Advanced/Retarded Eddington-Finkelstein coordinates & Black/White Holes

The Schwarzschild spacetime is described by $$ds^2=-(1-\frac{r^*}{r})c^2dt^2+(1-\frac{r^*}{r})^{-1}dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2,$$ where $$r^*$$ is the Schwarzschild radius.

The advanced Eddington-Finkelstein (EF) coordinate system is a transformation of the Schwarschild coordinates $$(t,r,\theta,\phi)$$ where the time coordinate is transformed using $$c\bar{t}=ct+r^*\ln|r-r^*|.$$ While the retarded EF coordinate system is transformed from Schwarzschild coordinates using $$c\bar{t}=ct-r^*\ln|r-r^*|.$$

I then read that advanced EF coordinate system describe black holes and retarded EF coordinate system describe white holes. Why is that so?

My understanding is that these two coordinate systems are just two convenient ways to describe Schwarzschild spacetime where the gravitational source has a radius smaller than the Schwarzschild radius. What is the connection to black holes and white holes?

• @safesphere in Schwarzschild spacetime the source of gravity is not the horizon either. The horizon is an outgoing light like surface so nothing on the horizon can affect anything outside the horizon
– Dale
Commented Nov 20, 2020 at 20:57
• @safesphere said “to be the source, the horizon does not need to affect anything outside”. If that were true then it would also apply to the singularity. You cannot forbid spacelike connections in one case and accept them in another. “per the Birkhoff theorem, gravity is already there even before the horizon forms” in Schwarzschild spacetime there is no before the horizon forms, but you are thinking along the right lines for Oppenheimer Snyder spacetime. But there it is not the horizon but the collapsing matter that is the source.
– Dale
Commented Nov 21, 2020 at 13:40
• @safesphere Birkhoff’s theorem is also not about sources of gravity. It neither states nor in any way implies your claim that the source of gravity is the horizon. In GR the source of gravity is the stress energy tensor, and the Schwarzschild horizon is vacuum, so it is not the source.
– Dale
Commented Nov 21, 2020 at 19:23
• @Dale I am sorry I replied to your comment, as I was well aware of your intolerance to opinions different from your misconceptions. Commented Nov 22, 2020 at 6:29
• @safesphere ad hominem is a fallacy, not a logical argument. The EFE says that the stress energy tensor is the source of gravitation. That is zero at the horizon in Schwarzschild spacetime. Therefore the horizon is not the source, regardless of how you characterize me
– Dale
Commented Nov 22, 2020 at 11:54

1 Answer

The different coordinate systems cover different parts of the full solution, which are relevant to different sorts of objects.

The full Schwarzschild vacuum solution has four 4D regions, and 3D boundaries (event horizons) between them. They're conventionally labeled with roman numerals:

Region II is the black hole interior, IV is the white hole interior, and I and III are exteriors. Only one of these exterior regions is present in black holes that form from collapsing matter, and it's arbitrarily taken to be I. A white hole that was simply a time reversal of a black hole would also have only one exterior region. An "eternal grey hole" is the only sort of object in which all four of these regions would actually be physically relevant at once.

The $$r>r^*$$ part of the Schwarzschild coordinates covers region I or III (normally taken to be I) and the $$0 part covers region II or IV (usually taken to be II, if this part of the coordinate map is used, which it typically isn't). The Schwarzschild coordinates don't cover the event horizons at all; $$r=r^*$$ is a coordinate singularity. The lowercase $$t$$ and $$r$$ coordinates shown in this image are Schwarzschild coordinates in units where $$r^*=1$$. ($$T$$ and $$X$$ are Kruskal-Szekeres coordinates.)

Ingoing coordinates (Eddington-Finkelstein and otherwise) cover regions I and II (or III and II) and the boundary between them. Outgoing coordinates cover regions I and IV (or III and IV) and the boundary between them.