Rate of Change of Bondi-Sachs Mass: Derivation Reference

[ergospherical]

If $(u,v)$ are null coordinates and $S(u,v)$ a two-surface of constant $u$ and $v$ then the Bondi-Sachs mass $M_{\mathrm{BS}}(u) = -\dfrac{1}{8\pi} \displaystyle{\lim_{v \rightarrow \infty}} \oint_{S(u,v)}(k-k_0) \sqrt{\sigma} d^2 \theta$ satisfies (Poisson, 2007)\begin{align*}
\dfrac{d\mathrm{M}_{BS}}{du} = - \lim_{v \rightarrow \infty} \oint_{S(u,v)} F\sqrt{\sigma} d^2 \theta
\end{align*}if $F$ is the outward radiative flux. Where can I find a derivation of this formula?

Poisson, E., 2007. A relativist's toolkit. Cambridge: Cambridge University Press, pp.116-117.

 

[Ambitwistor]

Hello! Thank you for your post. The derivation of this formula can be found in the book "A Relativist's Toolkit" by Eric Poisson, specifically in Chapter 4.3. In this chapter, Poisson discusses the Bondi-Sachs formalism and derives the Bondi-Sachs mass formula using the Newman-Penrose formalism. The formula you mentioned is derived in section 4.3.3.2, where Poisson shows that the Bondi-Sachs mass is equal to the negative limit of the outward radiative flux over a two-surface of constant u and v. I hope this helps!