general relativity - 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 group. In order to define the quantum theory on $AdS_3$, we must specify boundary conditions at infinity. These should be relaxed enough to allow finite mass excitations and the action of $SL(2, \mathbb{R})_{L} \times SL(2, \mathbb{R})_{R}$, but tight enough to allow a well-defined action of the diffeomorphism group.

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$SL(2, \mathbb{R})_{L} \times SL(2, \mathbb{R})_{R}$ encodes global transformations of $AdS_{3}$:

1. These transformations transform a physical state into a different physical state.


2. These transformations reach infinity.

Local spacetime diffeomorphisms of $AdS_{3}$ encode gauge transformations of $AdS_{3}$:

1. These transformations transform a physical state into itself.



2. These transformations do not reach infinity.

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Why must boundary conditions on a spacetime be relaxed enough to allow the action of global transformations, but tight enough to allow a well-defined action of the local diffeomorphism group.

I know that global transformations and the diffeomorphism group are definitely in tension, but I do not understand what the words relaxed enough, tight enough and well-defined mean.

Answer

By "boundary conditions" (BCs) in the AdS/CFT (or equivalently in the Graham-Fefferman) settings, we don't mean boundary conditions ON the boundary $r=\infty$, but rather fall-off conditions NEAR the boundary $r\to\infty$. One the GR side, one should specify fall-off conditions on the metric $g_{\mu\nu}$. The actual BCs are usually a result of somewhat messy calculations.

The BCs should for starters:

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  be relaxed enough to allow the group action of global asymptotic symmetry transformations & finite mass excitations, e.g. multiple stars & black holes, because we want the model to be able to accommodate and describe these.

  
  


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  be tight enough (i.e. fall-off fast enough for $r\to\infty$) for the Einstein-Hilbert action integral $S_{EH}[g]$ of the allowed metrics $g_{\mu\nu}$ to be well-defined with a finite value, possibly after renormalization.

  
  


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  be consistent with the EFE.