How did I get here?
While drafting my question, I found this very similar question on our site.
Three days ago, I happened upon the concept of supertranslations and superrotations in General Relativity while reading the comment section of a well-known blog. I decided to try to understand this idea. I even opened up a few books, looked up some references in the endeavor.
Can this be the case?
It seems to be the case that the following statements might be close to gospel truth.
Given a metric, we ask if it can be written in the form:
$g_{\mu \nu} = \eta_{\mu \nu} + O(\frac{1}{r})$
If this is the case, then it is asymptotically flat according to page 270 of Wald, chapter 11.
It may also be the case that given coordinates (t,x,y,z) we create null coordinates:
$u = t - r$
$v = t + r$
We might even be able to use lessens from Penrose to conformally compactify by defining:
$p = (\tan{u})^{-1}$
$q = (\tan{v})^{-1}$
To be very honest and straightforward it is then completely clear from the appendix on page 17 of this refence by Mu-Tao Wang that
$\bar{u} = u - f(\theta, \phi) + O\frac{1}{r}$
as defined in the reference.
Naturally, I reckon that
$f(\theta, \phi) = \sum_{\mathcal{l} = 0}^{\infty}\sum_{ m = \mathcal{-l} }^{\infty} C_l^m Y_l^m(\theta, \phi)$
I think I could probably absorb the math, I can see that the form matches assyptotic flatness and I can see a kind of translation.
My question
My question is probably very easy, soft and qualitative. What are the charges of these symmetries telling us about anything. What if null infinity is a place I can put a detector, and I have a charged black hole spinning somewhere, my thinking is that we can say something about a hypersurface somewhere near the boundary, but what can we say?
About my references
Asymptotic Symmetries in Gravitational Theory by R. Sachs DOI:https://doi.org/10.1103/PhysRev.128.2851
A RELATIVIST’S TOOLKIT by Poisson
Work in the Proceedings of the Royal Society by H. Bondi, M. G. J. van der Burg and A. W. K. Metzner
Wikipedia page on multipole expansion:
https://en.wikibooks.org/wiki/Mathematical_Methods_of_Physics/The_multipole_expansion
Other references embedded in the question itself
