Metric Definition and 1000 Threads

In the Wiki article about the Vaidya metric : http://en.wikipedia.org/wiki/Vaidya_metric there is mention of a "further generalisation" called the Kinnersley metric, without giving any details or even a reference. Is this a generalisation of the Vaidya metric to include angular momentum (...

Error below For a couple of years now, I have been attempting to solve for the values in GR of the time dilation z and the radial and tangent length contractions, L and L_t respectively, which form the metric c^2 dτ^2 = c^2 z^2 dt^2 - dr^2 / L^2 - d_θ^2 r^2 / L_t^2 (along a plane)...

if we know K^{a b}= (∇^a*ζ^b -∇^b*ζ^a)/2, ζ is a killing vector, under the variation of metric g_{a b}→g_{a b}+δ(g_{a b}) which preserves the Killing vector δ(ζ^a)=0, h_{a b} = δ(g_{a b}) = ∇^a*ζ^b +∇^b*ζ^a, how to prove δ(K^{a b})= ζ_c*∇^a*h^{b c} - h^{c a}*∇_c*ζ^b - (...

The killing form can be defined as the two-form ##K(X,Y) = \text{Tr} ad(X) \circ ad(Y)## and it has matrix components ##K_{ab} = c^{c}_{\ \ ad} c^{d}_{\ \ bc}##. I have often seen it stated that for a compact simple group we can normalize this metric by ##K_{ab} = k \delta_{ab}## for some...

The action for Yang-Mill's theory is often written as $$ S= \int \frac{1}{4}\text{Tr} (F^{\mu \nu} F_{\mu \nu})d^4 x = \int d^4 x\frac{1}{4} F^{k \mu \nu} F_{k \mu \nu}$$ where latin indices are indicies in the lie algebra, and the trace is taken with respect to the inner product...

Definition/Summary The metric tensor g_{\mu\nu} is a 4x4 matrix that is determined by the curvature and coordinate system of the spacetime Equations The proper time is given by the equation d\tau^2=dx^{\mu}dx^{\nu}g_{\mu\nu} using the Einstein summation convention It is a symmetric...

Hello. I am new to engineering and to imperial units, and currently learning by doing some exercises. I'm stuck on the following conversion: 0.04 g / min x m^3 -> lbm / hr x ft^3 I figured it like this: 0.04 g / min x m^3 x (60min/1hr) x (1m/35,314)^3 x (1 lbm / 454g) = 1,49x10^-4 lbm...

The killing form on a lie algebra is defined as $$B(X,Y) = \text{Tr}ad_X \circ ad_Y$$ where ##ad_X: \mathfrak{g} \to \mathfrak{g}## is given by ##ad_X(Y) = [X,Y]##, where the latter is the lie bracket between X and Y in ##\mathfrak{g}##. Expressed in terms of components on a basis on...

Dear all, As I was reading my book. It said that the line element of a particular coordinate system (spherical) in R^{3} is so and so. Then it said that the metric is flat. I don't get how the metric is flat in spherical coordinate. Could someone shed some light on this please? Thanks

Dear all, In my journey through learning General relativity. I have stumbled upon this problem. I have to calculate the geodesic equation for R^{3} in cylindrical polars. I am not sure how to use the metric connection. The indices confuse me. I would appreciate it if someone could shade some...

I have recently been studying the tensors on the left side of the Einstein field equations, but I have been studying and deriving them in 3-D. I would now like to move on to adding time into the mixture. I have some questions regarding the Minkowski metric \eta\mu\nu. First, I know that...

I recently derived a matrix which I believe to be the metric tensor in spherical polar coordinates in 3-D. Here were the components of the tensor that I derived. I will show my work afterwards: g11 = sin2(ø) + cos2(θ) g12 = -rsin(θ)cos(θ) g13 = rsin(ø)cos(ø) g21 = -rsin(θ)cos(θ)...

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?