All Questions
39
questions
23
votes
3
answers
2k
views
Why does the analogy between electromagnetism and general relativity differ if you consider them as gauge theories or fiber bundles?
Electromagnetism and general relativity can both be thought of as gauge theories, in which case there is a natural analogy between them:
(Strictly speaking, the gauge symmetry of diffeomorphism ...
- 48.4k
15
votes
1
answer
3k
views
Diffeomorphisms, Isometries And General Relativity
Apologies if this question is too naive, but it strikes at the heart of something that's been bothering me for a while.
Under a diffeomorphism $\phi$ we can push forward an arbitrary tensor field $F$ ...
- 7,936
12
votes
1
answer
3k
views
Infinitesimal transformations for a relativistic particle
The action of a free relativistic particles can be given by
$$S=\frac{1}{2}\int d\tau \left(e^{-1}(\tau)g_{\mu\nu}(X)X^\mu(\tau)X^\nu(\tau)-e(\tau)m^2\right),\tag{1.8}$$
with signature $(-,+,\ldots,+)$...
- 459
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
9
votes
2
answers
3k
views
What is the physical interpretation of harmonic coordinates?
When I see harmonic coordinates used somewhere, what should my association be?
Is there some general use or need to consider the harmonic coordinate condition?
I don't really see what's behind all ...
- 8,523
8
votes
2
answers
1k
views
Einstein-Yang-Mills Connections
I am playing around with coupling a classical $SU(2)$ Yang-Mills theory to Einstein's equations.
Assuming spherical symmetry, the $SU(2)$ connection can be written
\begin{equation}
A = \omega(r)\...
- 1,609
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
7
votes
2
answers
532
views
Is there a notion of torsion for Yang-Mills/gauge connection?
In theories of gravity, the Riemannian/metric connection, is allowed to have torsion, of which the Levi-Civita connection is the particular torsion-free case.
In the gauge theoretic description of ...
- 174
7
votes
2
answers
260
views
Akin to gauge field, why GR's lagrangian is not $R_{abcd}R^{abcd}$? What's the mathematical or physical meaning of $R_{abcd}R^{abcd}$?
For gauge field theory, the Lagrangian of the gauge field is $$\mathcal{L}=-\frac{1}{4}\mathrm{tr}(\mathcal{F}_{\mu\nu}\mathcal{F}^{\mu\nu})=-\frac{1}{8}F_{a\ \mu\nu}F^{a \ \mu\nu}$$
The field ...
- 5,971
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
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
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
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
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
5
votes
0
answers
118
views
What is the status of gauged gravity [duplicate]
The Standard Model of elementary particles is a gauge theory with gauge group $SU(3)\times SU(2)\times U(1)$, which is really a successful theory.
We might be able to quantize gravity similarly. ...
- 1,487