Questions tagged [differential-geometry]

Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.

Filter by
Sorted by
Tagged with
0
votes
2answers
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}{...
0
votes
0answers
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
votes
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 ...
0
votes
0answers
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 ...
0
votes
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}...
1
vote
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
votes
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
votes
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 ...
1
vote
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
votes
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 ...
0
votes
0answers
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 ...
1
vote
0answers
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 ...
0
votes
0answers
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
votes
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
vote
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 ...
0
votes
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 ${{\...
1
vote
0answers
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
votes
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 ...
1
vote
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 ...
0
votes
0answers
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
votes
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
vote
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
votes
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
votes
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
votes
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
vote
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
votes
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
votes
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?
0
votes
0answers
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
votes
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
votes
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}^{\...
1
vote
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
vote
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
votes
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 }\...
0
votes
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 ...
0
votes
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 ...
0
votes
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
votes
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
vote
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
votes
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
votes
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 ...
0
votes
0answers
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,...
0
votes
0answers
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?
0
votes
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
votes
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
votes
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
votes
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
vote
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 ...
0
votes
0answers
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
votes
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 ...