Metric Definition and 1000 Threads

I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand Example 3.10 (b) on page 95 ... ... Example 3.10 (b) reads as follows: My question is as...

Assume that we have a flat FRW metric expressed in conformal time ##\eta## so that the line element is $$ds^2=a^2(\eta)(d\eta^2-dx^2-dy^2-dz^2)\tag{1}$$ where ##a=1## at the present time ##\eta=0## and the speed of light ##c=1##. This metric has the following non-zero Christoffel symbols...

Can I say that the Lorentz metric is the specific form ##-c^2dt^2 + dx^2 + dy^2 + dz^2## whereas the Minkowski metric is the metric of Minkowski space which can take the Lorentz form I just gave, but can also, e.g., be written in spherical coordinates?

I have been teaching myself general relativity and wanted to see if I got these metric tensors right, I have a feeling I didn't.For the first one I get all my directional derivatives (0, 0): (0)i + (0)j (0, 1): (0)i + 2j (1, 0): 2i + (0)j (1, 1): 2i + 2j Then I square them (FOIL): (0, 0): (0)i...

I am trying to construct a particular manifold locally using a metric, Can I simply take the inner product of my basis vectors to first achieve some metric.

I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand the proof of Lemma 3.44 on page 105 ... ... Lemma 3.44 and its proof read as follows: In the above...

I brought up this subject here about a decade ago so this time I'll try to be more specific to avoid redundancy. In chapter five of Bernard F. Schutz's A First Course In General Relativity, he arrives at the conclusion that in flat space the covariant derivative of the metric tensor is zero...

Calculating the christoffel symbols requires taking the derivatives of the metric tensor. What are you taking derivatives of exactly? Are you taking the derivatives of the inner product of the basis vectors with respect to coordinates? In curvilinear coordinates, for instance curved spacetime in...

I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 3.6 on page 94 ... ... Theorem 3.6 and its proof read as follows: In the above...

Hello! I am a bit confused about the definition of the radius in Schwarzschild metric. In the Schutz book on GR (pg. 264, General rules for integrating the equations) he says: "A tiny sphere of radius ##r = \epsilon## has circumference ##2\pi\epsilon##, and proper radius...

In Special Relativity, you learn that invariant mass is computed by taking the difference between energy squared and momentum squared. (For simplicity, I'm saying c = 1). m^2 = E^2 - \vec{p}^2 This can also be written with the Minkowski metric as: m^2 = \eta_{\mu\nu} p^\mu p^\nu More...

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ... I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ... I need some help with an aspect of the proof of Theorem 11.1.11...

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ... I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ... I need some help with an aspect of the proof of Theorem 11.1.11 ...

I am trying to get an intuition of what a metric is. I understand the metric tensor has many functions and is fundamental to Relativity. I can understand the meaning of the flat space Minkowski metric ημν, but gμν isn't clear to me. The Minkowski metric has a trace -1,1,1,1 with the rest being...

Which properties of metric tensor are invariant of basevectors transforms? I know that metric tensor depends of basevectors, but are there properties of metric tensor, that are basevector invariant and describe space itself?

how to find tetrad of this metric the tetrad given is this one I m a newly born in General Relativity please help me out how this tetrad is derived

Our 4-D space is ##x^1,x^2,x^3 ,t##. Our sub-manifold is defined by ##(x^1,x^2,x^3)## Therefore for this sub-manifold to be maximally symmetric and for which the tangent vector ##\frac{∂}{∂t}(\hat t)## orthogonal to this sub-manifold The metric becomes...

From the geodesic equation d2xμ/dΓ2+Γμ00(dt/dΓ)2=0,for non-relativistic case ,where Γ is the proper time and vi<<c implying dxi/dΓ<<dt/dΓ. Now if we assume that the metric tensor doesn't evolve with time (e,g gij≠f(t) ) then Γμ00=-1/2gμs∂g00/∂xs. If we here assume that the metric components of...

Hello. I ask for solution help from the integral below, where y and x represent angles in a metric of a spherical, 2-D surface. He was studying how to obtain the geodesic curves on the spherical surface, the sphere of radius r = 1, to simplify. The integral is the end result. It is enough, now...

