All Questions
39
questions
1
vote
1
answer
73
views
Geometrical interpretation of gauge fields of spin other than 2
Gravitation can be interpreted as a gauge theory with a spin 2 graviton field. This graviton field in general relativity is also interpreter as a Riemannian metric. Do other gauge theories also have ...
0
votes
1
answer
66
views
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
$$D_{\mu}V^{\nu}=\partial_{...
- 101
2
votes
0
answers
41
views
Coframe fields and spin connection as gauge fields and gauge invariance of torsion 2-form
I have questions about differential geometry calculations. If there is any misunderstanding of mine in the contents below, please let me know and help me to fix it.
Let's consider a 3-dimensional ...
- 33
1
vote
0
answers
61
views
Reparametrization invariance of Einbein action [closed]
I'm going through David Tong's online lecture notes on String theory. At the end of section 1.1.2, where he introduces the einbein action
$$S=\frac{1}{2} \int d\tau (e^{-1}\dot{X}^2-em^2),\tag{1.8}$$
...
- 11
1
vote
1
answer
159
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Reasoning about spin coupling on curved space
In the course of QFT i learnt that the gauge field emerges from the need of a gauge invariance in the action, as we use the covariant derivative in minimal coupling. Now i'm studying how spin fields ...
- 167
2
votes
1
answer
186
views
Gauge-fixing condition invariant under auxiliary gauge transformation
In quantum gravity one usually splits the metric $g= \bar{g}+h$ into a background field $\bar{g}$ and a fluctuation field $h$. In order to obtain a propagator one has to gauge fix the action (e.g. ...
- 425
0
votes
1
answer
122
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Einstein field equations from covariant derivative of a general linear gauge transformation
A general linear transformation is given by
\begin{align}
\psi'(x) \to g \psi(x) g^{-1},
\end{align}
The gauge-covariant derivative associated with this transformation is
\begin{align}
D_\mu \psi=\...
- 1,548
7
votes
4
answers
1k
views
Question on connections in general relativity and particle physics
$1$ Introduction
It seems that when you learn General Relativity all the technology of bundles are irrelevant (at least in elementary discussions as $[1]$, $[2]$, $[3]$ and others). But, even the ...
- 3,085
1
vote
0
answers
104
views
Doubt on the gauge group of Gravitation
(I wrote the introduction section for the sake of completeness, notation and study. The question per se, is written in the section "My Question")
Introduction
On the one hand of nature, we ...
- 3,085
5
votes
0
answers
146
views
Complex, Holomorphic connection, and Symplectic — a not-so-Kähler manifold?
While studying mathematical physics, I wondered whether if there is a mathematical theory for a manifold that
has a complex structure [almost complex structure $J^\mu{}_\nu$ with vanishing Nijenhuis ...
- 564
1
vote
0
answers
106
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Is the Poincaré gauge theory a real gauge theory in the mathematical sense?
When studying Poincaré gauge theory using Milutin Blagojevich's book on "Gravitation and gauge symmetries" we find an interesting line of thought. But to get that I need to set some ...
- 113
6
votes
7
answers
908
views
What are the analogues of $F_{\mu\nu}$ in General Relativity?
In electromagnetism, the measurable gauge-invariant quantities are the electric and magnetic fields or the six independent components of the field strength tensor $F_{\mu\nu}$. What are the analogues ...
- 11.8k
0
votes
0
answers
248
views
Derivation of the transformation rules for vielbein and spin connection
I have been taking a course on General Relativity. Recently, I was given the following homework assignment, which reads
Derive the following transformation rules for vielbein and spin connection:
$$\...
- 159
4
votes
3
answers
373
views
Local Lorentz invariance of General Relativity
Consider the gravitational action as an integral over a differential 4-form $$ S = \int_{\mathcal{M}} \star F_{ab}\wedge e^a \wedge e^b$$ where $\star F_{ab} = \epsilon_{abcd} F^{cd}$ and $F$ is the ...
- 1,200
1
vote
1
answer
202
views
Gauge-fixing conditions in Einstein-Cartan gravity
What are the gauge-fixing conditions one needs to impose on the tetrad one-form $e^a$ and the spin-connection one-form $\omega^{ab}$ while working in the Einstein-Cartan formalism where both are ...
- 1,200
3
votes
0
answers
318
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GR as a gauge theory whose gauge group is $GL(4,ℝ)$ and whose gauge field are the Christoffel symbols $Γ_𝜇$ viewed as a $GL(4,ℝ)$-valued field
The answer to this question Is spacetime symmetry a gauge symmetry? makes the following claim:
One may indeed view general relativity as a gauge theory whose gauge group is $GL(4,ℝ)$ and whose gauge ...
