Metric Definition and 1000 Threads

Homework Statement We've got a line element ds^2 = f(x) du^2 + dx^2 From that we should find the geodesic equation Homework Equations Line Element: ds^2 = dq^j g_{jk} dq^k Geodesic Equation: \ddot{q}^j = -\Gamma_{km}^j \dot{q}^k \dot{q}^m Christoffel Symbol: \Gamma_{km}^j = \frac{g^{jl}}{2}...

Does there exist any 2D surface whose metric tensor is, ##\eta_{\mu\nu}= \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}##

How did Einstein compute the amount of light deviation due to the Earth's gravitational field when the Schwarzschild metric was not known yet?

When reading through this paper(http://www2.warwick.ac.uk/fac/sci/physics/current/teach/module_home/px436/notes/lecture20.pdf), I have trouble understanding some parts of it, 1. Is the r in ρ=Rr a unit vector? 2. it shows x^2 + y^2 + z^2 + w^2 = ρ^2 + w^2 = R^2 but isn't ρ=Rr, thus isn't p^2...

Hi, Does anybody know a link where the Einstein field equations are fully written out, i.e. in terms of only the coefficients of the metric tensor and derivatives on the left side? I'm just curious how huge this must be.

Hi, When I started learning about GR I wondered if it emerged from SR (which the name suggests) or if the connection between the two is mere technical. GR describes the behaviour of the metric of space-time, which is locally Minkowskian and therefore SR applies. But is a curvature-based theory...

Consider the following metric for a 2D spacetime: ##g_{tt} = -x ## ##g_{tx} = g_{xt} = 3## ##g_{xx} = 0## i.e. g_{\mu \nu} = \left( \begin{array}{cc} -x & 3\\ 3 & 0 \end{array} \right) Now, since the metric is independent of time (t), there is supposedly a conservation law containing...

Hello. I don't know exactly if my question can be treated physically but so... Let us have a 3D space with non-constant metric. We are in the first region with a euclidian metric. ds^2=dx^2+dy^2+dz^2 So the distance between two points is got through pythagorean theorem Then near us we have the...

I'm messing around with the swarzchild metric, and I keep getting errors. First, it was a memory, which I could have guessed, 10000x10000 array, so I lowered it to 1000x1000 and it moves past that point, now. However, this is where I'm getting my error: Gravity = zeros([1000,1000]) while i <...

Hi, I have this question: in the context of linear algebra, would it be correct to say that a metric is a kind of inner product?

suppose the AdS_3 metric is given by $$ds^2 =d\rho^2+cosh^2\rho d\psi^2 +sinh^2 \rho d\phi^2$$ what is the n-sheeted space of it? Can the n-sheeted BTZ be constructed from it by identifications as n=1 case? Thanks in advance.

If I am asked to show that the tt-component of the Einstein equation for the static metric ##ds^2 = (1-2\phi(r)) dt^2 - (1+2\phi(r)) dr^2 - r^2(d\theta^2 + sin^2(\theta) d\phi^2)##, where ##|\phi(r)| \ll1## reduces to the Newton's equation, what exactly am I supposed to prove?

Hi all, I need some help- I was reading Carroll's GR book, and on pages 71-71 he discusses the metric in curved spacetime. I have a few questions regarding this section: (1) He says In our discussion of path lengths in special relativity we (somewhat handwavingly) introduced the line element...

Q1: How do we prove that a Riemannian metric G (ex. on RxR) is invariant with respect to a change of coordinate, if all we have is G, and no coordinate transform? G = ( x2 -x1 ) ( -x1 x2 ) Q2: Since the distance ds has to be invariant, I understand that it has to be proved...

hello dear, I need to calculate the Weyl and Ricci scalars for a given metric. let's assume for a kerr metric. by using the grtensor-II package in maple i am not able to get the results. would anyone help me out. for ordinary metric like schwarzschild metric, However its not working for Kerr...

Homework Statement Write down an orthonormal basis of 1 forms for the rotating C-metric [/B] Use the result to find the corresponding dual basis of vectorsSee attached file for metric and appropriate equations The two equations on the left are for our vectors. the equations on the right...

Not a formal course - just a question I decided to try to tackle with what I've gleaned from Stanford's lectures on Youtube, but still putting this here on account of this. So, I've been watching the Stanford GR series, and I have two motivations for messing around with this type of metric; 1...

Homework Statement I'm doing a project on the redshift from a star system (I chose a binomial system because why not). I might be going a little overboard using topology to calculate redshift, but whatever. First off, can I just treat a binomial system as the superposition of 2 sources which...

Let me see if I understand this correctly. Using the metric to raise an index converts a vector into a one form and lowering the index converts a one form into a vector. The contraction on the indices is the dot product between the two. Am I correct so far? If so, here is my question. What is...