Since it's possible to choose a coordinate chart where the Schwarzschild metric components are dependent on time, why that's not done? Would'nt there be a scenario where such a choice would be useful?

Hello everyone, I have been studying perturbation theory in the context of FRW cosmologies, and so far have had a really hard time understanding why the SVT (Scalar, Vector, and Tensor) perturbations associated with the metric tensor "decouple" at first order in perturbation theory. All...

Homework Statement I'd like to calculate the form of Liouville operator in a Robertson Walker metric. Homework Equations The general form is $$ \mathbb{L} = \dfrac{\text{d} x^\mu}{\text{d} \lambda} \dfrac{\partial}{\partial x^\mu} - \Gamma^{\mu}_{\nu \rho} p^{\nu} p^{\rho}...

I'm wondering if the galactic rotation curves could be attributed to a deviation of the metric of spacetime from the ideal Schwarzschild metric. The Schwarzschild-metric is a well tested good approximation for the regions near the central mass - but at the outer space, far away from the...

Does anyone know how to get Maxima to solve the Kerr metric? I enter the terms for that metric that I found on Wikipedia. It tries to print out the Einstein tensor (covariant, leinstein(true)) and the expressions are so long that it literally locks up my computer. And isn’t the Kerr metric a...

Hi, I'm trying to deduce orbit velocity of a particle with mass from Schwarzschild metric. I know for Newtonian gravity it is: $$v^2=GM\left(\frac{2}{r}-\frac{1}{a}\right)$$ The so called vis-viva equation. Where ##a## is the length of the semi-major axis of the orbit. For Schwarzschild metric...

Let us suppose we have a metric in the form of, $$ds^2=-c^2dt^2+[(a^2(t)+b(r)e^{-lt})(dr^2+r^2d\Omega^2)]$$ Where scale factor is defined as ##(a^2(t)+b(r)e^{-lt})## Is this metric describes homogeneity and isotropy or not ? I think it cannot since there's an ##r## dependence, and there are...

Hi, Is there a simple explanation for the presence of a minus sign in the Minkovsky´s metric? Best wishes, DaTario

Let ##(x_1,x_2,x_3)=\vec{r}(\theta,\phi)## the parametrization of a usual sphere. If we consider a projection in two dimension ##(a,b)=\vec{f}(x_1,x_2,x_3)## Then I don't understand how to use the metric, since it is ##g_{ij}=\langle \frac{\partial\vec{f}}{\partial...

This is a problem from Tensor Calculus:Barry Spain on # 69 Prove that a space with Schwarzschild's metric is an Einstein space, but not a space of constant curvature. The metric as given in the book is $$d\sigma^2=-\bigg(1-\frac{2m}{c^2r}\bigg)^{-1}dr^2-r^2d\theta^2-r^2\sin^2 \theta...

I am reading Spacetime and Geometry : An Introduction to General Relativity – by Sean M Carroll and he writes: Quote: A useful characterisation of the metric is obtained by putting ##g_{\mu\nu}## into its canonical form. In this form the metric components become $$ g_{\mu\nu} = \rm{diag} (-1...

I'm trying to use Cartan's method to find the Schwarzschild metric components from Hughston and Tod's book 'An Introduction to General Relativity' (pages 89-90). I'm having problems calculating the components of the Ricci tensor. The given distance element is $$ ds^2 = e^{2 \lambda} dt^2 -...

Homework Statement Hi, I am just stuck in why / how we can write minkowski metric where I would usually write delta. I see that the product rule is used in the first term to cancel the terms in the second term since partials commute for a scalar and so we are left with the d rivative acting...

Homework Statement Prove Rijkl= k/R2 * (gik gjl-gil gjk) where gik is the 3 metric for FRW universe and K =0,+1,-1, and i,j=1,2,3, that is, spatial coordinates. . Homework Equations The Christoffel symbol definition: Γμνρ = ½gμσ(∂ρgνσ+∂νgρσ-∂σgνρ) and the Riemann tensor definition: Rμνσρ =...

