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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How are sources described in gauge theory?

Let's only assume the case of electromagnetism. If one varies the Yang-Mills-Functional one gets the Yang-Mills-equation $*d*F=0$. The whole theory can be geometrically described on principal bundles. ...
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Why isn't spacetime flat in Ehrenfest's Paradox?

In a previous question, the metric in the frame of reference of a rotating disc takes the following form: \begin{align} ds^2 &= - dt^2 + dx^2 + dy^2 + dz^2\\ &= -\left(1 - \omega^2 ({x'}^2 + {...
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Derivation of residual reparameterization symmetry equation

Here is my attempt to derive the residual reparameterization equation $$\partial_{\alpha}\xi_{\beta}+\partial_{\beta}\xi_{\alpha}=\Lambda \eta_{\alpha\beta}$$ For a reparameterization $x^{\alpha}\...
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Parallel transport and Geodesic deviation

We know that when we derive the Geodesic equation, we want to actually understand the geometrical meaning of the Riemann tensor. We see from the geodesic equation that the second derivative of the ...
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Extending $\mathbb R^3$ coordinate systems concepts

I was thinking about how to use different coordinate systems in 3D space and how to describe curved surfaces embedded in 3D space when I realized that all the notations I know make sense only if ...
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Covariant derivative of the spin connection

I wish to compute $$[\nabla_{\mu}, \nabla_{\nu}]e^{\lambda}_{~~a}. $$ To do so, I make use of $\nabla_{\nu}e^{\lambda}_{~~a} = \omega_{a~~~\nu}^{~~b}e^{\lambda}_{~~b}$, so that I may write $$\nabla_{\...
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1answer
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Christoffel Symbols and Metric Tensor

I know that the Christoffel Symbols are made out of the first derivative of Metric Tensor. Is there any relation between the number of metric components and Christoffel symbols? Is there any general ...
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How to check if a parametrized curve is a geodesic

Given some parametrization in the plane, e.g., $x = \sin \lambda$ and $y = \cos \lambda$, how can I know if that curve is a geodesic? $x$ and $y$ constitute a circle in the plane, centered at $(0,0)$ ...
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How to interprete gauge symmetry in following way?

From Wikipedia: In here, $G$-principal bundle $P$ on spacetime $\Sigma$ is given. Gauge transformation $\phi$ can be written in three ways, and it seems to use the third way. More specifically, \...
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Second derivative of a function in a manifold

Suppose we have a function curve $\gamma(t)$ on a manifold $M$ . Define the function $$f(t)=\frac{1}{2}g_{ab}\dot\gamma(t)^a\dot\gamma(t)^b$$ Introducing coordinates $x^i$ the first derivative of the ...
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Anticommutation of variation $\delta$ and differential $d$

In Quantum Fields and Strings: A Course for Mathematicians, it is said that variation $\delta$ and differential $d$ anticommute (this is only classical mechanics), which is very strange to me. This is ...
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“The moment map gives more than Noether's theorem”

I recently ran across this Not Even Wrong blog post which has the following passage The moment map however gives you much more [than Noether's theorem], with phase space providing structure that is ...
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Understanding the Israel Junction Conditions

The well known and frequently used Israel Junction conditions are the equivalent of Einstein's field equation on a membrane in the brane-world picture. All the sources have this notation: $K_{ab}^{(i)}...
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How do 'locally Euclidean' and 'Lorentzian' requirements in manifolds reconcile?

In GR, we define our manifolds to be locally Euclidean. However, we also demand that our metric tensor have a Lorentzian signature. Since the metric tensor is a measure of curvature, doesn't the first ...
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Covariant derivative and vector fields along a curve

I have to prove a relation that involves the covariant derivative induced by the Levi-Civita connection. With the reference on N. Straumann General Relativity Given $T=\frac{\partial x^\alpha}{\...
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1answer
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Interpretation of black hole metric with fractional $\kappa$ instead of the usual $\kappa\in\{-1,0,1\}$

The metric for a black hole can be written: $$d s^{2}=-\left(\kappa-\frac{2 M}{r}\right) d t^{2}+\left(\kappa-\frac{2 M}{r}\right)^{-1} d r^{2}+r^{2} d \Sigma_{2, k}^{2}$$ where $\kappa=-1,0,1$ ...
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Gravity action in terms of Null-tetrads

I have seen gravity action being written in terms of spin-connection and the vierbeins. Has anyone attempted to write the gravity action in terms of Null-tetrads?
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Cosmology and Spherical Coordinates

My question refers to page 10 of this document. Specifically, when using spherical polar coordinates in cosmology, why does the author of this work choose the origin of the coordinate system to be at ...
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Is this case a failure of Stokes' theorem?

