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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Vorticity: what is the difference between $\nabla \times f$ and $\nabla \wedge f$? [duplicate]
I have always thought that we may obtain information about the vorticity of a vector field $f$ by considering the cross product $\nabla \times f$. In higher level texts on fluid mechanics I see that ...
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Independence and ambiguity of holonomic constraints
I've got a couple of questions concerning constraint equations:
Suppose I've got $n$ holonomic constraint equations for a particle, how can I be sure those are all the ones there are and I didn't ...
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Doubt on the continuity factor of Dyson mega-spheres
I) Dyson Mega-Spheres
In a nice and cool recent paper, $[1]$, the authors constructed another interresting solution of general relativity; they constructed a thin-shell around a star: a dyson sphere ...
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Doubt on: $G = SU(2)_{L} \times U(1)_{Y}$ representations, the Chiral Spinor bundle and the "split" of covariant derivative for $G$
Firstly,
I've made two other questions $[1]$,$[2]$ concerning the same situation, but I think that this one will clarify better what I'm trying to understand.
I'm following the text book $[3]$ and I ...
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Time-dependent Christoffel symbols
I was studying the Schwarzschild metric, and I found out that all of the Christoffel symbols aren't time-dependent. This is because the nonzero Christoffel symbols of the Schwarzschild metric are:
$$\...
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Question on reciprocal metric tensor
First of all, to simplify we will assume 2 dimensional space with a symmetrical metric tensor $g_{\mu\nu}$
It's known that d'Alembert operator (we will use it for example) is defined as $\partial_\nu \...
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Chern-Simons forms: interpretation and generalizations
Studying again differential geometry, anomalies and topology, I wondered if there is ANY physical interpretations (in terms of QFT or even classical field theory) of the Chern-Simons forms, via vacuum,...
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Lie derivative acting on a function
I'm a little confused about Lie derivatives. In fact all that definitions of pull-back and push-forward and one-parameter family of diffeomorphisms and integral curves and so on seems strongly ...
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Local conformal group of the Euclidean plane and the (local) diffeomorphism group of locally Lorentzian 4-manifolds
Maybe this question is rather too general to mention, but I hope the community can help to make it more precise in case it's needed.
The question is: if there's any link between the local conformal ...
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Tensor that depends on first-order derivatives of the metric
The question is pretty straightforward. Just as all tensor fields depending on the metric's second derivative can be derived from the Riemann curvature tensor, is there any well-known tensor, such ...
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When connections are fundamental objects, is LQG the only possible theory?
In an interview Abhay Ashtekar claims:
if you have a background independence and if you want to have Einstein connections this (Loop Quantum Gravity) is the only thing you can write down.
Where the ...
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Cross product of basis vectors in general relativity
Say I wanted to find an expression for the differential surface trapped between two basis vectors (in 4-dimensional curved coordinates), one approach would be to take their cross product, and take the ...
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Variation of normal, coordinate and mean curvature with respect to metric
Using the divergence theorem, we can compute volume by integrating along the surface:
$$\mathrm{vol}(M)=\int_M\mathrm{d}V=\oint_{\partial M}\mathrm{d}S \,\vec{n}\cdot\vec{v},$$
where $\vec{n}$ is ...
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Riemann curvature tensor
I am little bit confused on Riemann curvature tensor,
Riemann curvature tensor written in component form as;
$$R^d_{cab}=\partial_a\Gamma^d_{bc}-\partial_b\Gamma^d_{ac}+\Gamma^i_{bc}\Gamma^d_{ai}-\...
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Can the metric tensor always be diagonalized?
I know a few examples of metric tensors (flat, spherical, hyperbolic, Schwartzchild, Kerr, Kerr-Newman) and there's a common property between them. All of them can be expressed as
\begin{equation}
g_{\...
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Variation of functional with area and volume term
Let $M$ be a closed manifold in $\mathbb{R}^3$ and $\partial M$ its surface. I want to find (in general terms) the manifold that minimizes a functional of the form
$$I[M]=\int_{\partial M}f\,\mathrm{d}...
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Action of Lie derivative on 1-forms
In Sean Carroll's spacetime and geometry appendix B he derives the action of the Lie derivative on 1-forms. He finds that $\mathcal{L}_X Y^\mu = [X, Y]^\mu$, which I believe is meant as $\mathcal{L}_X(...
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The Newtonian limit of torsion
Torsion is a common modification to general relativity. It is loosely described as gyroscopes twisting along geodesics.