3
votes
1
answer
4k
views
Transverse-traceless gauge: Why the traceless condition?
I'm right now following a course on GR and I arrived to the gravitational waves part. Letting the metric be that of the plane Minkowski space with a small perturbation:
$$g_{\mu\nu}=\eta_{\mu\nu}+h_{\...
- 577
2
votes
0
answers
54
views
Can the covariant derivative of General relativity be obtained from a $GL(4,\mathbb{R})$ transformation?
Is it possible to obtain general relativity as a gauge theory from the general linear group?
The starting point is:
$$
M'=GM
$$
where $M',M,G$ are elements of $GL(4,\mathbb{R})$.
I believe the second ...
- 1,548
6
votes
1
answer
887
views
What is the analogy between the gauge covariant derivative and the covariant derivative in General Relativity?
A particle in the Dirac field can be described with the following equation
$$i\gamma^\mu\partial_\mu\psi-m\psi=0$$
This is if the particle is non-interacting. However, if we impose a local $U(1)$ ...
- 775
1
vote
1
answer
442
views
Reparametrization invariance of the particle in GR
In the article Antibracket, Antifields and Gauge-Theory Quantization, the relativistic particle in spacetime is studied. Its action is
$$\int\text{d}\lambda\,\frac{1}{2}\left(\frac{1}{e}\dot{x}^2-m^2e\...
- 3,915
0
votes
1
answer
888
views
Kaluza-Klein metric and Ricci scalar?
The metric is
\begin{equation}
ds^2 = G^D_{MN}dx^M dx^N
= G_{\mu\nu}dx^\mu dx^\nu + G_{dd}(dx^d + A_\mu dx^\mu)^2.
\end{equation}
Then
\begin{equation}
G^D = \begin{bmatrix}
G_{\mu\nu} + G_{dd}A_\mu ...
- 333
5
votes
2
answers
327
views
Intuition behind bundle constructions in curved space-time and gauge theories
Let us assume that we have constructed a $G$-principal bundle $P$ over the manifold $M$ (for a curved space-time this is a $GL$-bundle, for a gauge theory I take $U(1)$ = electrodynamics) and the ...
- 306
4
votes
3
answers
372
views
Can we dispense with the Manifold in General Relativity?
I am studying Quantum Gravity by Rovelli. In chapter 2, the author describes the path that Einstein followed to arrive to General Relativity (GR). At the end of the discussion of the hole argument, ...
- 1,335
1
vote
0
answers
86
views
Non-mathematical description of manifolds and bundles in gauge theory [closed]
I am teaching myself gauge theory at the moment and occasionally I need to ask what may appear to be a very random or completely bizarre question that is way off, just in order to check if I have ...
- 79
4
votes
2
answers
301
views
Diffeomorphic but physically inequivalent spacetimes
In the last few years there has been a considerable endeavor in understanding the asymptotic symmetries of quantum gravity on Minkowski Spacetime. This has been tied to a study of the BMS group that ...
- 36.4k
4
votes
3
answers
1k
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 ...
- 3,085
1
vote
1
answer
803
views
Gauge dependence of the Einstein tensor and the Riemann/Ricci curvature tensors in non-linear general relativity
The Einstein field equations are given by (with assuming $\Lambda = 0$),
$$
R_{ab} - \frac{1}{2} R g_{ab} = \kappa T_{ab}.
$$
The principle of general covariance states that the form of these ...
- 11
6
votes
1
answer
485
views
Different types of covariant derivatives in Poincare' invariant differential geometry
I have been reading the lectures: http://www.damtp.cam.ac.uk/research/gr/members/gibbons/gwgPartIII_Supergravity.pdf about Poincare' gauge theory. The Poincare' group is considered as semidirect ...
- 9,778
10
votes
2
answers
598
views
GR as a gauge theory: there's a Lorentz-valued spin connection, but what about a translation-valued connection?
Given an internal symmetry group, we gauge it by promoting the exterior derivative to its covariant version:
$$
D = d+A,
$$
where $A=A^a T_a$ is a Lie algebra valued one-form known as the connection ...
- 281
0
votes
2
answers
74
views
Physical manifold with a natural linear connection on them
Of course in many situation a manifold raised from a physical situation (like spacetime or configuration manifold and so on) are really much more richer than an abstract manifold. for example phase ...
- 1,409