Equation (4) means that $\mathbf{g}$ and $\mathit{\upsilon }$ are Weyl-covariant objects of respective weights $-2$ and 1.

So far, we have only imposed the Einstein equations that provide, in Ashtekar’s terms [29, 70], the “kinematic arena” for formulating the holographic correspondence, i.e., the nature of the boundary (conformal Carroll) and the connection induced on the conformal boundary by the bulk physical Levi–Civita connection. These data must be specified prior to formulating any holographic theory that is dual to Einstein gravity. In AdS, for instance, one typically imposes, consistent with the Einstein equations, that the conformal boundary is Lorentzian, with an effective speed of light proportional to the cosmological constant, and that the boundary connection is the Weyl–Levi–Civita connection.

Note that this is analogous to the standard treatment of matter fields coupled to a background Lorentzian geometry, where the covariant (or Einstein–Hilbert) energy–momentum tensor is obtained by varying the matter action with respect to the background metric. Its conservation then follows from diffeomorphism invariance and the matter equations of motion. The only difference here is that the connection includes components not determined by the geometry, thereby increasing the number of background fields.

Note that flux-balance equations sharing the same structure as Eq. (20) appear while considering the constraints implied by diffeomorphism invariance for relativistic theories coupled with more general connections, i.e., without requiring metric compatibility or a vanishing torsion; see Refs. [73, 74] and more recently [95].

Performing a thorough analysis, one can check that the right side of Eq. (20) is a divergence if and only if the hypermomentum ${{\mathbf{\Omega }}_0}^a$ associated, in particular, with the “ambiguous” part of the boundary connection vanishes. As we show below, the latter is related to the news tensor which encodes the gravitational radiation: sending it to zero would therefore strongly reduce the bulk solution space.

Namely, if $r$ is affine, then $\beta =0$ everywhere and our gauge conditions are those fixing Newman–Unti coordinates [48, 99, 100]. Furthermore, if $r$ is Sachs’ luminosity distance (i.e., the radius of expanding null spheres), then our gauge fixing leads to Bondi–Sachs coordinates [17, 18].

This is a usual mechanism in Carroll physics; see e.g. Ref. [102] for a detailed explanation.

By this, we mean a transformation $\mathbf{k}\mapsto {\mathrm{e}}^W\mathbf{k}$, where the function $W$ vanishes at the boundary.

Equation (27) reproduces exactly Eq. (4.56) of [21], up to three amendements: the sign convention in the variation, the fact that $l=0$, as it corresponds to $\theta$ here.

See also Refs. [114, 115] for another proposal for the covariant BMS charges derived from a Wald–Zoupas prescription [30] and based on Geroch’s supermomentum [28].