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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Efficient method to evaluate the Christoffel symbols and Riemann tensor in Bondi-Sachs coordinates

In General Relativity we may employ the so-called Bondi-Sachs coordinates $(u,r,x^A)$ adapted to a null foliation. The level sets of $u$ are null hypersurfaces and $(r,x^A)$ are coordinates on the ...
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Willmore energy

Recently I saw an equation, which is called as Willmore energy, which gives an interpretation that nature tends to minimise this Willmore energy by changing its shape (elasticity). But, how does this ...
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$AdS_3$ in complex coordinates

I am looking for a 2d manifold parametrized by $z$ and $\bar z$ such that $z \bar z = 1$. Now, to see what manifold it leads to I write $$z = \frac{x+iy}{1+iz}$$ so that by imposing $z\bar z = 1$, I ...
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Problem solving Euler-Lagrange equations of a particle constrained to a spherical spiral

Problem I want to calculate the time it takes for a particle living in a spherical spiral to fall under de force of gravity down to the bottom. So far I've sketched the procedure but when I tried to ...
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What do Riemann curvature tensor, Einstein tensor, Ricci tensor, Weyl conformal tensor, Bianchi identities mean? [closed]

Gravitation page 325 section 13.5, From (1) the Riemann curvature tensor $$R^\alpha{}_{\beta\gamma\delta},$$ one could construct (2) the double dual of Riemann $$G^{\alpha\beta}{}_{\gamma\delta}\...
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Two solutions for a 4-velocity component given 3 other components?

The Setup Suppose I know, in some particular coordinate system, three components of the four-velocity vector $u^{\alpha}$ with $\alpha = \{0, 1, 2, 3\}$. For this question I'm going to assume the ...
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Doubt on proper time explicit integration

I have a doubt on explicit calculation of proper time. Considering that the metric is given by: $$ds^{2} = -Adt^{2} + B^{-1}dr^{2}+Cd\Omega^{2} -2Ddtd\phi \tag{1}$$ where $d\Omega^{2}$ is the ...
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Christoffel symbols and metric determinant

I am looking at a special class of space-time where the Christoffel symbols obey $$\Gamma^a_{bc} = g_{bc}g^{ad}\partial_d\ln \sqrt{g}$$ I wish to know if such class of space-times have been seen or ...
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What's the difference between the tetrad and vierbein fields (local inertial coordinates)?

I'm studying the formalism of gravity with torsion, the Einstein-Cartan (EC) theory, and i've encountered this book by H. Kleinert "Gauge fields in condensed matter", in which he derives the basic ...
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Can a null hypersurface be foliated by spacelike sections?

Let $(M,g)$ be a $d$-dimensional Lorentzian manifold and let $\Sigma \subset M$ be a null hypersurface, which therefore has dimension $(d-1)$. We know that its normal vector $k^\mu$ is null and since ...
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Spin connection in terms of the vielbein/tetrad and their derivatives

Mimicking the process for finding the Christoffel symbol in terms of the metric (and its derivatives), see box 17.4 on page 205 of Moore's GR workbook, we can use the torsion-free (gauge local ...
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Dimensions of EM potential/field tensor components in curvilinear coordinates

According to this resource on Maxwell theory in curved spacetime, the form of the field strength tensor is independent of the choice of coordinates: $$F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu ...
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Covariant Derivative: What does changing direction mean in curved space?

I am on my way to general relativity, but I am struggling with the covariant derivative. At this point I am trying to ignore the spacetime character of the world i.e. I am trying to understand what ...
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The proof in choosing coordinates to set the first order derivative of metric tensor zero in a spacetime point

I'm teaching myself general relativity by reading the notes of Leonard Susskind in his series of online lectures The Theoretical Minimum. He mentioned that we can choose some coordinates such that ...
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Index on a compact manifold

How can the integral of a topological term (like the Nieh-Yan term) on all of a compact manifold be nonzero whereas it's a total derivative and the manifold has no boundary? I assume the manifold can ...
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Energy and spacetime: a doubt on notation

In the reference $[1]$ I saw a very neat formula, given by: $$ \mathcal{E} =: \int_{\Sigma} d^{3}x T_{00} = \frac{1}{8\pi G}\int_{\Sigma} d^{3}x G_{00}. \tag{1}$$ The author stated that this is ...
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Definition of orbits of a Killing vector field

Given a Killing Horizon $\mathcal{N}$ of Killing vector field $\xi$, one can prove that the surface gravity $\kappa$ is constant on orbits of $\xi$, i.e $$\xi\cdot\partial\kappa^2 = 0.$$ What is the ...
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What's the significance of a Killing horizon?

