Metric Definition and 1000 Threads

Under a change of variables: x^{\mu} \rightarrow x^{\mu}+ \delta x^{\mu} How can I see how the determinant of the metric changes? \sqrt{|g(x)|}? Is it correct to see it as a function? f(x) \rightarrow f(x+ \delta x) = f(x) + \delta x^{\mu} \partial_{\mu} f(x) ?

I am reading this paper http://arxiv.org/pdf/1111.4837.pdf and I came across under eq12 that the new metric is degenerate... How can someone see that from the metric's form? Degeneracy for a metric means that it has at least 2 same eigenvalues (but isn't that the same for the Minkowski metric...

I keep trying to find a certain metric prefix but i can't seem to find it, i need to know what prefix makes 2mL into 200_L i believe it is 10^-5 but i can't find that on any charts. Any help is much appreciated!

Homework Statement I want to know the expression for the round metric of an n-sphere of radius r Homework Equations I have obtained this for a 3-sphere dS^{2}=dr^{2}+r^{2}(d\theta_{1}^{2}+sin^{2}\theta_{1}d\theta_{2}^{2} +sin^{2}\theta_{1}sin^{2}\theta_{2}d\theta_{3}^{2}) The...

Is it possible for a ball(with nonzero radius) to be empty in an arbitrary metric space?

Suppose we have a foliation of leaves (hypersurfaces) with codimension one of some Riemannian manifold ##M## with metric ##g##. For any point ##p## in ##M## we can then find some flat coordinate chart ##(U,\phi) = (U, (x^\mu, y))## such that setting ##y## to a constant locally labels each leaf...

Ansatz metric of the 4 dimensional spacetime: ds^2=a^2 g_{ij}dx^i dx^j + du^2 (1) where: Signature: - + + + Metric g_{ij} \equiv g_{ij} (x^i) describes 3 dimensional AdS spacetime i,j = 0,1,2 = 3 dimensional curved spacetime indices a(u)= warped factor u = x^D =...

In a Bianchi IX universe the metric must be invariant under the SO(3) group acting on the 3-sphere. Hence, the metric must be translation invariant in the spatial parts, where t=constant. This implies that the metric must take the form such that: ds^2 = dt^2 - g_ij(t)(x^i)(x^j), where g is a...

Homework Statement Show the space of all space of all continuous real-valued functions on the interval [0, a] with the metric d(x,y)=sup_{0\leq t\leq a}e^{-Lt}|x(t)-y(t)| is a complete metric space. The Attempt at a Solution Spent a few hours just thinking about this question, trying to prove...

Homework Statement Hi all, I'm having trouble evaluating the product g_{αβ}ϵ^{αβγδ}. Where the first term is the metric tensor and the second is the Levi-Civita pseudotensor. I know that it evaluates to 0, but I'm not sure how to arrive at that. The Attempt at a Solution My first thought...

I was trying to prove: d(A,B) = d( \overline{A}, \overline{B} ) I "proved" it using the following lemmas: Lemma 1: d(A,B) = \inf \{ d(x,B) \}_{x \in A} = \inf \{ d(A,y) \}_{y \in B} (By definition we have: d(A,B) = \inf \{ d(x,y) \}_{x \in A, y \in B} ) Lemma 2: d(x_{0},A) = d(x_{0}...

Hello, I am having a problem about the nature of the measurements of the intervals ds's forming out of infinitesimal displacements dx's of the coordinates and the actual meaning of the measurements of the same dx's, in flat metric spaces. I am certain that this must be a trivial problem...

From what I've understood, 1) the metric is a bilinear form on a space 2) the metric tensor is basically the same thing Is this correct? If so, how is the metric related to/different from the distance function in that space? Some other sources state that the metric is defined as the...

Hi everyone! There is something that I would like to ask you. Suppose you have \frac{1}{\sqrt{-g}} \frac{\delta (\sqrt{-g} (g^{ab} u_a u_b + 1))}{\delta g^{cd}} The outcome of this would be ##u_{c}u_{d}## or ##-u_{c} u_{d}## ? I am really confused.

Hello, if you could help, I will be glad. I am studying the Einsteins-Rosen bridge (a matematically solution of the black hole) and I thought that the Einsteins-Rosen bridge was what we found making the Schwarzschild metric a change in kruskal coordinates. But reading an scientific article it...

Hello, I've been struggling with the so often spoken idea that a metric tensor gives you all necessary information about the geometry of a given space. I accept that from the mathematical point of view as every important calculation (speaking as a physicist with respect to GTR rather than...