In the presence of a hypothetical magnetic point charge at the origin of coordinates, it turns out that an irremovable physical singularity of the vector potential ${\bf A}({\bf r})$ exists for any ...
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Why is covariant derivative of Null-tetrads not zero?

I think the null tetrads are just re-writing of the vierbiens. However, the vierbiens satisfy $\nabla_\mu e^{a}_{~~\nu} = 0$, but the same is not true for the null tetrads because the spin ...
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Is $g(\nabla_{e_0}X,)$ is equal to $\nabla_{e_0}g(X,)$?

Let $g$ be a Riemannian metric , $e_0$ and $X$ vector field with $g(e_0,e_0)=-1$ .From the compatibility condition of the metric we have for another field $Y$ we have $$\nabla_{e_0}g(X,Y)= g(\nabla_{...
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Building lightweight 1 km high fence [closed]

I would like to build a fence that is 5 km2 and height of 1km. This wouldn't be feasible as the material needed and cost implications plus environmental factors. Therefore I thought of finding the ...
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Is there a smooth, spherically symmetric, static, asymptotically de Sitter spacetime of constant scalar curvature?

...other than de Sitter space itself? Two clarifications: I am not talking about solutions of Einstein's equations, just spacetimes. So, general Lorentzian manifolds. I am not counting, for example,...
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Fiber bundles and relativity

In the book Geometrical Methods of mathematical physics, in the section 2.10, talking about fiber bundles, gives the example of the Newtonian physics fiber-bundle structure: The view of spacetime ...
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As a physicist, why are associated bundles important?

I have a good grasp on principal bundles as providing a lie group on some fibers of our field. So for example, the wavefunction tells us the phase of a particle in space and time, and this can be a ...
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Why and where Poincaré gauge theory fails?

I would like to post this question because I have seen no one post it in this explicit way. From what I have seen, Poincaré gauge theory uses connections (gauge fields) like the spin connection and ...
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1answer
37 views

Dimensions of velocity vector in differential geometry

If we have a velocity vector written in, say, cartesian coordinates: $\mathbf V$ = $\dot{x}$ i $+$ $\dot{y}$ j Note that Dim($\dot{x}$) = Dim($\dot{y}$) = $LT^{-1}$ which are the dimensions of speed ...
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Killing equation and normal coordinates

I have already seen a similar question but I have not sure to have understand completely so I hope you can help me. If I write the killing equation $\cal{L}_X g=0$ as $X_{\alpha;\beta}+X_{\beta;\alpha}...
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Can flat spacetime have curvature?

Hi I am new and I hope this can make sense. Is it true that flat spacetime can curve? For example, you have a flat piece of spacetime that curves at a constant rate of pi/2. the flat spacetime would ...
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When and why can the spin connection term of the Dirac Operator be omitted?

The Dirac Operator $D$ is defined by \begin{equation}\tag{1} D=i\gamma^a\nabla_a=i\gamma^a\nabla_{e_a}=i\underbrace{\gamma^a{e_a}^\mu}_{=\gamma^\mu}\nabla_{\partial_\mu}=i\gamma^\mu\nabla_\mu=i\gamma^\...
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Hyperbolic isometries in the context of General Relativity

In the context of hyperbolic geometry, it is possible to create a classification for isometries. I would like to know if these isometries have any particular meaning in the context of general ...
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The contravariant derivative of a substitution for the de Sitter metric

Consider the de Sitter metric: $$ds^2 = (1-\frac{r^2}{a^2})dt^2 -(1-\frac{r^2}{a^2})^{-1}dr^2-r^2d\Omega^2$$ I know that we rewrite the metric as $(u,r,\theta \phi)$ using the substitution $$u = t-...
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Property of Dirac operator in spinor frame bundle

