Questions tagged [differential-geometry]
Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.
3,885
questions
9
votes
1
answer
621
views
Topological/Geometrical justification for $\text{CFT}_2$ being special
It is known as a fact that conformal maps on $\mathbb{R}^n \rightarrow \mathbb{R}^n$ for $n>2$ are rotations, dilations, translations, and special transformations while conformal maps for $n=2$ are ...
- 188k
4
votes
0
answers
50
views
Coordinate bases: in what sense is $\partial_i=\frac{\partial}{\partial q^i}$ a basis vector? [migrated]
I have gotten comfortable working with coordinate bases but I still do not fully understand them.
Let $\mathbb{Q}$ be a smooth $n$-manifold and $q=(q^1,\dots,q^n):Q\to \mathbb{R}^n$ some local ...
- 10.6k
1
vote
1
answer
398
views
Levi-Civita symbol in 2-spinor notation
I'm reading An Introduction to Twistor Theory, by Huggett and Tod, and I don’t get the result we're being given page 17: the 2-spinor form of the 4 dimensional Levi-Civita symbol.
\begin{equation}
\...
CommunityBot
- 1
1
vote
1
answer
664
views
Finding the correct Christoffel-symbols in a 2+1D space-time
I'm trying to calculate the Christoffel Symbols in a 2+1D space-time with the following metric:
$$ds^2 = N^2(\vec r)c^2dt^2-\phi(\vec r)(dx^1)^2-\phi(\vec r)(dx^2)^2$$
To find the Christoffel ymbols ...
CommunityBot
- 1
-1
votes
0
answers
63
views
A solution for Maxwell field
I know $\nabla_\mu F^{\mu\nu}=0$. Do you think the $F^{\mu\nu}=(1/\sqrt{-g})(\epsilon^{\mu\nu\alpha\beta}\partial_{\alpha}A_{\beta})$ can be a solution of this equation? where $\epsilon^{\mu\nu\alpha\...
- 188k
4
votes
1
answer
651
views
Wave function as a section of a complex line bundle to do QM in polar coordinates
If you want to change the coordinates of a Wave function $\Psi$ in 2D QM from cartesian to polar coordinates in a naive way one encounters a problem, namely the (naively defined) radial momentum ...
5
votes
1
answer
180
views
What mathematical constraints does the equivalence principle impose on GR
I'm trying to determine exactly how the equivalence principle affects GR mathematically (rather than conceptually). I found this StackExchange post which more or less says that the equivalence ...
CommunityBot
- 1
1
vote
0
answers
62
views
How to show that covariant derivative of null vectors under variation of the null vectors themselves is preserved?
Having two null vectors with $$n^{a} l_{a}=-1, \\ g_{ab}=-(l_{a}n_{b}+n_{a}l_{b}),\\ n^{a}\nabla_{a}n^{b}=0$$ gives $$\nabla_{a}n_{b}=\kappa n_{a}n_{b}\\\nabla_{a}n^{a}=0\\\nabla_{a}l_{b}=-\kappa n_{a}...
- 188k
2
votes
0
answers
101
views
Physical meaning of the Riemann curvature tensor with all 4 lower indexes
Since Riemann with 1 upper and 3 lower index mean the changes of a vector after being parallel transported around a closed loop, does Riemann with all 4 lower indexes mean the changes of a co-vector ...
1
vote
1
answer
74
views
Classical systems with compact phase space
In the Hamiltonian formalism of classical mechanics, a system with configuration space $Q$ is represented by a symplectic manifold $(T^*Q,\omega^\mathrm{can})$ called the phase space. The dynamics are ...
- 2,352
2
votes
0
answers
43
views
Linearized gravity in BTZ black hole
I'm currently studying this article on traversable wormholes in an eternal BTZ black hole. In the first pages the authors say that is instructive to check that a small spherically symmetric ...
- 188k
2
votes
3
answers
155
views
What is Dirac's reasoning when showing the curvature vanishing implies we can choose rectilinear coordinates?
