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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72 views

Generating new solutions of the Einstein equation by active transformation, and the physical interpretation of the new ones

Given a manifold $\mathcal{M}$ with coordinates $\psi : \mathcal{M} \rightarrow \mathbb{R^4}$ , $\Psi(p)= (r,\theta ,\phi,t)$ for $ p \in \mathcal{M}$ Suppose we have the active transformation $F : \...
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Determining the manifold picture of a Lie group — and thus determining global properties

1) How do we determine a Lie group's global properties when the manifold that it represents is not immediately obvious? Allow me to give the definitions I am working with. A Lie group G is a ...
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Why covariant derivative is partial derivative in SR?

I'm new in these topics and i've been confuse at some relations between the limit of SR for GR. In cartesian coordinates, basis do not change, so \begin{equation}\Gamma^{\mu}_{\alpha\beta}=0 \quad \...
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Isometry of Riemann sphere?

The complex metric on the Riemann sphere is given in the Wikipedia article to be $$ds^2=\frac{4}{(1+\zeta\bar \zeta)^2}d\zeta d\bar \zeta$$ while the sphere should be mapped to itself under $SL(2,\...
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Geometrical interpretation of curvature invariants

Consider a Riemannian manifold. It is possible to describe it by curvature invariants. Now, is there any geometrical description (intuition) for simple invariants such as scalar curvature, Ricci ...
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Can a “time dimension” be part of a spherical topology?

I've heard it speculated that the spatial dimensions of the universe is a 3-sphere. Or a 3-torus. But usually, I guess, it's assumed that the "time" dimension just has its own geometry, like a line, ...
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Matrix Expression of the Maurer-Cartan Form

I'm looking for clarification re: the 'classical' matrix expression for the Maurer-Cartan form $$g^{-1} dg$$ (I have seen the related posts, they don't answer my specific question.) Specifically I ...
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Lagrangian form of acceleration

Reading the Wikipedia article on Lagrangian mechanics I have problems to understand one basic proposition in the motivation of Euler-Lagrange equation. The article says: For me is unclear how the ...
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A Potential Euler-Lagrange Equation Alternate Derivation?

Can the Euler Lagrange Equation be derived with this overall strategy? Step 1 – Define a geodesic in flat space to be $\frac{d}{dy} \frac{ds}{dx} = \frac{d}{dx} \frac{ds}{dy}$, where $ds$ represents ...
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Geodesic Equation from Coordinate Transformation

Let $\xi^a$ be the usual coordinates and $x^\mu$ the new coordinates, both flat. Now we know that since the metric is flat, $$ \frac{d^2\xi^a}{d\tau^2} = 0 $$ $$ \Rightarrow \ \frac{\partial}{\...
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Diffeomorphic but physically inequivalent spacetimes

In the last few years there has been a considerable endeavor in understanding the asymptotic symmetries of quantum gravity on Minkowski Spacetime. This has been tied to a study of the BMS group that ...
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Action in Electromagnetism expressed in differential geometry and tensor notation

$$ S = -\frac{1}{4} \int F_{\mu\nu}F^{\mu\nu} = -\frac{1}{2} \int F \wedge *F$$ Trying to figure out why this identity holds true and getting stuck.
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How to find a normal to an hypersurface

I have to apply the Israel junction conditions in a region in which a hypersurface with O(3) symmetry separates two spacetime with Schwarzschild metric (with masses $M_+$, the exterior one, and $M_-$, ...
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Counting independent components of the Riemann curvature tensor

In 4D spacetime, we may choose a locally inertial frame at point P, that is we always have a transformation such that $g_{{\mu'}{\nu'}}(P) = \eta_{{\mu'}{\nu'}}$ and its first derivatives vanish. ...
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Tensor analysis: confusion about notation, and contra/co-variance

I'm learning about tensors in the context of special relativity, and I'm a bit confused some notation. I understand a four-vector is a four dimensional vector, which is written in the form $(ct, x, y,...
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How can I show that the inverse of the induced metric $h_{\alpha \beta}$ is $h^{\alpha \beta}$?

