Questions tagged [differential-geometry]
Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.
2,165
questions
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63 views
Coordinate independence in spacetime
As far as I know, the $n$ coordinates $(x_1, x_2, ..., x_n)$ chosen to describe an $n-$manifold have to be mutually independent $\to$ the mutual derivatives must equal $0$ (for example, $\frac{dx_1}{...
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30 views
Christoffel symbols in general coordinates
In order to understand the meaning of covariant derivative, I have seen the following argument. Let us consider a covariant vector $V_\mu$. We would like to understand whether
$$T_{\mu\nu} = \frac{\...
2
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0answers
60 views
Almost complex structure $J_i^j$ from covariantly constant spinor $\eta$
In the context of superstring compactification on a 6-manifold which admits a covariantly conserved spinor $\eta$, which we normalize so that $\eta^{\dagger} \eta = 1$, I am trying to show that the ...
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22 views
Silly doubt about relationship between Levi-Civita Connections and Koszul Form
In this paper [1] the author wrote and interresting relationship between differential geometry objects (the Levi-civita connection and Koszul form) by means of a musical isomorphism [2] (roughly ...
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1answer
41 views
Transformation of four-vectors and basis vectors
I was taught that vectors are one rank tensors, so under diffeomorphism they do not change (their components do change but not them as tensors) they are contravariant, i.e.
\begin{equation}
A = A^{\mu}...
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1answer
44 views
Non-linear Perturbations of Minkowski Spacetime
I am reading some of the following paper on the bounded $L^2$ conjecture in general relativity where it mentions non-linear perturbations of the Minkowski metric in the context of quasilinear wave ...
2
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1answer
122 views
The equation of motion for a scalar field in curved spacetime in terms of the covariant derivative
The equation of motion for a scalar field in curved spacetime $$\frac{\partial\mathcal{L}}{\partial\phi}=\frac{1}{\sqrt{-g}}\partial_{\mu}\left[\sqrt{-g}\frac{\partial\mathcal{L}}{\partial\left(\...
2
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2answers
105 views
Yang-Mills Bianchi identity in tensor notation vs form notation
I've seen the Yang-Mills Bianchi identity written as both
$$0 = dF^a + f^{abc} A^b \wedge F^c$$
and, in tensor notation, as
$$\epsilon^{\mu\nu\lambda\sigma}D_{\nu} F^a_{\lambda\sigma} = 0.$$
Here ...
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0answers
40 views
Proof of equivalence of variational principle and Euler-Lagrange equations on a manifold [closed]
Let M be some manifold, and TM the tangent bundle. Let $\gamma : [a,b] \to M$ be a smooth curve on M defined on an interval on $\mathbb{R}$. Let $J$ be another interval in $\mathbb{R}$ containing 0. A ...
2
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2answers
84 views
Does $\mathcal{M} = AdS_2 \otimes S_2$ makes any sense as a manifold?
I'm not a topologist or a group theorist and I need a clarification about some notations.
Consider the Bertotti-Robinson metric in General Relativity (relativity students should study this metric, by ...
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77 views
On the Boundary term in $f(R,T)$ Gravity
In standard $f(R)$ gravity we consider the Lagrangian of the form $L=\frac{1}{16\pi G}f(R)\epsilon$, where $\epsilon$ is the spacetime volume form and similarly, we consider the boundary term to be of ...
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62 views
What parts of “Geometry, Topology and Physics” by Mikio Nakahara is typically studied in a 1 semester course in graduate school? [closed]
I have some months in my hand before i head to graduate school. I would like to learn and strengthen my grasp on mathematical physics. I would like to do high energy physics (not necessarily just ...
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23 views
Gradient of the null affine parameter
For a timelike geodesic, we have $\frac{d}{d\tau}\tau=V^a \partial_a \tau=1$, where $\tau,V^a$ is the proper time and four velocity. It is thus natural to identify $\partial_a \tau$ as $-V_a$.
My ...
2
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1answer
54 views
Orthogonality relation four-vectors
If $A,B,C,D$ are four-vectors and $A,B,C$ form an orthogonal hypersurface to $D$, and $det(g_{\mu\nu})=g$ where $g_{\mu\nu}$ are the metric components, then it is true that
\begin{equation}
\sqrt{-g}A^...
1
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1answer
94 views
Varying covariant derivatives
If we take a variation of a covariant derivative, we must take into account the connections, so we get:
$$ \delta (\nabla_\beta T_{\mu \nu}) = \nabla_\beta \delta(T_{\mu \nu}) -\delta (\Gamma_{\beta ...
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1answer
66 views
Is there a generalization of ${{\partial }_{\alpha }}\left( \sqrt{-g}{{V}^{a}} \right)=\sqrt{-g}{{\nabla }_{a}}{{V}^{a}}$ for arbitrary connection?
