Questions tagged [differential-geometry]
Mathematical discipline which studies some properties of smooth manifolds, which allow to generalize calculus to beyond $\mathbb{R}^n$. General relativity is written in this language.
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Notation for vector density in Lagrangian density
Consider a manifold $M$ and a Lagrangian density $\mathcal{L} \equiv \mathcal{L}(\phi, \nabla \phi)$. By varying the action, one obtains the equation
$$\int_M \, dV \; \Big( \frac{\partial \mathcal{L}}...
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Index theorem of Callias operator in physics
In the article "On the index type of Callias-type operator" (https://doi.org/10.1007/BF01896237) Anghel study the index of a Callias type operator over an odd dimensional complete ...
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How do I know if a motion is 1 dimensional or 2 dimensional?
If an object is moving in a straight line with an angle with x axis (it may be vertical or horizontal) , is it 1 dimensional or 2 dimensional?
The question was asked by my teacher and he himself gave ...
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Are de Sitter, Anti-de Sitter and Minkowski spaces spatially infinite?
I am not someone who has studied general relativity, however have recently developed an interest in it. From what I have seen online, de Sitter, Minkowski and Anti-de Sitter spaces are often compared ...
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Looking for video courses on general relativity, aimed at a mathematician crowd [closed]
I am a mathematician, working in symplectic geometry.
I am looking for online avalible video recordings of courses in general relativity, which are geared towards an audience of mathematicians.
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Boundary conditions on transition maps on general relativity
On the initial courses of topology and differential geometry, we learn again and again about charts, and atlas, and transition maps. I feel that transition maps are a very powerful idea, because they ...
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Are Christoffel symbols in Schwarzschild metric symmetric?
Schwarzschild metric is following
$$ ds^2 = (1-\frac{r_s}{r})c^2dt^2 -\frac{dr^2}{1-\frac{r_s}{r}} - r^2d\theta^2 - r^2\sin^2(\theta)d\phi^2
$$
Metric tensor has only diagonal terms non zero from this ...
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What kind of object is a function in the context of gauge theory?
In the context of differential geometry, we have the Levi-Civita connection that tells us how to take derivatives of tensors. Two examples of the covariant derivative are
$$\nabla_\mu \phi = \partial_\...
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Are Landau-Lifshitz equations equivalent to Hamilton's equations for classical spins?
Let $\boldsymbol{s}_1$ describe a "classical spin", i.e. a point on the surface of a unit sphere embedded in $\mathbb{R}^3$. It can be parametrized, for example, as
$$ \boldsymbol{s}_1 = \...
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GR and Riemann Surfaces -- does the complex plane have anything to do with it?
I have only the vaguest understanding of Riemann Surfaces -- my sense is that Einstein used them in General Relativity because of their shape.
But Riemann Surfaces I think are not just deformations of ...
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Under what circumstances can a 4D singularity occur in General Relativity?
I've tried to find on the literature about 4D (single point) singularities, but most of the theorems about singularities pertain to either space-like or time-like singularities, which always have some ...
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Cone vs. small circle parallel transport
I'm having trouble reconcile the following two seemingly contradicting conclusions (in 2d space):
A cone is flat, because you can unfold it and it's a flat 2d surface.
A cone as shown in the picture ...
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Orthogonal complement of null vector [closed]
I am trying to solve excercise 8.5 d) from Straumanns book on general relativity (here V is an n-dimensional Minkowski vector space):
Prove that the orthogonal complement of a null vector is an $(n-1)$...
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Is it possible to understand in simple terms what a Symplectic Structure is?
I would like to understand what a Symplectic Structure is, and its implications in Classical Mechanics (Phase Space), but in pre-grade terms (If that could be possible). I have not taken any ...
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Extrinsic Curvature Calculation on the Sphere
Given the following 2+1 dimensional metric:
$$ds^{2}=2k\left(dr^{2}+\left(1-\frac{2\sin\left(\chi\right)\sin\left(\chi-\psi\right)}{\Delta}\right)d\theta^{2}\right)-\frac{2\cos\left(\chi\right)\cos\...