So I was reading through Becker, Becker, Schwarz and there is a line in the second chapter that states that $h^{\alpha \beta} = (h_{\alpha \beta})^{-1}$ where $h_{\alpha \beta}$ is defined as: $$h_{\...
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Leibniz Rule for Covariant derivatives

I recently came across a video by prof Fredrick Schuller on general relativity where he defines the leibniz rule to be, $\nabla_X (T(\omega,Y))=\nabla_XT(\omega,Y)+T(\nabla_{X} \omega,Y)+T(\omega,\...
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Basic question about units of velocity and speed of a curve on a smooth manifold

Frederic Schuller says that velocity has units in Hertz in The WE-Heraeus International Winter School on Gravity and Light. He says: \begin{align} [v^a]&=\frac 1 T \\ [g_{ab}]&=L^2 \\ \Big[\...
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Notion of Present

Can't I sync all watches in spacetime and call this time slice the present? In Carlo Rovelli's book he tried to explain that the notion of the present is local only, which I could not follow.
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The flux of a vector field through a cylinder [closed]

The question is by using Gauss’ Theorem calculate the flux of the vector field $$\overrightarrow{F} = x \hat{i} + y \hat{j}+ z \hat{k}$$ through the surface of a cylinder of radius $A$ and ...
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Question about the Lie group $SU(3) \times SU(2) \times U(1)$ and the concept of manifold

I don't know if this question is a duplicate, so I'll delete if is. Well, I'm in the very beginning of the study of contemporary topics such as gauge theories, I would say that I'm in a "science ...
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Uniqueness of affine connections

This is a problem from Carmelli book on general relativity. the conceptual problem is, given a spacetime, and hence a metric, can there exist more than one affine connection for which one can take ...
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Question about computing Christoffel symbols

I am trying to calculate the Christoffel symbols in polar coordinates, and I am confused on one step. Given that I am here, for example: $$\Gamma_{r \theta}^{\theta}=\frac{1}{2} g^{\alpha \theta}\...
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Conceptional question about Quintic 3-fold and Calabi–Yau 3-fold

Has anyone an idea if I’m studying CY 3-fold in the context of string theory decomposition, for instance, from 10 dimensions to 4 dimensions supergravity , can I take the metric of the complex ...
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Transformation of metric by diffeomorphism: pushforward or pullback?

Let $(M,g)$ be a smooth manifold with a metric tensor of signature $(p,q)$. The signature isn't really important for this question so we leave it general. If $\Phi : M\to M$ is a diffeomorphism we ...
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How Quintic 3-fold is a Calabi–Yau manifold and has non-vanishing Ricci scalar?

It’s well known that quintic 3-fold is a Calabi-Yau manifold in the complex projective space $\mathbb{CP}^{n+1}$ , see for instance: https://en.wikipedia.org/wiki/Quintic_threefold Now the main ...
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What would torsion in spacetime look like? [duplicate]

Christoffel symbols explain the gravity we observe in our daily lives. Say I have a region of spacetime with high torsion, so general relativity isn't a good description of the region. In that region, ...
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Confusion re: geodesics, connections, and straightness

In my readings in GR I often come across geodesics characterized as "straightest possible curves." This characterization confuses me. I'd like some clarification as to whether I'm understanding the ...
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What do we mean by straight line in Inertia Law, autoparallel curve or extremal line?

As usually stated: Newton’s first law states that, if a body is at rest or moving at a constant speed in a straight line, it will remain at rest or keep moving in a straight line at constant speed ...
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How many different ways can Riemann-Christoffel Curvature Tensor can be derived? [closed]

In today's Relativity and Gravitation class, my prof was discussing about Riemann-Christoffel Tensor and he derived it. But in the end he told that there are many ways one can derive the Riemann ...
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Connected and disconnected dimensions

The usual way of determining the dimensionality of space is from the number of values needed to define a unique point. However, when choosing a ski, my body is defined by two numbers - my mass and ...
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Carroll's interpretation of 1-forms

Carroll writes in his Spacetime and Geometry book on page 68 that "[...] in fact, however, we could just as well have begun with an intrinsic definition of one-forms and used that to define vectors ...
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How to derive the angular velocity of circular orbits in Kerr geometry?

