Questions tagged [differential-geometry]
Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.
2,919
questions
3
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1answer
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How are sources described in gauge theory?
Let's only assume the case of electromagnetism. If one varies the Yang-Mills-Functional one gets the Yang-Mills-equation $*d*F=0$. The whole theory can be geometrically described on principal bundles. ...
0
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0answers
25 views
Why isn't spacetime flat in Ehrenfest's Paradox?
In a previous question, the metric in the frame of reference of a rotating disc takes the following form:
\begin{align}
ds^2 &= - dt^2 + dx^2 + dy^2 + dz^2\\
&= -\left(1 - \omega^2 ({x'}^2 + {...
1
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0answers
21 views
Derivation of residual reparameterization symmetry equation
Here is my attempt to derive the residual reparameterization equation $$\partial_{\alpha}\xi_{\beta}+\partial_{\beta}\xi_{\alpha}=\Lambda \eta_{\alpha\beta}$$
For a reparameterization $x^{\alpha}\...
0
votes
0answers
31 views
Parallel transport and Geodesic deviation
We know that when we derive the Geodesic equation, we want to actually understand the geometrical meaning of the Riemann tensor. We see from the geodesic equation that the second derivative of the ...
0
votes
1answer
42 views
Extending $\mathbb R^3$ coordinate systems concepts
I was thinking about how to use different coordinate systems in 3D space and how to describe curved surfaces embedded in 3D space when I realized that all the notations I know make sense only if ...
8
votes
2answers
225 views
Covariant derivative of the spin connection
I wish to compute $$[\nabla_{\mu}, \nabla_{\nu}]e^{\lambda}_{~~a}. $$
To do so, I make use of $\nabla_{\nu}e^{\lambda}_{~~a} = \omega_{a~~~\nu}^{~~b}e^{\lambda}_{~~b}$, so that I may write
$$\nabla_{\...
1
vote
1answer
76 views
Christoffel Symbols and Metric Tensor
I know that the Christoffel Symbols are made out of the first derivative of Metric Tensor. Is there any relation between the number of metric components and Christoffel symbols? Is there any general ...
0
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0answers
13 views
How to check if a parametrized curve is a geodesic
Given some parametrization in the plane, e.g., $x = \sin \lambda$ and $y = \cos \lambda$, how can I know if that curve is a geodesic? $x$ and $y$ constitute a circle in the plane, centered at $(0,0)$ ...
3
votes
2answers
78 views
How to interprete gauge symmetry in following way?
From Wikipedia:
In here, $G$-principal bundle $P$ on spacetime $\Sigma$ is given. Gauge transformation $\phi$ can be written in three ways, and it seems to use the third way. More specifically,
\...
2
votes
1answer
94 views
Second derivative of a function in a manifold
Suppose we have a function curve $\gamma(t)$ on a manifold $M$ . Define the function $$f(t)=\frac{1}{2}g_{ab}\dot\gamma(t)^a\dot\gamma(t)^b$$
Introducing coordinates $x^i$ the first derivative of the ...
9
votes
2answers
93 views
Anticommutation of variation $\delta$ and differential $d$
In Quantum Fields and Strings: A Course for Mathematicians, it is said that variation $\delta$ and differential $d$ anticommute (this is only classical mechanics), which is very strange to me. This is ...
3
votes
1answer
80 views
“The moment map gives more than Noether's theorem”
I recently ran across this Not Even Wrong blog post which has the following passage
The moment map however gives you much more [than Noether's theorem], with phase space providing structure that is ...
1
vote
2answers
61 views
Understanding the Israel Junction Conditions
The well known and frequently used Israel Junction conditions are the equivalent of Einstein's field equation on a membrane in the brane-world picture.
All the sources have this notation: $K_{ab}^{(i)}...
0
votes
1answer
42 views
How do 'locally Euclidean' and 'Lorentzian' requirements in manifolds reconcile?
In GR, we define our manifolds to be locally Euclidean. However, we also demand that our metric tensor have a Lorentzian signature. Since the metric tensor is a measure of curvature, doesn't the first ...
0
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0answers
41 views
Covariant derivative and vector fields along a curve
I have to prove a relation that involves the covariant derivative induced by the Levi-Civita connection.
With the reference on
N. Straumann General Relativity
Given $T=\frac{\partial x^\alpha}{\...
1
vote
1answer
84 views
Interpretation of black hole metric with fractional $\kappa$ instead of the usual $\kappa\in\{-1,0,1\}$
The metric for a black hole can be written:
$$d s^{2}=-\left(\kappa-\frac{2 M}{r}\right) d t^{2}+\left(\kappa-\frac{2 M}{r}\right)^{-1} d r^{2}+r^{2} d \Sigma_{2, k}^{2}$$
where $\kappa=-1,0,1$ ...
