Questions tagged [diffeomorphism-invariance]
The diffeomorphism-invariance tag has no usage guidance.
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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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1answer
73 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 : \...
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1answer
44 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 ...
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21 views
Supertranslations, superrotations and beyond
Is there any other hidden asymptotic symmetry beyond supertranslations and superrotations? What about superboosts or alike? And super-special transformations analogue to special conformal ...
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19 views
Superrotation equivalent definition
Just as supertranslations can be defined as certain diffeomorphic transformations such as angular-dependent time translations at the null infinity, what is the equivalent definition of superrotations?
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57 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 ...
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1answer
38 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 ...
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1answer
40 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 $F(\phi)$ ...
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48 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 ...
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56 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 ...
2
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1answer
37 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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1answer
57 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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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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2answers
235 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:
$$g'_{\mu \nu}(x'...
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1answer
77 views
Gauge dependence of the Einstein tensor and the Riemann/Ricci curvature tensors in non-linear general relativity
The Einstein field equations are given by (with assuming $\Lambda = 0$),
$$
R_{ab} - \frac{1}{2} R g_{ab} = \kappa T_{ab}.
$$
The principle of general covariance states that the form of these ...
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1answer
117 views
Why are the Klein-Gordon equations warranted from the conservation of the energy-momentum tensor?
If we have an action with a scalar field non-minimally coupled to the gravity:
$$\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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27 views
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
127 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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1answer
149 views
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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5answers
405 views
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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0answers
66 views
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
127 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 ...
2
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1answer
173 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
63 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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2answers
326 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
$$
is ...
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0answers
147 views
Diffeomorphism invariance & solutions to EFEs
I’ve read (in Sean Carroll’s GR notes, and several other places), that general relativity (GR) is diffeomorphism invariant. By this, it is meant that, if $\phi:M\rightarrow M$ is a diffeomorphism, ...
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1answer
310 views
Diffeomorphism & Weyl transformations in the 2D worldsheet of string theory and the existence of conformal gauge
D. Tong's notes on string theory, chapter 5 (PDF), feature the following in introducing the symmetries used in the Faddeev-Popov method:
We have two gauge symmetries: diffeomorphisms and Weyl ...
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2answers
185 views
Is there any example of a physical theory which isn't invariant under translations?
Isn't it trivial that all physical theories in spacetime are invariant under local translations? Is there an example of a theory which isn't invariant under translations?
Please, take note that I'm ...
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1answer
82 views
Prove that background independence and diffeomorphism invariance of a spacetime theory are equivalent
A theory's equations can generally be derived from an action along with a principle of least action ($\delta S=0$). The action is given by:
$$ S[f_1, f_2, ...]=\int_M \mathcal{L}(f_1, f_2, ..., g_1, ...
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132 views
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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1answer
607 views
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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0answers
118 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
158 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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0answers
133 views
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
197 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, $h_{\mu\...
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1answer
156 views
Gravitational wave or “pure gauge”?
Given the following linearized metric tensor ($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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90 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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1answer
75 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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1answer
455 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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3answers
846 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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2answers
863 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
85 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 ...
2
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0answers
149 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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0answers
237 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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261 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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3answers
201 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
563 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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1answer
802 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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2answers
442 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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66 views
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 (...