A gyroscope is basically a mass-spring system. The masses making up the spinning ...
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Showing the equivalence principle mathematically [duplicate]
Given the geodesic equation
$$\ddot{x}^{\mu}+\Gamma^{\mu}_{\nu\lambda}\dot{x}^{\nu}\dot{x}^{\lambda} = 0$$
I wish to find a co-ordinate system around a point $x_0$ such that the geodesic equation ...
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Index-free notation indulgence!
I have been try to find as an exercise for myself the most suitable coordinate-free form of the following equation found in Misner, Thorne, Wheeler's Gravitation (p. 84)
\begin{equation}
v^\ell = (F_{...
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Metric at the junction
In general relativity, while deriving junction conditions one works with the metric $$g_{\mu\nu} = \Theta(f) g^{+}_{\mu\nu} + \Theta(-f) g^{-}_{\mu\nu} \tag{1}$$
where $f = 0$ is the junction. Now, ...
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Removing a Coordinate Singularity of a 2D metric
While trying to find the null geodesics of the metric
$$
ds^2 = (r^2 - 1)dt^2 - (drdt + dtdr)
$$
gives $$\frac{dt}{dr} = \frac{2}{r^2-1}$$ which is singular at $r=1$. However, we know that this is a ...
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Infinitesimal geodesic motion directly from the metric?
How can I see---directly from the Schwarzschild metric---that initially stationary (w.r.t. Schwarzschild coordinates) inertial test clocks will begin to fall toward e.g. the Earth (i.e. far outside ...
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Doubt on the action of covariant derivative of $SU(2)_{L} \otimes U(1)_{Y}$ on a right-fermion
Most texts books says that when you act the covariant derivative in the triplet, $D_{\mu}\Psi$, some internal calculation "breaks" the single $D_{\mu}$ into two different ones:
$$D_{\mu}\Psi ...
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Is gauge covariant derivative an ordinary covariant derivative?
The formal treatment of the gauge covariant derivative in most reference texts for students is too informal and too ad hoc, so that some general issues remain unclear. For example, the gauge covariant ...
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What is the Lie derivative of the field describing the change of mass?
I'm trying to understand Ch. 3.2 of the paper On Bubble Rings and Ink Chandeliers by Padilla et al.. I'm trying to understand the derivation of equation (15). Right now I'm stuck at the point where ...
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Divergence of tensor fields
I have found numerous definitions for the divergence of a tensor which makes me confused as to trust which one to use.
In Itskov's Tensor Algebra and Tensor Analysis for Engineers, he begins with ...
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Extrinsic Trace and Time Reversal Symmetry
For states with time-reversal symmetry, will the extrinsic trace of the time-like Killing vector be zero? The context of the question comes from the Ryu-Takayanagi formula. On a constant time slice, a ...
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Twin paradox with black hole (based on Interstellar) [closed]
I'm an Undergrad student working on a summer project. I'm learning about Differential Geometry, Schwarzschild's Solution, General and Special Relativity. I want to include the twin paradox as well. In ...
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Can the path derivative be defined with a non-constant measure?
I am trying to make sense of the following functional integral in the continuous limit:
$$
G({\bf x},{\bf y})=\lim_{N \to \infty}\int \prod_{k=1}^{N-1} d^2 {\bf z}_k \prod_{n=1}^N dp_n \exp\Bigg\{i\...
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Riemann curvature tensor in real universe
The way Riemann Curvature tensors are usually introduced is as follows:
Take a vector v at point A, parallel transport it to B then to C then again back to A, the resulting vector v' will not point in ...
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How to get the contravariant magnetic vector potential from the covariant vector potential?
The vector potential is a divergence-free field. When it is subjected to the operation $\widetilde{F}=\nabla\vec{A}-\left(\nabla\vec{A}\right)^T$, you get the covariant Faraday tensor. The ...
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Kaluza-Klein reducing the Chern-Simons term in 11D supergravity
In 11D supergravity we have the Chern-Simons term of the 3-form field $C$
\begin{equation}
\int C \land d C \land d C
\end{equation}
I want to consider this on a spacetime $\mathbb{R}^{1,6}\times S$ ...
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Can the geodesic equation be understood in plain english to articulate the radial attraction of gravity?
I'm looking to gain an intuitive understanding of the geodesic equation (which incorporates the Christoffel symbol) and how it is used to calculate the radial attraction of gravity. In its native form ...