In section 12 of Dirac's book "General Theory of Relativity" he is justifying the name of the curvature tensor, which he has just defined as the difference between taking the covariant ...
- 1,248
0
votes
1
answer
55
views
Covariant and partial derivative of a vector field (not component)
Is the covariant derivative of a vector field (not the components of a vector) same as the partial derivative?
I am adding a screenshot from page 69 from General Relativity: An introduction for ...
- 549
1
vote
0
answers
36
views
Raychauduri Equation in The Large Scale Of Space-Time book by Hawking & Ellis
Previously to Raychauduri equation, Hawking-Ellis obtain equation (4.25) (pg. 84) namely
$$
\frac{d \theta_{\alpha\beta}}{ds} = - R_{\alpha 4 \beta 4} - \omega_{\alpha\gamma}
\omega_{\gamma\beta} ...
- 188k
4
votes
2
answers
202
views
Why if the metric tensor components are constant then SR applies?
In this paper, at the end of page 177, it says that if the metric tensor components $g_{\mu \nu}$ are constant, then
with a suitable choice of the system of reference, the special theory of ...
- 188k
1
vote
0
answers
51
views
Raising and Lowering Indices of Curvature 2-Forms
I am trying to understand the use of Cartan's formalism for calculating the Riemann and Ricci Tensors in differential geometry. My question is about the equation relating the curvature 2-forms with ...
- 13
1
vote
1
answer
215
views
Classification of connection coefficients
We always can define the connection coefficient using such a formula:
$$D_{ \mu} e_\nu(x)= \frac{\partial e_\nu(x)}{\partial x^\mu}-\Gamma^\rho_{\mu\nu}(x)e_\rho(x)=0$$
Here is a problem, the ...
CommunityBot
- 1
3
votes
2
answers
324
views
Partial derivatives vs Covariant derivatives in polar coordinates
Covariant derivatives take into account for both component and basis changes, thereby applicable for curved spaces - where partial derivatives only take component changes into account - is this ...
- 1,607
1
vote
1
answer
53
views
Difference between $R^{a}_{bcd}$ and $R_{abcd}$ Riemann tensor types
What is the intuitive, geometrical meaning regarding the usual mixed Riemann tensor $R^{a}{}_{bcd}$ with respect to its purely covariant counterpart $R_{abcd}$?
- 3,503
1
vote
0
answers
62
views
Confusions about partial and covariant derivatives
Let, we are in 1d cartesian space with metric $g_{xx} = x^2$. Let we have a vector $v = 1/x e_x$. Since the vector is designed to shrink its components as the basis grows - its total length will ...
- 188k
0
votes
1
answer
42
views
What is the difference between covariant and contravariant tensors? [duplicate]
What is the difference between covariant and contravariant tensors?
I have been seeing in a lot of problems but I´m not sure what is the difference or if is only a equivalent notation.
- 188k
1
vote
0
answers
48
views
Wald: 2-dim Covariant Derivative for Null Hypersurfaces
On pp. 221-222, Wald introduces the 2-dim "hatted" manifold of null geodesics. He moves from 9.2.30 to 9.2.31 and he is allowed to do so because the tensors have the special properties that ...
- 188k
0
votes
1
answer
51
views
Equating 2 sides of EFE
Can we say if the covariant derivative is zero, the partial derivative is also zero because locally covariant derivative reduces to partial derivatives (since locally spacetime is flat)? Because, in ...
- 188k
2
votes
1
answer
208
views
Variation of the Gauss-Bonnet action and Palatini identity for the purely covariant Riemann tensor
I'm taking the variation of the Gauss-Bonnet action
$$\mathcal{L}_{GB} = \frac{1}{2}\left(R^{2} - 4R^{\mu\nu}R_{\mu\nu} + R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}\right)$$
to obtain the equations of ...
CommunityBot
- 1
4
votes
1
answer
254
views
Pre-requisites for V.I. Arnold's mathematical methods for classical mechanics
I am an undergraduate, studying physics. I have studied maths courses like Groups, Linear Algebra, Real analysis, Differential geometry and probability. I wish to get into mathematical physics, ...
