Questions tagged [differential-geometry]

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

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Gauss divergence theorem

I was reading new edition of Arfeken's book on Mathematical Physics I came to a two line state in the new edition in gauss divergence theorem section which was not included in the old edition and I ...
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General Newton's second law [closed]

Thank you, I apologize: next time i will use LaTeX. I ask to Vincent Thacker: I read your post but I ask if it is possible to get your last equation (or similar) in classical mechanics starting by d2/...
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Differential forms in projective space

I am currently reading some paper about the Amplituhedron, and it is using projective geometric way to present amplitudes. How can we define forms in projective space to measure volume for a polytope?
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Definition of pull-back in gauge transformation

In an online lecture given by a famous physicist Youshi Wu, he says: Consider a map $g:\quad M\rightarrow G:x\rightarrow g(x)\in G$ ($G$ is a Lie-Group manifold, $M$ can be physical space or ...
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Coefficient of an 1-form in position-representation of momentum operator where configuration space is NOT $\mathbb{R}^m$

I found this in the book Geometric Phase in Quantum Systems by A. Bohm et al. Where the position space representation of the momentum operator carries a (Where exactly my doubt is) coefficient of 1-...
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What is the physical meaning of the exterior derivative of a non-holonomic constraint form?

Suppose we have a non-holonomic mechanical system, say Lagrangian, for example the Chaplygin sleigh is a model of a knife in the plane. Its configuation space is $Q = S^1\times \mathbb{R}^2$ with ...
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Show that $\chi=\frac{1}{4\pi}\int _M d^2 \sigma g^{1/2} R +\frac{1}{2\pi}\int_{\partial M}ds k$ is locally a total derivative in 2D [closed]

In Polchinski Page 83 mentioned that $$\chi=\frac{1}{4\pi}\int _M d^2 \sigma g^{1/2} R +\frac{1}{2\pi}\int_{\partial M}ds k$$ is locally a total derivative in two dimension $R$ was the Ricci scalar ...
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Looking for differential operators satisfying a specific commutation relation with the Laplace operator

Consider the Laplace operator on some manifold, $\Delta=(\det g)^{-1/2} \frac{\partial}{\partial x^j} \left( (\det g)^{1/2}g^{jk} \frac{\partial}{\partial x^k}\right) $. I am looking for differential ...
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Global (not locally defined) quantities in General relativity?

So I was reading about center of mass (relativistic) on Wikipedia and saw an interesting comment: Since the Poincaré generators depend on all the components of the isolated system even when they are ...
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Extrinsic curvature calculation

In this paper section 3.2 the author uses the second junction condition to derive equation 3.8 $$ K_{1ab} = \frac{1}{L_1}\tanh\left(\frac{\rho_1^*}{L_1}\right)h_{ab}\;\;\;\;\;\;\;\;\;\;\;K_{2ab} = -\...
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Euler Characteristic [migrated]

Can someone provide the exact formula for the Euler Characteristic in 4D? I know it goes as $\epsilon^{\mu\nu\rho\lambda}R_{\mu\nu}^{\alpha\beta}R_{\rho\lambda}^{\gamma\delta}\epsilon_{\alpha\beta\...
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Is my derivation of the line element in GR of a rotating disk correct? [duplicate]

EDIT: I get a different expression than those given in the referred questions that are supposed to answer it. Therefore I want to know if my result is correct. I have tried to find the line element of ...
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Is it always possible to decompose the metric tensor of a spacetime using an arbitrary well-defined vector field ( or rather flow)?

In his book, A relativist's toolkit, in the section where Poisson is talking about the expansion and sheer vectors, he uses deviation vectors to decompose the metric into two parts. But to me, it ...
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Deriving the transformation law for the Christoffel symbols

I am a first year undergraduate teaching myself General Relativity from the book by Bernard Schutz. In one of the problems he asks to derive the transformation law for the Christoffel symbols from the ...
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Landau Classical Fields theory argument for invariance of $ds^2$

In Landau's classical field theory, chapter 1, I'm getting mad with an argument used to derive the $ds^2$invariance from Einstein's postulate of the invariance of the speed of light. Landau says that ...
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Mass change of a black hole during the Penrose process

I have been reading about Kerr black holes, specifically via Hobson et. al, General Relativity: An Introduction for Physicists. When discussing the Penrose process, the book considers a particle A ...
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Solid angle subtended by a tilted surface [closed]