A Killing horizon is defined as a null hypersurface generated by a Killing vector, which is then null at that surface. Some often cited examples come from the Kerr spacetime, where the Killing vector $...
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What's the difference between locally lorentz and locally euclidean?

What's the difference between locally Lorentz and locally euclidean? Was the former (Lorentz) the hyperbolic surface restriction of the latter (Euclidean)?
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Commutator of a Killing vector and a proportional vector field

We have been given the following problem in our GR course. I have been able to develop part of it successfully, but at some point, I'm not sure if I'm doing it correctly; Let $ξ$ be a Killing vector ...
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Why can Stokes theorem be used in Aharonov-Bohm, eventhough there is a singualrity at the solenoid?

One can show that for the phase difference $\Delta$ between the two wave functions (slit 1 and slit 2) it holds the first equality on the LHS $$\Delta=\oint_{\partial\Omega} \vec{A}\overset{!}{=}\...
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Jacobi curvature operator v.s. Riemann curvature operator

On Gravitation Page 286 Exercise 11.7, it mentioned a very interesting operator called "Jacobi curvature operator" $${\cal J}(u,v) n\equiv \frac{1}{2} [{\cal R}(n,u)v +{\cal R}(n,v)u ]$$ where it "...
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Why $\Gamma^\mu_{\beta\delta;\gamma} =\Gamma^\mu_{\beta\delta,\gamma}$?

Gravitation Page 276 Exercise 11.3 solution indicated that $$\nabla_\gamma \nabla _\delta e_\beta =e_{\mu}\Gamma^\mu_{\beta\delta,\gamma} +(e_\nu\Gamma^\nu_{\mu\gamma}) \Gamma^\mu_{\beta\delta}$$ ...
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How to compare gaussian curvature and space-time curvature?

I'm reading a book where I encountered Gaussian curvature, which felt very nature, and it was well defined. However, in 4 space, people use Riemann curvature tensor, etc. There's one big ...
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Is it possible to construct a geodesic for an anholonomic system?

For anholonomic system, i.e. Gravitation Eq. 9.22 $$[e_\mu, e_\nu] =c_{\mu\nu}^\alpha e_\alpha$$ where $$[e_\mu, e_\nu]\neq 0$$ for some $\mu,\nu$, the states of the system dependent on its path. ...
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Linking the de Rham bundle/complex over spacetime to the gauge bundle

In some textbooks, the Maxwell equations are stated in a very simple mathematical form (up to multiplicative constants coming from the system of units being used): $$ \begin{array} \mbox{d}F =0, \\ \...
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Does modifying the geodesic Lagrangian $L$ with a smooth function $f(L)$ give the same geodesic curves as solutions?

Mathematical side of the problem Given the metric $$ds^2 = dr^2+r^2d\theta^2+r^2\sin^2\theta d\varphi^2$$ we can easily construct the action of a free particle $$S=\alpha \int d\tau \underbrace{\...
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Null-geodesics vs null-killing vectors

Consider a null-killing vector $\xi^{\mu}$. Now due to the killing equation we have $$\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu} = 0$$. Now I constract one of the index with $\xi^{\mu}$ to obtain $$\...
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A manifold that is not embedded? Manifold definition in general relativity by Robert Wald

I am starting to read "General Relativity" by Robert Wald. A little bit of my physics background: I am pursuing a mathematics major and I have not taken any physics courses, just Analysis and ...
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When is the scalar product of two normalizable 4-vectors, normalizable?

The specific dilemma that I'm encountering is as follows: Let $n_\alpha$ denote the normal to a hypersurface $\Sigma$. Let $u_\mu$ denote the velocity of a particle randomly crossing $\Sigma$. They ...
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Null Killing vectors constrain the space-time? [closed]

I have heard that spacetimes which admit null Killing vectors are sort of constrained. I wish to know how and why? What makes null Killing vectors so special?
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A doubt on Christoffel symbols: to be a tensor, or not to be, what's the answer to that question? [duplicate]

Following $[1]$ we realize that, in order to construct a covariant derivative, we must to compare two possible covariant derivatives such as: $(\bar{\nabla}_{a} - \nabla_{a})\Omega_{b}$, where $\...
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How to experimentally eliminate the dynamical phase from the evolved quantum state to measure the geometric phase?