Homework Statement Consider Minkowski space in the usual Cartesian coordinates ##x^{\mu}=(t,x,y,z)##. The line element is ##ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}=-dt^{2}+dx^{2}+dy^{2}+dz^{2}## in these coordinates. Consider a new coordinate system ##x^{\mu'}## which differs from these...

Homework Statement I have the metric ##ds^2 = -X^2dT^2 + dX^2## Find the coordinate transformation that reduces the metric to that of flat spacetime: ##ds^2 = -dt^2 + dx^2## Homework EquationsThe Attempt at a Solution I'm not sure there's a systematic way to solve this (or in general to...

I need help to visualize the geometry involved here, How can I visualize the last paragraph? Why is the surface of fixed r now an ellipsoid? Also for r = 0, it is already a disk? I've tried searching for the geometry of these but I can't find any image of the geometry that I can just stare...

Using Ray D'Inverno's Introducing Einstein's Relativity. Ex 6.31 Pg 90. I am trying to calculate the purely covariant Riemann Tensor, Rabcd, for the metric gab=diag(ev,-eλ,-r2,-r2sin2θ) where v=v(t,r) and λ=λ(t,r). I have calculated the Christoffel Symbols and I am now attempting the...

Homework Statement Given the line element ##ds^2## in some space, find the transformation relating the coordinates ##x,y ## and ##\bar x, \bar y##. Homework Equations ##ds^2 = (1 - \frac{y^2}{3}) dx^2 + (1 - \frac{x^2}{3}) dy^2 + \frac{2}{3}xy dxdy## ##ds^2 = (1 + (a\bar x + c\bar y)^2) d\bar...

Homework Statement This is a problem from A. Zee's book EInstein Gravity in a Nutshell, problem I.5.5 Consider the metric ##ds^2 = dr^2 + (rh(r))^2dθ^2## with θ and θ + 2π identified. For h(r) = 1, this is flat space. Let h(0) = 1. Show that the curvature at the origin is positive or negative...

Homework Statement The familiar Mercator map of the world is obtained by transforming spherical coordinates θ , ϕ to coordinates x , y given by ##x = \frac{W}{2π} φ, y = -\frac{W}{2π} log (tan (\frac{Θ}{2}))## Show that ##ds^2 = Ω^2(x,y) (dx^2 + dy^2)## and find ##Ω## Homework Equations...

I'm doing a thesis about wormhole and I would like to put a part in which I conjecture that a shuttle(precisely the Endurance from Interstellar) goes to Kepler 422-b using a Thorne-Miller wormhole. The problem is that I don't know hot to solve a difficult differential equations. Thank you very...

Is the metric tensor a tensor of rank two simply because the line element (or equivalent Pythagorean relation between differential distances) is "quadratic" in nature? This would be in opposition to say, the stress tensor being a tensor of rank two because it has to deal with "shear" forces. I...

Just a thought... Would there be any implicit differences between (A) a two-body metric where the two central masses are drawn ever further together, with angular momentum included, and (B) the Kerr metric? Angular momentum would still be part of the system, but it would be explained by a more...

I understand that in order to preserve the inner product of two four vectors under a change of coordinates x^{\mu}\rightarrow x^{\mu^{'}}=\Lambda^{\mu^{'}}_{\,\, \nu}x^{\nu} the Minkowski metric must transform as \eta_{\mu^{'}\nu^{'}}=\Lambda^{\alpha}_{\,\...

It is not clear for me why a positive definite metric is necessary to define a topology as noted in some textbooks like the one by Carroll. When we define a manifold we require that it locally looks like Euclidean. But even the Lorentzian metric in SR does not locally looks like Euclidean let...

I'm trying to get an intuitive feel for Minkowski space in the context of Special Relativity. I should mention that I have not studied (but hope to) the mathematics of topology, manifolds, curved spaced etc., but I'm loosely familiar with some of the basic concepts. I understand that spacetime...

I'm reading an introduction to relativity which uses different notation to the standard indices used in my college course. I came across: L(\nu)gL(\nu)g = 1 Where L is the Lorentz transformations four-vector and g is the metric. Without the indices, I'm a little lost. Is there some convention...

Hello! (Wave) I want to show that if $\rho(x,y)$ is a metric on $X$, then $\sigma (x,y)= \min \{ 1, \rho(x,y) \}$ is a metric. I have thought the following: $\rho(x,y)$ is a metric on $X$, so: $\rho(x,y) \geq 0, \forall x,y \in X$ $\rho(x,y)=0$ iff $x=y$ $\rho(x,y)=\rho(y,x) \forall x,y...

