Metric Definition and 1000 Threads

I was recently reading that the space of 3d rotations should have the topology of a real projective space. For confirmation, see wiki, https://en.wikipedia.org/wiki/3D_rotation_group. It seems to me that when we assign coordinates to this space (I was thinking of using the Euler angles, but...

Another GR question... in the thick of revision season. I would appreciate a sketch of how to approach the problem. You basically are given a metric, involving a positive function ##A(z)##, $$g = A(z)^2(-dt^2 + dx^2 + dy^2) + dz^2$$The game is to figure out somehow that the null-energy...

If a vector ##V(x)## being transported down a path ##l##, The vector field is described with equation: $$\partial_\mu V(x)=\Gamma_\mu V(x)$$ The solution of the equation can be described with parallel propagator ##P(x, x_0)##(in mathematics it is also called product integration): $$V(x)=P(x...

Hi, I'm not sure if I have calculated the task here correctly Task 4-4b looked like this I have now obtained the following with ##n=-v_p(x-y)## $$\displaystyle{\lim_{n \to \infty}} p^n= \infty$$ $$\sum\limits_{n=0}^{\infty} p^n=\frac{p}{p-1}$$ Is that correct?

What is the orbital period in General Relativity using the Schwarzschild metric? In classical mechanics, it is something like T=2pi(GnM/a3). Where a is the semi-major axis, this is for a small body orbiting a larger one. I think I have an idea but I am not 100% sure. I am interested in an...

I'm reading "Problem Book In Relativity and Gravitation". In this book there is a problem 7.5 Show that metric tensor is covariant constant. To prove it, authors suggest to use formulae for covariant derivative: Aαβ;γ=Aαβ,γ−AσαΓβγσ−AσβΓαγσ after that they write this formulae for tensor g and...

I am reading about musical isomorphisms and for the demonstration of the index raising operation from the sharp isomorphism, we have to multiply the equation by the inverse matrix of the metric. Can we assume that this inverse always exists? If so, how could I prove it?

Using an example of 1 space dimension and 1 time dimension, consider the metric ##d\tau^2 = a dt^2 - dx^2## near a heavy mass (##a>1##). From what I've read a clock ticks slower near a heavy mass, as viewed from an observer far away. A clock tick would be representative of ##d\tau## right...

Let ##\gamma > 1##. If ##(X,d)## is a metric space and ##f : \mathbb{R} \to X## satisfies ##d(f(x),f(y)) \le |x - y|^\gamma## for all ##x,y\in \mathbb{R}##, show that ##f## must be constant.

What is the metric for a bag-of-gold spacetime?

We all know that The gradient of a scalar-valued function ##f(x)## in ##IR^n## is a vector field ##V_\mu(x)=\partial_\mu f(x)##, Such a vector field is said to be conservative. Not all vector fields are conservative. A conservative vector field should meet certain constraints...

In this popular science article [1], they say that if our universe resulted to be non-uniform (that is highly anisotropic and inhomogeneous) then the fundamental laws of physics could change from place to place in the entire universe. And according to this paper [2] anisotropy in spacetime could...

I have the following question to solve:Use the metric: $$ds^2 = -dt^2 +dx^2 +2a^2(t)dxdy + dy^2 +dz^2$$ Test bodies are arranged in a circle on the metric at rest at $$t=0$$. The circle define as $$x^2 +y^2 \leq R^2$$ The bodies start to move on geodesic when we have $$a(0)=0$$ a. we have to...

Are there non-smooth metrics for spacetime (that don't involve singularities)? I found this statement in a discussion about the application of local Lorentz symmetry in spacetime metrics: Lorentz invariance holds locally in GR, but you're right that it no longer applies globally when gravity...

I started by expanding ##dx## and ##dt## using chain rule: $$dt = \frac{dt}{dX}dX+\frac{dt}{dT}dT$$ $$dx = \frac{dx}{dX}dX+\frac{dx}{dT}dT$$ and then expressing ##ds^2## as such: $$ds^2 =...

I was discussing this paper with a couple of physicists colleagues of mine (https://arxiv.org/abs/2011.12970) In the paper, the authors describe "spacetimes without symmetries". When I mentioned that, one of my friends said that no spacetime predicted or included in the theory of relativity...

Hi, I am reading through my lecture notes - I haven't formally covered killing vectors but it was introduced briefly in lectures. Reading through the notes has highlighted something I am not sure about when it comes to co-ordinate transformations. Q1.Can someone explain how to go from...

