# 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 ...
1 vote
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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, $$, 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 $$,$$ 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 $$ and I ...
1 vote
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555 views

### 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_{\...
68 views

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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 ...
120 views

### 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 ...
1 vote
104 views

### 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 ...
1 vote
56 views