## B8 Conformally invariant Dab operator

To help us to understand the physical implications for Vv, and to see how the Einstein equations for that region will operate, let us first examine Tab[Q] explicitly:

This is an interesting equation in that the 2nd-order operator

on the left, when acting on a scalar quantity of conformal weight 1 (where the extra symmetry over AB plays no role when the operator acts, as here, on a scalar), had been earlier pointed out to be conformally invariant by Eastwood and Rice.[B10] In tensor terms we can write this (with the sign conventions for Rab adopted here) as

The quantity a does indeed have conformal weight 1, since if gab is further rescaled according to gab — gab = ft2 gab then, taking the definition of a, for the g-metric, to mirror that of a, for the g-metric, gab = a2 gab to mirror gab = a2gab, we find a — a = fta (i.e. a has conformal weight 1). Thus,

We can write this conformal invariance in the operator form

Einstein's equations for the g-metric, written in the g-metric in the terms given above

Daba=4uGa3Tab, tell us that the quantity Daba must vanish to third order across kitself, when (as would be expected) Tab is smooth across k In particular, the fact that Daba = 0 on ktells us that

Va|(a-Vbobw ( = — wOaba-b-) = 0 on k, and we can rewrite this as

(with Nc=Vm, as in B4 above), which tells us that the normals to k are 'shear-free' at k, which is the condition for kto be 'umbilic' at every one of its points.[B11]