**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)

$$

{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:

\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)

I emailed Claude Semay about this to thank him. He replied very quickly

ReplyDeleteDear colleague,

Thank you very much for your interest in my work (the only one with a weak link to general relativity). The incorrect diagram is frequently seen in popular articles or books, even on Wikipedia. I'm glad your calculations confirm mine.

Best regards, CS