I watched a lecture that derived the robertson walker metric by creating a metric to describe a four dimensional sphere in three dimensions. Then from minkowski's equation-...

I am following Vaidya metric and how it is related to pure radiation from Wikipedia. But when it reaches the line where stress-energy tensor is equated to product of two four-vectors, I cannot follow from where they are assumed to be null vectors, and why if the stress-energy tensor is given in...

Let $d$ be a metric on $X$. Fix ${x}_{0}\in X$. Let ${d}_{\lambda}\left(x,y\right)=\frac{1}\lambda{}\left| x-y \right|$ and The two sets ${X}_{w}={X}_{w}\left({x}_{0}\right)=\left\{x\in X:{d}_{\lambda}\left(x,{x}_{0}\right)\to0\left( as \lambda\to\infty\right) \right \}$ and...

Given the metric c^2 d\tau^2 = c^2 B(r) dt^2 - A(r) dr^2 - C(r) r^2 d\phi^2 and solving only for a static, spherically symmetric vacuum spacetime, I want to reduce the number of coordinate functions A, B, and C from three to only one using the EFE's. We can then make a coordinate choice for...

What is the relation between metric spaces and normed spaces... What is the meaning of " metric spaces are seen as a nonlinear version of vector spaces endowed with a norm" ? Thank you for your attention...Best wishes...:)

When I was studying complex manifolds in a Freedman's SUGRA book, I ran across this. In a complex manifold, the metric is...

Let $X=R$ and ${d}_{1}\left(x,y\right)=\frac{1}{\eta}\left| x-y \right|$ $\eta\in \left(0,\infty\right)$ and ${d}_{2}\left(x,y\right)=\left| x-y \right|$..By using ${d}_{1}$ and ${d}_{2}$ please show that ${x}_{n}=\left(-1\right)^n$ is divergent and ${x}_{n}=\frac{1}{n}$ is convergent...

Homework Statement I've been working on Exercise 14.3 in MTW. This starts with the FLRW metric (see attachment) and asks that you find the connection coefficients and then produce the non-zero elements of the Riemann Tensor. The answer given is that there are only 2 non-zero elements vis...

At time 1:11:20, Lenny introduces the metric for ordinary flat space in the hyperbolic version of polar coordinates? Is that what he is doing here? d(tau)^2 = ρ^2 dω^2 - dρ^2. He then goes on to say that this metric is the hyperbolic version of the same formula for Cartesian space, i. e...

Homework Statement The aim is to find a solution for the scale factor in a Robertson Walker Metric with a scalar field and a Lagrange multiplier. Homework Equations I have this action S=-\frac{1}{2}\int...

Hey, I have not done any proper differential geometry before starting general relativity (from Sean Carroll's book: space time and geometry), so excuse me if this is a stupid question. The metric tensor can be written as $$ g = g_{\mu\nu} dx^{\mu} \otimes dx^{\nu}$$ and its also written as...

Homework Statement I'm trying to calculate the variation of the following term for the determinant of the metric in the polyakov action: $$h = det(h_{ab}) = \frac{1}{3!}\epsilon^{abc}\epsilon^{xyz}h_{ax}h_{by}h_{cz}$$ I know that there are some other ways to derive the variation of a metric...

Hello! (Wave) A metric space $(X, \rho)$ is called complete if every Caucy sequence on $X$ converges to an element of the space $X$ i.e. if $(x_n) \subset X, n=1,2, \dots$ such that for each $\epsilon>0$ exists $n_0 \in \mathbb{N}$ so that $\rho(x_n,x_m)< \epsilon$ for all $n,m \geq n_0$, then...

I have what I think is a basic question. Say I have a manifold and a metric. How do I write down the most general curve for some arbitrary parameter? For example in \mathbb{R}^2 with the Euclidean metric, I think I should write \gamma(\lambda) = x(\lambda)\hat{x} + y(\lambda)\hat{y} But what...

Homework Statement Hi all, I currently have a modified Einstein-Hilbert action, with extra terms coming from some vector field A_\mu = (A_0(t),0,0,0), given by \mathcal{L}_A = -\frac{1}{2} \nabla _\mu A_\nu \nabla ^\mu A ^\nu +\frac{1}{2} R_{\mu \nu} A^\mu A^\nu . The resulting field...

The metric $$ds^2=-R_1(r)dt^2+R_2(r)dr^2+R_3(r)r^2(d\theta^2+sin^2d\phi^2)$$ when changed to $$ds^2=-R_1(r)dt^2+R_2(r)(dr^2+r^2d\Omega^2)$$ upon setting ##R_2(r)=R_3(r)##, the later metric holds the name of isotropic metric. My question what is the difference between the first and the second...