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0answers
30 views

Are general coordinate transformations and diffeomorphisms the same? [duplicate]

Infinitesimal diffeomorphisms $x{}^\mu \rightarrow x{}^\mu + \xi{}^\mu$ (with $\xi{}^\mu \ll 1$) change geometric objects by means of the Lie derivative, that is, $X \rightarrow X + \mathcal{L}_\xi \, ...
3
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1answer
138 views

Diffeomorphism invariance and geodesic action

I'm trying to understand the role of diffeomorphism and isometry invariance in the geodesic action in GR: $$ S = \int_{\tau_1}^{\tau_2} \! d\tau~ g_{ab}(x(\tau)) \frac{dx^a}{d\tau} \frac{dx^a}{d\tau} ...
4
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1answer
320 views

What's the difference between the diffeomorphism invariance and reparametrization invariance?

Can somebody tell me what's the difference between the diffeomorphism invariance and reparametrization invariance?
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0answers
44 views

Gravity as a gauge theory - Cartan-Killing form?

First, let me state the form of Lagrangian for YM and GR \begin{align} L_{YM} = \alpha \textrm{tr}(F^2), \qquad L_{GR} = \beta R \end{align} I heard, YM is a gauge theory but GR isn't a really gauge ...
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3answers
137 views

Is the local Lorentz transformation a general coordinate transformation?

There is a saying in Nakahara's Geometry, Topology and Physics P371 about principal bundles and associated vector bundles: In general relativity, the right action corresponds to the local Lorentz ...
0
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1answer
110 views

Gauge transformations in gravity [duplicate]

The Maxwell equations are invariant under the transformation $$A_{\mu} \rightarrow A_{\mu} - \dfrac{1}{e}\partial_{\mu}\alpha(x)$$ where $\alpha(x)$ is a phase transformation varying from point to ...
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2answers
101 views

What does coordinate invariance mean?

I would like to really understand what the mathematical as well as Physical meaning of coordinate invariance is. I have pretended to know what this means, but upon thinking a little harder today, I am ...
3
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1answer
148 views

Diffeomorphism group vs. $GL(4,\mathbb{R})$ in General Relativity

I am quite confused with the groups Diff$(M)$ and $GL(4,\mathbb{R})$ in the context of general relativity. I understand that the symmetries of GR are the transformations that leave the equations ...
1
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1answer
76 views

Reparameterization invariance of strings

I'm working through Zweibach's First Course In String Theory. I'm stuck on a problem (16.1) where I need to show that the coupling Kalb-Ramond field to a worldsheet have the following transformation: ...
2
votes
2answers
58 views

Invariance under local diffeomorphisms

In the context of the Polyakov action, the action for a relativistic point particle $$ S_P = \frac{1}{2} \int \mathrm{d}\tau \, e(\tau) \left(\frac{1}{e^2(\tau)}\left(\frac{\mathrm{d} X^\mu(\tau)}{\...
2
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2answers
162 views

How are FRW metric and Minkowski metric physically different?

According to GR, matrices are coordinate invariant. Does this mean we can transform FRW metric to Minkowski metric with a coordinate transformation like $$dx'=dx\cdot a(t), dy' = dy\cdot a(t), dz' = ...
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0answers
69 views

Local translations included in covariant general coordinate transformation [duplicate]

It is known that if we use the constraint $R_{\mu\nu}(P^{a})=0$ , i.e. the curvature of local translations vanishes, then we can modify general coordinate transformations (gct) to a covariant ...
3
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1answer
90 views

Length path integral

Let's consider a 2-dimensional Euclidean plane. The length between two points $a$ and $b$ can be defined in the following way: $$ (ab) := \inf_{\gamma} \,\int_0^1 d\tau \,\sqrt{\delta_{ab} \,\dot{\...
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2answers
362 views

Is there an accepted axiomatic approach to general relativity?

I am reading Steven Weinberg's book Gravitation and Cosmology. He makes a big deal out of the equivalence principle and showed a bunch of deductions you can make based on it. This surprised me since ...
0
votes
1answer
55 views

Invariance of the low energy effective string action

It is well known that the action of General Relativity $$S = \frac{1}{16\pi G}\int R\;\sqrt{-g} d^D X$$ is invariant under "diffeomorphisms". The low energy effective action for bosonic strings is $...
1
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1answer
199 views

Can a spacetime solution in GR have no Killing vector fields?