I am trying to derive the angular velocity of a circular orbit in Kerr geometry, eqn.(2.16) in Bardeen et al (1972) which reads $$\Omega=\dfrac{1}{r^{3/2}+a}$$ (Note that I am using the units in which ...
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Derivation of Isometries of $AdS_3$ in Poincare Coordinates

We know that $SO(d,2)$ is the isometry group of $AdS_{d+1}$. Let's only consider $AdS_3$ in this question. In Poincare coordinates ($r,t,x)$, these can be grouped as follows : Two translations $$r'=...
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Deriving and plotting Quintic 3-fold metric

In this reference: https://people.maths.ox.ac.uk/delaossa/LecturesQuad.pdf Any help to know how the metric in figure 5 has been plotted ? Following the last equations in page ( 143) ? I don’t get ...
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Point of stokes theorem - conservative field

I am working through electrodynamics at the moment and I have a rather elementary question - which I apologize for. But after some research on google I am still not sure if I have understand "the ...
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In the most trivial spacetimes, is the existence of a null geodesic equivalent to horismos relations?

Take a globally hyperbolic topologically trivial spacetime $M \cong \mathbb{R} \times \Sigma$, $\Sigma \cong \mathbb{R}^{(n-1)}$. Given $p, q \in M$, such that there exists a future-directed null ...
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Flat spacetime with curved metric? Small change in metric drastically changes geometry? GR

I'm about to be taking an undergrad course in GR. I'm trying to get ahead of the game, and I've hit a problem. The metric for flat Minkowski space (using the $-+++$ signature and $c=1$) is: $$ds^2 = ...
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How to translate this equation into physicist's notation? [closed]

I asked this in math stackexchange but no one has answered there so I ask here. How to translate this equation into physicist's notation, i.e. tensors with indices? $$\left\langle R_{N}\left(u,v\...
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Pseudo-Riemannian 2D manifold (visualize time curvature)

My goal is to visualize somehow the curvature of time, as opposed to the curvature of space. I know that we generally talk about spacetime curvature altogether; however, the fact that spacetime has ...
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Derivation of radial momentum equation in Kerr geometry

I am trying to derive the radial momentum equation in the equatorial plane of Kerr geometry obtained by Lasota (1994) which reads (eqn. 6 in page-343; I am using units in which $M=1$) as follows: $$uu'...
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1answer
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Some clarifications about ADM Hamiltonian constraints

I have some trouble with refreshing ADM split and Hamilton formalism of GR in context of introducing Wheeler-de-Witt equation. Having Lagrangian in form: $$\mathcal{L}_{ADM}=\sqrt{h}N(G^{abcd}K_{ab}...
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Integral Over a two dimensional surface in 4 space

I am reading Landau and Lifshitz's book The Classical Theory of Fields. They make the following statement in the book. I am interested in trying to show that $df_{ik}df^{*ik}=0.$ I intuitively see ...
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Four-velocity of hypersurface in relativity

I was thinking about the boundary condition of intersection surface in special relativistic hydrodynamics, I tried to Establish boundary conditions on the moving interface,and then return to the ...
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Geodesic equation and spatial variation of time

I am trying understand the interpretation of geodesic equations. For simplicity, let us take a metric $$ds^2 = g_{00}(x)dt^2 + a(x,y,z)(dx^2 + dy^2 + dz^2).$$ I interpret the metric to be a spacetime,...
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Translation invariance of point particles as a field theory

The case of point particles, relativistic or not, can be treated as a field theory in general, ie for the $(1+1)$-dimensional case this is the theory of a field theory on the vector bundle $$\pi : \...
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Holonomic basis

Is the following definition correct? Given a differentiable manifold $M$ and an ordered basis $\{e_j^m\}$ of the tangent space $T_m M$ with $m\in M$ (they are vectors and not vector fields). An ...
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Christoffel symbols with mixed indices

I've just begun taking a graduate course in General Relativity and I came across the Christoffel symbols which we know are defined as: $$\Gamma_{cab}=\frac{1}{2}(g_{ca,b}+g_{cb,a}-g_{ab,c})$$ for a ...
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Projection tensor in General Relativity

In MTW "Gravitation", the projection tensor is defined as $$\boldsymbol{P} = \boldsymbol{g} + \boldsymbol{u}\otimes\boldsymbol{u}$$ And one exercise asks to prove that a tangent vector $\boldsymbol{...
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Does anyone have references on classical field theory that develops the differential form formalism?

I am familiar with the usual way of doing Classical Field Theory, but I am currently taking a course where the professor works with differential forms to teach the subject. I wonder if anyone knows ...