Questions tagged [diffeomorphism-invariance]
The diffeomorphism-invariance tag has no usage guidance.
77
questions
3
votes
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41 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 ...
0
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0answers
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 ...
2
votes
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 $g_{\mu\nu}=e^{2\phi}\eta_{\mu\nu}$....
0
votes
1answer
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 ...
1
vote
0answers
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 ...
0
votes
2answers
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:
$$g'_{\mu \nu}(x'...
0
votes
1answer
68 views
Gauge dependence of the Einstein tensor and the Riemann/Ricci curvature tensors in non-linear general relativity
The Einstein field equations are given by (with assuming $\Lambda = 0$),
$$
R_{ab} - \frac{1}{2} R g_{ab} = \kappa T_{ab}.
$$
The principle of general covariance states that the form of these ...
3
votes
1answer
94 views
Why are the Klein-Gordon equations warranted from the conservation of the energy-momentum tensor?
If we have an action with a scalar field non-minimally coupled to the gravity:
$$\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 ...
0
votes
0answers
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 ...
0
votes
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 ...
1
vote
0answers
109 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 ...
6
votes
5answers
355 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 ...
2
votes
0answers
64 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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
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
$$
is ...
1
vote
0answers
135 views
Diffeomorphism invariance & solutions to EFEs
I’ve read (in Sean Carroll’s GR notes, and several other places), that general relativity (GR) is diffeomorphism invariant. By this, it is meant that, if $\phi:M\rightarrow M$ is a diffeomorphism, ...
4
votes
1answer
266 views
Diffeomorphism & Weyl transformations in the 2D worldsheet of string theory and the existence of conformal gauge
D. Tong's notes on string theory, chapter 5 (PDF), feature the following in introducing the symmetries used in the Faddeev-Popov method:
We have two gauge symmetries: diffeomorphisms and Weyl ...
1
vote
2answers
181 views
Is there any example of a physical theory which isn't invariant under translations?
Isn't it trivial that all physical theories in spacetime are invariant under local translations? Is there an example of a theory which isn't invariant under translations?
Please, take note that I'm ...
2
votes
1answer
77 views
Prove that background independence and diffeomorphism invariance of a spacetime theory are equivalent
A theory's equations can generally be derived from an action along with a principle of least action ($\delta S=0$). The action is given by:
$$ S[f_1, f_2, ...]=\int_M \mathcal{L}(f_1, f_2, ..., g_1, ...
1
vote
0answers
128 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 :
...
7
votes
1answer
531 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 ...
1
vote
0answers
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), ...
2
votes
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 ...
2
votes
0answers
126 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);
...
2
votes
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, $h_{\mu\...
2
votes
1answer
155 views
Gravitational wave or “pure gauge”?
Given the following linearized metric tensor ($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 \\
...
1
vote
0answers
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 ...
0
votes
1answer
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 ...
5
votes
1answer
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 $\...
13
votes
3answers
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 ...
14
votes
2answers
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.) ...
2
votes
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 ...
2
votes
0answers
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 ...
2
votes
0answers
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 ...
1
vote
1answer
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 ...
0
votes
3answers
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 ...
5
votes
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 $\...
4
votes
1answer
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 ...
6
votes
2answers
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,$$ where $k=\sqrt G$, $g$ is the determinant of the metric and $R$ the Ricci scalar. It's ...
2
votes
0answers
64 views
Where is a proof that string field theory is generally covariant?
Given a space-time coordinate of a string $X^\mu(\sigma)$ dependent on the position $\sigma$ around the string. And a string field functional $\Phi[X]$, is there a proof that the equations of motion (...
6
votes
3answers
329 views
What is the purpose of emphasizing that an action is invariant under diffeomorphism?
When learning field theory and string theory, I always see physicists stress the fact that the action, which is an integral of the Lagrangian density $S(x)=\int L(x,\dot{x})dt$, is invariant under ...
4
votes
1answer
527 views
Diffeomorphism invariance and geodesic action
I'm trying to understand the role of diffeomorphism and isometry invariance in the geodesic action in GR:
$$ S = \int_{\tau_1}^{\tau_2} \! d\tau~ g_{ab}(x(\tau)) \frac{dx^a}{d\tau} \frac{dx^a}{d\tau} ...
5
votes
1answer
906 views
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?
0
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0answers
94 views
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 ...
1
vote
3answers
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 ...
0
votes
1answer
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 ...
1
vote
2answers
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 ...
11
votes
2answers
2k views
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 ...
