Metric Definition and 1000 Threads

I need help with the proof of Theorem 4.28 in Tom Apostol's book: Mathematical Analysis (2nd Edition). Theorem 4.28 reads as follows:In the proof of the above theorem, Apostol writes: " ... ... Let m = \text{ inf } f(X). Then m is adherent to f(X) ... ... " Can someone please explain to me...

Conventional GR is based on the Levi-Civita connection. From the fundamental theorem of Riemann geometry - that the metric tensor is covariantly constant, subject to the metric being symmetric, non-degenerate, and differential, and the connection associated is unique and torsion-free - the...

This may seem an odd question but it will clear something up for me. Are "The spacetime interval is invariant." and the "The spacetime metric is a tensor." exactly equivalent statements? Does one imply more or less information than the other? Thanks!

There are .67 gallons of paint in a can. A. How many cubic meters of paint are in the can? B. How many liters of paint are in the can? C. Imagine that all of this paint is used to apply a coat of uniform thickness to a wall of area 13m^2. What is the thickness of the layer of wet paint in metric...

Homework Statement Suppose everything is moving slowly, How can we find the metric tensor in GR in terms of the mass contained. Homework Equations I understand in case of everything moving slowly only below equation is relevant - R00 - ½g00R = 8πGT00 = 8πGmc2 The Attempt at a Solution None.

In considering special relativity as a limiting case of the general theory (without matter or curvature) the question arose as to whether the pseudo-riemann nature of the SR metric is actually an independant (essentially experimentally determined) assumption/property or derivable from the...

Hi guys. I am trying to understand einstein field equation and thus have started on learning tensor. For metric tensor, is it just comprised of two contra/covariant vectors tensor product among each other alone or does it requires an additional kronecker delta? I am confused about the idea...

I understand the Kerr metric has an off-diagonal term between the rotation and the time degrees-of-freedom? That a test mass falling straight down toward a large rotating mass from infinity will begin to pick up angular momentum? Is that what’s called “frame dragging”? Did the Gravity Probe B...

Does anyone know a reference with a discussion on the experimental determination of the metric tensor of spacetime? I only know the one in "The theory of relativity" by Møller, pages 237-240. https://archive.org/details/theoryofrelativi029229mbp

Some subtleties of the metric tensor are just becoming clear to me now. If I take ##g_{\mu\nu}=diag(+1,-1,-1,-1)## and want to write ##\partial_\mu\phi^\mu##, it would be ##\partial_0\phi^0 -\partial_i\phi^i##, correct? ##\phi## is a 4-vector.

Mod note: OP warned about not using the homework template. I have read that 'a(t) determines the value of the constant spatial curvature'.. Where a(t) is the scale factor, and we must have constant spatial curvature - this can be deduced from the isotropic at every point assumption. I'm trying...

Mod note: OP warned about not using the homework template. Let ##g_{ac}## be a 3-d metric. So the trace of a metric is equal to its dimension so I get ##g_{ac}g^{ac}=3## But I'm a tad confused with the expression : ##g^{ac}g_{ad}##=##delta^{c}_{d}## I thought it would be ##3delta^{c}_{d}###...

Context: Deriving the maximally symmetric- isotropic and homogenous- spatial metric I've seen a fair few sources state that the Rienamm tensor associated with the metric should take the form: * ##R_{abcd}=K(g_{ac}g_{bd}-g_{ad}g_{bc})## The arguing being that a maximally symmetric space has...

In schwarzschild metric: $$ds^2 = e^{v}dt^2 - e^{u}dr^2 - r^2(d\theta^2 +sin^2\theta d\phi^2)$$ where v and u are functions of r only when we calculate the Ricci tensor $R_{\mu\nu}$ the non vanishing ones will only be $$R_{tt}$$,$$R_{rr}$$, $$R_{\theta\theta}$$,$$R_{\phi\phi}$$ But when u and v...

Anyone noticed this paper: Square Root of Inverse Metric: The Geometry Background of Unified Theory? Authors: De-Sheng Li, arXiv:1412.2578 ? The author tries to construct the square root of the inverse metric, based on a product of a fermion field and a framefield. Somehow the Standard model...

When working with light-propagation in the FRW metric $$ds^2 = - dt^2 + a^2 ( d\chi^2 + S_k(\chi) d\Omega^2)$$ most texts just set $$ds^2 = 0$$ and obtain the equation $$\frac{d\chi}{dt} = - \frac{1}{a}$$ for a light-ray moving from the emitter to the observer. Question1: Do we not strictly...

Hello everyone, I already know that the solution to this question is obvious but I can't find it. Consider an expanding universe following the FRW metric ds^2=-dt^2-a^2(t)dx^2 (1 space dimension for simplicity). We know that the physical spatial distance x_p is related to the comoving spatial...

