Questions tagged [diffeomorphism-invariance]
The diffeomorphism-invariance tag has no usage guidance.
116
questions
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Weyl + Diffeomorphism invariance = Conformal symmety only in flat space?
Let $(M,g)$ be a Riemannian manifold and consider a classical field theory $\phi: M \to \mathbb{R}$ given by some action functional $S$ satisfying Weyl & diffeomorphism invariance
$$
S(\phi, e^\...
2
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1answer
115 views
What is the meaning of diffeomorphism invariance?
I have read a bunch of papers and I see "diffeomorphism invariance" and I cannot understand how it works.
For instance, in asymptotic safe quantum gravity, we make 2 assumptions:
...
1
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0answers
50 views
Gauge freedom when solving Einstein field equation
In Weinberg's Gravitation and Cosmology, Ch7, the Einstein field equations give us 6 independent equations (totally 10 equations but $∇^\mu G_ {\mu \nu} =0$ gives 4 constraints) while $g_{\mu \nu}$ ...
2
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3answers
198 views
Diffeomorphism invariance in GR
I apologize for asking the duplicate question, but I still get confused after reading the related post In general relativity, are two pseudo-Riemannian manifolds physically equivalent if they are ...
4
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1answer
113 views
Hamilton-Jacobi-Einstein equation
I have been looking at the Hamiltonian formalism of GR for some time and recently stumbled across the Hamilton-Jacobi-Einstein equation:
$$\frac{1}{\sqrt{g}} (\frac{1}{2}g_{pq}g_{rs} - g_{pr}g_{qs}) \...
5
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1answer
81 views
Diffeormophism invariance of a non-local series possible?
If I want to construct Lorentz invariant forms involving a scalar field $\phi(x)$ I could have non-local terms such as:
$$\int \phi(x)\frac{1}{|x-y|^2}\phi(y) dx^4 dy^4$$
or 'local' forms such as:
$$\...
2
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0answers
85 views
Trying to understand the conformal gauge "derivation" in Polyakov action symmetries
In section 2.3 of the book "Basic Concepts of String Theory" by Blumenhagen, Lüst, Theisen, 3 symmetries of Polyakov action are discussed: Poincarè invariance, diffeomorphism invariance and ...
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61 views
Are the Maxwell/YM quantum gauge equations background independent in any way?
I have certain confusion about the contrast between gauge theory of the Standard Model always referenced to the setting of static Minkowski spacetime as fixed background and claims about the gauge ...
3
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0answers
102 views
Linearised diffeomorphisms on arbitrary gravitational background Part 2
This question is a follow on from my recent post here, in the sense that I will use the notation introduced there. In that post, I considered infinitesimal diffeomorphisms of a metric $g_{\mu\nu}$ ...
3
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0answers
146 views
Linearised diffeomorphisms on arbitrary gravitational background Part 1
Consider some spacetime $\big(\mathcal{M},g_{\mu\nu}\big)$ parameterised by local coordinates $x^{\mu}$ ($\mathcal{M}$ is a smooth differentiable manifold equipped with a Lorentzian metric $g_{\mu\nu}$...
0
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1answer
47 views
Are fundamental symmetries thought to be exact or just an approximation?
As title states, I wish to know whether fundamental symmetries (in the most general sense of the word, e.g. gauge symmetries, Lorentz invariance, diffeomorphism invariance, not necessarily just global ...
2
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1answer
203 views
Why do we have residual symmetry when we already used all symmetry in gauge fixing of the worldsheet metric?
In Becker, Becker and Schwarz' book about string theory, the following symmetries are listed for the $\sigma$-model of the string:
Poincaré transformations
Reparametrizations $\sigma^{\alpha}\...
7
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0answers
93 views
Why is the group of gauge transformations on the frame bundle isomorphic to $\text{Diff}(M)$?
Consider the frame bundle $LM \to M$ for given Lorentzian manifold $M$. The group $\mathcal{G}$ of gauge transformations of the second kind are automorphisms $\phi:LM \to LM$ covering the identity $\...
0
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1answer
70 views
Why do we integrate up to the invariances
Following Witten's essay What every physicist should know about string theory I understood that in the Hilbert-Einstein action is invariant under diffeomorphism in 1D and under conformal mapping in 2D....
0
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0answers
57 views
Why is the proper time $\tau$ the only invariant under diffomorphizem?
I'm following Witten's essay: What Every Physicist Should Know About String Theory
. When applying path integral in 1D Witten mention:
Part of the process of evaluating the path integral in our ...
3
votes
1answer
85 views
Prerelativity physics, Special Relativity, and General Relativity formalisms summary [closed]
In order this have a better understanding of "the big picture", a tried to do the following summary, but I can't really complete it for GR. This based on the introduction chapter of Wald's ...
1
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1answer
110 views
General relativity from the general linear group
I am looking at this answer: https://physics.stackexchange.com/a/225417/747. It states:
Let $f\colon U\to V$ be any coordinates transformation on charts of a
manifold $U,V\subset\mathcal{M}$ (i. e. a ...
