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Asymptotic symmetry of space-time corresponds to diffeomorphism transformation of physical space-time, which manifests as isometry near Conformal Boundary. Asymptotic symmetry can be defined using various frame-work : such as Gauge fixing approach, geometric approach and Hamiltonian approach (see [1]). In the gauge fixing approach, one can find similarities between role played by electro-magnetic potentials in $U(1)$ theory and space-time metric in gravity. Actually, it is more transparent in principal Lie group bundle formalism where the diffeomorphism group for gravity $GL(3+1,R)$ plays the same role of gauge group as $SU(N)$ for Yang-Mills theory.

However, even in classical gravity, it is possible to construct toy models such as propagating shock waves (with reasonable energy conditions) which induces diffeomorphism transformation of physical co-ordinates (see [2],[3]), or consider any tensor field (e.g. velocity field), whose components will change under diffeomorphism. So observables will depend on choice of gauge, unlike in electro-magnetism.

In quantum field theory, it is known that the $S-$matrix for $U(1)$ theory is gauge invariant and is valid for all orders in perturbative expansion. However, for pure gravity (such as Christodoulou and Klainerman space-times), it has been shown that the quantum gravity $S-$ matrix is invariant under a certain subset of the total asymptotic group (see [4]). Based on similarities with $SU(N)$ theories in gauge theory, we should expect that the gravitational $S$-matrix to be invariant under total asymptotic group. While, the shock wave argument shows that it is possible to physically distinguish two choice of gauge in case of gravity, thus $S$-matrix need not be invariant under full symmetry group.

I apologize for the above convoluted arguments, however, I do not understand the frame-work based on which one can identify whether a certain quantity is observable or not. The close similarity b/w role played by diffeomorphism group and SU(N) group in principal lie group bundle formalism or in gauge fixing approach gives the impression that observables in gravity should not depend on a particular frame, while toy models such as shock waves and $S$-matrix argument implies that this conclusion is not true. The fact that only a subset of the total asymptotic group acts as symmetry group for $S$-matrix, it could imply that a certain subset of the total $GL(3+1,R)$ group should act as the unphysical gauge, on which our observables won't depend. How do we choose such a subset? Can there be a classical interpretation for choosing such a subset?

References:

[1]https://arxiv.org/abs/1910.08367

[2]https://ui.adsabs.harvard.edu/link_gateway/1985NuPhB.253..173D/doi:10.1016/0550-3213(85)90525-5

[3]https://arxiv.org/abs/1611.09175

[4]https://arxiv.org/abs/1312.2229

KP99
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    See https://physics.stackexchange.com/a/706483/50583 – ACuriousMind Oct 18 '22 at 19:37
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    The way to distinguish between trivial and non-trivial gauge transformations is that the trivial ones have vanishing charge and the non-trivial ones have non-vanishing charge. Whether one or the other happens depends essentially on the behavior of the gauge transformation near the boundary. The clearest approach to this question IMHO is the Covariant Phase Space. See e.g.: https://arxiv.org/abs/1801.07064 and https://arxiv.org/abs/2009.14334 and https://physics.stackexchange.com/questions/719053/can-residual-gauge-symmetries-have-compact-support/725592#725592. – Gold Oct 18 '22 at 20:10
  • @Gold Thank you for the links, I'll go through it. – KP99 Oct 18 '22 at 21:38
  • @ACuriousMind Thankyou for the link. I once came across the soldering form while learning about Principle lie group bundle, but back then I didn't quite paid attention to it. I will have a look at it. One more question, related to this post: is it possible to have some lie group bundle formalism, where the curvature 2-form in case of gravity is just the Weyl curvature and not the full Riemann tensor? There are lots of similarities b/w Weyl tensor and Maxwell tensor (https://physics.stackexchange.com/q/665326). That could be relevant while talking about invariance of S-matrix – KP99 Oct 18 '22 at 21:48

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