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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Regarding Derivation of Einstein Field Equations

In most sources I come across that try to justify the Einstein Field Equations outside the context of Einstein-Hilbert action, the argument goes mostly as follows: In analogy with the Poisson equation ...
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Covariant derivative for spin-2 field

I have mostly seen the concept of covariant derivative with regard to spin-1 fields. Is it possible to define the covariant derivative for spin-2 fields as well?
physics_2015's user avatar
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Kretschmann Invariant Cosmology

I am trying to understand the Wikipedia definition of the Kretschmann scalar for a cosmological solution. The metric is given by the standard FLRW: \begin{equation*} ds^2 = -dt^2 +a^2(t)\left(\frac{dr^...
LolloBoldo's user avatar
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Why is the 4-current a tensor rather than a tensor density?

I am trying to understand electromagnetism better in terms of tensors and differential geometry. First recall that (in the Lorenz gauge) the equation of motion for the four-potential $A^\mu$ is $$(-\...
Daniel Grimmer's user avatar
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Geodesic in flat space in spherical coordinates

let's consider the expression, where $u^\mu$ is the tangent vector to the geodesic $\theta = \nabla_\mu u^\mu$....scalar $\Rightarrow$ valid in every coordinate system So in flat space in Cartesian ...
Coderboy's user avatar
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Exponential of the metric tensor

Exponential of an arbitrary matrix can be written as $$e^A = \displaystyle\sum_{n=0}^\infty \dfrac{A^n}{n!}$$ In Einstein notation, how this expression will look like? In Einstein notation, what ...
SCh's user avatar
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Does the divergence theorem require the covariant derivative to be metric compatible?

I know this is more of a mathematical question, but it arises in the context of general relativity and uses its language so I thought it would be best to ask it here. I understand that the divergence ...
P. C. Spaniel's user avatar
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Ricci tensor in locally Lorentz frame

In the book A Relativist's Toolkit by Eric Poisson, section 4.1.4, page $123$, it is written that in a local Lorentz frame at a point $P$: $$\delta R_{\alpha \beta} \stackrel{*}{=} \delta\left(\Gamma^\...
darkphysics's user avatar
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How is this deduced? (Differentiation of tensors)

In Schutz's An Introduction to General Relativity, he talked about how to differentiate tensors. Here is a step that I cannot understand. $$\frac{d\mathbf{T}}{d\tau} = \left( T^{\alpha}_{\beta, \gamma}...
Gene's user avatar
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What is the relation between gauge field and Levi-Civita connection?

In field theory, covariant derivative is something like $$D_{\mu}\phi=(\partial_{\mu}-igA_{\mu})\phi$$ while in differential geometry, covariant derivative is something like $$D_{\mu}V^{\nu}=\partial_{...
Baoquan Feng's user avatar
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Compute the difference between the Christoffel symbols compatible with two different metric tensors

Imagine I have two metric tensors $g_{\alpha\beta}$ and $\hat{g}_{\alpha\beta}$ on the same manifold M and two metric-compatible, torsion free Christoffel symbols $\Gamma^{\mu}_{\alpha\beta}$ $$\Gamma^...
P. C. Spaniel's user avatar
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Help understanding gauge symmetry and principal bundles

i'm diving into gauge theories and i'm having a hard time understanding the concept of Gauge symmetry. What i understand is: gauge symmetry is the invariance of a field theory under a certain family ...
Tomás's user avatar
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Does the Weyl tensor amount to tidal effects of gravity?

The Ricci tensor, for the spacetime surrounding the Earth, is zero, so the spacetime around the Earth is Ricci-flat. The Riemann tensor though is not zero since spacetime certainly is curved. This ...
Il Guercio's user avatar
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How to Relate the Functional Derivative to Infinitesimal Change in Noether's Theorem

When the Euler-Lagrange equation or the expression for Noether current are derived the term infinitesimal change is often used. For example, we write $\phi\rightarrow \phi + \delta\phi$ and say that $\...
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Gauge transformation with harmonic one-form

The electromagnetic four-potential $A^{\mu}$ is not uniquely determined by the physical situation. We have the equation $$\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}=F^{\mu\nu}.$$ Here $F^{\nu\mu}$ is ...
Riemann's user avatar
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Stokes' theorem and vector continuity equations

I have been working with homogeneous continuity equations of the general form: $$\frac{\partial \rho}{\partial t}+\vec{\nabla}\cdot \vec{J}=0$$ This has me wondering whether we can formulate other ...
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Derivation of Wald's general relativity equation 7.5.8 ; conformal transformation of Riemann tensor

I am trying to derive some nice properties of conformal transformation of Riemann tensor. I found some formulas on appendix G on Carroll and appendix D in Wald, and recognize their starting point is ...
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How to understand the relationship between Weinhold geometry and Ruppeiner geometry in thermodynamic geometry? [closed]

