Metric Definition and 1000 Threads

Say we were given an expression for the energy-momentum tensor (also assuming a perfect fluid), without getting into an expression with multiple derivatives of the metric, are there any cases where it would be possible to deduce the form of the metric?

Homework Statement I've searched everywhere, and I cannot find an example of calculation of Lie derivation of a metric. If I have some vector field \alpha, and a metric g, a lie derivative is (by definition, if I understood it): \mathcal{L}_\alpha g=\nabla_\mu \alpha_\nu+\nabla_\nu...

In a paper about field theory in curved spacetime, an author says that the Lagrangian density for a free scalar particle is L = \sqrt{-g} ((\nabla_\mu \Phi)(\nabla^\mu \Phi) - m^2 \Phi^2) Is there a simple explanation for why this is scaled by \sqrt{-g}?

Homework Statement This is not homework but more like self-study - thought I'd post it here anyway. I'm taking the variation of the determinant of the metric tensor: \delta(det[g\mu\nu]). Homework Equations The answer is \delta(det[g\mu\nu]) =det[g\mu\nu] g\mu\nu...

Hi, If X \LARGE is a metric space and E \subset X is a discrete set then is E \LARGE open or closed or both? Here's my understanding: E \LARGE is closed relative to X \LARGE. proof: If p \subset E then by definition p \LARGE is an isolated point of E \LARGE, which implies that p...

This may be a stupid question, but why can't the expansion/contraction of spacetime from a gravitational wave be used to create the areas of expansion/contraction required in the Alcubierre metric, instead of using regions of positive/negative energy density? I saw on the forums about the...

Hi, In Baby Rudin, Thm 3.6 states that If p(n) is a sequence in a compact metric space X, then some subsequence of p(n) converges to a point in X. Why is it not the case that every subsequence of p(n) converges to a point in X? I would think a compact set would contain every sequence...

Hi, I'm reading Baby Rudin and have a quick question regarding topology. Given a nonempty subset E of a metric space X, is it true that the only points in E are either isolated points or limit points? (b/c all interior points are by definition limit points, but not all limit points are...

If you know that {{x}^{a}}{{g}_{ab}}={{x}_{b}} is it proper to say that you also know {{g}_{ab}}=\frac{{{x}_{b}}}{{{x}^{a}}}

Homework Statement show the set {f: ∫f(t)dt>1(integration from 0 to 1) } is an open set in the metric space ( C[0,1],||.||∞） and if A is the subset of C[0,1] defined by A={f:0<=f<=1} is closed in the norm ||.||∞ norm. Homework Equations C[0,1] is f is continuous from 0 to 1.and ||.||∞...

The usual proof of this theorem seems to assume that the topology of the metric space is the one generated by the metric. But if I use another topology, for example the trivial, the space need not be Hausdorff but the metric stays the same. Am I missing something or is the statement of the...

I need to build a tensor from the product of the metric components, like this (using three factors, not less, not more) : H^{\mu \nu \lambda \kappa \rho \sigma} = g^{\mu \nu} \, g^{\lambda \kappa} \, g^{\rho \sigma} + g^{\mu \lambda} \, g^{\nu \kappa} \, g^{\rho \sigma} + ..., however, that...

hi we know that our universe is homogenous and isotropic in large scale. the metric describe these conditions is FRW metric. In FRW, we have constant,k, that represent the surveture of space. it can be 1,0,-1. but the the Einstan Eq, Ricci scalar is obtained as function of time! and this...

I've read that the metric tensor is defined as {{g}^{ab}}={{e}^{a}}\cdot {{e}^{b}} so does that imply that? {{g}^{ab}}{{g}_{cd}}={{e}^{a}}{{e}^{b}}{{e}_{c}}{{e}_{d}}={{e}^{a}}{{e}_{c}}{{e}^{b}}{{e}_{d}}=g_{c}^{a}g_{d}^{b}

In my general relativity course we recently covered the definition of a killing vector and their importance. However, I am not completely comfortable calculating the killing vectors for a given metric (in a particular case, the 2-sphere), and would like to know if anyone knows of a good...