- 65.2k
1
vote
0
answers
53
views
Diffeomorphism Invariance of Terms in Lagrangian which use Gauge Fields
A term in a Lagrangian is gauge invariant if one makes sure to use quantities which transform in proper representations of the group of gauge transformations. This means that one cannot write terms ...
- 188k
0
votes
1
answer
312
views
Geometry of Null hypersurfaces
In Wald section 9.2 page 221 he says that
We turn our attention; now , to null geodesic congruences. Again, we
parameterize the geodesics by an affine parameter $\lambda$, but ,
unlike the timelike ...
CommunityBot
- 1
1
vote
1
answer
514
views
Lie derivative - Problem 8.5 from General Relativity by Hughston & Tod [closed]
I was trying to solve the problem 8.5 from the textbook An Introduction to General Relativity by Hughston & Tod. Given a Riemannian manifold and a vector field $V^a$ such that $\mathcal{L}_Vg_{ij}=...
CommunityBot
- 1
6
votes
1
answer
1k
views
Diffeomorphism & Weyl transformations in the 2D worldsheet of string theory and the existence of conformal gauge
D. Tong's notes on string theory (PDF), subsection 5.1.1, feature the following in introducing the symmetries used in the Faddeev-Popov method:
We have two gauge symmetries: diffeomorphisms and Weyl ...
- 188k
2
votes
1
answer
480
views
Proof for covariant vector transformation law
(I'm asking this on the physics exchange not on the math one because i don't need an extremely rigorous explanation)
I understand the derivation for the contravariant vector transformation law is ...
CommunityBot
- 1
0
votes
0
answers
101
views
Friedmann equation and the Ricci Flow
The Friedmamn equation I'll concern myself with is,
$\frac{\ddot a}{a}=-\frac{4\pi G}{3}\left(\rho+\frac{3p}{c^2}\right)$
Instead of setting $c=1$ a nice mathematical/dimensional trick is to just ...
0
votes
0
answers
49
views
Commuting material time derivative and material space derivative
Let's note $x$ the coordinates in the current configuration and $\nabla$ the associated gradient; similarly, let's note $X$ and $\nabla_0$ for the reference configuration. I will also note the ...
- 188k
0
votes
0
answers
49
views
Is the diameter an invariant in General Relativity?
How can you define diameter in General Relativity? How would you calculate it for a given fixed $r$ in spherical coordinates in Minkowski and Schwarzschild metrics?
I've seen somewhere that "In ...
- 163
3
votes
1
answer
85
views
How does the wavefunction transform under an arbitrary change of variables?
Suppose we have a variable $x$ and a probability density $\rho(x)$. The pushforward of this density under a bijective function $y = f(x)$ is given by
\begin{equation*}
\rho'(y) = \frac{\rho(f^{-1}(y))}...
- 188k
1
vote
0
answers
57
views
Relation between "method of moving frames", spin connection, Cartan forms, and classic rotational kinematics in $\mathbb{E}^n$
I want to know how the "method of moving frames" involving things like connection 1-forms, torsion 2-forms, spin connections, etc. are applied to basic rotational kinematics in flat 3-space (...
- 467
0
votes
1
answer
214
views
Mathematical description of a conical spacetime
In multiple explanations of general relativity, the curvature of a cone has been used to explain why objects fall in a gravitational field, like so:
It also appears in videos like this and this one ...
CommunityBot
- 1
9
votes
1
answer
570
views
How the Poisson bracket transform when we change coordinates?
I'm studying the book Geometric Mechanics by Darryl D. Holm and there's one exercise in the book I'm not quite getting what has to be done. The same discussion the author makes in the book is made on ...
CommunityBot
- 1
4
votes
1
answer
110
views
Does the sign of the connection coefficients $\Gamma^\lambda_{\mu\nu}$ change with the signature of a metric?
I think the answer is no. My reasoning is the following:
We have the Schwartzchild metric with signature $(+,-,-,-)$.