The Dirac operator is defined by $$d=g(e_\alpha,e_\beta) \gamma(e_\beta)\nabla_{e_\alpha} \tag1$$ Here $\nabla$ is the spin covariante derivative $e_\alpha$ basis of the tangent space and $\gamma$ the ...
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Surface swept out by tangent vector [migrated]

This question is from Novikov and Fomenko's Modern Geometry Part I. I quote the question here (problem 6 in Exercise 8.4) in full: Let $S$ denote the surface swept out (i.e. "generated") by ...
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Gauge transformation in a frame bundle

Let $\pi:P\longrightarrow M$ be the orthogonanal frame bundel. A gauge transformation is an automorphism $f:P\longrightarrow P$ such that $\pi(p)=\pi(f(p))$ that is fiber preserving automorphism. Now ...
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Clarification about local Lorentz transformation

Note there are others questions about local Lorentz transformations and global Lorentz transformations but they all concerned about mathematics. Here what I am trying to understand is the link ...
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What are physical evidence that we are living in affine space?

An affine space of dimension n on $\mathbb R$ is defined to be a non-empty set $E$ such that there exists a vector space $V$ of dimension n on $\mathbb R$ and a mapping $\phi:E \times E \rightarrow V,\...
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Equation of a conformal Killing vector

Question 10(a) on pages 469-470 in the book "Spacetime and Geometry" by Sean Carroll asks: Suppose that two metrics are related by an overall conformal transformation of the form: $$ \tilde{...
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Diffeomorphisms and coordinate changes

In this question: Simple conceptual question conformal field theory, an answer states that the invariance of the theory under the change of coordinates is a particular case of invariance under ...
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Calculating proper time from different perspective and obtaining different results

A $\textbf{clock}$ carried by a specific observer $(\gamma, e)$ will measure a $\textbf{time}$ $$ time := \frac{1}{c} \int_{\tau_0}^{\tau_1} d\tau \sqrt{ \eta(v_{\lambda}, v_{\lambda}) } \tag1$$ ...
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Is the set of spacetime manifold a set of event?

In general relativity spacetime is defined as manifold. Is this manifold the set of events? For example if our universe is constituted of only one non-interacting particle, should our manifold be a ...
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Identity regarding problem 2.4 of Walds General Relativity

I was going through some of the problems in Wald's General Relativity and in problem 4 chapter 2 I found something that confuses me. So, basically we are asked to show that in any coordinate chart $(...
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Tangent spaces at different points in special relativity

In the book Theoretical Physics by J. C. Dutailly the author says However because the manifolds are actually affine spaces, in SR and Galilean Geometry the tangent spaces at different points share ...
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Infinite differentiablity in Nature? [duplicate]

I'm working through understanding Clay Institute's paper addressing their Navier-Stokes millennium prize. I can't find any resources online stating if there are any infinitely differentiable examples ...
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Questions on Affine Parameter

According to p.75 of Hobson's General Relativity book, the author defines an affine parameter along a curve by a parameter that makes the length of the tangent vectors constant along the curve. By ...
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Transformation of Christoffel Symbol

I'm studying General Relativity by Hobson and got stuck in understanding the derivation of the transformation law of Christoffel symbols (p. 64). Let $x^a$ and $x'^a$ be two coordinate systems on a ...
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Lie brackets of vector fields along a geodesic to obtain the Jacobi equation

I have done this question in mathstack but someone has suggested me it is more appropriate to ask this here. With reference in https://archive.org/details/GeneralRelativity/page/n82/mode/1up, where it ...
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Transformation law of vector fields on $\mathbb{R}^n$

So suppose we have a function $F$ from $\mathbb R^2$ to $\mathbb R^2$ defined by $F(x,y) = (g(x,y),h(x,y))$ where $g$ and $h$ represent temperate and pressure respectively (the point is, they are both ...
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Transformations in General Theory of Relativity

In our lecture we dicussed that certain elements in GTR transform as certain other things, i.e. Christoffel symbol transforms as an affine connection covariant derivative of a contravariant vector ...
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Covariant gradient - What am I missing?

I know that the components of the gradient should transform covariantly, and have used this many times in special relativity, etc. However, I also know that covariant components transform like the ...

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