So, there's a paper here that I'm a bit confused about. On pages 3 and 4, it talks about energy density magnitude and York time. What I'm a bit confused about, is in the article that linked me to it, the scientist makes mention of eventually generating negative vacuum energy. However, from...

If I am not mistaken, the change of the minkowski metric to: n_{\mu\nu} \rightarrow g_{\mu\nu}(x) will violate the Poincare invariance of (example) the Electromagnetism Action. However it allows us to define a wider set of arbitrary transformations (coordinate transformations). The last...

Hi i am confused of the following question. Suppose we have a Metric Space (X,d), where d is the usual metric. Now are the following subsets complete, if so why?? 1.$$X=[0,1]$$ 2.$$X=[0,1)$$ 3.$$X=[0,\infty)$$ 4.$$(-\infty,0)$$

Homework Statement Consider the metric space (R^{n}, d_{∞}), where if \underline{x}=(x_{1}, x_{2}, x_{3},...,x_{n}) and \underline{y}=(y_{1}, y_{2}, y_{3},...,y_{n}) we define d_{∞}(\underline{x},\underline{y}) = max_{i=1,2,3...,n} |x_{i} - y_{i}| Assume that (R^{n}, d_{∞}) is...

Hello, I have a question, whether is possible to looking for Killing Vectors (KV) in this way (I know about general solution): From Schwarzschild metric I can see two KV \frac{\partial}{\partial t} and \frac{\partial}{\partial\phi} . Then I see that other trivial KV arent there. Metric...

I'm trying to work out: (∇f)^2 (f is just some function, its not really important) While working in curved space with a metric: ds^2 = α dt^2 + dr^2 + 2c√(α+1) dtdr I'm not really sure how to calculate a derivative in curved space, any help would be appreciated thanks

In an expanding universe that is modeled by the FRW metric we assume that scale factor of the "present epoch" is unity which is equivalent to a zero redshift. Therefore, most observed galaxies with nonzero redshifts are in our past light cone. But it is unclear to me how much back in time or...

Hello All, Sorry if my question seems to be elementary. I am trying to find out a little bit details of the Riemann metric tensor but not too much in details. I found out the metric (g11, g12, g13, g14...). It provides information on the manifold and those parameters have the information...

Homework Statement Given that f is continuous and strictly increasing, f:[0, ∞)->[0, ∞), f(0)=0, and d(x,y) is the standard metric on the real number line, Is there a function f such that d'(x,y)=f(d(x,y)) is not a metric on the real number line? The Attempt at a Solution The standard...

Homework Statement Let (X,d) is a metric space. Show that d_1=log(1+d) is a metric space. The Attempt at a Solution (it's not stated what d is so I'm assumed d=|x-y|) I've checked positivity and symmetry but am having trouble with showing the triangle inequality holds. i.e. log(1+|x-y|)...

Let $$(E=]-1,0]\cup\left\{1\right\},d) $$ metric space with $$d$$ metric given by $$d(x,y)=|x-y|$$, and $$||$$absolute value. How I can find open sets of E explicitly? Thanks in advance.

I am trying to self-study FLRW and I hope someone cares to answer a simple question regarding this explanation:http://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric#Newtonian_interpretation If I got it right the expanding matter is contrasted by...

Reading over Alcubierre's paper on his "warp" drive (http://arxiv.org/abs/gr-qc/0009013), the metric in equation 3 has a velocity term, v, that doesn't seem to be needed anywhere. Even in the one spot where it seems potentially valuable, equation 12, he just call it =1 and essentially ignores...

I don't know very much about differential geometry but from the things I know I think that the metric is somehow the quantity which specifies what kind of a geometry we're talking about(Though not sure about this because different coordinate systems on the same manifold can lead to different...

One of the properties of the unique Levi-Civita connection is that it preserves the metric tensor at each point's tangent space, allowing the definition of invariant intervals between points in the manifold. I'd be interested in clarifying: when the metric preserved by the L-C connection is a...

Hi all. I'm taking a course in GR and trying to get my intuition and mathematical techniques up to speed. I've been trying to derive the velocity addition formula in Minkowski space, but for some reason I can't do it. Here's what I have: I'll use the Minkowski metric of signature...

I am trying to write the Einstein field equations $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} R=\frac{8\pi G}{c^4}T_{\mu\nu}$$ in such a way that the Ricci curvature tensor $$R_{\mu\nu}$$ and scalar curvature $$R$$ are replaced with an explicit expression involving the metric tensor $$g_{\mu\nu}$$...

