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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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Must the electromagnetic 2-form be harmonic in vacuum?

The Maxwell equations in vacuum are $dF=0$ and $d*F=0$. Is this not the same as saying $F$ is both closed and co-closed, and hence harmonic? But Hodge's theorem says the space of harmonic $p$-forms on ...
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Existence of closed and intersecting geodesics for a classical mechanical system

Consider the configuration space of a double pendulum - it's a torus, $T^2$. Some well-known theorems say that on such closed manifold there exists at least one (closed) geodetic for every closed ...
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2answers
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Seems like the coordinate independent 1-form transforms like a scalar

I'm studying general relativity and and learning up on tensors through a lecture series. It says that $\omega_\mu$ represents a 1-form in the $x^\mu$ coordinate system. The coordinate independent 1-...
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1answer
46 views

Vectors transforming under change of coordinates

I was watching a lecture on tensors and the professor said that a defining feature of a vector $v$ is that it transforms under a coordinate transformation $x^{\mu} \rightarrow x^{\mu'}$ as $$v^{\mu'}...
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Extensors in mathematics and in physics [on hold]

Could someone explain in a simple but accurate manner what extensors are as mathematical entities and how they are used? How do extensors essentially differ from tensors? Are there or could there be ...
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1answer
80 views

How do the properties of a Lie group (represented as a manifold) manifest in the metric tensor of that manifold?

I know this is a math question; however, physicists are more likely to be familiar with what I'm asking (also, I'm directly trying to utilize it in the context of general relativity). I may have ...
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2answers
50 views

Rindler Metric and Minkowski metric

I am trying to understand why the Rindler Metric line-element and Minkowski metric line-element represent the same spacetime. Could someone help me understand that?
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How exactly does a spin TQFT depend on the spin structure?

Take a spin Chern-Simons TQFT, such as $U(N)$ or $SO(N)$ with odd level. Such system depends on the spin structure of the underlying manifold. But how exactly does the theory depend on the spin ...
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Geometry of Affine Kac-Moody Algebras

One can reconstruct the unitary irreducible representations of compact Lie groups very beautifully in geometric quantization, using the Kähler structure of various $G/H$ spaces. Can one perform a ...
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Given a black hole spacetime, how do we actualy find the event horizon?

As for a definition, there are quite precise ones for what an event horizon is. One can define it as the boundary of the causal past of future null infinity, i.e., $\mathcal{H}=\partial J^-(\mathscr{I}...
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1answer
53 views

Are there timelike 3D surfaces in special relativity

I am reading Scharf's 'finite QED' and I am puzzled at the beginning. He first introduces Minkowski space with $(+,-,-,-)$ signature, and here is a definition I find difficult: A three-...
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Integral over an area of spacetime [on hold]

Is it possible to evaluate this integral in spacetime? $$\int_{\Sigma} \frac{dydz}{[a_{o}(y^{2}+z^{2})+2f_{o}y+2g_{o}z+c_{o}]^{2}}$$ where $$a_{o}c_{o}-f_{o}^{2}-g_{0}^{2}=\frac{1}{4}.$$ If it is ...
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Intepreting Fermions as Differential Forms?

In this paper on path-integral quantization of Chern-Simons theory, on page 434 (equation 4.17), the authors used fermions to interpret wedge product and contractions of differential forms. Let $M$ ...
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4answers
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What is the geometric interpretation of the Einstein tensor $R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R$

The Riemann curvature tensor $R_{\mu \nu \rho \sigma}$ has the geometric interpretation of giving how much parallel transport fails to close around tiny loops. The Ricci tensor $R_{\mu \nu}$ the ...
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1answer
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Derivation of Hamilton-Jacobi theory using canonical transformations

The derivation of the Hamilton-Jacobi equation using canonical transformations is typically done involving a type-2 generating function. Is it possible to use a another type of generating function, ...
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3answers
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Doubt about the vacua equations of General Relativity

I'm facing a quite annoying conceptual problem concerning the Einstein Field Equations (EFE) in so called "vacuum". This problem is both physical and mathematical. So, in a elementary point of view, ...
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2answers
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Determinant of the metric tensor

After a change of coordinate system on flat space from $x\rightarrow y$, we have the metric tensor: $$g_{\mu \nu} = \frac{\partial y^{\alpha}}{\partial x^{\mu}} \frac{\partial y^{\beta}}{\partial x^{\...
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Riemann curvature in orthonormal frame and Lorentz transformations

I have problem with understading how Riemann tensor in orthonormal frame transforms using Lorentz transformation of frames. I was reading Morris Thorne paper from 1988 (American Journal of Physics 56, ...
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2answers
75 views

How can the $v$ coordinate be null if $g_{vv}\neq 0$?

