Use this for questions pertaining to curvature of manifolds. Does not need to be specific to general relativity, but also for curvature of e.g. a [tag:calabi-yau] manifold.

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2answers
66 views

Physical meaning of harmonic function?

In complex numbers, we define a harmonic function as a twice continuously differentiable function such that the Laplace operator acting on it gives zero. Can anybody explain me the physical ...
3
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1answer
219 views

Detail of deriving Berry Curvature From Berry Connection

The Berry curvature of the $n^{\mathrm{th}}$ eigenstate of Hamiltonian $H$ for the vector of external parameters $\vec{R}$ can be derived in part by writing the following two lines: $B^n(\vec{R}) ...
2
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1answer
139 views

How would it be to live in a very small universe, let´s say 20x20 square meters?

Let´s consider a curved universe that is very small, say 20 square meters and not expanding. If you stood at the middle of this tiny universe and looked forward, you wouldn´t see any walls, since it ...
6
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0answers
101 views

gravitational convergence of light

light has a non-zero energy-stress tensor, so a flux of radiation will slightly affect curvature of spacetime Question: assume a flux of radiation in the $z$ direction, in flat Minkowski space it ...
5
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0answers
118 views

Euclidean black hole extrinsic curvature

I have read that the extrinsic curvature at the horizon of a euclidean black hole is zero? Does anybody know how this can be shown?
5
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0answers
177 views

Why does the overhand knot jam but the figure-8 knot doesn't?

After tensioning a rope with an overhand knot in it, it is often very hard if not impossible to untie it; a figure-8 knot, on the other hand, still releases easily. Why is that so? Most "knot and ...
4
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0answers
142 views

Tricks for Computing Riemann Curvature Tensor with Levi-Civita connection

I am new to differential geometry, so far it seems to me that computing the Riemann tensor tends to be a rather tedious task, I wanted to know whether there are some tricks that I am missing. In ...
4
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0answers
417 views

How to prove that Weyl tensor is invariant under conformal transformations?

I need to verify that the solution for vanishing Weyl tensor is conformally flat metric $g_{\mu\nu} = e^{2\varphi}\eta_{\mu\nu}$. The most convenient way to show this is to prove that Weyl tensor is ...
3
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0answers
80 views

What Would Negative Mass Do To Spacetime?

It's known that positive mass bends space-time to create a curvature. But if something had negative mass what would it do? Make it flat or like a crest?
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0answers
50 views

Rigid rectangle in Schwarzschild

Say I build a perfect rectangle. Side lengths $l_1$ and $l_2$ and perfect right angles. I am on earth and the metric is given by the Schwarzschild metric. Setting $dt=0$ leads to the spatial ...
3
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0answers
108 views

Computing the Einstein tensor for a spherically symmetrical metric using the tetrad formalism

I am having some trouble understanding how to use the tetrad formalism. I will start with what I have so far, my question will be after that. I begin with the metric $$ \text{d}s^2 = e^{2a} \text{ ...
3
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0answers
208 views

Gauss-Bonnet term in Physics

Given a 4-dimensional compact manifold (torsion free), the Euler characteristic is defined as: $$E_4 ~=~ \int \epsilon_{abcd}R^{ab} \wedge R^{cd}$$ with $R^{ab}$ is the curvature 2-form. Perturb the ...
3
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0answers
125 views

Curvature and spacetime

Suppose that it is given that the Riemann curvature tensor in a special kind of spacetime of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a ...
2
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0answers
48 views

Examples of manifolds (not) being: flat, homogeneous and isotropic

I am looking for (at least) one example of the following manifolds: Flat, homogeneous and isotropic Curved, homogeneous and isotropic Flat, non-homogeneous and isotropic Flat, homogeneous and ...
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0answers
59 views

Geodesic Deviation between Test Particles from Gravitational Wave

I'm having trouble understanding how Carroll (Spacetime and Geometry p.296) explains the effect of a passing gravitational wave on test particles. If we have two geodesics with tangents $\vec{U}$, ...
2
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0answers
546 views

de Sitter and anti de Sitter metric

Is the following correct for the distance $d$ from the origin $(0,0)$ to point $(t,x)$ in the 2-dimensional de-Sitter and anti de-Sitter spaces? Here, $t$ is time and the distance may be called the ...
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0answers
60 views

(Scalar) Ricci flatness of a metric

What is the physical meaning to vanishing Ricci scalar $R=0$ of a metric in general relativity? Note that this is not the same questions as the geometric meaning of $R_{\mu\nu}=0$ which has been asked ...
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0answers
38 views

Conditions for a diagonal induced metric?

