1 is the gravitational potential. Using the standard form for the Christoffel symbols have found: \Gamma^0_{00}=\phi_{,0}...

I have started with the space-time metric in a weak gravitational field (with the assumption of low velocity): ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2) Where \phi<<1 is the gravitational potential. Using the standard form for the Christoffel symbols have found...

Homework Statement Prove that lim_{n} p_{n}= p iff the sequence of real numbers {d{p,p_{n}}} satisfies lim_{n}d(p,p_{n})=0 Homework Equations The Attempt at a Solution I think I can get the first implication. If lim_{n} p_{n}= p, then we know that d(p,p_{n}) = d(p_{n},p) <...

This is kind of a weird question. I like to think about how I would explain things to other people, and I realized that I don't know a great way to explain in general how terms defined in the context of metric spaces are generalized to the context of topological spaces. It's not at all difficult...

Hi, The typical representation of the Dirac gamma matrices are designed for the +--- metric. For example /gamma^0 = [1 & 0 \\ 0 & -1] , /gamma^i = [0 & /sigma^i \\ - /sigma^i & 0] this corresponds to the metric +--- Does anyone know a representation of the gamma matrices for -+++...

I haven't learned much of advanced mathematics. It seems that we can use metric tensors to lower or raise index of christoffel symbols. But isn't christoffel symbols made of metric tensors and derivatives of metric tensors? How can we contract indices of a derivative directly with metric tensors...

Homework Statement Let \left (X,d \right) be a metric space, and let \left\{ x_n \right\} and \left\{ y_n \right\} be sequences that converge to x and y. Let \left\{ z_n \right\} be a secuence defined as z_n = d(x_n, y_n). Show that \left\{ z_n \right\} is convergent with the limit d(x,y)...

I'm trying to verify that if (M,d) is a metric space, then (N,e) is a metric space where e(a,b) = d(a,b) / (1 + d(a,b)). Everything was easy to verify except the triangle inequality. All I need is to show that: a <= b + c implies a / (1 + a) <= (b / (1 + b)) + (c / (1 + c) Any help...

Hi guys, two problems, first one I understand for the most part, the second one, I do not know how to set up and solve. Homework Statement Let X = R^{n} for x = (a_{1},...,a_{n}) and y = (b_{1},...,b_{n}), define d_{\infty}(x,y) = max {|a_{1}-b_{1}|,...,|a_{n}-b_{n}|}. Prove that this is a...

Suppose (X,d) is a metric space and A, a subset of X, is closed and nonempty. For x in X, define d(x,A) = infa in A{d(x,a)} Show that d(x,A) < infinity. I really don't have much of an idea on how to show it must be finite. An obvious thought comes to mind, namely that a metric is...

Homework Statement Consider a submarine cruising 32 ft below the free surface of seawater whose density is 64 lbm / (ft^3). What is the increase in the pressure in psi exerted on the submarine when it dives to a depth of 172 ft below the free surface? Assume that the acceleration due to...

I have never been formally trained in GR, and have a question regarding the basics of how to calculate the energy density from a metric. This question arises from thought experiments involving a field with a negative energy density. This is important only because I expect the energy density...

I've encountered the term Hausdorff space in an introductory book about Topology. I was thinking how a topological space can be non-Hausdorff because I believe every metric space must be Hausdorff and metric spaces are the only topological spaces that I'm familiar with. my argument is, take two...

I've worked through a common-sense argumenthttp://www.mathpages.com/rr/s8-09/8-09.htm" showing the time-time component of the Schwarzschild metric g_{tt} = \left (\frac{\partial \tau}{\partial r} \right )^2\approx 1-\frac{2 G M}{r c^2} On the other hand, I've not worked through any...

I'm currently reading Ross's Elementary Analysis, which presents the definition of continuity as such: (not verbatim) Let x be a point in the domain of f. If every sequence (xn) in the domain of f that converges to x has the property that: lim f(xn) = f(x) then we say that f is...

Suppose we're in a general normed space, and we're considering a sequence \{x_n\} which is bounded in norm: \|x_n\| \leq M for some M > 0. Do we know that \{x_n\} has a convergent subsequence? Why or why not? I know this is true in \mathbb R^n, but is it true in an arbitrary normed space? In...