Sometimes Killing vector fields in a given spacetime are described as giving information about a symmetry of that particular spacetime solution. If I look at the requirement of a Killing vector field ...
4
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3answers
172 views

The Hole Argument

I have read explanations of this but haven't really understood. Given a spacetime $(M,g)$ I have read that if I represent the metric in some coordinates $(x,y,z,t)$ as $g(x,y,z,t)$ and then in another ...
5
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1answer
115 views

Kac-Moody algebras in 5 dimensional Kaluza-Klein theory

I am trying to make sense to the issue of how does the Kac-Moody algebra encode the symmetries of the non-truncated theory. Let's contextualize a little bit. Ok, so in the 5 dimensional Kaluza-Klein ...
3
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1answer
420 views

Are diffeomorphisms a proper subgroup of conformal transformations?

The title sums it pretty much. Are all diffeomorphism transformations also conformal transformations? If the answer is that they are not, what are called the set of diffeomorphisms that are not ...
5
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1answer
180 views

Supergravity, torsion and diffeomorphism invariance

The action for $N=1$ supergravity in an $4$ spacetime dimenions is $$ S= \int e\left( R + \overline{\psi}_a \gamma^{abc} D_b \psi_c \right) $$ Here $R$ is the scalar curvature, $e=\det(e_{a\mu})$, ...
2
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0answers
82 views

General covariance and global Poincaré algebras

Reading an article (page 7) I read this: Just as ordinary general covariance may be regarded as the local gauge symmetry corresponding to the global Poincare algebra and local gauge invariance ...
4
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2answers
234 views

Hilbert action's invariance under general coordinate changes

In an article, when considering invariance of the Hilbert action under a general coordinate change this formula appears for how the metric changes $$\delta{}g_{\mu\nu}=\partial_{\mu}\xi^{\rho}g_{\rho\...
2
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2answers
570 views

What transformation is the metric of general relativity invariant under?

My limited understanding of metrics comes from Cartan. From there, I understand that a metric is something invariant under certain transformations, e.g. Lorentz in special relativity. But with the ...
4
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1answer
217 views

Field action of linearized gravity associated with spin-2 particle in Thorne book

In MTW book there is one exercise in which there was proposed to discuss linearized tensor gravity, which is represented as $$ g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}, \quad \eta_{\mu \nu} = diag(1,...
5
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1answer
572 views

6 independent Einstein field equations?

I can't understand the comment on page 409, Gravitation, by Misner, Thorne, Wheeler It follows that the ten components $G_{\alpha\beta} =8\pi T_{\alpha\beta}$ of the field equation must not ...
3
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1answer
209 views

Are background independence and diffeomorphism invariance the same?

What is the difference between background independence, which loop quantum gravity formalism claims to have, and diffeomorphism invariance?
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2answers
2k views

Diffeomorphism Invariance of General Relativity

I'm sorry I know this has been asked before, but I'm still a bit confused. I understand that an active diffeomorphism $\varphi:M\to M$ can be equivalently viewed as a coordinate transformation so that ...
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2answers
2k views

The Role of Active and Passive Diffeomorphism Invariance in GR

I'd like some clarification regarding the roles of active and passive diffeomorphism invariance in GR between these possibly conflicting sources. 1) Wald writes, after explaining that passive ...
7
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0answers
191 views

Time diffeomorphisms breaking in inflation

I am currently working on the topic of inflation. It seems that at the stage of inflation, the universe can be described as a de Sitter space. In such a space, all spacetime diffeomorphisms are ...
7
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1answer
178 views

Relations between diffeomorphism symmetry theories and invariant $SU(N), N \rightarrow \infty$ theories

Is it possible to have, an exhaustive panorama (as much as possible), about the relations between theories having a diffeomorphism symmetry, and theories having a $SU(N), N\rightarrow\infty$ ...
13
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2answers
764 views

Gauge invariance and diffeomorphism invariance in Chern-Simons theory

I have studied Chern-Simons (CS) theory somewhat and I am puzzled by the question of how diff. and gauge invariance in CS theory are related, e.g. in $SU(2)$ CS theory. In particular, I would like to ...
5
votes
1answer
752 views

failing to see the conundrum in the Einstein hole argument

I've been reading about the Einstein hole argument, and i fail to understand what makes active diffeomorphisms "special" compared to passive diffeomorphismsm also known as good old coordinate ...
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2answers
966 views

argument about fallacy of diff(M) being a gauge group for general relativity

I want to outline a solid argument (or bulletpoints) to show how weak is the idea of diff(M) being the gauge group of general relativity. basically i have these points that in my view are very solid ...
2
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0answers
282 views

composition of space expansion and movement as a gauge invariance

suppose i have a space-time where we have one point-like object* which we will call movement space probe or $\mathbf{M}_{A}$ for short, and it will be moving with constant velocity $V^A_{\mu}$ in ...
4
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2answers
551 views

Diff(M) and requirements on GR observables

This question is kind of inspired in this one: Diff(M) as a gauge group and local observables in theories with gravity The conundrum i'm trying to understand is how is derived the (quite) ...