I'm probably missing something very basic here. As far as I know, a coordinate is called null when its coordinate lines are null. This that if $(M,g)$ is spacetime and $x^\mu$ a coordinate chart, the ...
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0answers
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Significance of geodesics of a Hamiltonian surface in condensed matter physics?

Many Hamiltonians in 2D quantum systems can be parameterized as a surface (such as the Bloch sphere) by their k-space coordinates. Another example is given by the (kx,ky) points of the Brillouin torus ...
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2answers
92 views

Silly question about “full” abstract structure of a Spacetime

My question is simple. Consider then a vector space: $$\mathfrak {V} \equiv [\mathcal{V},(\mathbb{K},+_{\mathbb{K}},\cdot_{\mathbb{K}}),\boxplus_{\mathfrak{V}},\boxdot_{\mathfrak{V}}]$$ Where $\...
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1answer
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What exactly are the sections in gauge theories?

In trying to understand precisely how fiber bundle theory maps to physical models, I came across this quotation: We can think of the elements of the principal bundle as generalized frames for the ...
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1answer
296 views

Can solutions of GR have non-zero genus?

Imagine there is "cavity" in one's locally Lorentzian manifold (the manifold has non-zero genus). Have these kind of solutions in general relativity been considered? If so, where can I read more about ...
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0answers
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Divergence of rank 2 tensor in cylindrical coordinates

I am having trouble calculating the divergence of a rank 2 tensor $\sigma_{\mu \nu}$ in cylindrical coordinates using the Christoffel symbols. Given: $Div(\sigma)_\nu = \nabla_{\mu}\sigma_{\mu\nu} = ...
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1answer
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Relation between curvature in orthonormal basis and in “standard” metric form

Im familiar with both formulations of GR - standard with metric and connection coefficients and that based on orthonormal frames and differential forms (Cartan's structure eqns) in solving Einstein's ...
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2answers
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GR as a gauge theory: there's a Lorentz-valued spin connection, but what about a translation-valued connection?

Given an internal symmetry group, we gauge it by promoting the exterior derivative to its covariant version: $$ D = d+A, $$ where $A=A^a T_a$ is a Lie algebra valued one-form known as the connection ...
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Can one find Dirac matrices for any spacetime metric?

For any metric $$g_{μν}$$ is there always a linearly independant spacetime algebra satisfying $$\{γ_μ,γ_ν\} = 2 g_{μν} I?$$ For a diagonal metric I was able to work out that $$\bar{γ}_μ=\sqrt{n_{μμ}*...
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1answer
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Can GR be reformulated in terms of invariant observables?

Question So recently I was thinking about this: How many scalars are available in $4$ dimensions in General Relativity (without being redundant)? For example, with metric we can construct the ...
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2answers
121 views

What is the meaning of the Equation of Geodesic Deviation?

I've seen the Equation of Geodesic Deviation stated in several text as:$$\frac{D\xi^2}{dt^2}+Riemann(\textbf U,\xi,\textbf U)=0$$ but I haven't seen a real good explanation of why it works or the ...
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1answer
55 views

D'Alembert Operator

In which book or where can I find the derivation of the d'Alembert operator? \begin{equation} \Box \psi= \frac{1}{\sqrt{-g}}\partial_\mu \left( \sqrt{-g}\partial^\mu \psi \right) \end{equation}
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1answer
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Why is Penrose's diagrammatic notation for tensor operations not widely used? [closed]

Strictly speaking this is a mathematics question rather than a physics question, but since it is about a way of dealing with tensor bundles that is very remote from what is done in math, and very ...
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Mathematics of general relativity? [duplicate]

I need help. I have some difficulties with the mathematical side of general relativity. What books or other resources would help me?
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Fermat principle [closed]

According to Fermat's principle, the actual path taken by a light ray in space locally minimizes the "optical length". It is possible to use the optical length (for some given function n) to defined ...
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1answer
45 views

Isometry definition

I work in holography and I'm trying to get my feet when in non-relativistic holography. Can someone explain exactly what an "isometry" is in this context? "the correspondence can be extended to a non-...
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1answer
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Silly question about kinematics and Christoffel symbols

An interresting "method" that allows you to know the acceleration vector with respect to any coordinate system is just a matter of recognize some key formulas. 1) Given the metric of a particlar line ...
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0answers
178 views

Periodicity trick for Kerr Black Holes

I am slightly confused concerning the euclidean section of a Kerr black hole. In page 5 of the following paper https://arxiv.org/abs/hep-th/9908022 it is said that in order to get the euclidean ...
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How does the fiber bundle around a single point charge concretely look like?