Dear PF Forum, It's been a while since I logged in here. And I really do appreciate all the answers that I've been getting here. Now, I wonder. Is there any standardization for 1 light year distance? Is it 10 trillion kilometers, or 299,792,458 * 60 * 60 * 24 * 365.256 = ...

Consider a hypothetical five dimensional flat spacetime ##\mathbb{R}^5## with coordinates ##x, y, z, w, t##. Now imagine that the hypersurface ##\Sigma =\mathbb{R}^3## of ##x, y, z## moves with constant rate ##r## along the coordinate ##w##, i.e. ##dw/dt=r##. Assuming that ##t \in (-\infty, +...

In Dirac's "General Theory of Relativity", he begins Chap 16, with "Let us consider a static gravitational field and refer it to a static coordinate system. The ##g_{\mu\nu}## are then constant in time, ##g_{\mu\nu,0}=0##. Further, we must have ##g_{m0} = 0, (m=1,2,3)##." It's obvious that...

Once having converted the FLRW metric from comoving coordinates ##ds^2=-dt^2+a^2(t)(dr^2+r^2d\phi^2)## to "conformal" coordinates ##ds^2=a^2(n)(-dn^2+dr^2+r^2d\phi^2)##, is there a way to facilitate solving for general geodesics that would otherwise be difficult, such as cases with motion in...

How do I calculate the unit normal vector for any metric tensor?

"Spherical symmetry requires that the line element does not vary when##\theta## and##\phi## are varied,so that ##\theta##and ##\phi##only occur in the line element in the form(##d\theta^2+\sin^{2}\theta d\phi^2)##" I wonder why: "the line element does not vary when##\theta## and##\phi## are...

From Wikipedia article about Hyperbolic motion, I have the following coordinate equations of motion for Bob in his accelerated frame: ##t(T)=\frac{c}{g} \cdot \ln{(\sqrt{1+(\frac{g \cdot T}{c})^2}+\frac{g \cdot T}{c})} \quad (1)## ##x(T)=\frac{c^2}{g} \cdot (\sqrt{1+(\frac{g \cdot T}{c})^2}-1)...

In describing the spacetime around a massive, spherical object, one would use the Schwarzschild Metric. What metric would instead be used to describe the spacetime around multiple massive bodies? Say, for example, you want to calculate the Gravitational Time Dilation experienced by a rocket ship...

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I've been trying to find a way to calculate Gaussian curvature from a 4D metric tensor. I found a program that does this in Mathematica using the Brioschi formula. However, this only seems to work for a 2D metric or formula (I would need to use something with more dimensions). I've found...

I'm wondering if there is a way to find the proper volume of the warped region of the Alcubierre spacetime for a constant ##t## hypersurface. I can do a coordinate transformation ##t=τ+G(x)##, where ##G(x)=\int \frac{-vf}{1-v^2f^2}dx##. This eliminates the diagonal and makes it so that the...

Q. Calculate the linearised metric of a spherically symmetric body ##\epsilon M## at the origin. The energy momentum tensor is ##T_{ab} = \epsilon M \delta(\mathbf{r}) u_a u_b##. In the harmonic (de Donder) gauge ##\square \bar{h}_{ab} = -16\pi G \epsilon^{-1} T_{ab}## (proved in previous...

The Robertson-Walker-Metric is given by To calculate the Friedmann equations ist is choosed with despite Minkowskis, Schwarzschilds and Kerrs What is the reason for this difference? Tanu

The FLRW metric has been introduced to characterize the homogeneity and isotropy of the Universe and accordingly to obtain "easy" manageable solutions in Friedmann equations. The FLWR metric is where the LHS can be written as where is the proper time (despite we know that time is...

I'm interested in describing a 6-dimensional space of which three are compactified to small circles. Globally this space looks 3-dimensional, like a 2-dimensional cylinder looks 1-dimensional globally. Kaluza and Klein did a similar thing in the context of 4-dimensional spacetime. They extended...

In the absence of a metric, we can not raise and lower indices at will. There are two sorts of Christoffel symbols, Christoffels of the first kind, ##\Gamma^a{}_{bc}## in component notation, and Christoffel symbols of the second kind, ##\Gamma_{abc}##. What's the relationship between the two...

5/18/22 I am an MS in physics. I need to find out if the following CONFORMAL METRIC produces zero or nonzero curvature? I suspect the curvature is zero, but others have said it's probably not? MAXIMA sometimes says it is, and other times produces a Ricci scalar that looks like the FRW scalar...