I am studying general relativity and I am trying to understand how to perform variation of the Einstein–Hilbert action with respect to the metric ${{g}_{\mu \nu }}$ and an arbitrary connection ${{\...
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63 views
Volume element if given metric tensor in spherical coordinates
Given the spatial part of the metric tensor
$$g^{\mu\nu} =\begin{bmatrix}
g^{rr}&0&0\\
0&r^{2}&0\\
0&0&r^{2}sin^2 \theta
\end{bmatrix} \tag{1}$$
What would be the ...
3
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3answers
300 views
How to write down various metrics without coordinates?
For example Schwarzschild metric, or Alcubierre metric, but using only intrinsic (natural, canonical, etc.) physical objects (like length, angles, etc.) for relations between natural objects on ...
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2answers
115 views
Spacetime around a massive cylinder
I'm trying to find the metric describing the spacetime around an infinite cylinder of radius $\rho$ and mass density $m$. Since the spacetime is static and cylindrically symmetric, the metric must be ...
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63 views
Line element in a non-static system of coordinates
So I am attempting to find the line element in a non-static system of coordinates $r^{\prime}, \theta, \phi, t^{\prime}$ in vacuum, where the transformations are
\begin{equation}
r=(9 m / 2)^{1/3}\...
2
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0answers
25 views
Information length and heat dissipation
I am trying to relate the information length of a path on a statistical manifold (according to the Fisher-Rao metric) to the heat dissipated, or work done, by moving a probability distribution along ...
1
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3answers
98 views
The necessity of using tangent space as the vectors in general relativity
I’ve recently learnt what manifolds are to prepare myself for a course in GR. My relevant mathematical background is linear algebra (abstract, proof-ish course) and multivariable/vector calculus (...
2
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1answer
123 views
How do you work out the coefficients of the metric tensor?
The definitions of covariant and contravariant tensor quantities are that they transform as $A' ^i=\frac{\partial x_j}{\partial x'_i} A^j$ and $A'_i=\frac{\partial x'_i}{\partial x_j}A_j$ respectively,...
3
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2answers
56 views
Globally constant vector field in a curved spacetime
Is it possible to define a globally constant vector field in a curved spacetime, that is a vector field for which the covariant derivative vanishes along every world line? The vector field $V^{\mu}=0$ ...
4
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1answer
41 views
What “luminosity distance” means in a general spacetime?
In the paper "Asymptotic Symmetries in Gravitational Theory" by R. Sachs from 1962, the author says the following:
In analyzing gravitational fields it is sometimes useful to introduce coordinates ...
1
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1answer
75 views
Yang-Mills Action for Non-Trivial Bundle
Suppose we have a principal $G$ bundle $(P,M,π)$ where $M$ is a 4-dimenational manifold and $G$ a Lie group (and $\mathfrak{g}$ its Lie algebra).The Yang Mills action is a functional of the gauge ...
6
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1answer
103 views
Are there good reasons why special relativity should motivate geometrised gravity in a way that Newtonian mechanics does not?
I have studied a bit of Newton Cartan theory recently, the geometrised version of Newtonian gravity in which gravity is due to the curvature of spacetime, but is Newtonian (simultaneity is absolute).
...
3
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1answer
43 views
Derivative with respect to vector
How in Lagrangian and Hamiltonian mechanics we take derivatives with respect to velocity and momentum respectively if they are vectors? Can we take derivative with respect to a vector?
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73 views
Do the Christoffel symbols $\Gamma_{rn}^w\partial_sV_w = \Gamma_{sn}^w\partial_rV_w$?
In lecture 3 (about 97 min into the lecture) of Leonard Susskind's general relativity course, he suggests finding the Riemann curvature tensor in terms of the Christoffel symbols as an exercise.
I ...
2
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3answers
156 views
How to prove that the covariant derivative obeys the product rule [closed]
In General Relativity the covariant derivative of contravariant vectors $A^\mu$ is:
\begin{equation}
\nabla_\mu A^\nu=\partial_\mu A^\nu+\Gamma^\nu_{\mu\alpha}A^\alpha
\end{equation}
where $\Gamma^\...
0
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1answer
47 views
Ricci scalar in terms of vierbein and spin connection
I have been trying to derive the following form for the Ricci scalar in terms of vierbein and spin connection
$$R=(e^{\mu a}e^{\nu b}-e^{\mu b}e^{\nu a})(\partial_\mu \omega_{\nu ab}+\omega_{\mu a}^{\...
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0answers
29 views
Spin connection Kruskal space-time
Would anyone give me some references where I can find the components of the spin connection in Kruskal space-time?
1
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1answer
57 views
Deriving Kretschmann scalar for Schwarzschild solution
I'm trying to derive kretschmann scalar for schwarzschild solution, which is
\begin{equation}
K=\frac{48M^{2}}{r^{6}}
\end{equation}
I know I have to compute $R_{abcd}R^{abcd}$, but it seems like an ...
