Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.

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1answer
252 views

The Weyl tensor and gravitational waves

How exactly is the Weyl tensor is connected with information about gravitational waves? And what are physical reasons for that?
2
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1answer
136 views

Is any apparent horizon a minimal surface?

I faced "any apparent horizon is a minimal surface", but I don't know how I can relate a physical concept (apparent horizon) to pure mathematical concept (minimal surface). How can I prove it?
4
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1answer
82 views

Question about exterior derivatives

I know from Carroll that the integration in GR is basically a mapping from n-form to the real number. And it's given that $$d^nx=dx^0\wedge\ldots\wedge ...
3
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1answer
82 views

Question about simple permutation of covariant derivatives

I must to compute value $$ [[D_{\mu}, D_{\nu}],D_{\lambda}]A^{\rho}. $$ It is equal to $$ [D_{\mu}, D_{\nu}]D_{\lambda}A^{\rho} - D_{\lambda} ([D_{\mu}, D_{\nu}]])A^{\rho} - [D_{\mu}, ...
2
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2answers
176 views

Why do we must know the Weyl tensor for 4-dimensional space-time?

I heard that we must know the Weyl tensor for fully describing the curvature of the 4-dimensional space-time (in space-time with less dimensions it vanishes, so I don't interesting in cases of less ...
4
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0answers
363 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 ...
6
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6answers
613 views

Physical meaning of non-trivial solutions of vacuum Einstein's field equations

According to Einstein, the space-time is curved and the origin of the curvature is the presence of matter i.e. the presence of the energy-momentum tensor $T_{ab}$ in Einstein's field equations. If our ...
3
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0answers
176 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 ...
7
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1answer
128 views

Betti multiplets in Kaluza Klein compactifications

It is well known that if the compactification manifold of a supergravity theory has non-zero Betti numbers, this may lead to the so called Betti multiplets in the spectrum of the low dimensional ...
1
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1answer
46 views

Pre-gauge-fixed superspace action of the RNS superstring

When writing down the the action of the RNS superstring in superspace, all of the sources I have checked (BBS, GSW, Polchinski) seem to just write down the action in conformal gauge, that is $$ ...
3
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1answer
205 views

Etale bundles and sheaves

Before answering, please see our policy on resource recommendation questions. Please try to give substantial answers that detail the style, content, and prerequisites of the book or paper (or ...
8
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3answers
3k views

Maxwell's Equations using Differential Forms

Maxwell's Equations written with usual vector calculus are $$\nabla \cdot E=\rho/\epsilon_0 \qquad \nabla \cdot B=0$$ $$\nabla\times E=-\dfrac{\partial B}{\partial t} \qquad\nabla\times ...
3
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1answer
115 views

What spacetimes satisfy this identity?

What spacetimes satisfy $R^{\mu\nu} R_{\mu\nu} =\alpha R^2$, where $R = g^{\mu\nu}R_{\mu\nu}$ is the Ricci scalar, and $\alpha$ is some constant? A follow-up question: in what spacetimes does ...
2
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1answer
76 views

What do physicists mean by ${g^{i}}_j$?

Maybe this is an idiot question, but in relativity I see a lot of ${g^{i}}_j$ for a metric tensor $g$. Is this just $$\delta^i_j ~=~ g(dx^i \sharp, \partial_{ x^j})~?$$
3
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0answers
439 views

Killing vectors for 2-sphere as generators of $SO(3)$ symmetry

How to get Killing vectors in a form of generators of $SO(3)$ group symmetry? By using Killing equations for metric $ds^{2} = d\theta^{2} + \sin^{2}(\theta^{2}) d\varphi^{2}$ I got $$ ...
4
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1answer
92 views

Metric of a manifold foliated by maximally symmetric submanifold

I am reading the last chapter (Schwarzchild solution and Black Holes) of Sean Caroll's GR notes (http://arxiv.org/abs/gr-qc/9712019). While talking about spherical symmetry, he says how the ...
5
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1answer
248 views

A simple conjecture on the Chern number of a 2-level Hamiltonian $H(\mathbf{k})$?

For example, let's consider a quadratic fermionic Hamiltonian on a 2D lattice with translation symmetry, and assume that the Fourier transformed Hamiltonian is described by a $2\times2$ Hermitian ...
4
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1answer
227 views

What are the generators of spherical symmetry?

