Is this in fact an equation for the normal vector of the null surface?

I'm reading the paper "Area, Entanglement Entropy and Supertranslations at Null Infinity" and there is a point that I have a doubt on what the authors are actually doing. First, let me summarize the context. They are considering asymptotically flat spacetimes on which we introduce retarded coordinates $$(u,r,z,\bar{z})$$ near $$\mathcal{I}^+$$ so that the metric becomes $$ds^2=-du^2-2dudr+2r^2\gamma_{z\bar{z}}dzd\bar{z}+\frac{2m_B}{r}du^2+rC_{zz}dz^2+rC_{\bar{z}\bar{z}}d\bar{z}^2+D^zC_{zz}dudz+D^{\bar{z}}C_{\bar{z}\bar{z}}dud\bar{z}+\dots\tag{3}$$ In the above notation $$D_A$$ is the covariant derivative on the unit sphere and $$\gamma_{AB}$$ its corresponding round metric. The coordinates $$z,\bar{z}$$ are holomorphic coordinates on the sphere.

Now, in section 3, page 5, the authors say:

We wish to study the area of a cut $$\Sigma$$ of $$\mathcal{I}^+$$ defined by $$u=u_\Sigma(z,\bar{z})\tag{9}$$ in the geometry (3) and also to study its variation under supertranslations of $$\Sigma$$ $$u_\Sigma\to u_\Sigma+f\tag{10}$$ with the geometry held fixed. Of course this area is infinite so we must introduce both a regulator and a subtraction. We regulate the area by the replacement of $$\mathcal{I}^+$$ wth the past lightcone of a point which approaches $$i^+$$. For the flat Minkowski metric the null hypersurface $$r=-\frac{1}{2}(u-u_0)\tag{11}$$ approaches $$\mathcal{I}^+$$ for $$u_0\to +\infty$$ with $$u$$ held fixed. More generally we solve the ODE $$\left(1-\frac{2m_B}{r}\right)du+2dr=0\tag{12}$$ which guarantees that the surface is null, and choose the integration constants at each $$(z,\bar{z})$$ so that the surface lies at large radius, approaching infinity, for finite $$u$$.

If I understood, they wish to define a familly of surfaces at finite radius such approach $$\mathcal{I}^+$$ in the large $$r$$ limit.

In Minkowski spacetime they do it directly by defining, in Eq. (11) the surfaces as level sets of a function.

My issue is with Eq. (12). Where it comes from? How do we derive it? I believe what the authors want is to define the normal covector $$n_a$$ of the surface. In that case is Eq. (12) just the null covector condition for $$n_a$$ written in Bondi coordinates dropping corrections proportional to $$1/r$$? If not, where the equation comes from?

• Minor comment to the post (v1): In the future please link to abstract pages rather than pdf files. – Qmechanic Jun 29 at 5:16

(12) is an equation for a null hypersurface, just like (11) is, this hypersurface is a past lightcone of a very late-time event, serving as an approximation of future null infinity $$\mathcal{I}^{+}$$. Only (11) is for a Minkowski spacetime while (12) is for asymptotically flat spacetime in Bondi gauge and so it is written as a differential equation. This equation should be understood as following: if a point $$p$$ with coordinates $$(u,r,z,\bar{z})$$ is on a hypersurface, then a nearby point $$p'$$ with coordinates $$(u+du,r+dr,z+dz,\bar{z}+d\bar{z})$$ is also on a hypersurface only if (12) is satisfied for the coordinate differentials.
This equation may be interpreted as $$n_{\mu} d x^\mu=0$$, where $$n^\mu$$ is a normal to this hypersurface. Since this is a null hypersurface this normal is a null vector, $$n_\mu n^\mu = 0$$ and is also tangent to the hypersurface.
• So, we are describing the hypersurface using a one-form $\omega$ such that $\omega(X)=0$ for all $X\in T\Sigma$. For level sets $f = c$ as in (11), the appropriate form is $\omega = df$. Now in the general case, one is defining the form - hence the distribution of tangent spaces to the hypersurface - and getting the surface out of it, which by Frobenius' theorem works out when $\omega \wedge d\omega = 0$. Is this the overall idea? – user1620696 Jun 30 at 15:30