If $u$ and $v$ are lightlike coordinates for 1+1-dimensional Minkowski space ($ds^2=dudv$), then we can compactify with a coordinate transformation of the form

$$U=f(u)$$ $$V=f(v),$$

where $f$ is a nonlinear function that takes $(-\infty,\infty)$ to an interval of the real line. In principle I don't think there are too many specific requirements for $f$, except that it should be a smooth, odd function with a smooth inverse. Two choices I've seen have been $\tan^{-1}$ and $\tanh$.

However, $\tan^{-1}$ seems to be by far the most popular choice when people feel the need to commit to a particular choice, and my understanding is that this is because then you can tile a cylinder representing the static Einstein universe with diamonds for copies of Minkowski space, and angles then have a natural interpretable on the cylinder. I assume that when we see illustrations of Penrose diagrams in books with things like geodesics drawn in, they were probably done with this choice of $f$.

For Minkowski space, it's trivial to find parametric descriptions of the geodesics, which are, e.g., in the case of a timelike geodesic, of the form $U=f(\alpha+(1+\beta )t)$, $V=f(\alpha+(1-\beta) t)$.

Now consider the case of the Schwarzschild spacetime. Suppose we start with Kruskal-Szekeres coordinates $(T,X)$, so that $U=f(T+X)$ and $V=f(T-X)$, and

$$ds^2 = \frac{32}{r} e^{-r/2} g(U)g(V)dUdV+r^2d\Omega^2,$$

where $g=(f^{-1})'$. (Don't trust me on the details of this expression, especially factors of 2.) Is there any nice closed-form description of any interesting class of geodesics? Null radial geodesics are trivial. I guess circular orbits would come out nice, although they wouldn't look like geodesics on the Penrose diagram because the angular variables are suppressed. For cases where we know a closed-form equation for a geodesic in Schwarzschild coordinates (e.g., radial infall from rest of a massive particle), I guess we get expressions simply by plugging in to the transformation equations, but the results seem like they'd be very ugly. Are there any cases where the expressions turn out simple? Can we make them be simpler with a tricky choice of $f$?

It seems possible that there are examples that can be expressed in closed form on the Penrose diagram, but that can't be expressed in Schwarzschild coordinates in closed form in terms of elementary functions. This possibility arises because the inverse transformation from K-S coordinates to Schwarzschild involves a Lambert W function.


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