0
votes
0answers
37 views
Gravity action in terms of Null-tetrads
I have seen gravity action being written in terms of spin-connection and the vierbeins. Has anyone attempted to write the gravity action in terms of Null-tetrads?
0
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2answers
47 views
Cosmology and Spherical Coordinates
My question refers to page 10 of this document. Specifically, when using spherical polar coordinates in cosmology, why does the author of this work choose the origin of the coordinate system to be at ...
1
vote
1answer
95 views
Is this case a failure of Stokes' theorem?
In the presence of a hypothetical magnetic point charge at the origin of coordinates, it turns out that an irremovable physical singularity of the vector potential ${\bf A}({\bf r})$ exists for any ...
3
votes
1answer
76 views
Why is covariant derivative of Null-tetrads not zero?
I think the null tetrads are just re-writing of the vierbiens. However, the vierbiens satisfy $\nabla_\mu e^{a}_{~~\nu} = 0$, but the same is not true for the null tetrads because the spin ...
0
votes
3answers
75 views
Is $g(\nabla_{e_0}X,)$ is equal to $\nabla_{e_0}g(X,)$?
Let $g$ be a Riemannian metric , $e_0$ and $X$ vector field with $g(e_0,e_0)=-1$ .From the compatibility condition of the metric we have for another field $Y$ we have
$$\nabla_{e_0}g(X,Y)= g(\nabla_{...
0
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0answers
24 views
Building lightweight 1 km high fence [closed]
I would like to build a fence that is 5 km2 and height of 1km. This wouldn't be feasible as the material needed and cost implications plus environmental factors. Therefore I thought of finding the ...
1
vote
0answers
41 views
Is there a smooth, spherically symmetric, static, asymptotically de Sitter spacetime of constant scalar curvature?
...other than de Sitter space itself? Two clarifications:
I am not talking about solutions of Einstein's equations, just spacetimes. So, general Lorentzian manifolds.
I am not counting, for example,...
2
votes
1answer
66 views
Fiber bundles and relativity
In the book Geometrical Methods of mathematical physics, in the section 2.10, talking about fiber bundles, gives the example of the Newtonian physics fiber-bundle structure:
The view of spacetime ...
2
votes
1answer
59 views
As a physicist, why are associated bundles important?
I have a good grasp on principal bundles as providing a lie group on some fibers of our field. So for example, the wavefunction tells us the phase of a particle in space and time, and this can be a ...
0
votes
0answers
49 views
Why and where Poincaré gauge theory fails?
I would like to post this question because I have seen no one post it in this explicit way.
From what I have seen, Poincaré gauge theory uses connections (gauge fields) like the spin connection and ...
1
vote
1answer
37 views
Dimensions of velocity vector in differential geometry
If we have a velocity vector written in, say, cartesian coordinates:
$\mathbf V$ = $\dot{x}$ i $+$ $\dot{y}$ j
Note that Dim($\dot{x}$) = Dim($\dot{y}$) = $LT^{-1}$ which are the dimensions of speed ...
0
votes
1answer
37 views
Killing equation and normal coordinates
I have already seen a similar question but I have not sure to have understand completely so I hope you can help me.
If I write the killing equation $\cal{L}_X g=0$ as $X_{\alpha;\beta}+X_{\beta;\alpha}...
0
votes
2answers
77 views
Can flat spacetime have curvature?
Hi I am new and I hope this can make sense. Is it true that flat spacetime can curve? For example, you have a flat piece of spacetime that curves at a constant rate of pi/2. the flat spacetime would ...
3
votes
1answer
73 views
When and why can the spin connection term of the Dirac Operator be omitted?
The Dirac Operator $D$ is defined by
\begin{equation}\tag{1}
D=i\gamma^a\nabla_a=i\gamma^a\nabla_{e_a}=i\underbrace{\gamma^a{e_a}^\mu}_{=\gamma^\mu}\nabla_{\partial_\mu}=i\gamma^\mu\nabla_\mu=i\gamma^\...
1
vote
1answer
49 views
Hyperbolic isometries in the context of General Relativity
In the context of hyperbolic geometry, it is possible to create a classification for isometries.
I would like to know if these isometries have any particular meaning in the context of general ...
0
votes
1answer
41 views
The contravariant derivative of a substitution for the de Sitter metric
Consider the de Sitter metric:
$$ds^2 = (1-\frac{r^2}{a^2})dt^2 -(1-\frac{r^2}{a^2})^{-1}dr^2-r^2d\Omega^2$$
I know that we rewrite the metric as $(u,r,\theta \phi)$ using the substitution $$u = t-...