For simplicity, let's work with the gauge group $U(1)$ and let's assume there is no external field. Without any charge then, our fiber bundle is flat: In the presence of an electric charge the ...
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Orthogonal vector to $t=$const. hypersurface in Kerr-Schild coordinates

How do I find the orthogonal vector for the t= const. hypersurface in the Kerr - Schild metric? I know that I start by dt=0 but where do I go from there?
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1answer
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Interpretation of Ricci rotation coefficients in tetrad formalism

Given an orthonormal frame (tetrad in 4 dimensions, vielbein more generally) $\{(e_{\mu})^{a}\}$ with $g(e_{\mu}, e_{\nu}) = \eta_{\mu\nu}$, the Ricci rotation coefficients are defined as $$\omega_{\...
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Is the obeserver moving in Ehrenfest paradox?

On wikipedia https://en.wikipedia.org/wiki/Ehrenfest_paradox, it seems like it indicates there are two observers in Ehrenfest paradox. The first one is stationary in the center of the disk, this one ...
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2answers
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What is the tangent vector representing rotation?

I am reading Mathematics for physics: A guided tour for graduate students by Michael Stone. On the page 379, the book says The surface of the unit sphere is a manifold...We may label its points ...
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1answer
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How to measure proper distance?

The proper distance from $R$ to $R+\Delta R$ in Schwarzschild metric is given by $$\displaystyle l = \int_R^{R+\Delta R} \frac{1}{\sqrt{1-\frac{r_s}{r}}} \; dr.$$ If static observer whose radial ...
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Contracted Bianchi Identity for FLRW Metric

I am trying to verify that the Contracted Bianchi Identity $\nabla_\mu G^{\mu\nu}=0$ holds for the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric, $$g_{00}=-1,g_{i0}=0,g_{ij}=a^2(t)\delta_{ij}$$ ...
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2answers
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Is $∂_\mu + i e A_\mu$ a “covariant derivative” in the differential geometry sense?

I have heard the expression "$∂_\mu + i e A_\mu$" referred to as a "covariant derivative" in the context of quantum field theory. But in differential geometry, covariant derivatives have an ostensibly ...
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2answers
318 views

Puzzle concerning the Divergence Theorem

Something is puzzling me concerning the divergence theorem. Usually, one writes the divergence theorem as \begin{equation} \int_\mathcal{M} d^4x \sqrt{-g} \nabla_\mu v^\mu=\int_{\partial \mathcal{M}} ...
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1answer
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How would General Relativity be different if we assumed Galilean instead of Lorentz transformations?

If we assume a universe where Galilean transformations are the correct transformations between inertial reference frames, would GR be any different ? Gravitational and inertial mass would still be ...
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What is the difficulty in extending geometrodynamics to non-abelian fields?

In an attempt to widen my own horizons I've decided to educate myself in Wheeler's Geometrodynamics. In the so-called "already unified theory" one can essentially reproduce an electromagnetic field ...
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Are there more general gauge transformations than simple phase shifts?

Usually, in the context of a $U(1)$ gauge theory, we only consider gauge transformations of the form \begin{equation} \Psi(x) \to \mathrm{e}^{i\epsilon(x)} \Psi(x) \, . \end{equation} Are there ...
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1answer
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Are Minkowski and Schwarzschild spacetimes diffeomorphic?

Another mathematical question, arising from GR. Some days ago I wrote, in an answer to 1, that they are. Then @magma commented they are not. He promised a proof, but none appeared. After magma's ...
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Killing Spinor Equation in 4 Dimensions

From https://en.wikipedia.org/wiki/Killing_spinor we see that a Killing Spinor $\epsilon$ is defined as a solution of the following equation: $$\nabla_{X}\epsilon = \lambda X * \epsilon \quad , \quad ...