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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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General Relativity and cosmology [duplicate]

What is the physical meaning of Ricci scalar is a covariantly constant?
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Mathematical software for calculating Christoffel symbols, curvature, etc [duplicate]

I'm looking for a Mathematical software that can do the following: I put in a metric (some kind of a two-dimensional matrix), and the software calculates Christoffel symbols, Ricci curvatures, ...
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$* d * $ operator — Digest the (differential/geometry) meaning

I like to digest better: the $* d * $ operator in Maxwell differential form equation the $* D * $ operator in Yang-Mills differential form equation We already knew that in Maxwell differential ...
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Simplify Yang-Mills Equation of Motion in the 1-form gauge field $A$

We know the Yang-Mills theory Equation of Motion (eom) without source $$ * D * F = * (d (* F ) + [A, (* F )])= 0. $$ My question is that what are the most simple form we can boil down this ...
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Geodesics from variational principle with respect to coordinate?

I know you can find geodesic equations with respect to proper time $\tau $ using the variational principle, i.e. using Euler-Lagrange equations $$ \frac{\partial}{\partial x^{\mu}}L-\frac{d}{d\tau}\...
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1answer
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Tangent space of pseudo-Euclidean manifold

My textbook says (without any justification that I can see) that "since Minkowski spacetime is pseudo-Euclidean, the tangent space $T_P$ at any point $P$ coincides with the manifold itself". My ...
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1answer
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Line Element Transformation

This is just something that I've made up to see if I understand the method. If I have the line element: $$ds^2 = dr^2 + r^2\,d\phi^2$$ and I want to carry out a transformation with $r = \dfrac{...
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1answer
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Mistake or Rewriting of Yang-Mills in Nakahara

I am familiar with Yang-Mills equation of motion E.O.M. (without matter or source fields) in differential form. $$ D * F =0 $$ and Bianchi identity $$ D F=0 $$ where $F= dA + A \wedge A$ and $D=d + [...
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1answer
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Witten Index of Riemannian Manifold

Consider a system on a Riemannian manifold with the Lagrangian $$L = \frac{1}{2}g_{IJ} \dot{\phi}^I \dot{\phi}^J + \frac{i}{2}g_{IJ}(\overline{\psi}^I D_t \psi^J - D_t \overline{\psi}^I \psi^J) - \...
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Why do we visualize space time as trampoline surface? [duplicate]

It's a silly question probably. But the thing is that what is the fascinating about fabric surface. I am so sorry for this silly question in advance. I read a lot articles regarding my question. But ...
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Coordinate transformations [migrated]

I have two scalar functions of $x$ and $y$ that I can define: $$f(x,y)=x^2+y^2\qquad \text{and}\qquad g(x,y)=x^2 + \sin^2(x) y^2.$$ Is it true that there is literally no coordinate change that will ...
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Write electromagnetic field tensor in terms of four-vector potential

How can we know that the electromagnetic tensor $F_{\mu\nu}$ can be written in terms of a four-vector potential $A_{\mu}$ as $F_{\mu \nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}$? In the ...
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Differential geometry: If $\vec v = v^i \vec e_i$, then why is $\vec r = r \vec e_r$ in spherical coordinates?

In differential geometry (and later carried over to GR) any abstract vector $\vec v$, exists on its own vector space. We can then choose to represent this vector in a coordinate basis $\vec v = v^i ...
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2answers
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Metric tensor: Why relate it to Cartesian/Minkowski coordinates?

Why does the metric tensor always relate to cartesian coordinates? Let's take the simple case for the metric tensor in 3D-space without a time dimension, $g_{ij}= \begin{bmatrix} 1 & 0 &...
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Writing the curvature tensor in terms of spin connection

I am trying to know what are the missing steps in deriving the following expression for the Riemann curvature tensor in terms of the spin connection $\omega$ and the triad $e_\mu^j$. Where the ...
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1answer
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Whats the quickest way to compute the Ricci tensor?