According to the content of the following paper On the relation between entropy and energy versions of thermodynamic length The second derivative matrix $D^2 U$ of the internal energy may be used to ...
000 666's user avatar
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Lie derivative: moving boat on a flowing river

Lie derivatives signifies how much a vector (Tensor) changes if flown in the direction of some other vector. I am thinking the typical moving boat on a flowing river problem where the river is flowing ...
spacetime's user avatar
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How to decompse kinetic term operator in string compactification

In general textbook, when we want to calculate the dimension of moduli space of string compactification, i.e. calculate the number of massless modes after dimension reduction, we use the following ...
AlphaNotKnows's user avatar
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Does the term $d ( \omega_{ab} \wedge \theta^a \wedge \theta^b )$ have any significance?

If $\omega_{ab}$ is the spin connection 1-form, and $\theta^a$ are the tetrad 1-forms, then one has the equality \begin{equation} \int \, d ( \epsilon_{abcd} \omega^{ab} \wedge \theta^c \wedge \theta^...
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Why future infinity have no future end points?

I am studying Hawking's area theorem from the book the large scale structure of spacetime by Hawking and Ellis. At the end of page#318, it said: null geodesic generators of future infinity have no ...
Talha Ahmed's user avatar
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Prove that spinors satisfying the Killing equation lives on a sphere

Considering that the covariant derivative $D_{\mu}$ acting on spinors be given by $$D_{\mu} \eta = \pm \frac{i}{2} \gamma_{\mu} \eta$$ It is claimed that theses spinors lives on a constant curvature ...
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(Time-)Orientability in the Language of Fiber Bundles

I'm currently studying spin geometry through Hamilton's book Mathematical Gauge Theory. At a given point, Hamilton considers a pseudo-Riemannian manifold, which I'll take to be Lorentzian in $d=3+1$ ...
Níckolas Alves's user avatar
5 votes
4 answers
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Contracting the metric tensor with its inverse yields Kronecker delta

It's probably straightforward, but I would like to see the proof of the identity: $$g_{\mu\nu}g^{\nu\alpha}=\delta^\alpha_\mu.$$ In the book 'Spacetime and Geometry' by Carroll, this identity is the ...
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Five-form flux in Giddings-Kachru-Polchinski (GKP)

I'm studying the work of Giddings-Kachru-Polchinski (GKP) for hierarchies in string theory and I came across the five-form flux defined in eq. 2.9. Now, if one calculates the Ricci tensor for the ...
Fredrigo6's user avatar
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Terms in the Israel Junction Conditions

I'm confused about the Israel Junction Conditions. I've seen them written several different ways so far, but here I'll use: $$K^-_{ij}-K^+_{ij}=8\pi(S_{ij}-\frac{1}{2}g_{ij}S).$$ My understanding is ...
user345249's user avatar
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Could the universe have a form of a $T^3$-torus?

Cosmological measurements suggest that we live in a flat universe. However, what might be less clear is its topology. So could the flat universe have the form of a $T^3$-torus, i.e. the torus whose ...
Frederic Thomas's user avatar
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Maxwell's equations with differential form formalism

I've been reading Sean Carroll's book on GR and I stumbled upon an exercise on EM using $p$-forms. I think I've solved the problem correctly but I am having problems with my answers. I'll provide the ...
user20046481's user avatar
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What actually is Boyer-Lindquist coordinates?

I want to know the difference between spherical and Boyer-Lindquist coordinates. Don't they both use $r, \theta, \phi$ parameters? I've searched books and sources on the internet and there's none that ...
posfn0319's user avatar
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Einstein's gravity Lagrangian invariance under the change of differential structure

I came across an article claiming the appearance of singularities in the energy-momentum tensor $T_{\mu \nu}$ as a result of changing the differential structure: I wonder what symmetry or current (in ...
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Reaching a turning point in photon trajectory

Given the geodesic equations for a photon in a Schwarzchild or Kerr metric (provided by a near BH for example), the radial equation has usually two possible signs: \begin{equation} \dfrac{dr}{d\tau}= ...
gravitone123's user avatar
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Orthogonal self-intersection of geodesics

I learnt that geodesics parallel transport their velocity vectors. Does that mean a geodesic cannot intersect itself orthogonally?
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Are pseudo Riemannian manifolds with identical Wilson loops isometric?