6
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0answers
95 views
No particle states in quantum gravity?
It is established that the only states in the Hilbert space of a gauge invariant theory are states which are gauge invariant. This seems like a redundant sentence, but it has nontrivial consequences ...
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2answers
166 views
Symmetry and Coordinate invariance
How are spacetime symmetries different from simple general coordinate invariance?
Physical laws should be coordinate independent. Are Poincare invariances not simply changing coordinates?
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0answers
54 views
Cosmology without introducing a metric in a manifestly diffeo-invariant way
I have a question that bothers me for quite some time:
Can cosmology be done without introducing a metric explicitly, and in a manifestly diffeomorphism-invariant way? Assuming that we are in the ...
3
votes
1answer
133 views
Noether's theorems and General covariance
Given the action
$$ A = \int_{M} d^{4}x \ \mathcal{L}(\phi, \nabla \phi) $$
where $\mathcal{L}$ is a lagrangian density, or if you prefer $\mathcal{L} = \sqrt{-g} \mathcal{\tilde{L}}$ and $\mathcal{\...
0
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1answer
23 views
What special symmetries is this subset of diffeomorphisms?
To make a general diffeomorphism I can make an infinitessimal transformation to the coordiante $x^\mu$ to $x^\mu + \delta f^\mu(x)$.
Now I want to restrict it so I only allow transformations such that:...
1
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1answer
125 views
Proof of volume density transformation under infinitesimal diffeomorphisms using Levi-civita/ determinant
Given I diffeomorphism $x^\mu \rightarrow y^\mu = y^\mu(x) $, I want to show that the volume density is invariant, i.e. $ \sqrt{-g(x)}\,\mathrm d^4x \rightarrow \sqrt{-g(y)}\,\mathrm d^4y $. The ...
0
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1answer
306 views
Understanding the metric transformation under infinitesimal diffeomorphism
In my general relativity course, we are discussing infinitesimal diffeomorphisms defined by $x^{\mu}\rightarrow y^{\mu}(x) = x^{\mu} + \xi^{\mu}(x)$. We have been examining how different objects ...
6
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274 views
Weyl transformation vs diffeomorphism; conformal invariant vs general in/covariant
Background info:
My understanding:
1.
Weyl transformation is a local rescaling of the metric tensor
$$
g_{ab}\rightarrow e^{-2\omega(x)}g_{ab}
$$
A theory invariant under this Weyl transformation is ...
1
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0answers
78 views
Generalizing a flat-spacetime-approach for time dilation to curved spacetimes
I would like to discuss an idea to generalize a flat-spacetime-approach for time dilation to arbitrary curved spacetimes.
Starting Point
Suppose we have - in flat spacetime - one inertial observer ...
1
vote
1answer
128 views
Do all continuous gauge theories use Lie groups?
The article about gauge theory on Wikipedia contains the sentence "Lie group". How can we prove that the gauge transformations that are given in an article form a Lie group?
I give you an ...
7
votes
3answers
275 views
Covariance in special and general relativity
I am self-studying SR and GR and need to make sense of the covariance principle. I understand the idea that physical principles should have no preference in coordinates and therefore must be expressed ...
6
votes
2answers
466 views
Is the FRW metric physically distinguishable from a metric with a speed of light that changes over time?
There are many questions on this site that ask whether the expansion of space could instead be interpreted as a speed of light that changes over time, e.g.:
Has the speed of light changed over time?
...
2
votes
1answer
164 views
Diffeomorphism invariance and energy momentum conservation
I was reading Sean Carroll book "Space-Time and geometry", in the appendix B he derives the energy momentum conservation from the diffeomorphism invariance of the action, however I don't understand a ...
1
vote
2answers
126 views
Gauge symmetries not from promotion of global symmetries
The most intuitive example of a gauge symmetry is such where you take a theory that has some global symmetry, and ask what needs to be done for this symmetry to be local. This procedure involves the ...
2
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0answers
191 views
Do Lorentz rotations transform the Gamma matrices $\gamma_a$?
Do local Lorentz rotations (see below definition) actually transform the Dirac Gamma matrices? If so, how can they collude with coordinate transformations to make the Gamma matrices
$\gamma_a$ ...
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0answers
232 views
If gravity is a gauge theory, what is the Lie group? [duplicate]
Here I asked a question. In one curious comment, I see a statement that gravity is a gauge theory. However, my definition (based on what I read till date) of a gauge theory is a field theory which is ...
0
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1answer
85 views
General gauge algebra identity
In https://arxiv.org/abs/1011.1145 the following rather general gauge algebra identity (2.4) is used
$$\delta_{gct}(\xi) B_\mu^{\>\>A} + \xi^\lambda R_{\mu\lambda}^{\quad\! A}
-\sum_{\{C\}}\...