Homework Statement For a metric space (X,d) and a subset E of X, define each of the terms: (i) the ball B(p,r), where pεX and r > 0 (ii) p is an interior point of E (iii) p is a limit point of E Homework Equations The Attempt at a Solution i) Br(p) = {xεX: d(x.p)≤r}...

I am currently studying special relativity on my own and I am looking into space time and space time diagrams. While reading through various sources I came across what seemed to be two methods to describe space time. X0, X1, X2, X3 (ct, x,y,z) -> Lorentz Metric X1, X2, X3, X4 (x,y,z,ict)...

Homework Statement The metric for this surface is ds^2 = dr^2 + r^2\omega^2d\phi^2, where \omega = sin(\theta_0). Solve the Euler-Lagrange equation for phi to show that \dot{\phi} = \frac{k}{\omega^2r^2}. Then sub back into the metric to get \dot{r} Homework Equations L = 1/2 g_{ab}...

I've been watching the Stanford lectures on GR by Leonard Susskind and according to him the metric tensor is not constant in polar coordinates. To me this seems wrong as I thought the metric tensor would be given by: g^{\mu \nu} = \begin{pmatrix} 1 & 0\\ 0 & 0\\ \end{pmatrix} Since...

What does one mean when one says that a certain manifold 'admits' a certain metric?

Hi, I'm doing the master in science and one of things that I have to study is the Gödel metric. His paper have a high level for me and I'm seeking theses and dissertations about the Gödel universe. At moment I got three theses about the subject in english and portuguese-BR and one dissertation...

Homework Statement A sequence \{x_{n}\} of real numbers is called bounded if there is a number M such that |x_{n}| ≤M for all n. Let X be the set of all bounded sequences, show that d(\{x_{n}\},\{y_{n}\})=sup \{|x_{n}-y_{n}| :n \in N \} is a metric on X.The only part I am struggling with is the...

Hi In the Schwarzschild metric, the proper time is given by c^{2}dτ^{2} = (1- \frac{2\Phi}{c^2})c^2 dt^2 - r^2 dθ^2 with where \Phi is the gravitational potential. I have left out the d\phi and dr terms. If there is a particle moving in a circle of radius R at constant angular velocity ω...

How do you find the inverse of metric tensor when there are off-diagonals? More specifivally, given the (Kerr) metric, $$ d \tau^2 = g_{tt} dt^2 + 2g_{t \phi} dt d\phi +g_{rr} dr^2 + g_{\theta \theta} d \theta^2 + g_{\phi \phi} d \phi^2 + $$ we have the metric tensor; $$ g_{\mu \nu} =...

Hi, could anyone help me out? The FLRW metric in spherical coordinates is: \;\; ds2 = dt2 - a(t)2(dr2 + r2dΩ2) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (1) I am considering a similar metric of the format: \;\; ds2 = \frac{1}{a(t')^{2}}dt'2 - a(t')2(dr2 + r2dΩ2)...

Hi, In his original paper, Schwarzschild set the "'equation of the determinant" to be: |g|=-1. In other words, he imposed the determinant of the metric to be equal to minus one when solving the Einstein's equations. Must we impose this equality systematically in general relativity and why...

Suppose we have some two-dimensional Riemannian manifold ##M^2## with a metric tensor ##g##. Initially it is always locally possible to transform away the off-diagonal elements of ##g##. Suppose now by choosing the appropriate equivalence relation and with a corresponding surjection we construct...

Does \eta^{\alpha \beta}=\eta_{\alpha \beta} in all coordinate systems or just inertial coordinate systems?

Please take a look at the proof I added, there are some things I do not understand with this proof. 1. Does it really show that |f(x)-f(y)|≤d(x,y) for all x and y? Or does it only show that if there is an ball with radius r around x, and this ball is contained in an O in the open covering, and...

I'm working on a problem that requires me to take the cartesian metric in 2D [1 0;0 1] and convert (using the transformation equations b/w polar and cartesian coords) it to the polar metric. I have done this without issue using the partial derivatives of the transformation equations and have...

I'm trying to clarify for myself the relation between the stress-energy tensor and the mass scalar term in metric solutions to Einstein's equations. Maybe I should also say I'm trying to understand the energy tensor better, or how it relates to boundary conditions on the solutions. My...

In another thread WannabeNewton mentioned: and gave this reference: Until WBN mentioned it, I had never given any thought to the difference between these methods of measuring rotation, so I would like to explore those ideas further here, particularly in relation to the Kerr metric. Consider...