Conformal wavefunctions for graviton amplitudes

Chang Liu,David A Lowe

doi:10.1007/jhep01(2022)148

Chang Liu, David A Lowe

Open Access

https://doi.org/10.1007/jhep01(2022)148

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Abstract

The extended-BMS algebra of asymptotically flat spacetime contains an SO(3, 1) subgroup that acts by conformal transformations on the celestial sphere. It is of interest to study the representations of this subgroup associated with gravitons. To reduce the equation of motion to a Schrodinger-like equation it is necessary to impose a non-covariant gauge condition. Using these solutions, leading-order gauge invariant Weyl scalars are then computed and decomposed into families of unitary principal series representations. An invertible holographic mapping is constructed between these unitary principal series operators and massless spin-2 perturbations of flat spacetime.

Highlights

For Minkowski spacetime in four dimensions, the gravitational perturbations can be expanded in terms of both the scalar and the vector spherical harmonics defined on the 2-sphere
We consider the Newman-Penrose formalism [5] of general relativity and construct leading-order gauge-invariant scalars known as the Weyl scalars
A gauge choice allows us to express the metric perturbation hμν, expanded in terms of the spherical harmonics Ylm, as hab = fabYlm, hai = 0, hij = f γij Ylm where fab and f are functions that are related to a master function φ(t, r)

Summary

Scalar perturbations

For the scalar perturbation ref. [9] eq 88 has shown that for hωμνlm and hωμνl m , derived from the master function φωl and φω l via eq (2.1) respectively, the conserved inner product is. [9] eq 88 has shown that for hωμνlm and hωμνl m , derived from the master function φωl and φω l via eq (2.1) respectively, the conserved inner product is. Hωlm, hω l m = −i dr δll δmm J 0 where the orbit spacetime current Ja is given by. Ω hωlm, hω l m = l(l − 1)(l + 1)(l + 2)δ(ω − ω )δll δmm. We see that in order to normalize the metric perturbations to have hωlm, hω l m = δ(ω − ω )δll δmm we will perform a change of variable hμν → [l(l − 1)(l + 1)(l + 2)]−1/2hμν. In the interest of notational clarity we will assume this has been done and will continue to use hμν to denote the normalized scalar metric perturbation

Vector perturbations

Klein-Gordon inner product

Gauge invariant observables

Holographic mapping to the celestial sphere

Discussion

B Weyl scalars for vector perturbation