I want to compute the solid angle subtended by the surface $$\begin{pmatrix} u\\ v\\ r \end{pmatrix},\quad -1\leq u\leq 1, \quad -1\leq v \leq 1$$ tilted by an angle $\theta$ from to the $z$ axis, ...
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A coordinate-free understanding of the space-time manifold

I study dynamics and continuum mechanics. Over the years I've gotten used to the coordinate-free, or geometric, way of thinking. A velocity vector, for example, is a tensor. It is the same object when ...
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Physical meaning of the exterior derivative of the first law of thermodynamics

We know, $$ dU = d \overline{q} - d \overline{W}.$$ suppose we took the exterior derivative on both sides, then: $$ 0= d( d \overline{q}) - d( d \overline{W})$$ This means, $$ d^2 \overline{q} = d^2 \...
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Terminology: “Sum over Geometries” vs. “Sum over Topologies” in Quantum Gravity

I am currently studying loop quantum gravity and related approaches like simplicial quantum gravity, spin foam models, tensor models and group field theories. In texts about this topics, one often ...
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Riemann tensor in Newman-Penrose formalism [closed]

In the NP (Newmann-Penrose) formalism, Riemann tensor is retarded time function $u=t-z$ $$R_{abcd}(u)$$ and Riemann tensor can take this form $$R_{abcd,p}(u)=0$$ I can't understand why this equation ...
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Why in the curved space-time, the double derivatives of the position vector is symmetric but any other vector is not symmetric?

The double derivatives of the position vector (see image eq. (1)), connecting the two points in a curved space-time defined by the Schwarzschild metric, are symmetric under no torsion condition. This ...
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How to write the wedge product in terms of Levi-Civita symbol [migrated]

Suppose we are in 3 dimensional space and that we have a one form $E_i$ and a two form $\omega_{jk}$.The wedge product between these forms is $$(E\wedge \omega ) _{ijk}=E_i \omega_{jk}-E_j \omega_{...
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Why is the curl of a vector field a pseudovector? [migrated]

There is an invariant definition of curl operation: $rotA = (*d(A^b))^{\text#}$ , where: A - vector field, * - hodge star, and $^\text#$ / $^b$ - lowering/raising the index. Acording this, $rotA$ is ...
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Rewriting the Yang-Mills action

I am reading about Yang-Mills theory in Section $10.5.4$ of Geometry, Topology and Physics by Nakahara. In Equation $10.108$ he gives two different forms for the Yang-Mills action, and I am having ...
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Can anyone help me derive and/or define the infinitesimal generators of a Lie group? [migrated]

I'm doing a project involving Lie groups in Physics, and a part of the project involves generators. I initially used Robert Gilmore's "Lie Groups, Lie Algebras, and some of their Applications&...
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Riemannian metric induced by $n$ point sets in Hilbert spaces?

Excuse my naive question: In quantum mechanics the mathematical description is through Hilbert spaces. Suppose we have $n$ points $x_1,\cdots,x_n$ in this Hilbert-Space. Then we get the Gramian matrix:...
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The wave equation for electromagnetic wave in a non-uniform region

I am looking for the wave equation for an electromagnetic wave in a non-uniform region but there is a really a few resources online, here is some: What is the difference between a uniform and an non-...
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Reidemeister Torsion

Can somebody in a layman's language explain what is a Reidemeister Torsion? This seems to play an important role in path integral of 2+1-gravity as demonstrated here in arXiv:gr-qc/9406006. This is ...
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Evaluating the $A \land A \land A$ in the Chern-Simons action

I am trying to evaluate $A \land A \land A$, but I am a bit confused on how exactly to do it and produce the usual notation used in physics. I am trying to use the definition of the wedge product of ...
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The spin-connection under general coordinate transformation

How does the spin-connection $\omega_{\mu}^{~~ab}$ transform under general coordinate transformation? Is it a tensor with respect to the $\mu$ index?
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1answer
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Has the Klein-Gordon equation in curved spacetimes the same form as in flat ones? [duplicate]

The KG equation in curved geometries has the following form: $$\frac{1}{\sqrt{-g}}\partial_\mu(\sqrt{-g}~g^{\mu \nu}\partial_\nu\phi) + m^2\phi = 0,$$ where $g$ is the determinant of the metric tensor ...
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1answer
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Proper time in General Relativity and change of coordinates

Let $M$ be the spacetime manifold and let us consider a local coordinate system \begin{align} \varphi_i:\,U_i&\subset M\to \varphi_i(U_i)\subset \mathbb R^n, \end{align} which associates $p\in ...
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2answers
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Confusion regarding the equivalence principle