Let $M$ be a manifold of dimension $n$ with a (say Lorentzian) metric $g$, that is diagonal in some choice of local coordinates. Let $S$ be manifold of dimension $k<n$ , embedded in $M$ by some ...
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0answers
62 views

Riemann curvature of a unit sphere

The Riemann curvature of a unit sphere is shown in many textbooks to be sine-squared theta where theta is the azimuthal angle of spherical co-ordinates. But what is the significance of the angle and ...
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0answers
80 views

Higher-Dimensional Metrics in (Hyper)-Spherical Coordinates

I want to compute the components of the Riemann curvature tensor (for a case similar to the Schwarzschild solution) in 4 + 1 dimensions, but I want to use a higher-dimensional analogue of spherical ...
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0answers
76 views

About the proof of the second Bianchi Identity

The second Bianchi Identity is $$ \nabla_{[a}R_{bc]de}=0 $$ As far as I know, the proof (say, Walfram Mathword) start by stating the representation of Riemann tensor in local inertial coordinates $$ ...
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0answers
80 views

Scattering theory of Dirac equation in curved space-time in presence of a strong magnetic field

What is the exact solution of the Dirac equation in curved space-time in the presence of a strong magnetic field? The solution should be in momentum space for simplicity to calculate scattering cross ...
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0answers
43 views

Physical applications of the mathematical curvature

I was studying multivariable calculus last semester and had one of the topics talking about a curvature, but we had no applications on it. So how does it help in physics? E.g. curvature of curve: ...
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0answers
85 views

Ricci scalar higher dimensions

I was wondering if there is a straightforward way to compute the Ricci curvature of a metric that has the form (à la Kaluza-Klein): $g_{MM}\equiv\begin{pmatrix}g_{\mu\nu}&g_{\mu ...
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0answers
51 views

How to prove the derive the expression for space part of Riemann tensor for homogeneous and isotropic space-time?

It's not a homework!! For spheric, hyperbolic and flat case $$ dl^{2} = R^{2}\left(d \psi^{2} + sin^{2}(\psi )(d \theta^{2} + sin^{2}(\theta )d \varphi^{2})\right), $$ $$ dl^{2} = R^{2}\left(d ...
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0answers
29 views

Uniqueness of the Einstein tensor

This is related with an exercise 17.4-a in MTW Here what i want to show is the Einstein tensor $G_{\alpha\beta} = R_{\alpha\beta} - \frac{1}{2} R g_{\alpha \beta}$ is the only second-rank, symmetric ...
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0answers
40 views

Background field expansion in normal coordinates

Background field expansion following form $Y= X+\pi$ where $X$ is my background field and $\pi$ is the fluctuation. From the Normal coordinates we have the expansion of $\pi^{\mu} = ...
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0answers
63 views

de Sitter–Schwarzschild metric in the Kretschmann Gravity?

The Kretschmann Gravity (Gauss-Bonnet Gravity,Lovelock Gravity) results when the Ricci scalar is replaced by the Kretschmann invariant in the Lagrangian of the General Relativity. We consider here a ...
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0answers
29 views

Relation between the curvature of a manifold and the number of covariantly constant vector fields that it admits

Suppose that on a four dimensional manifold we are able to explicitly construct four linearly independent covariantly constant vector fields $K^a_{\mu}$: $$D_{\mu}K^a_{\nu}=0,$$ $a=1,2,3,4$ then it ...
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0answers
94 views

Curvature based derivation of Schwarzchild Metric

I'm a third year maths undergrad and I'm trying to find (and follow) a curvature based derivation of the Schwarzchild metric, if there exists such a proof?
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0answers
50 views

Curved space to flat space calculation

When changing the curved space co-ordinate into a flat space co-ordinate if a cone. I got the result transformation that i cannot get a transformation at the vertex(apex) why?