I am still learning general relativity (GR). I know we can find the path of a test particle by solving geodesic equations. I am wondering if it is possible to derive/convert metric tensor to gravity, under weak approximation, and vice versa. Thanks!

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.353.633&rep=rep1&type=pdf <Moderator's note: Quotation added to make clear, what the author meant by "Hausdorff metric".> The article above explores the notion of the Hausdorff metric. In the beginning of the article it describes how, if...

Homework Statement Identify ##\bar{c}##, ##\bar{c_0}## and ##\bar{c_{00}}## in the metric spaces ##(\ell^\infty,d_\infty)##. Homework Equations The ##\ell^\infty## sequence space is $$ \ell^\infty:=\left\{\{x_n\}_{n=1}^\infty\in\mathbb{R}^\mathbb{N}\,:\...

Is every homeomorphism between a metric space X and a topological Y equivalent to an isometry? I think it is, but I need to confirm.

Hello, I’m new to the cosmological inflation so in this paper: https://arxiv.org/abs/1809.09975 Has some one an idea how to make the Weyl transformation of the metric ## g_{\mu\nu}## Equation (3) , and how to get the potential (4) from the action (3) by this transformation as explained after...

I'm beginning to study analysis beyond real numbers, but I am a but confused. What is the relation between topology, metric spaces, and analysis? From what it seems, it's that metric space theory forms a subset of topology, and that analysis uses the metric space notion of distance to describe...

Is it possible to introduce the concept of a gradient vector on a manifold without a metric?

In the recent thread about the gravitational field of an infinite flat wall PeterDonis posted (indirectly) a link to a mathpages analysis of the scenario. That page (http://www.mathpages.com/home/kmath530/kmath530.htm) produces an ansatz for the metric as follows (I had to re-type the LaTeX -...

Homework Statement If ##\lim_{n \rightarrow \infty} x_n = L## then ##\lim_{n\rightarrow\infty}cx_n = cL## where ##x_n## is a sequence in ##\mathbb{C}## and ##L, c \epsilon \mathbb{C}##. Homework Equations ##\lim_{n\rightarrow\infty} cx_n = cL## iff for all ##\varepsilon > 0##, there exists...

From what I have read the twin paradox can be resolved with the Rindler metric and without the need to bring in general relativity. Special relativity will suffice. But how does the Rindler metric get derived in the context of a constant accelerating reference frame. I haven't seen anything in...

I am really confused about coordinate transformations right now, specifically, from cartesian to polar coordinates. A vector in cartesian coordinates is given by ##x=x^i \partial_i## with ##\partial_x, \partial_y \in T_p \mathcal{M}## of some manifold ##\mathcal{M}## and and ##x^i## being some...

Hi PF! Let's say a boundary condition for an ODE is ##f'(1)+f(1) = 0##. If we solve the ODE numerically, how can I tell if this BC is satisfied "good enough". Let's suppose the numerics generate ##f'(1)+f(1) = 0.134##; is this close enough to zero?

So I have a question regarding the Alcubierre metric and the phenomena of stars on outer edges of galaxies moving at higher velocities than their orbital calculations state they should. When taking the accelerating expansion of space into account due to dark energy, could a sub-luminal...

I've noticed that a very easy way to generate the Lorentz transformation is to draw Cartesian coordinate axes in a plane, label then ix and ct, rotate them clockwise some angle \theta producing axes ix' and ct', use the simple rotation transformation to produce ix' and ct', then just divide...

My question is, is it forbidden to have a connection not compatible with the metric?

Homework Statement The book I'm using provided a proof, however I'd like to try my hand on it and I came up with a different argument. I feel that something might be wrong. Proposition: Let ##<X,d>## be a metric space, ##<Y,D>## a complete metric space. Then ##<C(X,Y), \sup D>## is a complete...

Homework Statement Let ##x,y\in X## such that ##X## is a metric space. Let ##d(x,y)=0## if and only if ##x=y## and ##d(x,y)=1## if and only if ##x\neq y## Homework Equations N/A The Attempt at a Solution I have already seen various approaches in proving this. Although, I just want to know if...