I have been going through exam papers and often they ask us to calculate ricci tensor components and affine connections from a given metric. They seem to take far too long for the time you are ...
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Rigorous procedure of gluing together two spacetimes

There seems to exist a procedure of "gluing two spacetimes together". In particular I've seem this mentioned in the context of gravitational collapse. The examples I've seem where that of gluing ...
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1answer
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Coordinate transformation of basis vectors

The question Let $e_a$ be the coordinate basis vectors in a manifold described by coordinate system $x^a$. The vector displacement between two nearby points is given by \begin{equation} ds=dx^ae_a=dx'...
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Why is it that the equation of a massless scalar field *must* be conformal invariant?

I'm reading a paper [1], p.111 where it is said that: However, the equation of scalar field with zero mass must be conformal invariant while equation $\square\varphi=0$ does not satisfy this ...
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1answer
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How to calculate $F^{\mu\nu}F_{\mu\nu}$ efficiently?

The quantity $\frac{1}{2}F^{\mu\nu}F_{\mu\nu}$ is equal to $B^2-E^2$. I can show this pretty laboriously via matrix multiplication, and slightly faster via the index notation. Is there an elegant ...
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Does the Einstein field equation uniquely determines the topology of spacetime? [duplicate]

I am trying to understand whether the Einstein field equation uniquely determines the topology of spacetime. As far as I know, given a metric, we can always find the induced topology. However, I was ...
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2answers
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ADM formulation of GR derivative on the 3-metric

In the ADM formalism where the projector is given by ${P^\mu}_\alpha={\delta^\mu}_\alpha+n^\mu{n}_\alpha$ and $n^\alpha$ is a future pointing normal vector to the constant time hypersurface $\Sigma$. ...
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1answer
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Do Bianchi identities hold in all coordinates?

I understand by expanding out the Riemann tensor, that the Bianchi identities can be derived within a local inertial frame (LIF) by taking the partial derivatives of the Riemann tensor relations in a ...
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2answers
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Finding the dimensions of a spacetime given the Riemann tensor

The question is: For a spacetime the Riemann tensor is given below: $$R_{\mu \nu \rho \sigma} = \frac{R}{6} (g_{\mu \rho} g_{\nu \sigma} - g_{\mu \sigma} g_{ \nu \rho} )$$ What is the dimension of ...
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2answers
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Electromagnetism and differential forms

I am currently writing a Bachelor's thesis in theoretical physics, and since I like the interplay between mathematics and theoretical physics, I am writing about Maxwell's law in terms of differential ...
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Critical points of vector field with zeros in the magnitude

I am studying a vector field which has critical points (sources, sinks, saddle points and centers). The magnitude of the vector field goes to zero smoothly in these points, however. Contrast that to ...
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1answer
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Confusion Einstein notation polar coordinates

I'm having issues using Einstein notation in polar coordinates in flat space, I must be missing something basic. Consider the following example. Take the following metric on a 2+1 spacetime; $ds^2 = ...
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4answers
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Curved space-time and metric tensor

I'm studying about curved spaces and I read that a manifold is flat if there a coordinate system such that the metric tensor is constant everywhere. Then I also read that when the space-time tensor ...
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1answer
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Berry phase: Spin in a magnetic field parameter space manifold

Canonical example for Abelian Bery phase is a spin in a magnetic filed, e.g.. Usually authors calculate spin eigenstates, conclude that they don't depend on B in spherical components and so deduce ...
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1answer
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The WKB approximation and the Cotangent bundle

When we say (see pag. 9 of Lectures on the Geometry of Quantization) that the image of the differential of the phase function lies in the level set of the classical Hamiltonian is it simply ...
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1answer
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Equality between derivatives of the metric

In one of my lecture, it is said: Let us use the freedom of the choice of parametrization to demand that the variation of $\lambda$ after a small displacement along the curve is proportional to the ...
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1answer
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General Relativity - (numerically) compute the metric from the stress-energy tensor?

I am new to GR and I am having trouble understanding how one goes back and forth between the metric $g_{\mu\nu}$ and the stress-energy tensor $T_{\mu\nu}$. First, have a look at the following post. ...
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Why isn't the scalar product of a covector and vector symmetric? [migrated]

In tensor math, how come the scalar product of a co-vector (co-variant vector) with contra-variant vector, as written between angle bracket separated by comma, $\langle x, a \rangle$, is not symmetric?...
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2answers
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Is the determinant of metric tensor stationary wrt. proper time for a particle moving along its world line?