In a paper published in Reviews of Modern Physics in 1949, http://journals.aps.org/rmp/pdf/10.1103/RevModPhys.21.378 , H.P. Robertson provided an analysis of the physical implications of the Michelson/Morley, Kennedy and Thorndike, and Ives and Stilwell experiments which seems definitive with...

I have been recently working with Schwarzschild's solution: ds2= - (1- (2GM/rc2))c2dt2 + dr2/(1-2GM/rc2) + r2(dθ2 + sin2(θ)d∅2) Now, when deriving the various general relativistic tensors for this metric such as the Ricci tensor, I found the calculations to be painfully tedious and monstrous...

I have a doubt since I see the next equation and the corresponding matrix: $$ ds^2 = \Bigg( \frac{1-\frac{r_s}{4\rho}}{1+\frac{r_s}{4\rho}}\Bigg)^2 dt^2 - \Big(1+\frac{r_s}{4\rho}\Big)^4 (d\rho^2 + p^2 d\Omega_2^2) $$$$ g_{\mu\nu} = \left( \begin{array}{ccc} \Bigg(...

Hello, this is the metric I am talking about: $$ ds^2= (dt - A_idx^i)^2 - a^2(t)\delta_{ij}dx^idx^j $$ I never see one like this. How the metric tensor matrix would be?

Is there any book or reference perhaps on string theory or superstring theory or even advanced general relativity that treats the Israel Wilson Perjes metric using the tetrad formalism in details, i.e, 1-forms and so? (Not spinors methos) I have ran across many papers that just place the spin...

In SRT, the line element is ##c^2ds^2 = c^2dt^2 - dx^2 -dy^2-dz^2## and ##g_{00} = 1## (or ##-1## depending on sign conventions). In the Schwarzschild metric we have g_{00}=(c^2-\frac{2 GM}{r}) . So in the first example, ##g_{00}## is constant, in the second it depends on another coordinate...

Hey! :o In a space of finite measure, if $f$ and $g$ are measurable we set $\rho (f,g)=\int \frac{|f-g|}{1+|f-g|}d \mu$. Show that $\rho$ is metric and that $f_n \rightarrow f$ as for $\rho$ if and only if $\forall c>0$ we have that $\mu(\{|f_n-f|>c\})\rightarrow 0$.What does "$f_n \rightarrow...

General Relativity... Non-rotational spherically symmetric body of isotropic perfect fluid Einstein tensor metric element functions: g_{\mu \nu} = \left( \begin{array}{ccccc} \; & dt & dr & d \theta & d\phi \\ dt & g_{tt} & 0 & 0 & 0 \\ dr & 0 & g_{rr} & 0 & 0 \\ d\theta & 0 & 0 & g_{\theta...

1. "William Tell is said to have shot an apple off his son's head with an arrow. If the arrow was shot with an initial speed of 55m/s and the boy was 15m away, at what launch angle did Bill aim the Arrow? (Assume that the arrow and the apple are initially at the same height above the...

\mathcal{L}_M(g_{kn}) = -\frac{1}{4\mu{0}}g_{kj} g_{nl} F^{kn} F^{jl} \\ \frac{\partial{\mathcal{L}_M}}{\partial{g_{kn}}}=-\frac{1}{4\mu_0}F^{pq}F^{jl} \frac{\partial}{\partial{g_{kn}}}(g_{pj}g_{ql})=+\frac{1}{4\mu_0} F^{pq} F^{lj} 2 \delta^k_p \delta^n_j g_{ql} I need to know how...

Suppose, I know the metric tensor of a 2D space. for example, the metric tensor of a sphere of radius R, gij = ##\begin{pmatrix} R^2 & 0 \\ 0 & R^2\cdot sin^2\theta \end{pmatrix}## ,and I just know the metric tensor, but don't know that it is of a sphere. Now I want to draw a 2D space(surface)...

How does one get time dilation, length contraction, and E=mc^2 from the spacetime metric? Suppose all that you are given is x12 + x22 + x32 - c2t2 = s2 How do you derive time dilation, length contraction, and E=mc^2 from this? What is the most direct way to do this?

I've never seen a satisfactory explanation of the metrics used in a calculation of distance in Minkowski space. In Euclidean space, the distance is: ds^2 = dx^2 + dy^2 + dz^2 But in Minkowski space, the distance is ds^2 = (dt * c)^2 - dx^2 - dy^2 - dz^2 Why are the signs reversed? This implies...

This is a question inspired by the "Golf Ball" thread, which is no longer open for comments, I guess. For a black hole of constant mass, the metric external to the black hole can be written in Schwarzschild metric, which is characterized by the constant M, and the corresponding radius 2 M...

A metric space is considered complete if all Cauchy sequences converge within the metric space. I was just curious if you could have a case of a metric space that doesn't have any Cauchy sequences in it. Wouldn't it be complete by default? When trying to think of a space with no cauchy...

Hello, I have a simple question about deriving $$g_{ij} \frac{\partial x^i}{\partial t}\frac{\partial x^j}{\partial t}$$ with respect to time t. I have noticed that the first term after derivation turns out to be$$ \frac{\partial g_{ij}}{\partial x^k} \frac{\partial x^k}{\partial...

