Abstract
In a companion paper [1] we showed that the symmetry group mathfrak{G} of non-expanding horizons (NEHs) is a 1-dimensional extension of the Bondi-Metzner-Sachs group mathfrak{B} at mathcal{I} +. For each infinitesimal generator of mathfrak{G} , we now define a charge and a flux on NEHs as well as perturbed NEHs. The procedure uses the covariant phase space framework in presence of internal null boundaries mathcal{N} along the lines of [2–6]. However, mathcal{N} is required to be an NEH or a perturbed NEH. Consequently, charges and fluxes associated with generators of mathfrak{G} are free of physically unsatisfactory features that can arise if mathcal{N} is allowed to be a general null boundary. In particular, all fluxes vanish if mathcal{N} is an NEH, just as one would hope; and fluxes associated with symmetries representing ‘time-translations’ are positive definite on perturbed NEHs. These results hold for zero as well as non-zero cosmological constant. In the asymptotically flat case, as noted in [1], mathcal{I} ±are NEHs in the conformally completed space-time but with an extra structure that reduces mathfrak{G} to mathfrak{B} . The flux expressions at mathcal{N} reflect this synergy between NEHs and mathcal{I} +. In a forthcoming paper, this close relation between NEHs and mathcal{I} + will be used to develop gravitational wave tomography, enabling one to deduce horizon dynamics directly from the waveforms at mathcal{I} +.
Highlights
Of infall of matter and radiation in the distant future, growth and dynamics of QLHs is governed by local physics in their immediate vicinity [16]
In a companion paper [1] we showed that the symmetry group G of nonexpanding horizons (NEHs) is a 1-dimensional extension of the Bondi-Metzner-Sachs group B at I+
All fluxes vanish if N is an NEH, just as one would hope; and fluxes associated with symmetries representing ‘time-translations’ are positive definite on perturbed NEHs
Summary
In the asymptotically flat case, a covariant phase space framework for general relativity has been available in the literature for quite some time [32–34]. That framework focused on charges and fluxes associated with asymptotic symmetries and did not consider space-times with internal boundaries. We will be primarily interested in the sub-manifold ΓNcoEvH of Γcov consisting of solutions gab for which N is an NEH ∆ (see figure 1), and perturbations δgab (around metrics gab in ΓNcoEvH) that represent weak gravitational waves carrying non-zero fluxes across this ∆. Thanks to this restriction, charges and fluxes will have physically expected properties
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