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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Solutions to the Geodesic deviation equation
It’s my first post so sorry if it’s in the wrong place.
I’m currently doing a project looking at using geodesic deviation to classify black holes. Obviously solving the deviation equation in the ...
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Do we need the Second Law of Thermodynamics to give physical meaning to the quantities appearing in the First Law?
My question is not about the mathematical formalism itself, but rather about its physical interpretation in a simple case.
In the contact-geometric formulation of thermodynamics, the phase space $M$ ...
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Christoffel symbols for Schwarzschild metric in cartesian coordinates
Some time ago I found the Christoffel symbols for Schwarzschild metric not only in the usual spherical coordinates but also in cylindrical coordinates and in cartesian ones. But I did not download ...
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Perturbative Expansion in the Vielbein Formalism
Most of perturbation theory in GR is done by perturbing the metric, i.e. writing $g = g_B + \alpha h$, where $\alpha$ is our expansion parameter and $h$ is our perturbation, and then developing either ...
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Level Manifolds and Conserved Quantities
Define a dynamical system D, upon its manifold M. Say we have three quantities for D: $F_{1,2,3}$. Why is it that when taking a subsection of M as a level manifold $\text{M}_{f}$ defined by $F_{1} = ...
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$3 + 1$ formalism using differential forms
As the title says, I'm interested in knowing how the $3 + 1$ formalism works with differential forms.
In standard metric formalism with signature $\eta_{\mu \nu} \rightarrow (- \, + \, + \, +)$, the ...
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Has anyone seen a 1D “ridge” (near-null Jacobian direction) in inverse problems like the seesaw?
I’m working on inverse problems in neutrino phenomenology (minimal Type-I seesaw), and I keep running into something that looks very robust, but I don’t know if it’s well known or if I’ve just ...
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Shear propagation equation
I'm working through Wald's General Relativity and have encountered the propagation equation for the shear (9.2.12). I understand that this expression is simply the trace-free symmetric part of ...
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System of three-spaces and rotation of matter relative to the compass of inertia
In the very first paragraph of Godel's 1949 paper (PDF), it is stated that
It is easily seen that the non-existence of such a system of three-spaces is equivalent with a rotation of matter relative ...
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Varying Gibbons-Hawking-York Boundary Term with respect to the boundary metric
For a theory of AdS gravity, because this theory contains boundary, Gibbons-Hawking-York Boundary term is needed to make the variation principle well-posed. Typically, we end up with a theory with ...
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Kaluza-Klein and Symplectic forms: physical role of extra dimensions
The Kaluza-Klein theory will be viewed below as the enrichment of the spacetime manifold with an additional dimension, where $(M,g)$ is defined over some Riemannian space $V:\dim V \geq 4$ with ...
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Meaning of coordinate system 'varies smoothly' between neighbouring hypersurfaces
I am reading Gourgoulhon's 3+1 Formalism in General Relativity. In section 5.2, titled 'Coordinates Adapted to the Foliation', Gourgoulhon introduces coordinates on the spacetime manifold $\mathcal{M}...
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What is the flat holomorphic connection for defining the projective line bundle in Friedan-Shenker formulation of CFT?
In Friedan-Shenker formulation of CFT a projective line bundle $E_c$ on the Deligne Mumford compactification of the moduli space $\overline{\mathcal{M}}_g$ is constructed out of $c$-connections which ...
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In differential geometry, is it possible to differentiate a basis without assuming a connection?
In classical mechanics and differential geometry, the notion of differentiation is often used in contexts where its underlying assumptions remain implicit. In particular, one frequently encounters the ...
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Why are vectors written out in a basis of covectors in general relativity? [duplicate]
I am following a course on general relativity and we are now talking about vectors and covectors. Covectors have lower indices and vectors have upper indices. In my book, they say that any vector $V$ ...
