Questions tagged [diffeomorphism-invariance]

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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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39 views

Diffeomorphisms on Polyakov action

The Polyakov action is invariant under arbitrary transformations of the sort: $$ \sigma^{\alpha} \rightarrow \tilde{\sigma}^{\alpha}(\sigma). $$ How do I show that the metric will transform like a ...
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1answer
34 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 $g_{\mu\nu}=e^{2\phi}\eta_{\mu\nu}$....
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49 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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40 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 ...
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189 views

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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1answer
68 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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Trouble in proving geodesic action invariance under diffeomorphism

In this post Diffeomorphism invariance and geodesic action it is said: You found (by computing in local coordinates) that this is invariant under a diffeomorphism $\phi: M \to M$. This statement ...
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1answer
101 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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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 ...
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In general relativity, are two pseudo-Riemannian manifolds physically equivalent if they are isometric, or just diffeomorphic?

In Carroll's Appendix B, he says You will often hear it proclaimed that GR is a "diffeomorphism invariant" theory. What this means is that, if the universe is represented by a manifold $M$ with ...
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Is is true Superstring theory does not satisify diffeomorphism invariance?

So it is well known that string theory contains General Relativity in the classical limit. And assuming the spin-2 fields all couple correctly to the other fields this means it is diffeomorphism ...
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1answer
115 views

Why the volume of a region is not a diffeomorphism invariant? (LQG)

In loop quantum gravity, the volume operator for a given region is not a diffeomorphism invariant. But classically we know that volume is a scalar quantity under a diffeomorphism even if we take the ...
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1answer
162 views

How can you show diffeomorphism invariance of closed string field theory?

String Field theory if it predicts General Relativity should have 26D space-time diffeomorphism invariance (presumably). How can one show that Closed String Field Theory has this symmetry? (Besides ...
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1answer
62 views

Question on how to think about diffeomorphism invariance

I know that GR must be diffeomorphism invariant, which (in my own words) means that GR, and by extension any observable, should not care about what coordinate system one chooses to use. Suppose I ...
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1answer
257 views

Is general relativity resulted from diffeomorphism invariance?

Any action expressed as the integral of a 4-form in 4-dimensional spacetime is diffeomorphism invariant. For example the following 4-form topological (Pontryagin) action $$ S = \int F\wedge F $$ is ...
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135 views

Diffeomorphism invariance & solutions to EFEs

I’ve read (in Sean Carroll’s GR notes, and several other places), that general relativity (GR) is diffeomorphism invariant. By this, it is meant that, if $\phi:M\rightarrow M$ is a diffeomorphism, ...
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Diffeomorphism & Weyl transformations in the 2D worldsheet of string theory and the existence of conformal gauge

D. Tong's notes on string theory, chapter 5 (PDF), feature the following in introducing the symmetries used in the Faddeev-Popov method: We have two gauge symmetries: diffeomorphisms and Weyl ...
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181 views

Is there any example of a physical theory which isn't invariant under translations?

Isn't it trivial that all physical theories in spacetime are invariant under local translations? Is there an example of a theory which isn't invariant under translations? Please, take note that I'm ...
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1answer
77 views

Prove that background independence and diffeomorphism invariance of a spacetime theory are equivalent

A theory's equations can generally be derived from an action along with a principle of least action ($\delta S=0$). The action is given by: $$ S[f_1, f_2, ...]=\int_M \mathcal{L}(f_1, f_2, ..., g_1, ...
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What is the mathematical formulation of the universality of physics in spacetime?

Consider a general spacetime manifold $\mathcal{M}$ of a given dimension (usually $D = 4$). I call two physical constraints that should be imposed on any reasonable classical theory of physics : ...
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Is local Lorentz + diffeomorphism invariance equivalent to full local Poincaré invariance?

Consider classical General Relativity without the torsion field (the affine connection is already assumed to be symmetric from the start). It is well known that this theory is independent of the ...
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108 views

Conformal transformation of fields

I am confused about the field transformation under conformal transformation. Consider the scale transformation of field $\phi$ (not necessarily scalar) In CFT of Francesco et al, formula (2.121), ...
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2answers
144 views

Laws of physics and diffeomorphism covariance

Up to my limited understanding, diffeomorphisms on a space-time manifold can be viewed as changes of coordinates. While studying general relativty, I read that the theory has diffemorphism covariance ...
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Showing Electromagnetism (QED) is not invariant under active diffeomorphisms

I recently gave a presentation on gauge invariance in general relativity that was, in a small part based upon [https://arxiv.org/pdf/gr-qc/9910079v2.pdf]. In this the authors state (top of page 30); ...
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1answer
189 views

Gauge invariance of the quadratic graviton Lagrangian

I've been reading up on how the linearised Einstein field equations can be derived from a field theoretic perspective, by constructing the most general Lagrangian for a massless spin-2 field, $h_{\mu\...
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1answer
155 views

Gravitational wave or “pure gauge”?

Given the following linearized metric tensor ($c=1$) $$h_{\mu\nu}=\begin{bmatrix} 0 & f(t-z) & 0 & 0 \\ f(t-z) & 0 & 0 & -f(t-z) \\ 0 & 0 & 0 & 0 \\ ...
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86 views

What is a diffeomorphism invariant action made from a single vector field?

Given an single vector field $A_\mu(x)$ is it possible to make a diffeomorphism invariant action in 4 dimensions? In the same way that General Relativity is diffeomorphism invariant? My first guess ...
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73 views

Why don't we write the symmetric term $\eta_{\mu\nu}\partial_\rho\xi^\rho$ in a change of coordinates in linearized gravity?