5
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0answers
51 views
Minimum required initial conditions to uniquely solve geodesic equation
The geodesic equation is a 2nd order differential equation given as
$$\frac{\mathrm{d}^2 x^\alpha}{\mathrm{d} \lambda^2 }+\Gamma^\alpha_{\beta\gamma}\frac{\mathrm{d} x^\beta}{\mathrm{d} \lambda }\...
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0answers
51 views
Why is the Christoffel symbol in the geodesic equation for a test particle negative?
The geodesic equation is
$$ {d^2 x^\mu \over {ds}^2}+\Gamma^\mu {}_{\alpha \beta}{d x^\alpha \over ds}{d x^\beta \over ds}=0\ $$
for some scalar parameter of motion s and connection coefficients of ...
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0answers
21 views
Scalar coupled to Gauss-Bonnet invariant vs Horndeski theory
So here it is a somewhat tormenting question. The first statement will be a little specific but then I will make clear what the jargon indicates.
How can we show that a Lagrangian made of a scalar ...
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0answers
40 views
Covariant derivative with respect to commutator
I have some confusion with the notion of $\nabla_{[A, B]}\bf{v}$, that expression, with a commutator of vector fields as the subindex of the connection appears for instance in the definition of the ...
2
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0answers
24 views
What is the relation between “projective geometry” and “Projective geometry of paths”?
this is my first time posting so please point out if I am doing it wrong.
So as the title says my question is,
What is the relation between
"projective geometry" and "projective geometry of paths"?
...
1
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1answer
61 views
Diffeomorphism in static spherically symmetric space-time
In a static, spherically symmetric space time we can choose the coordinates so that the metric takes the form:
$$ds^2=-A(r)dt^2+\frac{dr^2}{B(r)}+C(r)\,[d\theta^2+\sin^2\theta\,d\varphi^2]$$
Sometimes ...
0
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1answer
89 views
What are Connections in physics?
This question arises from a personal misunderstanding about a conversation with a friend of mine. He asked me a question about the "truly nature" of spinors, i.e., he asked a question to me about what ...
2
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0answers
39 views
Introduction to the black holes and mass positive theorem in a geometric point of view
I'm a math student with no physics background who want understand more about black holes and mass positive theorem in order to understand better the motivations of the Inverse Mean Curvature Flow (my ...
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42 views
Electromagnetism on 3 torus
We all know Maxwell equations in 3+1 spacetime, where the "space" is $\mathbb{R}^3$ and time is $\mathbb{R}$. Moreover, it is easy to construct (using differential forms) the corresponding theory in a,...
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63 views
What's the position vector for an ant on a sphere?
Imagine an ant on a sphere that perceives only two dimensions. Is there a coordinate system that allows the ant to describe the position with the position vector?
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2answers
49 views
Confusion regarding Ricci Scalar
Source: Thomas Moore's A General Relativity Workbook
Equation 1: $R= g^{\mu\nu}R_{\mu\nu} = R^\nu{}_\nu$
Equation 2: $R^{\mu\nu}=g^{\mu\beta}g^{\nu\sigma}R_\beta\sigma$
Question: Does $R$ also ...
3
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1answer
75 views
General Relativity - Confusion between choosing basis (orthonormal & coordinate) and coordinate transformations
I am reading the book 'Gravity' by Hartle and presently I am at the section discussing orthonormal and coordinate bases. I am confused about a few points I had read previously and can't exactly ...
0
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1answer
35 views
Pseudotensors for describing physical quantities
I have been reading about tensors from Mathematical methods for Physics and Engineering, by K.F. Riley, M.P. Hobson and S.J. Bence.
And there are a couple of things i am not getting. On page 949 (...
4
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1answer
145 views
About physical meaning of Hausdorff, Second Countable and Paracompact conditions of Manifold Theory
I would like to ask you, specially for the people here who deals with General Relativity/Differential Geometry, physical implications about Manifolds.
Well, the most intuitive notion about a manifold ...
1
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1answer
64 views
Exact differentials in thermodynamics
I was reading the book "Thermodynamics and Statistical Mechanics" of Allis and Herlin and I have a doubt in one part of the first chapter. Consider the differential $$df = A(x,y)dx + B(x,y) dy.$$The ...
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30 views
Torsion form and exterior covariant derivative
The torsion form can be defined as the exterior covariant derivative of a solder form, $\Theta=d_\omega\theta$. This derivative is always in the fundamental representation of the algebra $\mathfrak g$ ...
4
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1answer
70 views
Chern-Simons theory on a plane/sphere with a single charge insertion
Consider the pure Chern-Simons theory on the plane $\mathbb{R}^2$ with a single charge insertion in some representation $\rho$ of the group $G$. What does the Hilbert space look like? Is it null or ...