While writing the expression for stress energy tensor of a free massive particle moving along its world-line some authors take out of the integral sign, the $\sqrt{-g}$ where $g$ is the metric tensor ...
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Geometry of the Earth [migrated]

Is this true that every simple closed curve on the earth can be deformed continuously to a point without leaving the earth? Is the earth compact? Now if we consider the earth as a 2-manifold, can we ...
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Configuration space VS Euclidean space [closed]

Why Configuration space is non-Euclidean space? Why it doesn't look like an n-dimensional Euclidean Space in General? Thanks
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4answers
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Why does nature favour the Laplacian?

The three-dimensional Laplacian can be defined as $$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}.$$ Expressed in spherical coordinates, it ...
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Symplectic Manifolds in General Relativity for Integrable Systems

To solve the geodesic equations for a specific metric in General Relativity I can find conserved quantities $F = \xi_{\mu}\frac{\mathrm{d}x^{\mu}}{\mathrm{d}\lambda}$ along geodesics by using Killing ...
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1answer
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Is curvature the exterior covariant derivative of the connection?

Let $P\to M$ be a $G$-principal bundle, $G$ a topological group, $\omega$ the connection and $V$ a vector space. We define $d_\omega: \Omega^k_G(P, V)\to\Omega^{k+1}_G(P, V)$ the exterior covariant ...
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1answer
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De Sitter spacetime affine parameter

I am reading Chapter 8 in Carroll's "Spacetime and geometry " textbook and I was lead to exercise 8.2, given as: Consider de Sitter space in coordinates where the metric takes the form $$ds^{2} =...
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1answer
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Christoffel's Symbol's relation to the Metric Tensor

In chapter 9.2 of "Tensors, Relativity and Cosmology", the contracted Christoffel symbol of the second kind as a function of the metric tensor was defined as: $$\Gamma_{nm}^m=\frac{1}{2}\left(g^{mk}\...
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2answers
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Covariant surface vector

On pg 74 of Dalarsson's Tensors, Relativity and Cosmology (The Integral theorems for tensor field chapter), the covariant surface vector was defined as: $$dS_k=\frac{1}{2}\epsilon_{kmn}dx^mdx^n=\frac{...
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2answers
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There are two velocities on the sphere. How to sum them on the sphere?

As shown in the figure, there are two velocities on the sphere, one is the velocity along the meridian direction, the arc length is its size, the other is the velocity along the equatorial direction, ...
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0answers
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Earth's surface area

Here we are trying to calculate the earth's surface area via geodetic coordinates: \begin{align} x&=(R\,p(\lambda)+h)\sin\lambda\,\cos\phi\\ y&=(R\,p(\lambda)+h)\sin\lambda\,\sin\phi\\ z&...
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1answer
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Curvature scalars and singularities

Without resorting to the singularity theorems, can we say that there is a singularity at a particular $r=\textrm{constant}$ if the value of the Ricci and Kretschmann scalars get infinitely large at ...
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1answer
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Proving that $dS$ is an exact differential mathematically

OK...so I hope this is not too dumb a question: We know that we can express $dS$ as $$dS=\frac{dQ}{T}=\frac{C_v}{T}dT+\frac{R}{V}dV,$$ where $C_v$ is the thermal capacity at constant volume and $R$ ...
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Is there a useful relationship between connection on space coordinates and material derivative?

I am referring to an important part of the question Relationship between Connection and Material Derivative. Here is a paste and cut of the relevant part. That is the directional derivative along $...
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1answer
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What is a coordinate-free version of Noethers theorem? [closed]

What are some examples and derivations of some basic symmetries (not coordinate symmetries)? For example I remember a sufficient condition for being a symmetry of the lagrangian system is being an ...
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Is intrinsic curvature of an embedded surface a covariant quantity from the embedding space point of view?

Suppose I have a $(d+1)$-dimensional manifold with metric $g_{\mu\nu}$. In it I have an embedded codimension-$1$ surface, $\Gamma$, with induced metric $\gamma_{ab}$. Is Ricci scalar defined in terms ...