I am studying Kruskal coordinates for my General Relativity course. On the book Spacetime and Geometry: An introduction to General Relativity by Sean Carroll, the author gives the metric in Kruskal coordinates $$\mathrm{d}s^2=\frac{32G^3M^3}{r}e^{-r/2GM}(-\mathrm{d}T^2+\mathrm{d}R^2)+r^2\mathrm{d}\Omega^2$$ and the Kruskal diagram with $\theta$ and $\phi$ suppressed. How can I relate this diagram with a Minkowski diagram? What does the origin of the Kruskal diagram mean? The author then says that every point is a 2-sphere. How can I imagine this? Can I draw this diagram restoring, let's say, the $\phi$ coordinate in order to get a three-dimensional spacetime diagram?


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You can relate the Kruskal diagram to the Minkowski diagram by realising that, because of the (-+++) signature, $dT^{2}$ takes the role of the time-like coordinate, while $dR^{2}$ takes the role of the space-like coordinate. You can interpret the Kruskal diagram as a "Miskowski diagram of curved spacetime".

You can therefore also introduce lightcones to the diagram by considering the $ds^{2}=0$ null geodesics. It is here where $\theta$ and $\phi$ are surpressed by choosing radial light ($d\phi=0$) in the equatorial plane ($\theta=\pi/2$). By setting these variables to a constant you are limiting the diagram to a 2-dimensional slice of the actual geometry. Though, the surpressed degrees of freedom still exist. For every point, you have neglected the 2-sphere-spanning coordinates. In other words: every point is a 2-sphere.

You can restore $\phi$ by rotating the 2D cut light cones to represent actual cones. Just like with flat space Minkowski diagrams.


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