The title says it all. I think this should be a pretty simple question but I just couldn't find the answer. Ok -- I'll give a bit more context to my question. I'm encountering this in the context of ...
3
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1answer
345 views

What exactly is the connection between the Jacobi and Bianchi identities

While reviewing some basic field theory, I once again encountered the Bianchi identity (in the context of electromagnetism). It can be written as $$\partial_{[\lambda}\partial_{[\mu}A_{\nu]]}=0$$ ...
3
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1answer
110 views

Hyperkahler manifolds and their use in theoretical physics

Just as the title says: What is the easiest definition of a Hyperkahler Manifold? Could you give some examples of Hyperkahler manifolds, and manifolds which fail to be hyperkahler? Why are such ...
8
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2answers
267 views

What are orbifolds and why are they useful and interesting for physics?

Just what the title says. What's the basic definition of an orbifold? How do they arise in physics and why are they interesting?
12
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1answer
448 views

Asymptotic symmetry algebra

So after a lot of research, and tons and tons of papers that I've went through, I finally have some idea how to solve the equations that will give me candidates for the asymptotic symmetry group for ...
0
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1answer
661 views

General Relativity: Christoffel symbol identity

I want to show that $$\Gamma ^{\mu}_{\mu \nu}=\partial _\nu (\ln \sqrt{|g|}) .$$ (Here $|g|$ denotes the determinant of the metric.) Working out the left hand side:\begin{align} \Gamma ^{\mu}_{\mu ...
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2answers
479 views

General relativity: Induced metric and Killing vector fields

Assume that in spacetime ($M,g_{ab}$) there is a hypersurface generated by a set of independent one-parameter transformations acting on one single point, the generators of these transformations being ...
6
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1answer
356 views

Two formulas for a particle's acceleration

While on a class my teacher was taking about particle's motion in space. At some point she said the following: Consider that the particle's path is described by a curve in space defined by the ...
4
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2answers
800 views

What's the idea behind the Riemann curvature tensor?

The Riemann curvature tensor can be expressed using the Christoffel symbols like this: $R^m{}_{jkl} = \partial_k\Gamma^m{}_{lj} - \partial_l\Gamma^m{}_{kj} + \Gamma^m{}_{ki}\Gamma^i{}_{lj} ...
4
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0answers
49 views

Does heat kernel factorize on product spaces?

I have a doubt regarding whether the trace of the vector heat kernel on a product space factorizes into the corresponding heat kernel traces on each manifold in the product space. I know this holds ...
2
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1answer
199 views

Christoffel symbols and Dirac matrices mathematical similarities?

Maybe mine is a silly question, but are there mathematical similarities or common roots between the Christoffel symbols: $ \nabla - \partial = \Gamma $ and the Dirac matrices $ ( \gamma^\mu ...
3
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1answer
218 views

Riemann tensor notation and Christoffel symbol notation

In paper by Barnich and Brandt Covariant theory of asymptotic symmetries, conservation laws and central charges they defined the Riemann tensor like this: $$R_{\rho\mu\nu}^{\quad \ \ ...
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0answers
144 views

N=2 Dualities; k-differentials on the riemann sphere and a spectral curve

Currently I am working on my masters thesis about dualities in QFT and their geometric realizations. As of now, I am trying to understand the article 'N=2 Dualities" by Davide Gaiotto. On the ...
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0answers
60 views

Heat Kernel on Sphere for Vector Laplacian

Can somebody please provide me with a reference where the method of evaluating the heat kernel for spin-1 fields,i.e. vector laplacians on a 2-sphere is given? Please suggest some references ...
1
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1answer
151 views

Transformation rule of a partial derivative

We know the following transformation rule: $$ \partial'_b = \frac{\partial}{\partial x'^b} = \frac{\partial x^c}{\partial x'^b} \, \frac{\partial}{\partial x^c} = \frac{\partial x^c}{\partial x'^b} ...
7
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3answers
640 views

Covariant and contravariant vectors

Reading Weinberg's "gravitation and cosmology" I came across the sentence "The partial derivative operator $\partial/\partial x^\mu$ is a covariant vector, or in other words a 1-form,..." (p.115, ...
2
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1answer
269 views

Check it the Killing vectors satisfy Killing equation or not

I am going through Kerr/CFT correspondence paper again, and I am at the section where authors specify Killing vectors for near horizon extreme Kerr metric (shortly NHEK). The metric is ...
0
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2answers
158 views

Raising indices in Killing equation or not?

