# Questions tagged [diffeomorphism-invariance]

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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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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: $$\... • 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 ... • 778 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} ... 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}... 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 ... 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}\... • 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 \... • 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.... • 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 ... • 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 ... • 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 ... • 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 ... • 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? 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 ... 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) $$where \mathcal{L} is a lagrangian density, or if you prefer \mathcal{L} = \sqrt{-g} \mathcal{\tilde{L}} and \mathcal{\... 0 votes 1 answer 24 views ### What special symmetries is this subset of diffeomorphisms? To make a general diffeomorphism I can make an infinitessimal transformation to the coordiante x^\mu to x^\mu + \delta f^\mu(x). Now I want to restrict it so I only allow transformations such that:... • 6,080 1 vote 1 answer 152 views ### Proof of volume density transformation under infinitesimal diffeomorphisms using Levi-civita/ determinant Given I diffeomorphism x^\mu \rightarrow y^\mu = y^\mu(x) , I want to show that the volume density is invariant, i.e.  \sqrt{-g(x)}\,\mathrm d^4x \rightarrow \sqrt{-g(y)}\,\mathrm d^4y . The ... • 194 0 votes 1 answer 485 views ### Understanding the metric transformation under infinitesimal diffeomorphism In my general relativity course, we are discussing infinitesimal diffeomorphisms defined by x^{\mu}\rightarrow y^{\mu}(x) = x^{\mu} + \xi^{\mu}(x). We have been examining how different objects ... • 194 6 votes 0 answers 414 views ### Weyl transformation vs diffeomorphism; conformal invariant vs general in/covariant Background info: My understanding: 1. Weyl transformation is a local rescaling of the metric tensor$$ g_{ab}\rightarrow e^{-2\omega(x)}g_{ab} $$A theory invariant under this Weyl transformation is ... • 3,888 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 ... • 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 ... • 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 ... • 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? ... • 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 ... • 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 ... • 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 ... • 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 ... • 9,475 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\}}\...
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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 $p \in M$ ....