Metric Definition and 1000 Threads

I have another question, if I am given a metric and I want to find the constants of motion of that system, then how do I do it? Thanks!

I have the metric ##ds^2=dudv+F(y,z)du^2+dy^2+dz^2##. I have shown that for ##F(y,z)=0## it's Minkowski metric, but for ##F(y,z)\ne 0 ## I want to calculate Christoffel Symbols, Ricci tensor and scalar, the problem is that the metric ##g_{\mu\nu}## is not diagonal, and I need to find...

Some models of gravity, inspired by the main theme of spacetime fabric of Classical GR, treat the metric of the manifold and the connection as independent entities. I want to study this theory further but I am unable to find any paper on this, on ariXiv atleast. I will be very thankful if...

My discussion of the Friedmann metric comes from the derivation presented in section 4.2.1 of the reference: https://www1.maths.leeds.ac.uk/~serguei/teaching/cosmology.pdf I have a couple of simple questions on the derivation. The are placed at points during the derivation.I note the...

I am having trouble understanding the Kerr metric. One of the things which helped me understand the Schwarzschild metric is the Kruskal–Szekeres coordinates. In particular, the fact that light cones were still at 45 degrees was very helpful, and it was helpful to see that the singularity was a...

What is a good simple metric for a single photon?

Working through an online course "Introduction to General Relativity." They give the metric for, Kottler-Moller coordinates, i.e. $$ds^2=(1+ah)^2d\tau^2-dh^2-dy^2$$ and say that it "covers" the Rindler wedge in flat space time, which is defined by $$0<x<\infty,-x<t<x$$ I am having difficulty...

Homework Statement This is Problem 2 from Chapter 1, Section V of A. Zee's Einstein Gravity in a Nutshell. Zee asks us to imagine a colony of "eskimo mites" that live at the north pole. The geometers of the colony have measured the following metric of their world to second order (with the...

Homework Statement This is a problem from Munkres(Topology): Show that a connected metric space ##M## having having more than one point is uncountable. Homework Equations A theorem of that section of the book states: Let ##X## be a nonempty compact Hausdorff space. If no singleton in ##X## is...

I am trying to understand the solution to exercise 7.10(e) on pages 175-176 of Robert Scott's student's manual to Schutz's textbook. He writes the following: I don't understand how to find ##S^x, S^z## or ##T^x,T^z## from the metric or from the cartesian representation of the rotation...

Hi PF! I'm trying to compute $$ \frac{1}{\sqrt g}\frac{\partial}{\partial u^\mu}\left( \sqrt g g^{\mu v} \frac{\partial \eta(s,\phi))}{\partial u^v} \right) $$ where I found $$ \sqrt g = \csc^2\alpha \sin s\\ g = \begin{bmatrix} \csc^2\alpha &0\\ 0 & \csc^2\alpha\sin^2 s \end{bmatrix}...

Let M = {p, x1, x2, x3, ...} be a metric space with no isolated points. f: M → M is continuous with f(xn) = xn+1, and f(p) = p. We say f separates if ∃ δ > 0, ∋ for any y and z there is some n with |fn(y) - fn(z)| > δ, where fn+1(y) = f(fn(y)). QUESTION: Does f separate?

I have the metric: $$ds^2 = −(1 − \Omega^2 (x^2 + y^2 ))dt^2 + 2\Omega(ydx − xdy)dt + dx^2 + dy^2 + dz^2$$ In order to calculate the Christoffel symbol I need to compute the inverse matrix of the above, is there some computational shortcut besides the joke:" using mathematica"? Thanks!

Homework Statement [/B] For any metric ##g_{ab}## show that ##g_{ab} g^{ab} = N## where ##N## is the dimension of the manifold.Homework Equations $$ g_{ab} = \mathbf {e_a} \cdot \mathbf {e_b} $$ $$ g^{ab} = \mathbf {e^a} \cdot \mathbf {e^b} $$ The Attempt at a Solution When I substitute the...

Is there an easy example of a closed and bounded set in a metric space which is not compact. Accoding to the Heine-Borel theorem such an example cannot be found in ##R^n(n\geq 1)## with the usual topology.

What is g11? I am very curious, can someone briefly describe what the metric tensor is, please?