I'm having issues with computation of Killing equation. I'm using Mathematica to check if the given vectors are Killing vectors or not, and by hand for simple vector like $\xi=\partial_t$ I get the ...
6
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2answers
382 views

Parallel Transport of a 4-vector

Why does the parallel transported $4$-vector change from $X^a(x)$ to $X^a(x) + \bar{\delta}X^a(x)$ ? This is also discussed in D'Inverno's relativity book [page - 72]; but the reason is not clear.
2
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2answers
331 views

Weight of a tensor density

Is there any freedom in choosing the weight of a tensor density? I have seen in some papers that they introduce a tensor density made from metric with a special weight. There is a tensor density with ...
2
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1answer
166 views

Can two parallel lines meet? [closed]

My physics teacher talked about the meeting of 2 parallel lines, and he said that it may occur in the infinity or something. I know that 2 parallel lines can meet in spherical geometry, (thanks to ...
2
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1answer
187 views

Showing that the variation of an affine connection is a tensor

How can i show that $\delta\Gamma_{\mu\nu}^{\rho}$ transforms like a tensor? Metric compatibility is not assumed here. That means 1) First i need to compute $\delta\Gamma$ first. To do that i need ...
2
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2answers
3k views

Derive vector gradient in spherical coordinates from first principles

Trying to understand where the $\frac{1}{r sin(\theta)}$ and $1/r$ bits come in the definition of gradient. I've derived the spherical unit vectors but now I don't understand how to transform ...
2
votes
1answer
568 views

'Easy way' of finding out the Killing vector fields?

Is there a way for calculating the Killing vector fields of a given metric in a quick way? Sure I can guess looking at the metric at the symmetries, and then guess some of them, but, for instance, in ...
10
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1answer
413 views

Can masses move in 2+1 gravity?

I would like to understand basic concepts of the general relativity in 2+1 spacetime. As far as I know, GR predicts that such a spacetime is flat everywhere except for the point masses which create ...
3
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1answer
1k views

Riemann tensor in 2d and 3d

Ok so I seem to be missing something here. I know that the number of independent coefficients of the Riemann tensor is $\frac{1}{12} n^2 (n^2-1)$, which means in 2d it's 1 (i.e. Riemann tensor given ...
1
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1answer
119 views

Does Clifford algebra depend on the topology of manifold?

We know the greatest feature of Clifford algebra is coordinate-free. One can do vector operations without knowing the representation of vectors. And due to its very characteristc, Clifford or ...
2
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1answer
70 views

Getting diffeomorphisms from boundary conditions in $AdS_3$

As usual I'm asking a question about boundary conditions for AdS${}_3$, based on the thesis by Porfyriadis. He is solving equations $\mathcal{L}_\xi g_{\mu\nu}$ for AdS${}_3$ metric, with a given ...
3
votes
2answers
553 views

Coordinate Transformation of Scalar Fields in QFT

By definition scalar fields are independent of coordinate system, thus I would expect a scalar field $\psi [x]$ would not change under the transformation $x^\mu \to x^\mu + \epsilon^\mu $. Correct? ...
35
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2answers
1k views

Intuitively, why are bundles so important in Physics?

This question probably seems silly and I don't really know if it fits properly here, but the point is the following: I've seem the notion of bundles, fiber bundles, connections on bundles and so on ...
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0answers
60 views

When is the Fermi surface a surface of constant mean curvature?

Fermi surfaces are surfaces of constant energy in reciprocal space. They provide information about the properties of a material in solid state physics. Constant mean curvature surfaces are a superset ...
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1answer
64 views

Finding superpotentials and central charges in $AdS_3$

In text "Covariant theory of asymptotic symmetries, conservation laws and central charges" is given an example of finding central charges and superpotential (among other things). I am interested in ...
4
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0answers
92 views

Path Integral on Einstein Cartan Manifold

In condensed matter, crystal with disclination and dislocation has both curvature and torsion. I am looking for a reference in which path integral quantization of Dirac equation on manifold with ...