Then we can use the Lagrange-Euler equations
$$\frac{d}{dt}\frac{\partial L}{\...
- 4,154
0
votes
1
answer
380
views
Derivative of the Chern-Simons form
I want to verify the relation of the Chern-Simons form
$$ d \, \text{tr} (AdA+ \frac{2}{3} AAA) = \text{tr} FF$$
where $\omega \mu\equiv \omega \wedge \mu$ and $F=dA+AA$.
Using the property $d\, \...
CommunityBot
- 1
2
votes
4
answers
11k
views
Non-zero components of the Riemann tensor for the Schwarzschild metric
Can anyone tell me which are the non-zero components of the Riemann tensor for the Schwarzschild metric?
I've been searching for these components for about 2 weeks, and I've found a few sites, but ...
- 1
36
votes
8
answers
5k
views
Interval preserving transformations are linear in special relativity
In almost all proofs I've seen of the Lorentz transformations one starts on the assumption that the required transformations are linear. I'm wondering if there is a way to prove the linearity:
Prove ...
- 1,248
3
votes
0
answers
78
views
Lie derivative of surface gravity on killing horizon
Carroll's 'spacetime and geometry' says surface gravity $\kappa$ is defined by $K^\mu \nabla_\mu K=-\kappa K$ where $\nabla$ and $K$ each denote covariant derivative and Killing vector field.
This is ...
- 188k
1
vote
0
answers
53
views
Curvature and stability
In Topological methods in hydrodynamics 1 mentioned that "The Riemannian curvature of a manifold has a profound impact on the behavior of geodesics on it. If the Riemannian curvature of a ...
- 188k
5
votes
2
answers
306
views
Why can infinite planes be approximated as Gaussian surfaces?
A little background: I'm an undergraduate studying Electrodynamics, currently in Chapter 8 of Griffiths.
A question I came across (8.4 part a for those curious) asks for a calculation of the force ...
CommunityBot
- 1
0
votes
0
answers
42
views
Vanishing metric component?
Consider a metric $g_{\mu \nu} = \text{diag}(g_{xx},g_{yy},g_{zz})$.
Now consider another metric $h_{\mu \nu}=\partial_x g_{\mu \nu} = \text{diag}(\partial_x g_{xx} ,\partial_xg_{yy},\partial_x g_{zz})...
- 188k
0
votes
1
answer
88
views
What does it mean for a quantum field theory to "live" on a manifold?
I was attending lectures om holography where the lecturer kept on mentioning that a QFT lives on a Cauchy slice. What does that mean?
Is it such that each point of the slice is associated to a unique ...
- 188k
1
vote
1
answer
129
views
Precise definition of a string worldsheet as a manifold in string theory
I've spent some time studying some definition in smooth manifolds theory in order to give a proper definition of a worldsheet in classical string theory at least. My attempt is the following:
...
- 1,046
1
vote
1
answer
71
views
Simple distance calculation in General Relativity
So imagine a spacetime with the Schwarzschild metric:
$$ds^2=-c^2\left(1-\dfrac{2GM}{c^2r}\right)dt\otimes dt+\dfrac{1}{\left(1-\dfrac{2GM}{c^2r}\right)}dr\otimes dr+r^2\left(d\theta\otimes d\theta+\...
- 163
2
votes
0
answers
41
views
Gauss-Bonnet term in tetrad formalism
In eq. 4 of https://arxiv.org/abs/hep-th/9508128 it is stated that the Gauss-Bonnet term for gravity in 4d can be written as
$$
S=\frac{1}{4}\int d^{4}x\,\epsilon^{\mu\nu\alpha\beta}\epsilon_{abcd}R_{\...
- 188k
1
vote
2
answers
80
views
Covariant derivative of gauge theory in curved space
I am reading Witten's article and have a basic question about gauge theory in curved space.
In ordinary flat space (Euclidean space or Minkowski spacetime), covariant derivative of a gauge field $A_{\...
- 10.3k