Homework Statement Calculate the volume of a sphere of radius ##r## in the Schwarzschild metric. Homework Equations I know that \begin{align} dV&=\sqrt{g_\text{11}g_\text{22}g_\text{33}}dx^1dx^2dx^3 \nonumber \\ &= \sqrt{(1-r_s/r)^{-1}(r^2)(r^2\sin^2\theta)} \nonumber \end{align} in the...

This has been bugging me for a while, and I feel like I’m missing or misunderstanding some crucial piece of information, so please advise me: the scalar product of two vectors (say ##\mathbf v## and ##\mathbf w##) is given using the metric: ##g_{\alpha \beta} \mathrm v^{\alpha} \mathrm...

I have only seen scenarios so far where the elements are all along the diagonal, but what are some known cases where there are off-diagonal elements? Thank you.

Right now, I am studying Advanced Calculus (proof based), and it is hard thinking through some of the definitions without first thinking about it concretely (meaning how to visualize it better geometrically, if that makes any sense?). I need help with this definition. Definition Let X be a...

Homework Statement attached: I am stuck on question 2, and give my working to question 1 - the ##B(r) ## part I am fine with the ##A(r)## part which clearly is the same in both regions seen by looking at ##G_{rr}## , and attempt, however I assume I have gone wrong in 1 please see below for...

Hello. I am looking for help in establishing all the consequences of a modified Scwazschild metric where the length contraction is removed. ds^2=(1-rs/r)c^2dt^2-dr^2-r^2(... ) Thanks

Proposition: Consider an ##n + 1##-dimensional metric with the following product structure: $$ g=\underbrace{g_{rr}(t,r)\mathrm{d}r^2+2g_{rt}(t,r)\mathrm{d}t\mathrm{d}r+g_{tt}(t,r)\mathrm{d}t^2}_{:=^2g}+\underbrace{h_{AB}(t,r,x^A)\mathrm{d}x^A\mathrm{d}x^B}_{:=h} $$ where ##h## is a Riemannian...

Imagine that the CMB did not exist. What observational evidence exists to support the theory of the metric expansion of spacetime, as opposed to having a static spacetime and it's the matter distribution that is expanding - as it would in an explosion?

Hi, how can I prove that any 2-dim Lorentzian metric can locally be brought to the form $$g=2 g_{uv}(u,v) \mathrm{d}u \mathrm{d}v=2 g_{uv}(-\mathrm{d}t^2+dr^2)$$ in which the light-cones have slopes one? Thanks!

Homework Statement My Teacher says that in the Schwarzschild metric he uses natural units, where he writes ##g_{rr}=1-2M/R## He says that for one neutron star ##R=5## corresponds to approx 13 KM. Homework Equations ##1l_p=1,616 \cdot 10^{-35}m## The Attempt at a Solution Unfortunately he does...

I would like to ask what I hope are two simple questions about what I recognize to be a complicated subject. I did make an effort to search the Internet for the answers, but the two most promising looking sources I found did not help...

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 Corollary 11.1.5 ... Corollary 11.1.5 reads...

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 Corollary 11.1.5 ... Corollary 11.1.5 reads as...

I don't understand the reasoning for any of the three constraints imposed. why would ##dtdx^i## terms indicate a preferred direction? what if there was identical terms for each ##x^i## would there still be a specified or preferred direction? (or is it that in this case we could rename ##t## to...

If I understand it correctly, the proper time differential for a photon in flat space is zero. That is evident if the velocity of light is equal to c, so the right hand side of the Minkowski metric is equal to zero. Therefore the left side must also be zero. My question: Is the same true for...

Defining dS2 as gijdxidxj and given dS2 = (dx1)2 + (dx2)2 + 4(dx1)(dx2). Find gijNow here is my take on the solution: Since the cross terms are present in the line element equation, we can say that it's a non-orthogonal system. So what I did was express the metric tensor in form of a 2x2...

I am reading a book of General Relativity and I am stuck on a demonstration. If I consider the FLRW metric as : ##\text{d}\tau^2=\text{d}t^2-a(t)^2\bigg[\dfrac{\text{d}r^2}{1-kr^2}+r^2(\text{d}\theta^2+\text{sin}^2\theta\text{d}\phi^2)\bigg]## with ##g_{tt}=1##, ##\quad...