1
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1answer
250 views
Diffeomorphism invariance in special relativity
Suppose space time is the manifold $M $ isomorphic $ \mathbb{R^4}$ whit the metric $-\eta_{00}=\eta_{11}=\eta_{22}=\eta_{33}=1$ in the Cartesian coordinates $\Psi(p)=(x^0,x^1,x^2,x^3)$ for $p \in M $ ....
1
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1answer
84 views
Generating new solutions of the Einstein equation by active transformation, and the physical interpretation of the new ones
Given a manifold $\mathcal{M}$ with coordinates $\psi : \mathcal{M} \rightarrow \mathbb{R^4}$ , $\Psi(p)= (r,\theta ,\phi,t)$ for $ p \in \mathcal{M}$
Suppose we have the active transformation $F : \...
4
votes
1answer
155 views
Diffeomorphism invariance for QFT in curved space
My understanding is that one of the driving forces behind the LQG approach is to maintain diffeomorphism invariance inherent in GR on the way to a quantum theory of gravity. Along the way it's often ...
3
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1answer
505 views
Transformation of metric by diffeomorphism: pushforward or pullback?
Let $(M,g)$ be a smooth manifold with a metric tensor of signature $(p,q)$. The signature isn't really important for this question so we leave it general.
If $\Phi : M\to M$ is a diffeomorphism we ...
2
votes
1answer
137 views
Gauge fixing of Polyakov Action
In the Gauge fixing of Polyakov action we do general coordinate transformation where we take the transformation stated below
$$h_{\alpha\beta} = e^{\phi(\sigma)}\eta_{\alpha\beta}.$$
But here the ...
1
vote
1answer
54 views
Field diffeomorphisms preserve the equations of motion
In this paper (Field Diffeomorphisms and the Algebraic Structure of Perturbative Expansion, by Kreimer & Velenich), the authors claim in section 3, page 3, that the field diffeomorphism $F(\phi)$ ...
3
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0answers
83 views
Why is hydrodynamics a diffeomorphism invariant theory?
The fact that general relativity is a diffeomorphism invariant theory means that there is no preferred co-ordinate system in GR. How is it possible to understand this in the context of relativistic ...
6
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2answers
246 views
How is the Ricci scalar of a conformally flat metric non-zero?
I am having trouble with a contradiction arising from some computation, and I cannot figure out at which point I make a mistake.
Consider a conformally flat metric $g_{\mu\nu}=e^{2\phi}\eta_{\mu\nu}$. ...
0
votes
1answer
130 views
Choice of metric breaks diffeomorphism invariance?
In Weinberg's paper on the cosmological constant problem (CCP), he states that diffeomorphism invariance is always broken by the presence of any given metric $g_{\mu\nu}$. He then goes on to say that ...
1
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0answers
84 views
Variation under infinitesimal reparametrization
Say that under the infinitesimal reparametrization $\sigma \rightarrow \sigma' = \sigma-\xi(\sigma)$, $x^\mu$ transforms as a scalar, i.e. $x'^\mu(\sigma')=x^\mu(\sigma)$. I would like to show the ...
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3answers
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Conformal transformation vs diffeomorphisms
I am reading Di Francesco's "Conformal Field Theory" and in page 95 he defines a conformal transformation as a mapping $x \mapsto x'$ such that the metric is invariant up to scale:
$$g'_{\mu \nu}(x'...
1
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1answer
335 views
Gauge dependence of the Einstein tensor and the Riemann/Ricci curvature tensors in non-linear general relativity
The Einstein field equations are given by (with assuming $\Lambda = 0$),
$$
R_{ab} - \frac{1}{2} R g_{ab} = \kappa T_{ab}.
$$
The principle of general covariance states that the form of these ...
3
votes
1answer
328 views
Why are the Klein-Gordon equations warranted from the conservation of the energy-momentum tensor?
If we have an action with a scalar field non-minimally coupled to the gravity:
$$\int dx^4 \sqrt{-g}(-\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{2}\zeta R\phi^2-V(\phi)).....(1)$$
varying ...
1
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1answer
369 views
Special relativity and diffeomorphism invariance
In studying general relativity (GR) we learn that the Einstein-Hilbert (EH) action $S_{EH}=\int_{M}\mathrm{d}v_{g}R$ (where $\mathrm{d}v_{g}=\mathrm{d}^{4}x\sqrt{-g}$, with $g$ the metric tensor) is ...
9
votes
3answers
813 views
Conformal symmetry, Weyl symmetry, and a traceless energy-momentum tensor
I'm trying to drill down the exact relation between conformal symmetry, Weyl symmetry, and tracelessness of the energy-momentum tensor. However, I'm getting quite confused because every book I can ...
3
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1answer
228 views
Diffeomorphism invariance of scattering amplitude in bosonic string theory
It is mentioned in Polchinski's book (vol 1) that the diffeomorphism invariance of the scattering amplitude (see Polchinski, vol 1, eq 5.3.9) follows from the equation of motion of $b_{ab}$ (see ...
