1. ### I Argument for existence of normal coordinates at a given point

Hey there, I've been recently been going back over the basics of GR, differential geometry in particular. I was watching one of Susskind's lectures and did not understand the argument made here (26:33 - 35:40). In short, the argument goes as follows (I think): we have some generic metric ##{ g...
2. ### A Anisotropic Universe and Friedmann Equations

The Friedman Equations is based on the cosmological principle, which states that the universe at sufficiently large scale is homogeneous and isotropic. But what if, as an hypothesis, the universe was anisotropic and the clustering of masses are aligned to an arbitrary axis (axial pole), how...
3. ### Did I Get These Metric Tensors Right?

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)...
4. ### I Invariant properties of metric tensor

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?
5. ### I Convert metric tensor to gravity

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!
6. ### I Gradient vector without a metric

Is it possible to introduce the concept of a gradient vector on a manifold without a metric?
7. ### I Coordinates for diagonal metric tensors

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 -...
8. ### B Density of the early Universe contributing to the red-shift?

Does the relative density of the early universe contribute to the red-shift of distant galaxies? If so, by how much? How would this be calculated? Asked another way : Assuming both the early universe and the current universe are flat, could the relative difference of their space time metric...
9. ### A On metric and connection independence

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...
10. ### A Normal to coordinate curve

Let $$\phi(x^1,x^2....,x^n) =c$$ be a surface. What is unit Normal to the surface? I know how to find equation of normal to a surface. It is given by: $$\hat{e_{n}}=\frac{\nabla\phi}{|\nabla\phi|}$$ However the answer is given using metric tensor which I am not able to derive. Here is the...
11. ### A Causal structure of metric

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...
12. ### Finding the inverse metric tensor from a given line element

Defining dS2 as gijdxidxj and given dS2 = (dx1)2 + (dx2)2 + 4(dx1)(dx2). Find gij Now 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...
13. ### I Lie derivative of a metric determinant

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...
14. ### A Trouble with Peskin QFT textbook

I'm trying to work through a scattering calculation in the Peskin QFT textbook in chapter 5, specifically getting equation 5.10. They take two bracketed terms 4[p'^{\mu}p^{\nu}+p'^{\nu}p^{\mu}-g^{\mu\nu}(p \cdot p'+m_e^2)] and 4[k_{\mu}k'_{\nu}+k_{\nu}k'_{\mu}-g_{\mu\nu}(k \cdot...
15. ### A Simple metric tensor question

Good Day, Another fundamentally simple question... if I go here; http://www-hep.physics.uiowa.edu/~vincent/courses/29273/metric.pdf I see how to calculate the metric tensor. The process is totally clear to me. My question involves LANGUAGE and the ORIGIN LANGUAGE: Does one say "one...
16. ### I What constrains the metric tensor field in GR?

Do the field equations themselves constrain the metric tensor? or do they just translate external constraints on the stress-energy tensor into constraints on the metric tensor? another way to ask the question is, if I generated an arbitrary differentiable metric tensor field, would it translate...
17. ### I Perturbative expansion of the metric and its inverse

As I understand it, in the context of cosmological perturbation theory, one expands the metric tensor around a background metric (in this case Minkowski spacetime) as $$g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}$$ where ##h_{\mu\nu}## is a metric tensor and ##\kappa <<1##. My question is, how...
18. ### I Metric Tensor as Simplest Math Object for Describing Space

I've been reading Fleisch's "A Student's Guide to Vectors and Tensors" as a self-study, and watched this helpful video also by Fleisch: Suddenly co-vectors and one-forms make more sense than they did when I tried to learn the from Schutz's GR book many years ago. Especially in the video...
19. ### A Simple 1D kinematic exercises with metric tensor

Hi All I would like to know if there is a way to produce simple one dimensional kinematic exercises with space-time metric tensor different from the Euclidean metric. Examples, if possible, are welcome. Best wishes, DaTario
20. ### I Metric tensor derived from a geodesic

Let we have a 2D manifold. We choose a coordinate system where we can construct all geodesics through any point. Is it enough to derive a metric from geodesic equation? Or do we need to define something else for the manifold?
21. ### I Gravitational redshift

I am trying to find a derivation of gravitational redshift from a static metric that does not depend on the equivalence principle and is not a heuristic Newtonian derivation. Any suggestions?
22. ### I Two metric tensors describing same geometry

Consider two coordinate systems on a sphere. The metric tensors of the two coordinate systems are given. Now how can I check that both coordinate systems describe the same geometry (in this case spherical geometry)? (I used spherical geometry as an example. I would like to know the process in...
23. ### I Special Relativity Approximation of Gravitation

Hey there, I have two questions - the first is about an approximation of a central gravitational force on a particle (of small mass) based on special relativity, and the second is about the legitimacy of a Lagrangian I'm using to calculate the motion of a particle in the Schwarzchild metric...
24. ### A Riemann tensor equation

The Riemann-Christoffel Tensor (##R^{k}_{\cdot n i j}##) is defined as: $$R^{k}_{\cdot n i j}= \frac{\delta \Gamma^{k}_{j n}}{\delta Z^{i}} - \frac{\delta \Gamma^{k}_{i n}}{\delta Z^{j}}+ \Gamma^{k}_{i l} \Gamma^{l}_{j n}- \Gamma^{k}_{j l} \Gamma^{l}_{i n}$$ My question is that it seems that...
25. ### A Metric with Harmonic Coefficient and General Relativity

Goodmorning everyone, is there any implies to use in general relativity a metric whose coefficients are harmonic functions? For example in (1+1)-dimensions, is there any implies for using a metric ds2=E(du2+dv2) with E a harmonic function? In (1+1)-dimensions is well-know that the Einstein...
26. ### I Is there any 2D surface whose metric tensor is eta?

Does there exist any 2D surface whose metric tensor is, ##\eta_{\mu\nu}= \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}##
27. ### I The Order and Valence of Tensors

I'm having a bit of trouble understanding the nature of tensors (which is pretty central to the gen rel course I'm currently taking). I understand that the order (or rank) of a tensor is the dimensionality of the array required to describe it's components, i.e. a 0 rank tensor is a scalar, a 1...
28. ### Metric tensor

Hello, since gμν gμν = 4 where g = diag[1,-1,-1,-1], see: https://www.physicsforums.com/threads/questions-about-tensors-in-gr.39158/ Is the following equation correct? xμ xμ = gμνxν gμνxν = gμν gμνxν xν= 4 xμ xμ If not, where is the problem? Cheers, Adam
29. ### Non-Euclidean geometry and the equivalence principle

As I understand it, a Cartesian coordinate map (a coordinate map for which the line element takes the simple form ##ds^{2}=(dx^{1})^{2}+ (dx^{2})^{2}+\cdots +(dx^{n})^{2}##, and for which the coordinate basis ##\lbrace\frac{\partial}{\partial x^{\mu}}\rbrace## is orthonormal) can only be...
30. ### Varying The Gibbons-Hawking Term

The Gibbons Hawking boundary term is given as ##S_{GHY} = -\frac{1}{8 \pi G} \int_{\partial M} d^dx \sqrt{-\gamma} \Theta##. I want to calculate its variation with respect to the induced boundary metric, ##h_{\mu \nu}##. The answer (given in eqns 6&7 of...