Questions tagged [diffeomorphism-invariance]

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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^\...
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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: ...
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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 $g_{\mu \nu}$ ...
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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 ...
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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}) \...
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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: $$\...
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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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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 ...
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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 $g_{\mu\nu}$ ...
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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 $g_{\mu\nu}$...
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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 ...
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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}\...
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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 $\...
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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....
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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 ...
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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 ...
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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 ...
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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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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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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 ...
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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{\...
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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:...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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? ...
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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 ...
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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 ...
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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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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 ...
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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\}}\...
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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 $ ....
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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 : \...
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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 ...
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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 ...
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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 ...
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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 $F(\phi)$ ...
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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 ...
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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 $g_{\mu\nu}=e^{2\phi}\eta_{\mu\nu}$. ...
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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 ...
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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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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'...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...