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Tensor symmetry: A symmetric mixed $(1,1)$ tensor is necessarily a multiple of the identity tensor
I found this observation in a book. "It is not possible to have an invariant definition of symmetry in one contravariant and one covariant index". That's all right, my problem is that to ...
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Doubt on $SU(2)_{L} \times U(1)_{Y}$ covariant derivative and its action on a fermion
I) Introduction
I.1)
The mathematical structure is quite clear: given a spacetime $M$, and a Lie group $G$ (the gauge group), we can construct the Principal bundle $P^{G}_{M}$. The connection $1$-form ...
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Invariance under change of coordinates and the relation to a connection on the principle frame bundle
Any meaningful theory ought not to depend on the choice of coordinates. A seemingly simple intuitive fact takes a rather complicated form - at least when looking at quantum mechanics on arbitrary ...
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$g_{\mu\nu}$ metric and Urbantke metric
If $B^{IJ}=e^{I}\wedge e^{J}$, with $e^{I}$ the standard tetrad field, I need to prove that the expression \begin{equation}f_{ijk}\epsilon^{\alpha\beta\gamma\delta}B_{\mu\alpha}^{i}B_{\beta\gamma}^{j}...
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Coulomb gauge choice: Does $A_0=0$ imply that we also need to choose $\nabla \cdot \vec{A} =0$ from the EOM of $A_0$?
How to justify the Coulomb gauge fixing condition choice with
$$
A_0=0, \quad \nabla \cdot \vec{A} =0?
$$
Below in the text image, I find a text explaining that imposing $A_0=0$ is always possible ...
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Why is it useful to learn about Hamiltonian Mechanics in the framework of Symplectic Geometry?
...Other than providing a deeper insight into the mathematical background of dynamical systems.
Does casting certain classes of problems in terms of symplectic geometry make solving them easier/...
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Is it "mathematically wrong" to ignore dual spaces, 1-forms, and covariant/contravariant indices in classical mechanics?
If everything you are working with is in Euclidean 3-space (or $n$-space) equipped with the dot product, is there any reason to bother with distinguishing between 1-forms and vectors? or between ...
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Changing Coordinates, and the Geodesic Equation in GR
I have a question about doing Lorentz-like coordinate transformations in general relativity. I will try not to get into too much detail about what exactly I am trying to do to not muddy the waters.
...
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Why do we construct Lagrangian submanifolds after symplectic reductions
I am learning about Hamilton-Jacobi actions, symplectic reductions and Lagrangian submanifolds and I am trying to understand the relation between these concepts.
I have read that Lagrangian ...
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Can you convert the Christoffel Symbol to the form of a scalar?
Given some tensor $T_{\mu v}$, you can use the metric tensor to contract its indices, converting it into the form of a scalar:
$$g^{\mu v}T_{\mu v}=T$$
Even though the Christoffel Symbol is not a ...
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Taking the covariant derivative of the derivative of the metric tensor
Does the covariant derivative of the derivative of the metric tensor exist? if so, how do you evaluate it?
$$\nabla_a (\partial_b g^{\mu v})=?$$
It would seem natural to assume that this cannot exist, ...
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How does this scalar transformation law make sense?
A scalar field doesn't transform under a change of co-ordinates. Therefore, a scalar field $\phi(x)$ transforms to $\phi'(y)$ under the coordinate transformation $y^{\mu} = x^{\mu} + \epsilon^{\mu}(x)$...
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What are some good textbooks to learn EM with differential forms? (With focus on condensed matter)
As the title suggests, does anyone have recommendations for learning EM with differential forms? Both undergraduate and graduate level textbook suggestions are welcomed. I know there's Lindell's ...
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What is the physical interpretation and differences between static and stationary observers in Kerr geometry?
The geometry around a rotating uncharged axially-symmetric black hole is described by the Kerr metric. Being stationary and axisymmetric, the Kerr metric admits two Killing vector fields:
$$\partial_t=...
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Different interpretations of (bundle-theoretic) gauge transformations
The physical minimal coupling procedure is usually expressed mathematically in the language of fibre bundles where instead of local presentations we deal with global objects - gauge fields are ...
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Why does this falsely lead to $g_{\mu \nu}$ always being the identity map?
So I posted a question about the tetrad basis but later realised that there is a more fundamental, underlying question that is better suited here.
I’m using abstract index notation to denote all ...
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