Many textbooks use the space (spacetime, actually, but for now only space is good enough) around a spherically symmetrical Schwarzschild object to demonstrate curvature of space due to gravity. Let’s consider two shells around such a Schwarzschild object (say a neutron star of 1 solar mass)...

Greg Bernhardt submitted a new PF Insights post This article is part of our student writer series. The writer Arman777, is an undergraduate physics student at METU A Journey Into the Cosmos - FLRW Metric and The Friedmann Equation Continue reading the Original PF Insights Post.

Hi, I think this is a topology question, but I'm honestly not 100% sure. Feel free to move. On a number line, I am trying to find the minimum number of moves to get from one point to another, given certain allowed step sizes. For example, My allowed steps might be +/-2, +/-3, and I want to move...

I have a surface defined by the quadratic relation:$$0=\phi^2t^4-x^2-y^2-z^2$$Where ##\phi## is a constant with units of ##km## ##s^{-2}##, ##t## is units of ##s## (time) and x, y and z are units of ##km## (space). The surface looks like this: Since the formula depends on the absolute value of...

Hello, Is the Berry connection compatible with the metric(covariant derivative of metric vanishes) in the same way that the Levi-Civita connection is compatible with the metric(as in Riemannanian Geometry and General Relativity)? Also, does it have torsion? It must either have torsion or not be...

Greg Bernhardt submitted a new PF Insights post The Schwarzschild Metric: Part 3, A Newtonian Comparison Continue reading the Original PF Insights Post.

In FLWR metric or in Minkowski metric or in any general metric can we say that ##ds^2=0## cause speed of light should be constant to all observers ? Or there's another reason ?

1. The metric ##g_{\mu \nu}## of spacetime shall be constructed from tensor products of vectors (relevant are the unit vectors in the respective directions). One such vector shall be called ##A##. Homework Equations ##g_{\mu \nu} = \lambda \frac{A_\mu A_\nu}{g^{\alpha \beta} A_\alpha A_\beta}...

Greg Bernhardt submitted a new PF Insights post The Schwarzschild Metric: Part 2, The Photon Sphere Continue reading the Original PF Insights Post.

Greg Bernhardt submitted a new PF Insights post The Schwarzschild Metric: Part 1, GPS Satellites Continue reading the Original PF Insights Post.

I am trying to understand how to define a metric for a positively curved two-dimensional space.I am reading a book and in there it says, On the surface of a sphere, we can set up a polar coordinate system by picking a pair of antipodal points to be the “north pole” and “south pole” and by...

I’m hoping to clear up some confusion I have over what the Lie derivative of a metric determinant is. Consider a 4-dimensional (pseudo-) Riemannian manifold, with metric ##g_{\mu\nu}##. The determinant of this metric is given by ##g:=\text{det}(g_{\mu\nu})##. Given this, now consider the...

Hi, I have seen the general form for the metric tensor in general relativity, but I don't understand how that math would create a Minkowski metric with the diagonal matrix {-1 +1 +1 +1}. I assume that using the kronecker delta to create the metric would produce a matrix that has all positive 1s...

I am a little bit confused about the metric tensor and would like some feedback before I proceed with my learning of GR. So I understand that metric tensor describes the geometry of the space itself. I also understand that the components of the metric tensor (any tensor for that matter) come...

I am reading John B. Conway's book: Ä First Course in Analysis and am focused on Chapter 5: Metric and Euclidean Spaces ... and in particular I am focused on Section 5.2: Sequences and Completeness ... I need some help/clarification with Conway's defintion of completeness of a metric space ...

I am trying to learn GR. In two of the books on tensors, there is an example of evaluating the inertia tensor in a primed coordinate system (for example, a rotated one) from that in an unprimed coordinate system using the eqn. ##I’ = R I R^{-1}## where R is the transformation matrix and...

Homework Statement Consider following transformation: Transformation: $$X^{\mu}\rightarrow \tilde{X^{\mu}}= X^{\mu}+\xi^{\mu}(\eta, \vec{x})$$ where ##\xi^0=T, \xi^i=L_i## Show transformation of metric perturbation ##B_i\rightarrow \tilde{B_i}=B_i+\partial_iT-\partial_{\eta}L_i## Homework...