It is well established that in gauge theory, the Wilson loops of the theory determine the gauge potential up a gauge transformation. That is, two gauge potentials $A_\mu$ and $B_\mu $ produce the same ...
Trevor Scheopner's user avatar
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Why are the zero modes of the below operator Killing vectors? (2+1 dimensional gravity)

I'm trying to understand the eigenmodes of the following operator: $$(\Delta_{(1)}^{L L}-\frac{2}{3} R)V_\nu \equiv -\nabla^\mu \nabla_\mu V_\nu+R_{\nu \mu} V^\mu -\frac{2}{3} RV_\nu $$ Where $R_{\mu\...
faker 23's user avatar
6 votes
1 answer
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Are worldlines towards the origin with the Schwarzschild metric finite in time and length?

A recent spacetime video about Kerr's objection to the existence of singularities has made want to clear up something about geodesics towards the origin in the Schwarzschild solution. It is said that ...
John's user avatar
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Looking for textbook on differential geometry [duplicate]

I am looking for a textbook(s) that discusses differential geometry, smooth manifolds etc. More precisely, I have been trying to find a textbook that covers the following topics: Differential ...
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2 answers
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Is $dJ(V,V)=0$? where $J$ is a 1-form?

So is this always 0?( Where $dJ$ is the exterior derivative and $V$ a vectorial field) \begin{align} dJ(V,V)=\partial_jJ_i(dx^j\wedge dx^i)(V,V)=\\ \partial_j J_i (v^kdx^j(\partial_k)v^ldx^i(\...
Guillermo Fuentes Morales's user avatar
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1 answer
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Is wedge product a tensor or a pseudo tensor?

I'm doing an exercise where $J$ is a 1-form on a manifold $M$ of dimension $N$. The exercise ask me to calculate $J∧(*J)$ with $J=dx^0+2dx^1$ in a minkowski space with metric =(-1,1,1,1) where $*J$ is ...
Guillermo Fuentes Morales's user avatar
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Coupling torsion to electromagnetism and torsion tensor decomposition

When extending general relativity to include electromagnetism, several authors (e.g. Novello, Sabbata ecc.) assume that the traceless part of the torsion tensor vanishes or is deliberately set to zero....
user3764418's user avatar
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Disagreement of Ricci scalars

This question pertains a particular example, but I think the answer should be a general statement on a broader misundertanding I must have. Consider a 3-sphere with metric $g=d\theta^2+\sin^2\theta d\...
user984949's user avatar
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How to describe the gravitational field generated by a moving particle?

Which means the formula in general relativity similar to the use of Lienard-Wiechert potentials in electromagnetism.
xieqi's user avatar
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Is there a coordinate-independent representation for the Lienard-Wiechert potentials?

Treat particle trajectories as embedded submanifolds in the pseudo-Euclidean space, and treat the four-dimensional potential as a 1-form.Is there a statement for the Lienard-Wiechert potentials that ...
xieqi's user avatar
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The explaination of Einstein-Hilbert action

I've recently been studying about the General relativity and Einstein field equation. When I reading of the derivation of the field equation, I encounterd a method called Einstein-Hilbert action. This ...
PermQi's user avatar
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Calculation of Cartan structure constants [migrated]

I'm trying to figure out the step from equation (19) to equation (20) in this document when $\mathcal{F} = 0$ y $\mathcal{A} = 0$. In this case, equation (19) reduces to $$ - e^{\alpha \phi} \hat{\...
David Lazaro's user avatar
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1 answer
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Maxwell equations for $F^{\mu \nu \rho}F_{\mu \nu \rho}$

I am trying to find equivalent of Maxwell equations for 2 form gauge theory. My question is Do we get $\partial_{\mu}F^{\mu \nu\ \rho}$ using Euler-Lagrange equation of motions? How do we get the ...
trying's user avatar
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Obtaining the topological charge

I want to obtain the topological charge or winding number of the map $$ f_n(\mathbf{r})=(\sin \theta \cos (n \varphi), \sin \theta \sin (n \varphi), \cos \theta) $$ and my lecture notes say that it is ...
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The Role of the Kaehler Manifold in Supergravity

Actually, I already asked a similar question Coupling of supergravity to matter which has remained unanswered. So this time I will be less general. In the very interesting paper arXiv:2212.10044 [...
Frederic Thomas's user avatar
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How was the $\Gamma _\mu$ be used as a gauge condition in the Generalized Harmonic formulation $R_{\mu\nu}$

I'm watching a video(ICTP-SAIFR Numerical Relativity by Sascha Husa) where he mentioned that $$R_{\mu\nu} =-\frac{1}{2} g^{\lambda \rho} g_{\mu\nu,\lambda \rho} +\nabla_{ (\mu }\Gamma_{\nu)} +\...
ShoutOutAndCalculate's user avatar
2 votes
1 answer
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Reducing Tensor-rank by fixing an argument

Assume for example that you are given a (2,0) tensor $T^{\mu\nu}$ and you want to create a vector, i.e., a (1,0) tensor out of it. Is it possible to just fix an index of $T^{\mu\nu}$ while keeping the ...
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