[SIZE="3"]hello Whic one of these to metric are Minkowski metric ds^2 =-(cdt)^2+(dX)^2 ds^2 =(cdt)^2-(dX)^2 and what about timelike (ds^2<0) and spacelike (ds^2>0) for each metric? With my appreciation to those who answer

Homework Statement Let X and Y be metric spaces such that X is complete. Show that if {fα(x) : α ∈ A} is a bounded subset of Y for each x ∈ X, then there exists a nonempty open subset U of X such that {fα(x) : α ∈ A, x ∈ U} is a bounded subset of Y. Homework Equations Definition of...

Hello everybody! I have some questions concerning the structure of the Schwarzschild metric, which is given by $$ ds^2=-(1- \frac{2GM}{r})dt^2+(1-\frac{2GM}{r})^{-1}dr^2+ r^2(d\theta^2+ \sin^2(\theta) d\phi^2) $$ where we set $c=1$. I would like to know the following: \\ \\ 1. Why is it...

In the Wiki article http://en.wikipedia.org/wiki/Vaidya_metric it states that but it looks to me as if the 'particles' are traveling at the speed of light ( null propagation vector field ) and so must have zero rest mass. Is this a typo or have I misunderstood something ?

Hello, can anyone suggest a geometric interpretation of the metric tensor? I am also interested to know how we could "derive" the metric tensor (i.e. the matrix <ai,aj>) from some geometric considerations that we impose.

Double contraction of curvature tensor --> Ricci scalar times metric I'm trying to follow the derivation of the Einstein tensor through double contraction of the covariant derivative of the Bianchi identity. (Carroll presentation.) Only one step in this derivation still puzzles me. What I...

Hello guys , beside Einstein's General Theory of Relativity by hervik , is there any lecture or book about this case ? Thx

Hey guys! I am considering a space with a diagonal metric, which is maximally symmetric. It can be proven that in that case of a diagonal metric the following equations for the Christoffel symbols hold: \Gamma^{\gamma}_{\alpha \beta} = 0 \Gamma^{\beta}_{\alpha \alpha} =...

Hi all, S. Martin's Supersymmetry primer (http://arxiv.org/abs/hep-ph/9709356) is a wonderful source from which to learn SUSY. But, what really causes me (and others around me) huge consternation is Martin's use of mostly plus metric, when particle physicists use the mostly minus metric...

Kerr–Newman metric: c^{2} d\tau^{2} = - \left(\frac{dr^2}{\Delta} + d\theta^2 \right) \rho^2 + (c \; dt - \alpha \sin^2 \theta \; d\phi)^2 \frac{\Delta}{\rho^2} - ((r^2 + \alpha^2) d\phi - \alpha c \; dt)^2 \frac{\sin^2 \theta}{\rho^2} I used the Kerr–Newman metric equation form listed on...

yeh well, I once understood this, but now looking at it today I can't get it to make sense intuitively. The quantity: dx2+dy2+dy2-c2dt2 is the same for every intertial frame in SR - just like length is the same in all inertial frames in classical mechanics. Now I am not sure that I...

Hi all, I am trying to understand geometric flows, and in particular the Ricci flow. I understand how to calculate the metric tensor from the parametrization of a surface, but I am facing a problems in the concept phase. A metric tensor's purpose is to provide a coordinate invariant...

Y ⊂ X where X is a metric space with the function d. Prove that (Y,d) is a metric space with the same function d. The metric function d: X x X -> R. I know that the function for Y is: d* : Y x Y -> R How do I show that d is the same as d*.

Like any mathematical relativity between them as per General Relativity?

http://imageshack.us/a/img12/8381/37753570.jpg I am having trouble with this question, like I do with most analysis questions haha. It seems like I must show that the maximum exists. So E is compact -> E is closed To me having E closed seems like it is clear that a maximum distance...

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...