First of all, the metric I am referring to is this one: ds2= -c2dt2 + dl2 + (k2 + l2)(dᶿ2 + sin2(ᶿ)dø2) where k is the radius of the throat of the wormhole. (sorry for the small Greek letters) Now I have two questions about this solution to Einstein's equations: 1. What does the coordinate l...

OK. A problem is simple and probably very stupid... In many (most of) maths books things like Euclidean metric, norm, scalar product etc. are defined in Cartesian coordinate system (that is one which is orthonormal). But what would happen if one were to define all these quantities in...

Homework Statement I am confused if there is any standard way to check what should be the line element $$ds^{2}$$ when the dimensions are more than three ( since we don't have the option to draw things as we usually do in case of 3d or less dimensional cases. I am following Hobson's book. I am...

What is the most general method of obtaining the event-horizon from the given black hole metric. Let us consider Kerr black hole in Kerr coordinates given by ds^2 = -\frac{\Delta-a^2sin^2\theta}{\Sigma}dv^2+2dvdr -\frac{2asin^2\theta(r^2+a^2-\Delta)}{\Sigma}dvd\chi-2asin^2\theta d\chi dr +...

As you may know, the metric tensor for 3D spherical coordinates is as follows: g11= 1 g22= r2 g33= r2sin2(θ) Now, the Minkowski metric tensor for spherical coordinates is this: g00= -1 g11= 1 g22= r2 g33= r2sin2(θ) In both of these metric tensors, all other elements are 0. Now...

I am attempting to read my first book in QFT, and got stuck. A Lorentz transformation that preserves the Minkowski metric \eta_{\mu \nu} is given by x^{\mu} \rightarrow {x'}^{\mu} = {\Lambda}^\mu_\nu x^\nu . This means \eta_{\mu \nu} x^\mu x^\nu = \eta_{\mu \nu}x'^\mu x'^\nu for all x...

If I create a bijective map between the open balls of two metric spaces, does that automatically imply that this map is a homemorphism?

After my recent studies of the curvature of the 2- sphere, I would like to move on to Minkowski space. However, I can not seem to find the metric tensor of the 4 sphere on line, nor can I seem to think of the vector of transformation properties that I would use to derive the metric tensor of the...

I have one question, which I don't know if I should post here again, but I found it in GR... When you have a metric tensor with components: g_{\mu \nu} = \eta _{\mu \nu} + h_{\mu \nu}, ~~ |h|<<1 (weak field approximation). Then the inverse is: g^{\mu \nu} = \eta^{\mu \nu} - h^{\mu \nu}...

Homework Statement Consider the Schwarschield Metric in four dimensional spacetime (M is a constant): ds2 = -(1-(2M/r))dt2 + dr2/(1-(2M/r)) + r2(dθ2 + sin2(θ)dø2) a.) Write down the non zero components of the metric tensor, and find the inverse metric tensor. b.) find all the...

I was recently trying to test something out with the Riemann tensor. I used only 2 dimensions for simplicity sake. As I was deriving the Riemann tensor, I noticed that it looked as if all of the elements were going to come out to be 0 (which they all did). Therefore, this coordinate system is...

Can the above logic be applied to Schw. Metric as well? Suppose I have an object moving with a radial velocity v=const, then can I do the same to derive the Schwarchild time dilation as in the Minkowski? dr = v ~ dt ds^{2} = [K - \frac{v^2}{K} ] dt^2 So \gamma ^{-1} = \sqrt{K} [1 -...

Hello, I am trying to understand Kaluza Klein theory on the five dimensional unification. It was mentioned over there: " Of the 15 components of gαβ, five had to get a new physical interpretation, i.e. gα5 and g55; the components gik, i,k = 1,...,4, were to describe the gravitational field...

The Schwarzschild Metric has a form: ##ds^2 = Kdt^2 - 1/K dr^2 - r^2dO^2## where: K = 1 - a/r; There is a time scaled by K, but a space radially by 1/K. This is a typical time dilation and a space contraction, which is known from SR, but the Schwarzschild metrics is spherically...

Hello, I'm wondering what the exact definition of a local conformal transformation is, in the context of General Relativity (/Shape Dynamics) To be more precise: 1. Are local conformal transformations coordinate transformations or scalar transformations of the metric? 2. If they are...

Alright, so I was reading up on tensors and such with non-Cartesian coordinate systems all day but now I'm a bit tired an confused so you'll have to forgive me if it's a stupid question. So to express the dot product in some coordinate system, it's: g(\vec{A}\,,\vec{B})=A^aB^bg_{ab} And, if...

I think this will be a quick question... If the Schw's metric is a solution of the vacuum, then what does the mass M_0 in the metric correspond to? I thought it was the mass of the star... but if that's true then why is it a vacuum solution? Or is it vacuum because it describes the regions...