Hi. I've got a quick question on Penrose diagrams for the Schwarzschild space-time that I'd appreciate some comments on. In standard [tex](t,r,\theta,\phi)[/tex] coordinates the Schwarzschild metric is(adsbygoogle = window.adsbygoogle || []).push({});

[tex]ds^2 = -\left(1-\frac{2M}{r}\right)dt^2 + \left(1-\frac{2M}{r}\right)^{-1}dr^2 + r^2 d\Omega^2.[/tex]

The immediate difficulty with this is that if one looks at radial null curves one finds that the light cones appear to close up as [tex]r\to 2M[/tex]. The standard way of getting around this is to define a tortoise radial coordinate according to

[tex]r^* \equiv r + 2M\log|r-2M|.[/tex]

Then the metric in [tex](t,r^*,\theta,\phi)[/tex] coordinates becomes simply

[tex]ds^2 = \left(1-\frac{2M}{r}\right)(-dt^2 + dr^{*2}) + r^2 d\Omega^2.[/tex]

The thing about this radial coordinate is that the light cones don't close up anywhere. The downside, however, is that the surface [tex]r=2M[/tex] has now been pushed to [tex]r^*\to-\infty[/tex]. So, we introduce advanced and retarded Eddington-Finkelstein coordinates by defining

[tex]v \equiv t + r^*,[/tex]

[tex]u \equiv t - r^*.[/tex]

Then, for example, the metric in [tex](u,r,\theta,\phi)[/tex] coordinates becomes

[tex]ds^2 = -\left(1-\frac{2M}{r}\right)du^2 - 2dudr + r^2d\Omega^2.[/tex]

(I think this is correct.) Now to my question. Is there any way that I can conformally compactify this metric so that I can draw a Penrose diagram of it? I know that I can go to Kruskal-Szekeres coordinates and construct a Penrose diagram there, but surely it's possible to construct Penrose diagrams using just [tex](u,r,\theta,\phi)[/tex] or [tex](v,r,\theta,\phi)[/tex] coordinates. My problem is that I can't readily see what coordinate transformations I should take so as to get to a finite coordinate range from the infinite ranges of [tex]u,v,r[/tex]. Can anyone shed some light on this for me?

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# Penrose Diagram for Schwarzschild

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