Questions tagged [diffeomorphism-invariance]

Filter by
Sorted by
Tagged with
0 votes
1 answer
78 views

Mathematical characterisation of diffeomorphisms in General Relativity

Considering the diffeomorphism covariance/invariance of General Relativity, is it possible to characterise mathematically the various kinds of possible transformations $x'^{\mu} = f^{\mu}(x)$? All of ...
user avatar
  • 9
0 votes
1 answer
78 views

State of the art on Modified gravity : going beyond the 2nd order differential equations, diffeomorphism invariance breaking, extra degrees of freedom

I am going to do a state of the art on Modified gravity models. I have found a talk that presents the problematic. In particular, it is said the following things : Modifying General Relativity How to ...
user avatar
  • 1
6 votes
1 answer
292 views

String theory: Conformal invariance and Conformal Killing Vectors

I am confused by the relation between the invariance of the Polyakov action under conformal transformations and the Conformal Killing Vectors (CKVs) appearing during the process of quantization. Let ...
user avatar
  • 165
2 votes
0 answers
46 views

Does compactification of a Nambu-Goto string in one direction break Diff invariance?

Assume we have a Nambu-Goto action, in phase space, for a closed string. If I compactify one coordinate of the target space, do I reduce the diff invariance of the system. We have $$S=\int d^2\sigma \...
user avatar
  • 1,203
3 votes
0 answers
95 views

Why string theory action needs to be manifestly diffeomorphism invariant?

A typical QFT action is obviously not diff invariant, in other words, the integration parameters have physical significance as time and position. For example action for massless free scalar field on ...
user avatar
  • 1,203
0 votes
0 answers
20 views

Term violating diff invariance for Wess-Zumino action in higher loops while maintaining conformal symmetry

Conventional quantization for 2-D requires either maintaining diff invariance and sacrificing conformal invariance outlined in the paper:(https://arxiv.org/abs/2010.06771v2) Diffeomorphisms demand ...
user avatar
2 votes
1 answer
88 views

A Question about Diffeomorphism Invariant Action

I remember that the canonical Hamiltonian of a diffeomorphism-invariant theory, in general, is zero. For example, the geodesic equation is derived from the action of arc length $$S[g(\tau)]=\int_{a}^{...
user avatar
1 vote
1 answer
68 views

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^\...
user avatar
  • 161
2 votes
1 answer
162 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: ...
user avatar
1 vote
0 answers
83 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 $g_{\mu \nu}$ ...
user avatar
2 votes
3 answers
304 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 ...
user avatar
  • 109
4 votes
1 answer
164 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}) \...
user avatar
  • 461
5 votes
1 answer
85 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: $$\...
user avatar
  • 6,080
2 votes
0 answers
127 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 ...
user avatar
3 votes
0 answers
122 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 $g_{\mu\nu}$ ...
user avatar
3 votes
0 answers
154 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 $g_{\mu\nu}$...
user avatar
0 votes
1 answer
53 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 ...
user avatar
5 votes
1 answer
314 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}\...
user avatar
  • 2,815
9 votes
1 answer
135 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 $\...
user avatar
  • 91
0 votes
1 answer
72 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....
user avatar
  • 1,295
0 votes
0 answers
62 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 ...
user avatar
  • 1,295
3 votes
1 answer
90 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 ...
user avatar
  • 923
1 vote
1 answer
133 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 ...
user avatar
  • 1,440
7 votes
0 answers
103 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 ...
user avatar
  • 3,324
0 votes
2 answers
192 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?
user avatar
1 vote
0 answers
55 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 ...
user avatar
3 votes
1 answer
181 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{\...
user avatar
0 votes
1 answer
24 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:...
user avatar
  • 6,080
1 vote
1 answer
152 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 ...
user avatar
  • 194
0 votes
1 answer
485 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 ...
user avatar
  • 194
6 votes
0 answers
414 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 ...
user avatar
1 vote
0 answers
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 ...
user avatar
  • 583
1 vote
1 answer
151 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 ...
user avatar
  • 638
7 votes
3 answers
367 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 ...
user avatar
  • 183
7 votes
2 answers
513 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? ...
user avatar
  • 41.5k
3 votes
1 answer
225 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 ...
user avatar
  • 429
2 votes
2 answers
147 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 ...
user avatar
  • 1,180
2 votes
0 answers
212 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 $\gamma_a$ ...
user avatar
  • 3,078
1 vote
0 answers
274 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 ...
user avatar
0 votes
1 answer
95 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\}}\...
user avatar
1 vote
1 answer
281 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 $ ....
user avatar
1 vote
1 answer
87 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 : \...
user avatar
4 votes
1 answer
191 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 ...
user avatar
  • 193
3 votes
1 answer
654 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 ...
user avatar
  • 29.5k
2 votes
1 answer
155 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 ...
user avatar
1 vote
1 answer
57 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 $F(\phi)$ ...
user avatar
  • 1,493
3 votes
0 answers
86 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 ...
user avatar
  • 81
6 votes
2 answers
311 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 $g_{\mu\nu}=e^{2\phi}\eta_{\mu\nu}$. ...
user avatar
  • 1,921
0 votes
1 answer
145 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 ...
user avatar
  • 2,773
1 vote
0 answers
106 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 ...
user avatar
  • 1,493