Conformal Diagrams

Continuing my studies of conformal transformations and diagrams I move on to appendix H, follow Carroll's logic carefully and attempt to plot his conformal diagram of Minkowski space which he shows in Fig H.4 and I have copied above in the centre. My effort is on the right. The diagrams are similar except that the curves of constant $t$, the Minkowski coordinate, have gradient 0 nowhere on his diagram and twice on mine. And for lines of constant $r$ the score is 1,3 (gradient $\infty$). I was distressed. Carroll does not give explicit equations for the curves so there is quite a long chain of calculation to get them plotted. I triple checked it and could find no error so I ransacked the internet and found the short paper from from 2008 by Claude Semay, title "Penrose-Carter diagram for an uniformly accelerated observer". The first part is only about an inertial observer and Semay draws a conformal diagram for her with lines of constant $t,r$ just like mine. I have reproduced half his diagram on the left. So I think Carroll has made a mistake in his Figure H.4 - perhaps he just guessed at the curves!

Carroll lists the important parts of the diagram

$i^+=$ future timelike infinity ($T=\pi,R=0$)

$i^0=$ spatial infinity ($T=0,R=\pi$)

$i^-=$ past timelike infinity ($T=-\pi,R=0$)

$J^+=$ future null infinity ($T=\pi-R,0<R<\pi$)

$J^-=$ past null infinity ($T=-\pi+R,0<R<\pi$)

Carroll use a symbol like $\mathcal {J}$ not $J$ which he calls "scri". It is hard to reproduce.

Conformal diagrams are spacetime diagrams with coordinates such that the whole of spacetime fits on a piece of paper and moreover light cones are at 45° everywhere. The latter makes it easy to visualize causality. Since Minkowski spacetime has 45° light cones, if coordinates can be found which have a metric which is a conformal transformation of the Minkowski metric, the job is done. The first part of appendix H is devoted to finding conformal coordinates for flat Minkowski spacetime, expressed in polar coordinates - presumably to ease our work later in spherically symmetrical manifolds such as Schwarzschild. So we start from that metric: $${ds}^2=-{dt}^2+{dr}^2+r^2\left({d\theta}^2+\sin^2{\theta}{d\phi}^2\right)$$

On the way to finding conformal coordinates we tried coordinates $$\bar{t}=\arctan{t}\ \ ,\ \ \bar{r}=\arctan{r}$$ which certainly pack spacetime into the range $$-\frac{\pi}{2}<\bar{t}<\frac{\pi}{2}\ ,\ 0\le\bar{r}<\frac{\pi}{2}$$ as you will see below if you press the button. Carroll says it might be fun to draw the light cones on that, so I made a movie:

Light cone at various $\bar{r}$

Carroll's Figure H.2 is also quite confusing. It does not show the $u,v$ axes and I naturally assumed that the $u$ axis pointed down and to the right. It does not. It does the opposite.

Read all the details at Commentary App H Conformal Diagrams.pdf (12 pages)