If the quantum mechanical state f a given system fulfills the requirements to have a nonzero geometric phase, then how can one separate this geometric phase from the total phase measured in an ...
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On TQFT and theories without propagating degrees of freedom

Maybe not a very sensible question, but I would like to know, whether there exist topological field theories (TQFT) with propagating degrees of freedom, or, conversely, theories without propagating ...
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Coordinate vector fields associated to normal coordinates

Let $(M,g)$ be a pseudo-Riemann manifold, $p\in M$ and $\mathcal{B}_p=\{X_i^p \,:\,i=1,\ldots,n\}\in T_pM$ an orthonormal basis of the tangent space of the point $p \in M$. Attached to this basis $\...
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How can even there be a non-zero BMS vector field with zero asymptotic data?

Following the BMS approach, one spacetime $(M,g)$ is asymptotically flat when: We can find a Bondi gauge set of coordinates $(u,r,x^A)$ characterized by $$g_{rr}=g_{rA}=0,\quad \partial_r\det\left(\...
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Determining global hyperbolicity

Given a spacetime metric, how can one show that the spacetime is globally hyperbolic? I know that a globally hyperbolic metric has a Cauchy surface, but how can we determine the existence of a Cauchy ...
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Writing down a metric tensor given parametric equation of the surface

Let me begin by saying this question isn't related to GR. I'm reading a paper (see https://arxiv.org/abs/0903.0798v1) that talks about deriving a Schrodinger equation for an electron confined on a ...
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Does knowing the Yang-Mills field give us the principal bundle structure

Although the principal bundle view of Yang-Mills theory is very satisfactory geometrically, I find it hard to relate with the experience of actually computing potentials as we do in physics. Say ...
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Is Ricci's theorem can be simply deduced using covariant derivatives of fundamental tensors? [duplicate]

Well Ricci's theorem is given by: $$\mathrm{D}g_{ij}=\mathrm{D}g^{ij}=0$$ I was wondering that if the theorem can be proved using covariant derivatives of $\delta_i^k$, $g_{ij}$ and $g^{ik}$. I ...
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How do units 'change' when we move to the language of differential forms?

Consider a 2D Euclidean vector as we're taught in first year: $p = x \>\hat{i} + y \> \hat{j} $ where x and y are in meters. If one goes looking for the units of a unit vector they will be told ...
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Eddington-Finkelstein coordinates not well defined?

Consider the Schwarschild solution $$d s^{2}=-\left(1-\frac{2 m}{r}\right) d t^{2}+\frac{d r^{2}}{1-\frac{2 m}{r}}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \varphi^{2}\right) $$ and the radial null ...
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Physical interpretation of the symplectic property for particle systems

This this Wikipedia page states that symplecticness (being symplectic) is a property of particle systems governed by Hamilton's equation. I am thinking of classical-mechanics point particles described ...
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Are there two elements in each fiber in the fiber bundle of the tangent vectors of a circle, or infinitely many?

I'm watching this video (Frederic Schuller) and, at timestamp 9:50 have become confused about fiber bundles: https://youtu.be/UbQS40KHkH0?t=587 He says you can imagine 'turning the tangent vectors ...
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How to Calculate the Apparent Altitude from Earth for a Specific Zoom Level on Map

I am developing a mapping application in Java, using NASA WorldWind, and also OpenStreetMap, and am attempting to develop a method to show how far away I am at a particular zoom level. WorldWind does ...
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Doubt on a (“straightfoward”) derivation of Weyl tensor

I) My doubt: After Kruskal coordinates, we can introduce penrose diagrams after a quick talk about conformal metric tensors. Then, after the study of penrose diagrams the student should know that a ...
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Lie derivative of the non-coordinate metric being 0

I'm trying to answer a question about a the Lie derivative of a metric in a non-coordinate basis. Here, the $C^{a}_{bc}$ are from the Lie derivative of one basis vector with respect to another, or ...
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The physical meaning of maximal non-integrability of the contact structure

So, basically integrability is equivalent to the existence of an integral manifold of the distribution and I guess, the integral manifold is like a plane of motion where state moves in physical sense. ...
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Intuition behind bundle constructions in curved space-time and gauge theories

Let us assume that we have constructed a $G$-principal bundle $P$ over the manifold $M$ (for a curved space-time this is a $GL$-bundle, for a gauge theory I take $U(1)$ = electrodynamics) and the ...
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Replacing infinitesimals with full vectors in a differential relation. Is it legit?

I'm reading Leonard Susskind's "Special Relativity and Classical Field Theory". On pg. 138 he generalizes a differential relation by replacing infinitesimals with full vectors like so: Is this ...

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