In section V.2 of Prof. A. Zee's book Einstein Gravity in a Nutshell, it is given that to get the action of a point particle in a gravitational field from that of the action in SR, one just replaces $\...
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Confusion with certain aspects of the Equivalence Principle

I took a course on GR and I am revising it after a while. I am heavily confused about the equivalence principle. Consider the following two statements: On a Riemannian manifold, we can always choose ...
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Interpretation for other characteristic classes [migrated]

The Euler characteristics ($\chi$) is perhaps the most well-known in physics and it counts the number of holes in a given manifold or more technically its genus. But there are other characteristic ...
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Geodesic deviation equation in flat spacetime $\sim$ divergence of geodesics

Consider the above two neighbouring geodesics $\mathcal{Y}$ given by $x^{\alpha}(\sigma)$ and $\mathcal{\tilde{Y}}$ by $\tilde{x}^{\alpha}(\sigma)$ for top and bottom curves respectively. Vector ...
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Transformation behaviour of partial derivatives of vectors in curvilinear coordinates

It seems to be rather straight-forward calculation, but it is not (neither is this a homework question). Actually I'd like to show that $A_{[i,k]}$ transforms like a tensor in curvilinear coordinates ...
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1answer
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Four vector for dual field-strength tensor

To generate the electromagnetic field strength tensor, one can use the electromagnetic four-vector using by $F_{\alpha\beta}=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}$. Is there a similar ...
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A formulation of (classical) electrodynamics using fiber bundles

I have studied the mathematical foundations of gauge theories - in particular, fiber bundles and connection 1-forms - and knowing that classical mechanics can be formulated with fiber bundles$^1$, I ...
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Lie Derivative - Obtaining Equations for Tensors and Vectors

I am writing a code to calculate the Lie Derivatives, and so far, I have defined the Covariant derivative for scalar function; $$\nabla_a\phi \equiv \partial_a\phi~~(1)$$ for vectors; $$\nabla_bV^...
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Trying to derive the Bianchi identities $R_{[abc]d}=0$ and $\nabla_{[a}R_{bc]}^{de}=0$

I am not sure if we are working under the vielbein formalism. Anyway, I am trying to show $$R_{[abc]d}=0\text{ and }\nabla_{[a}R_{bc]}^{de}=0.$$ Is it helpful to plug $$[\nabla_a,\nabla_b]=\frac{1}{2}...
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Multiple skew brackets in tensor summation, e.g. $U_{[ab]}V^{[ab]}$

I'm wondering what does $U_{[ab]}V^{[ab]}$ or $U_{[ab]}V^{(ab)}$ usually mean? I thought the double brackets meant to apply the $\text{sgn}(\pi)$ twice so $$U_{[ab]}V^{[ab]}=\frac{1}{2}\sum_{a,b}\left(...
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Vectors as functions?

In my study of general relativity, I came across tensors. First, I realized that vectors (and covectors and tensors) are objects whose components transform in a certain way (such that the underlying ...
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Poincaré bundle, a way of understanding it

So I will call the Poincaré bundle $(FM,\pi, M)$ the principal fiber bundle that has the Poincaré group as a structure group, the space of linear frames as total space and $M$ as the Riemann-Cartan ...
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How can the integrability condition $G_{i\,\,;k}^k=0$ for Einstein's Field Equations be maintained in case of torsion?

In case of coupling of the Einstein's Field Equations to fermionic matter torsion is generated. However for a consistent set of EFEs it is required that the Einstein tensor $$G_{ik} = R_{ik} -\frac{1}{...
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Tensor notation of covariant derivative

I'm trying to apply Wald's General Relativity equation $3.1.14$: $$\nabla_a{T^{b_1\dots b_k}}_{c_1\dots c_{\ell}}=\overline{\nabla}_a{T^{b_1\dots b_k}}_{c_1\dots c_{\ell}}+\sum_i{C^{b_i}}_{ad}{T^{b_1\...
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86 views

Change of coordinate vs change of reference axes

Does basis vectors change opposite to coordinate scaling ? For example, suppose I have some oblique coordinate system, and I decide to scale up both 'axes' by a factor of $a$ and $b$ respectively. The ...
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Covariant or contravariant nature of Gradient

I've been having this confusion regarding the gradient being a covariant vector. Intuitively I seem to have understood the concept. However, mathematically, I'm unable to show this, in a single ...
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1answer
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Physical meaning of the vector Laplace operator

I have seen here a question asking for the physical interpretation of the Laplace operator for a scalar field. However, there is also a vectorial version of this operator, the vector laplace operator, ...

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