<p>What a black hole looks like is independent of coordinates - what an observer &quot;sees&quot; or measures does not depend on what coordinate system they choose to make their measurements in.</p>
<p>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 &quot;true&quot; reference frame.</p>
<p>Thus a distant observer &quot;sees&quot; (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.</p>
<p>Ditto, a falling observer will fall through the event horizon in a finite <em>proper</em> time, no matter what coordinate system they choose to use.</p>

CC BY-SA 4.0
2022-06-15T16:48:16.043Z
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?
Felicia
CC BY-SA 4.0
2022-06-15T16:49:26.653Z
Maybe I should have said "different states  of motion".
Felicia
CC BY-SA 4.0
2022-06-15T16:49:28.623Z
@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.
ProfRob
CC BY-SA 4.0
2022-06-15T16:56:40.937Z
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.
Felicia
CC BY-SA 4.0
2022-06-15T16:41:19.223Z
2022-06-15T16:41:19.223Z
ProfRob
<p>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).</p>
<p>Which raises the question, what is the true, uncoordinated, spacetime manìfold? Or are we implicitly always coordinating?</p>

CC BY-SA 4.0
2022-06-15T08:51:56.097Z
TBH I'm not sure what you are asking. Can you please elaborate? The nature of an object does not depend on how I draw it? In order to depict something you always have to choose some notion of space, thus define your coordinate system. To me the question reads like "how can I bath without getting wet" - so it seems I'm missing quite a few preconditions or implicit assumptions of your question?
planetmaker
CC BY-SA 4.0
2022-06-15T08:55:31.517Z
@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...
Felicia
CC BY-SA 4.0
2022-06-15T08:57:25.087Z
Are you sure you are not mixing up "unit free" with "coordinate independent"?
planetmaker
CC BY-SA 4.0
2022-06-15T09:19:10.113Z
@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?
Felicia
CC BY-SA 4.0
2022-06-15T11:56:18.263Z
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 $R \longrightarrow R/R_0$)
planetmaker
CC BY-SA 4.0
2022-06-15T16:33:15.680Z
What a black hole looks like *is* independent of coordinates, as is all covariant physics.
ProfRob
CC BY-SA 4.0
2022-06-15T16:38:51.627Z
@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?
Felicia
CC BY-SA 4.0
2022-06-15T16:41:59.137Z
@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.
Felicia
CC BY-SA 4.0
2022-06-15T16:51:27.177Z
@planetmaker I think "different states of motion" is more appropiate. How can a hole have an interior for the infalling observer, but none for someone stationary above the horizon?
Felicia
<black-hole><coordinate>
How does a black hole look like independently of coordinates?
Felicia
# How does a black hole look like independently of coordinates?

(Asked by: username_0 on 2022-06-15)

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?

**Tags:** black-hole, coordinate
> **Comments:**
> **username_1**: TBH I'm not sure what you are asking. Can you please elaborate? The nature of an object does not depend on how I draw it? In order to depict something you always have to choose some notion of space, thus define your coordinate system. To me the question reads like "how can I bath without getting wet" - so it seems I'm missing quite a few preconditions or implicit assumptions of your question?  
> **username_0**: @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...  
> **username_1**: Are you sure you are not mixing up "unit free" with "coordinate independent"?  
> **username_0**: @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?  
> **username_1**: 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 $R \longrightarrow R/R_0$)  
> **username_2**: What a black hole looks like *is* independent of coordinates, as is all covariant physics.  
> **username_0**: @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?  
> **username_0**: @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.  
> **username_0**: @planetmaker I think "different states of motion" is more appropiate. How can a hole have an interior for the infalling observer, but none for someone stationary above the horizon?  

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## 1 answers

### Answer by username_2 on 2022-06-15. Score: 1

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.

> **Comments:**
> **username_0**: 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?  
> **username_0**: Maybe I should have said "different states  of motion".  
> **username_2**: @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.  
> **username_0**: 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.  

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