# How does a black hole look like independently of coordinates?

Observing a black hole from far away, far it looks as if infalling matter accumulates on the horizon. If one falls in though, there is no horizon at all (or it always stays in front of you).

Which raises the question, what is the true, uncoordinated, spacetime manìfold? Or are we implicitly always coordinating?

• @planetmaker Haha! Yes, you always get wet. Well, tensors are coordinate independent, but how you show that in a coordinate free picture? Without getting wet... Jun 15, 2022 at 8:55
• @planetmaker In a fixed coordinate system you can use different units. Or adapt the coordinates to units. But you stay in the same frame. Now is there a frame from which we see the true hole? No. What's the true hole then? Jun 15, 2022 at 9:19
• I fail to follow your argument. I can describe a circle as cartesian $P(x,y): \{x^2 + y^2 = R^2\}$. Or shifted: $P(x,y): \{(x-x_0)^2 + (y-y_0)^2 = R^2\}$. Or I can describe it as polar $P(r,\phi) = \{r=R,\phi\in(0°;360°]\}$. (Or any other frame) It's different coordinate systems. It's the same circle. And I can choose whatever units (or even no units at all, if I norm it by whatever default radius $R \longrightarrow R/R_0$) Jun 15, 2022 at 11:56
• @ProfRob Yes, okay, but isn't the description dependent on the coordinates? It looks different in the coordinates used faraway. The hole has no interior as seen from there. But if you fall in it has an interior. Now how does it look in spacetime, Independent of coordinates? Jun 15, 2022 at 16:38
• @planetmaker Yes, of course, but a circle is a static spatial structure to which you look, in your example, in it's restframe. But has a hole a restframe? If you fall in it has a centre, if you're outside of it only a horizon. Jun 15, 2022 at 16:41

## 1 Answer

What a black hole looks like is independent of coordinates - what an observer "sees" or measures does not depend on what coordinate system they choose to make their measurements in.

However, many measurements (with notable exceptions like the spacetime interval) are not invariants for observers in different frames of reference. That is not a problem and there is no "true" reference frame.

Thus a distant observer "sees" (actually they don't because of redshift) material accumulate towards the event horizon and that is true whether they adopt Droste, Kruskal-Szekeres, Gullstrand-Painleve or any other coordinate system.

Ditto, a falling observer will fall through the event horizon in a finite proper time, no matter what coordinate system they choose to use.

• But an observer stationary above the horizon sees a different hole than someone falling in. For someone outside there is no interior, for someone falling in there is. Has the hole an interior or not? Jun 15, 2022 at 16:48
• Maybe I should have said "different states of motion". Jun 15, 2022 at 16:49
• @Felicia You seem to be asking "who is correct"? The answer is both are. In the framework of General Relativity we must say there is an interior according to both observers, but only one of them is going to find out if GR is correct on that point. Jun 15, 2022 at 16:49
• Precisely, that's what I ask. But normally, an object in its restframe is considered the true form. Though that's in SR. What is the restframe of the hole? Outside it? Stationary above the horizon? Yes, obviously... I get it. Jun 15, 2022 at 16:56