The Schwarzschild metric implies a potential different from that of Newtonian gravity. Is there a relationship between it and the process by which particles can be absorbed by other particles? (I haven't studied QFT yet)

I'm working through some things with general relativity, and am trying to solve for my equations of motion from the Schwarzschild Metric. I'm new to nonlinear pde, so am not really sure what things to try. I have 2 out of my 3 equations, for t and r (theta taken to be constant). At first glance...

I just need to compare my results of the Ricci and Riemann Tensors of FRW metric, but only considering the spatial coordinates.

For the dimensions of a right cylinder, I am given three significant digits for the diameter (17.4 mm) and the height (50.3 mm). The formula for the volume of a right cylinder is V = Pi x r^2 x h, which would lead here to Pi x (17.4 mm / 2)^2 x 50.3 mm = 11,960.69354 mm^3 before rounding to 3...

Homework Statement:: Please see below. Relevant Equations:: Please see below. I am trying to find a reference to a textbook or a paper that details the following time-invariance Kaluza-Klein metric: \begin{equation}...

##d'## is a metric on ##X## because it satisfies the axioms of metrics: Identity of indiscernibles: ##x=y\Longleftrightarrow d(x,y)=0\Longleftrightarrow \sqrt{d(x,y)}=\sqrt{0}## Symmetry: ##d(x,y)=d(y,x)\Longrightarrow \sqrt{d(x,y)}=\sqrt{d(y,x)}## Triangle inequality: ##d(x,z)\leq...

So, there are a fair amount of metrics designed with a zero value for the cosmological constant in mind. I was wondering if there was some method to modify metrics to account for a nonzero cosmological constant. Say, for instance, the Schwarzschild metric due to its relative simplicity. A...

Am refreshing on Metric spaces been a while... Consider the axioms below; 1. ##d(x,y)≥0## ∀ ##x, y ∈ X## - distance between two points 2. ## d(x,y) =0## iff ##x=y##, ∀ ##x,y ∈ X## 3.##d(x,y)=d(y,x)## ∀##x, y ∈ X## - symmetry 3. ##d(x,y)≤d(x,z)+d(z,y)## ∀##x, y,z ∈ X## - triangle inequality...

M. Blennow's book has problem 2.18: Show that the contracted Christoffel symbols ##\Gamma_{ab}^b## can be written in terms of a partial derivative of the logarithm of the square root of the metric tensor $$\Gamma_{ab}^b=\partial_a\ln{\sqrt g}$$I think that means square root of the determinant of...

I know some basic GR and encountered the Schwarzschild metric as well as the Riemann tensor. It is known that for maximally symmetric spaces there is a corresponding Riemann tensor and thus Ricci scalar. Question. How do you calculate the Ricci scalar ##R## and cosmological constant ##\Lambda##...

I'm still confused about the notation used for operations involving tensors. Consider the following simple example: $$\eta^{\mu \sigma} A_{\mu \nu} = A_{\mu \nu} \eta^{\mu \sigma}$$ Using the rules for raising an index through the (inverse) metric tensor ##\eta^{\mu \sigma}## we get...

I am having trouble calculating the extrinsic curvature (12) in the following paper: https://arxiv.org/pdf/gr-qc/0310107.pdf Specifically, I am unsure of what term to plug in for the induced metric h_{ab} in (8). If I am calculating the \sigma term in (12) is h_{ab} all of (4)? Also I would like...

I can't figure out how to transform the coordinates to get to the given metric \begin{align*}ds^2 = \cos x (dy^2 - dx^2) + 2\sin x dx dy \end{align*} for a 2-torus. Initially I parameterised it by two angles ##\theta## (around the ##z## axis) and ##\phi## (around the torus axis), to write...

In the Reissner–Nordström metric, the charge ##Q## of the central body enters only as its square ##Q^2##. The same is true for the Kerr-Schild form. This would seem to imply that all effects are even functions of ##Q##. For example, the gravitational time dilation is often written as $$\gamma =...

We know that the cofactor of determinant ##A##, is $$\frac{\partial A}{\partial a^{r}_{i}} = A^{i}_{r} = \frac{1}{2 !}\delta^{ijk}_{rst} a^{s}_{j} a^{t}_{k} = \frac{1}{2 !}e^{ijk} e_{rst} a^{s}_{j} a^{t}_{k}$$ By analogy, $$\frac{\partial Z}{\partial Z_{ij}} = \frac{1}{2 !}e^{ikl} e^{jmn}...

if a metric like ##ds^2=dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2 ## is given, we know it corresponds to a sphere in spherical coordinates . if you are given an arbitrary metric with two variables for example ##ds^2=\frac{du^2}{u}+dv^2## is ther guarenteed to be a surface embedded in ##R^3##...