Reading through this lecture, section (5.3.1): In linearized gravity, when we transform from one set of coordinates to another, the background metric remains Minkowski, but we may think of a metric ...
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407 views

How to show that the Einstein-Hilbert action is diffeomorphism invariant?

It is often stated in texts on general relativity that the theory is diffeomorphism invariant (N.B., I am considering active diffeomorphisms), i.e. if the universe is represented by a manifold $\...
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781 views

Why are the metric and the Levi-Civita tensor the only invariant tensors?

The only numerical tensors that are invariant under some relevant symmetry group (the Euclidean group in Newtonian mechanics, the Poincare group in special relativity, and the diffeomorphism group in ...
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794 views

Which global symmetry of Minkowski space (if any) gets gauged to the diffeomorphism invariance of general relativity?

Minkowski space has both translational and Lorentz symmetry, which together give Poincare symmetry. (It also has some discrete symmetries like parity and time-reversal that I won't be concerned with.) ...
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1answer
82 views

Boundary conditions due to local and global diffeomorphisms

Consider the following extract from page 2 of this paper. $AdS_3$ is the $SL(2, \mathbb{R})$ group manifold and accordingly has an $SL(2, \mathbb{R})_{L} \times SL(2, \mathbb{R})_{R}$ isometry ...
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142 views

Diffeomorphic manifolds of inequivalent smooth structures

Given two smooth structures $\mathcal A_1$ and $\mathcal A_2$ of a spacetime manifold $\mathcal M$, we say that they are equivalent if $\mathcal A_1 \cup \mathcal A_2$ is itself a smooth structure (or ...
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231 views

Classical analogue of the theorem of equivalence of the S-matrix

In quantum field theory there is a statement called the equivalence theorem of the S-matrix. S-matrix is invariant under reparametrization of the field. Is there in classical mechanics, the analogous ...
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241 views

Diffeomorphism invariance and correlation functions

Consider the following paragraph taken from page 15 of Thomas Hartman's lecture notes on Quantum Gravity: In gravity, local diffeomorphisms are gauge symmetries. They are redundancies. This means ...
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190 views

Diffeomorphism and causality

Say you have to events A and B such that A is in the future light cone of B. Now say you apply a diffeomorphism transformation $x\rightarrow f(x)$ on the space-time, smoothly transforming the ...
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2answers
514 views

How do coordinate-dependent functions (not dynamical fields) transform under diffeomorphisms?

Scalar fields transform under a diffeomorphism $x\to\tilde x$ under the rule $\phi(x)\to\tilde \phi (\tilde x) = \phi (x)$. They are often said to be invariant, though only kinda of. For example, if $\...
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718 views

Difference between gauge invariance and diffeomorphism invariance

What is the difference between gauge invariance and diffeomorphism invariance? The two seem very similar, but is the distinction between them that a gauge transformation changes the field variables ...
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430 views

Breaking of diffeomorphism invariance after fixing a background metric

The Lagrangian for the gravitational field in absence of matter is the following $$L=1/k\int dx^4 \sqrt g R,$$ where $k=\sqrt G$, $g$ is the determinant of the metric and $R$ the Ricci scalar. It's ...
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Where is a proof that string field theory is generally covariant?

Given a space-time coordinate of a string $X^\mu(\sigma)$ dependent on the position $\sigma$ around the string. And a string field functional $\Phi[X]$, is there a proof that the equations of motion (...
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329 views

What is the purpose of emphasizing that an action is invariant under diffeomorphism?

When learning field theory and string theory, I always see physicists stress the fact that the action, which is an integral of the Lagrangian density $S(x)=\int L(x,\dot{x})dt$, is invariant under ...
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527 views

Diffeomorphism invariance and geodesic action

I'm trying to understand the role of diffeomorphism and isometry invariance in the geodesic action in GR: $$ S = \int_{\tau_1}^{\tau_2} \! d\tau~ g_{ab}(x(\tau)) \frac{dx^a}{d\tau} \frac{dx^a}{d\tau} ...
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What's the difference between the diffeomorphism invariance and reparametrization invariance?

Can somebody tell me what's the difference between the diffeomorphism invariance and reparametrization invariance?
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Gravity as a gauge theory - Cartan-Killing form?

First, let me state the form of Lagrangian for YM and GR \begin{align} L_{YM} = \alpha \textrm{tr}(F^2), \qquad L_{GR} = \beta R \end{align} I heard, YM is a gauge theory but GR isn't a really gauge ...
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501 views

Is the local Lorentz transformation a general coordinate transformation?

There is a saying in Nakahara's Geometry, Topology and Physics P371 about principal bundles and associated vector bundles: In general relativity, the right action corresponds to the local Lorentz ...
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1k views

Gauge transformations in gravity [duplicate]

The Maxwell equations are invariant under the transformation $$A_{\mu} \rightarrow A_{\mu} - \dfrac{1}{e}\partial_{\mu}\alpha(x)$$ where $\alpha(x)$ is a phase transformation varying from point to ...
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676 views

What does coordinate invariance mean?

I would like to really understand what the mathematical as well as Physical meaning of coordinate invariance is. I have pretended to know what this means, but upon thinking a little harder today, I am ...
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Diffeomorphism group vs. $GL(4,\mathbb{R})$ in General Relativity

I am quite confused with the groups Diff$(M)$ and $GL(4,\mathbb{R})$ in the context of general relativity. I understand that the symmetries of GR are the transformations that leave the equations ...