# 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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### Topological/Geometrical justification for $\text{CFT}_2$ being special

It is known as a fact that conformal maps on $\mathbb{R}^n \rightarrow \mathbb{R}^n$ for $n>2$ are rotations, dilations, translations, and special transformations while conformal maps for $n=2$ are ...
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### Coordinate bases: in what sense is $\partial_i=\frac{\partial}{\partial q^i}$ a basis vector? [migrated]

I have gotten comfortable working with coordinate bases but I still do not fully understand them. Let $\mathbb{Q}$ be a smooth $n$-manifold and $q=(q^1,\dots,q^n):Q\to \mathbb{R}^n$ some local ...
1 vote
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### Levi-Civita symbol in 2-spinor notation

I'm reading An Introduction to Twistor Theory, by Huggett and Tod, and I don’t get the result we're being given page 17: the 2-spinor form of the 4 dimensional Levi-Civita symbol. \begin{equation} \...
1 vote
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### Finding the correct Christoffel-symbols in a 2+1D space-time

I'm trying to calculate the Christoffel Symbols in a 2+1D space-time with the following metric: $$ds^2 = N^2(\vec r)c^2dt^2-\phi(\vec r)(dx^1)^2-\phi(\vec r)(dx^2)^2$$ To find the Christoffel ymbols ...
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### Diffeomorphism & Weyl transformations in the 2D worldsheet of string theory and the existence of conformal gauge

D. Tong's notes on string theory (PDF), subsection 5.1.1, feature the following in introducing the symmetries used in the Faddeev-Popov method: We have two gauge symmetries: diffeomorphisms and Weyl ...
480 views

### Proof for covariant vector transformation law

(I'm asking this on the physics exchange not on the math one because i don't need an extremely rigorous explanation) I understand the derivation for the contravariant vector transformation law is ...
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### Friedmann equation and the Ricci Flow

The Friedmamn equation I'll concern myself with is, $\frac{\ddot a}{a}=-\frac{4\pi G}{3}\left(\rho+\frac{3p}{c^2}\right)$ Instead of setting $c=1$ a nice mathematical/dimensional trick is to just ...
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### Commuting material time derivative and material space derivative

Let's note $x$ the coordinates in the current configuration and $\nabla$ the associated gradient; similarly, let's note $X$ and $\nabla_0$ for the reference configuration. I will also note the ...
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### Is the diameter an invariant in General Relativity?

How can you define diameter in General Relativity? How would you calculate it for a given fixed $r$ in spherical coordinates in Minkowski and Schwarzschild metrics? I've seen somewhere that "In ...
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### How does the wavefunction transform under an arbitrary change of variables?

Suppose we have a variable $x$ and a probability density $\rho(x)$. The pushforward of this density under a bijective function $y = f(x)$ is given by \begin{equation*} \rho'(y) = \frac{\rho(f^{-1}(y))}...
1 vote
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### Relation between "method of moving frames", spin connection, Cartan forms, and classic rotational kinematics in $\mathbb{E}^n$

I want to know how the "method of moving frames" involving things like connection 1-forms, torsion 2-forms, spin connections, etc. are applied to basic rotational kinematics in flat 3-space (...
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### Mathematical description of a conical spacetime

In multiple explanations of general relativity, the curvature of a cone has been used to explain why objects fall in a gravitational field, like so: It also appears in videos like this and this one ...
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### How the Poisson bracket transform when we change coordinates?

I'm studying the book Geometric Mechanics by Darryl D. Holm and there's one exercise in the book I'm not quite getting what has to be done. The same discussion the author makes in the book is made on ...
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I think the answer is no. My reasoning is the following: We have the Schwartzchild metric with signature $(+,-,-,-)$. Then we can use the Lagrange-Euler equations $$\frac{d}{dt}\frac{\partial L}{\... 0 votes 1 answer 380 views ### Derivative of the Chern-Simons form I want to verify the relation of the Chern-Simons form$$ d \, \text{tr} (AdA+ \frac{2}{3} AAA) = \text{tr} FF$$where \omega \mu\equiv \omega \wedge \mu and F=dA+AA. Using the property d\, \... 2 votes 4 answers 11k views ### Non-zero components of the Riemann tensor for the Schwarzschild metric Can anyone tell me which are the non-zero components of the Riemann tensor for the Schwarzschild metric? I've been searching for these components for about 2 weeks, and I've found a few sites, but ... 36 votes 8 answers 5k views ### Interval preserving transformations are linear in special relativity In almost all proofs I've seen of the Lorentz transformations one starts on the assumption that the required transformations are linear. I'm wondering if there is a way to prove the linearity: Prove ... 3 votes 0 answers 78 views ### Lie derivative of surface gravity on killing horizon Carroll's 'spacetime and geometry' says surface gravity \kappa is defined by K^\mu \nabla_\mu K=-\kappa K where \nabla and K each denote covariant derivative and Killing vector field. This is ... 1 vote 0 answers 53 views ### Curvature and stability In Topological methods in hydrodynamics 1 mentioned that "The Riemannian curvature of a manifold has a profound impact on the behavior of geodesics on it. If the Riemannian curvature of a ... 5 votes 2 answers 306 views ### Why can infinite planes be approximated as Gaussian surfaces? A little background: I'm an undergraduate studying Electrodynamics, currently in Chapter 8 of Griffiths. A question I came across (8.4 part a for those curious) asks for a calculation of the force ... 0 votes 0 answers 42 views ### Vanishing metric component? Consider a metric g_{\mu \nu} = \text{diag}(g_{xx},g_{yy},g_{zz}). Now consider another metric h_{\mu \nu}=\partial_x g_{\mu \nu} = \text{diag}(\partial_x g_{xx} ,\partial_xg_{yy},\partial_x g_{zz})... 0 votes 1 answer 88 views ### What does it mean for a quantum field theory to "live" on a manifold? I was attending lectures om holography where the lecturer kept on mentioning that a QFT lives on a Cauchy slice. What does that mean? Is it such that each point of the slice is associated to a unique ... 1 vote 1 answer 129 views ### Precise definition of a string worldsheet as a manifold in string theory I've spent some time studying some definition in smooth manifolds theory in order to give a proper definition of a worldsheet in classical string theory at least. My attempt is the following: ... 1 vote 1 answer 71 views ### Simple distance calculation in General Relativity So imagine a spacetime with the Schwarzschild metric:$$ds^2=-c^2\left(1-\dfrac{2GM}{c^2r}\right)dt\otimes dt+\dfrac{1}{\left(1-\dfrac{2GM}{c^2r}\right)}dr\otimes dr+r^2\left(d\theta\otimes d\theta+\...
In eq. 4 of https://arxiv.org/abs/hep-th/9508128 it is stated that the Gauss-Bonnet term for gravity in 4d can be written as  S=\frac{1}{4}\int d^{4}x\,\epsilon^{\mu\nu\alpha\beta}\epsilon_{abcd}R_{\...