All Questions
39
questions
3
votes
0
answers
318
views
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