Let us consider the General Relativity metric: {ds}^{2}{=}{g}_{00}{dt}^{2}{-}{g}_{11}{{dx}_{1}}^{2}{-}{g}_{22}{{dx}_{2}}^{2}{-}{{g}_{33}}{{dx}_{3}}^{2} ---------------- (1) Using the substitutions: {dT}{=}\sqrt{{g}_{00}}{dt} {dX}_{1}{=}\sqrt{{g}_{11}}{dx}_{1}...

When reading about the ideas behind ricci flow, I've often read that the ricci tensor is proportional to the laplacian of the metric, but only in harmonic coordinates. Can someone explain this to me? What laplacian operator would one use to show this as there are many different laplacians in...

I'm currently working through General Relativity and I'm wondering how you would express the variation of a general metric tensor, or similarly, how you would write the total differential of a metric tensor (analogous to how you would write the total derivative for a function)? Also, on a...

Question: have some sense that in a space time with metric ds^2 = g_{tt}dt^2+ g_{xx}dx^2+ g_{yy}dy^2+g_{zz}dz^2, the coordinates x,y,z \in ]-\infty, \infty[ , but t \in [0, \infty[ ?

Homework Statement Having a little trouble on number 24 of Chapter 3 in Rudin's Principles of Mathematical Analysis. How do I prove that the completion of a metric space is complete? Homework Equations X is the original metric space, X^* is the completion, or the set of...

Dear readers, Let X be the product space of a countable family \{X_n:n\in\mathbb{N}\} of separable metric spaces. If X is endowed with the product topology, we know that it is again separable. Are there other topologies for X such that is separable? Is there a natural metric on X such that X...

Hi, I'm trying to attack a problem where the Riemannian metric depends explicitly on time, and is therefore a time-dependent assignment of an inner product to the tangent space of each point on the manifold. Specifically, in coordinates I encounter a term which looks like...

any help would be appreciated. not sure where to start.

When I was studying general relativity, I learned that to change a vector into a covector (or vice versa), one used the metric tensor. When I started quantum mechanics, I learned that the difference between a vector in Hilbert space and its dual is that each element of one is the complex...

Hi there, I came across the following problem and I hope somebody can help me: I have some complete metric space (X,d) (non-compact) and its product with the reals (R\times X, D) with the metric D just being D((t,x),(s,y))=|s-t|+d(x,y) for x,y\in X; s,t\in R. Then I have some sequences...

From the definition of an open set as a set containing at least one neighborhood of each of its points, and a closed set being a set containing all its limit points, how can we show that the complement of an open set is a closed set (and vice versa)? Usually this is taken as a definition, but...

Hi, Is a metric signature of (-,-,-,-) unphysical? Thanks,

This is an interesting hypothesis that doesn't seem to have been discussed yet. What are its flaws? Mark Hadley at the University of Warwick argues that galactic rotation causes gravitational frame-dragging sufficient to put a local asymmetric twist into spacetime and explain observed CP...

Do we need a metric field on a manifold so as to specify a coordinate system on it?

I am learning about General Relativity. The planetary orbits can be calculated with more precision especially Mercury. I am stuck on how to get from the Schwarzschild Metric: a four variable Differential Equation to a radius(r,theta,phi,t) and velocity(r,theta,phi,t) of a single planet...

Homework Statement T is a compact metric space with metric d. f:T->T is continuous and for every x in T f(x)=x. Need to show g:T->R is continous, g(x)=d(f(x),x). Homework Equations The Attempt at a Solution f is continuous for all a in T if given any epsilon>0 there is a delta>0...

Homework Statement Show That every metric space is first countable. Hence show that every SUBSET of a metric space is the intersection of a countable family of open sets. Homework Equations no equation The Attempt at a Solution its easy to show that it is first countable, because for every...

Hi, All: Let X be a metric space and let A be a compact subset of X, B a closed subset of X. I am trying to show this implies that d(A,B)=0. Please critique my proof: First, we define d(A,B) as inf{d(a,b): a in A, b in B}. We then show that compactness of A forces the existence of a in A...

Hi This might be a stupid question, so I hope you are patient with me. When I look for the definition of the energy-momentum tensor in terms of the Lagrangian density, I find two different (?) definitions: {T^\mu}_\nu = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\partial_\nu...

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...

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 Gaussian curvature in r-theta...

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...