0
votes
0answers
43 views
Property of Dirac operator in spinor frame bundle
The Dirac operator is defined by
$$d=g(e_\alpha,e_\beta) \gamma(e_\beta)\nabla_{e_\alpha} \tag1$$
Here $\nabla$ is the spin covariante derivative $e_\alpha$ basis of the tangent space and $\gamma$ the ...
0
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0answers
45 views
Surface swept out by tangent vector [migrated]
This question is from Novikov and Fomenko's Modern Geometry Part I. I quote the question here (problem 6 in Exercise 8.4) in full:
Let $S$ denote the surface swept out (i.e. "generated") by ...
-1
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0answers
22 views
Gauge transformation in a frame bundle
Let $\pi:P\longrightarrow M$ be the orthogonanal frame bundel. A gauge transformation is an automorphism
$f:P\longrightarrow P$ such that $\pi(p)=\pi(f(p))$ that is fiber preserving automorphism.
Now ...
2
votes
2answers
132 views
Clarification about local Lorentz transformation
Note there are others questions about local Lorentz transformations and global Lorentz transformations but they all concerned about mathematics. Here what I am trying to understand is the link ...
0
votes
1answer
66 views
What are physical evidence that we are living in affine space?
An affine space of dimension n on $\mathbb R$ is defined to be a non-empty set $E$ such that there exists a vector space $V$ of dimension n on $\mathbb R$ and a mapping
$\phi:E \times E \rightarrow V,\...
0
votes
0answers
72 views
Equation of a conformal Killing vector
Question 10(a) on pages 469-470 in the book "Spacetime and Geometry" by Sean Carroll asks:
Suppose that two metrics are related by an overall conformal transformation of the form:
$$ \tilde{...
0
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0answers
42 views
Diffeomorphisms and coordinate changes
In this question: Simple conceptual question conformal field theory, an answer states that the invariance of the theory under the change of coordinates is a particular case of invariance under ...
0
votes
2answers
80 views
Calculating proper time from different perspective and obtaining different results
A $\textbf{clock}$ carried by a specific observer $(\gamma, e)$ will measure a $\textbf{time}$
$$
time := \frac{1}{c} \int_{\tau_0}^{\tau_1} d\tau \sqrt{ \eta(v_{\lambda}, v_{\lambda}) }
\tag1$$
...
0
votes
4answers
73 views
Is the set of spacetime manifold a set of event?
In general relativity spacetime is defined as manifold.
Is this manifold the set of events? For example if our universe is constituted of only one non-interacting particle, should our manifold be a ...
1
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0answers
46 views
Identity regarding problem 2.4 of Walds General Relativity
I was going through some of the problems in Wald's General Relativity and in problem 4 chapter 2 I found something that confuses me. So, basically we are asked to show that in any coordinate chart $(...
0
votes
1answer
54 views
Tangent spaces at different points in special relativity
In the book Theoretical Physics by J. C. Dutailly the author says
However because the manifolds are actually affine spaces, in SR and
Galilean Geometry the tangent spaces at different points share ...
0
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0answers
18 views
Infinite differentiablity in Nature? [duplicate]
I'm working through understanding Clay Institute's paper addressing their Navier-Stokes millennium prize.
I can't find any resources online stating if there are any infinitely differentiable examples ...
0
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0answers
35 views
Questions on Affine Parameter
According to p.75 of Hobson's General Relativity book, the author defines an affine parameter along a curve by a parameter that makes the length of the tangent vectors constant along the curve. By ...
0
votes
0answers
39 views
Transformation of Christoffel Symbol
I'm studying General Relativity by Hobson and got stuck in understanding the derivation of the transformation law of Christoffel symbols (p. 64). Let $x^a$ and $x'^a$ be two coordinate systems on a ...
7
votes
1answer
132 views
Lie brackets of vector fields along a geodesic to obtain the Jacobi equation
I have done this question in mathstack but someone has suggested me it is more appropriate to ask this here.
With reference in https://archive.org/details/GeneralRelativity/page/n82/mode/1up, where it ...
0
votes
1answer
62 views
Transformation law of vector fields on $\mathbb{R}^n$
So suppose we have a function $F$ from $\mathbb R^2$ to $\mathbb R^2$ defined by $F(x,y) = (g(x,y),h(x,y))$ where $g$ and $h$ represent temperate and pressure respectively (the point is, they are both ...
0
votes
0answers
61 views
Transformations in General Theory of Relativity
In our lecture we dicussed that certain elements in GTR transform as certain other things, i.e.
Christoffel symbol transforms as an affine connection
covariant derivative of a contravariant vector ...
1
vote
3answers
147 views
Covariant gradient - What am I missing?
I know that the components of the gradient should transform covariantly, and have used this many times in special relativity, etc. However, I also know that covariant components transform like the ...
