Questions tagged [diffeomorphism-invariance]
The diffeomorphism-invariance tag has no usage guidance.
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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 ...
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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 ...
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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 ...
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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
$$S=\int d^2\sigma \...
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Why string theory action needs to be manifestly diffeomorphism invariant?
A typical QFT action is obviously not diff invariant, in other words, the integration parameters have physical significance as time and position. For example action for massless free scalar field on ...
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Term violating diff invariance for Wess-Zumino action in higher loops while maintaining conformal symmetry
Conventional quantization for 2-D requires either maintaining diff invariance and sacrificing conformal invariance outlined in the paper:(https://arxiv.org/abs/2010.06771v2) Diffeomorphisms demand ...
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A Question about Diffeomorphism Invariant Action
I remember that the canonical Hamiltonian of a diffeomorphism-invariant theory, in general, is zero. For example, the geodesic equation is derived from the action of arc length $$S[g(\tau)]=\int_{a}^{...
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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
$$
S(\phi, e^\...
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What is the meaning of diffeomorphism invariance?
I have read a bunch of papers and I see "diffeomorphism invariance" and I cannot understand how it works.
For instance, in asymptotic safe quantum gravity, we make 2 assumptions:
...
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Gauge freedom when solving Einstein field equation
In Weinberg's Gravitation and Cosmology, Ch7, the Einstein field equations give us 6 independent equations (totally 10 equations but $∇^\mu G_ {\mu \nu} =0$ gives 4 constraints) while $g_{\mu \nu}$ ...
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Diffeomorphism invariance in GR
I apologize for asking the duplicate question, but I still get confused after reading the related post In general relativity, are two pseudo-Riemannian manifolds physically equivalent if they are ...
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Hamilton-Jacobi-Einstein equation
I have been looking at the Hamiltonian formalism of GR for some time and recently stumbled across the Hamilton-Jacobi-Einstein equation:
$$\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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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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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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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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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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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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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) $$
where $\mathcal{L}$ is a lagrangian density, or if you prefer $\mathcal{L} = \sqrt{-g} \mathcal{\tilde{L}}$ and $\mathcal{\...
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What special symmetries is this subset of diffeomorphisms?
To make a general diffeomorphism I can make an infinitessimal transformation to the coordiante $x^\mu$ to $x^\mu + \delta f^\mu(x)$.
Now I want to restrict it so I only allow transformations such that:...
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Proof of volume density transformation under infinitesimal diffeomorphisms using Levi-civita/ determinant
Given I diffeomorphism $x^\mu \rightarrow y^\mu = y^\mu(x) $, I want to show that the volume density is invariant, i.e. $ \sqrt{-g(x)}\,\mathrm d^4x \rightarrow \sqrt{-g(y)}\,\mathrm d^4y $. The ...
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Understanding the metric transformation under infinitesimal diffeomorphism
In my general relativity course, we are discussing infinitesimal diffeomorphisms defined by $x^{\mu}\rightarrow y^{\mu}(x) = x^{\mu} + \xi^{\mu}(x)$. We have been examining how different objects ...
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Weyl transformation vs diffeomorphism; conformal invariant vs general in/covariant
Background info:
My understanding:
1.
Weyl transformation is a local rescaling of the metric tensor
$$
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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