Questions tagged [diffeomorphism-invariance]
The diffeomorphism-invariance tag has no usage guidance.
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How does the renormalization process differ between actions with and without diffeomorphism invariance?
If I have an action that is not invariant under a change of coordinates. Does this effect the renormalizability of the theory?
The procedure of renormalizing a field theory essentially boils down to ...
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Does the action have to be coordinate invariant?
Is the Lagrangian density $\mathcal{L}$ in the action $S=\int d^4x \sqrt{-g}\mathcal{L}$ required to be coordinate independent?
If so, and a set of EOM are not invariant under coordinate ...
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Diffeomorphism invariance for derivative of scalar fields [closed]
In GR, it is well-known that the gravitational stress-energy tensor is a pseudotensor, i.e. it is not gauge-invariant. To make it gauge-invariant one needs to take it under average integral $\langle \...
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Diffeomorphism Invariance of Terms in Lagrangian which use Gauge Fields
A term in a Lagrangian is gauge invariant if one makes sure to use quantities which transform in proper representations of the group of gauge transformations. This means that one cannot write terms ...
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Diffeomorphism invariance in 2 dimensional metric [duplicate]
I am reading "String Theory and M-Theory" by Becker, Becker and Schwarz. In Chapter 2, the authors try to gauge fix the auxiliary field. They start from the general expression
$$h_{\alpha\...
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Spatial translational invariance for de Sitter spacetime
Consider de Sitter spacetime in the 'flat slicing' i.e. in the coordinates $$ds^2 = a(\tau)^2\left[-d\tau^2 + dx^2 + dy^2 + dz^2\right],$$
where $x,y,z$ are Cartesian coordinates for space and $\tau$ ...
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Spin Connection, Killing Equation and Spinors under Diffeomorphism
Under a diffeomorphism $dx'^{\mu}=\frac{\partial x'^{\mu}}{\partial x^{\nu}}dx^{\nu}$ we have that the components $A_{\mu}$ of every 1-form $A=A_{\mu}dx^{\mu}$ transform as: $A'_{\mu}=\frac{\partial x^...
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Gauge invariance or diffeomorphism invariance in GR of observables?
I am confused by the definitions of a gauge transformation, a coordinate transformation and a diffeomorphism. In particular, should observables in GR be fundamentally invariant under gauge ...
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How do general coordinate transformations fix components of the metric?
In studying string theory, one sees the argument that we can gauge-fix the worldsheet metric to become the Minkowski metric. The argument usually goes that general coordinate transformations (GCTs) ...
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Current of conformal transformation
Suppose we have a theory with conformal invariance that has been extended to a diffeomorphism invariant theory in a way that the resulting energy-momentum tensor is traceless on-shell (which can ...
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Composition of diffeomorphisms in the ADM formalism?
In the ADM formalism there are 4 constraints: $C_\mu(x)$, which are known as the Hamiltonian and spatial diffeomorphism constraints. In the quantum theory, $C_\mu(x)$ are the generators of ...
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Can any useful physical theories other than TQFTs be formulated on a smooth manifold without a metric structure?
The vast majority of physical theories are formulated on a spacetime that is mathematically represented by a pseudo-Riemannian manifold, i.e. a smooth manifold with a metric tensor structure. The ...
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Is the massive Fierz-Pauli action invariant under changes of coordinates?
The action for linearlized gravity on a curved background has the form
$S_{LinGrav}=\frac{1}{16 \pi G} \int d^{4} x \;\sqrt{-g}\;\left[-\frac{1}{4} \nabla_{\rho} h_{\mu \nu} \nabla^{\rho} h^{\mu \nu} +...
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How to define volume of Weyl transformation and diffeomorphism groups?
I have a trouble in defining Weyl transformation and diffeomorphism group volumes in the formal expression of string partition function on some manifold $M$:
\begin{eqnarray}
Z_M=\frac{\int Dg [DX]_g}{...
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Corrected Poisson's equation and the preferred time direction of the cosmological constant? [closed]
So if I correct (equation 2.6) the Poisson's equation with the cosmological constant. Then, I get
$$ \Delta \Phi = (4 \pi G_N) \rho - \Lambda \tag{1}$$
But when I think of this from the Einstein-...
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Are different "topological" AdS spacetimes isometric?
In this paper https://arxiv.org/abs/hep-th/9808032 the author says that black holes in AdS spacetimes can have horizons with different topologies. In particular, when the black hole mass $M$ vanishes, ...
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How is the Virasoro symmetry realised on $AdS_3$?
In the context of the Holographic correspondence, $AdS_{n}/CFT_{n-1}$, it is often cited as a "confirmation" that the two symmetry groups of the theories correspond. Indeed, in dimensions $n&...
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How does an axial vector field transform under a diffeomorphism?
Does it behave like a normal vector field?
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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 ...
user87745
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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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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$$
or 'local' forms such as:
$$\...
user84158
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Trying to understand the conformal gauge "derivation" in Polyakov action symmetries [duplicate]
In section 2.3 on p. 16 of the book "Basic Concepts of String Theory" by Blumenhagen, Lüst, Theisen, 3 symmetries of Polyakov action are discussed: Poincarè invariance, diffeomorphism ...
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Linearised diffeomorphisms on an 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 ...
user290954
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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:...
user84158
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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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