# Event horizon in different coordinate systems

1. Jan 13, 2015

### smoothoperator

Hi guys,

I have a GR question. It is usually said that black holes have event horizons in which time freezes/stops relative to an outside observer. This happens in the Schwarzchild coordinate system. But are there any coordinate systems in which the coordinate time of the black hole and its event horizon does not slow down so enourmosly that it stops, so we can track the timelike events on the worldline of the black hole? If there are, then how is this related to the gravitational time dilation? I will try to explain the last question with an example. If we use the Scwarzschild Coordinates here on Earth and we conclude that no time has passed relative to us (time is 'frozen') on a black hole, how can we use another coordinate system and conclude that some time has elapsed on a black hole relative to us?

2. Jan 13, 2015

### Matterwave

You are putting too much emphasis on coordinate systems. You should think more physically. We can never get any signals sent to us from the Event Horizon itself (just outside, it's possible, but the signals will be very red shifted). That is physical reality. How then, if we can never receive signals from the Event Horizon, do we conclude that "no time is passing on a black hole"?

The Schwarzschild metric is singular at the Event Horizon. It's certainly possible to construct coordinate systems which are not singular at the Event Horizon. You can see e.g. Eddington-Finklestein coordinates or Kruskal-Szekeres coordinates.

3. Jan 13, 2015

### smoothoperator

Ok, so does time one a black hole 'pass' relative to an outside observer in those coordinate systems you mentioned and how does that correlate to the concept of gravitational time dilation? I mean, in one coordinate system the time at some point is frozen and in others it isn't. So in those coordinate systems you mentioned time does pass faster on a black hole than in Schwarzchild coordinates, of course relative to an outside observer?

4. Jan 13, 2015

### Matterwave

Like I said, you are putting too much emphasis on coordinate systems. Here's a quick example in Minkowski space-time (which will be similar when Eddington Finkelstein coordinates are used in Schwarzschild space time). In Minkowski spacetime we can use spherical coordinates $(t,r,\theta,\phi)$. In this coordinate system, $t$ is associated with time. We can; however, transform to an equally valid coordinate system $(u,v,\theta,\phi)$ where $u\equiv t-r$ and $v\equiv t+r$. These are so called "light cone coordinates". Now, in this coordinate system, which coordinate do you want to associate with time?

5. Jan 13, 2015

### m4r35n357

You might find this (Gullstrand-Painleve coordinates) easier to digest than the other systems (it's the theory behind the "river model", so it has a simple physical interpretation), but it's still quite mathematical.

6. Jan 14, 2015

### Staff: Mentor

There is no single "worldline of the black hole". The black hole is not a point. It's a region of spacetime that can't send light signals out to infinity. The event horizon is the boundary of this region.

Gravitational time dilation is only well-defined outside the event horizon. It is not well-defined on or inside the event horizon. This is true regardless of what coordinates you adopt.

As Matterwave said, you are putting too much emphasis on coordinates. However, even if you stop emphasizing coordinates and look at the physics, you won't find any unique answer to the question you're asking, because there isn't one. There is no unique way to specify "how much time has passed" at or inside the horizon of a black hole, compared to an observer outside the hole.

7. Jan 15, 2015

### bcrowell

Staff Emeritus
Matterwave's #2 and Peter Donis's #6 are good answers. A couple of additional points:

The definition of the event horizon is that it's a surface from which it's not possible to send signals to arbitrarily large distances. This definition is completely coordinate-independent.

One way of seeing that we can't define gravitational time dilation at or inside the event horizon is that gravitational time dilation is defined for a static observer, i.e., an observer who is hovering. You can't hover at or inside the horizon.

8. Jan 15, 2015

### smoothoperator

@PeterDonis, bcrowell

So regardless of what coordinate system we use, there is no way to track down the passing of the coordinate time on the worldtube of the black hole relative to an outside observer? If the case is such, why don't all coordinate systems have a singularity at the event horizon beyond which the space-time isn't covered?
I know that I'm focused too much on the coordinate systems, but the event horizon fascinates me and I want to know what property of the black hole, if any, is shared between different coordinate systems.

9. Jan 15, 2015

### George Jones

Staff Emeritus
It is possible, using appropriate coordinates, for an observer crossing the event horizon and inside a black hole to measure gravitational time dilation relative to an observer outside, i.e., the observer inside can watch the watch of an observer outside.

10. Jan 15, 2015

### George Jones

Staff Emeritus
A better wording would be

"It is possible to analyze, using appropriate coordinates, the gravitational time dilation that an observer crossing the event horizon and inside a black hole measures relative to an observer outside, i.e., the observer inside can watch the watch of an observer outside."

11. Jan 15, 2015

### smoothoperator

But no other way around, no matter what coordinates we use? So an outside observer cannot track down events on the world-tube of a black hole. I've red some opinions where it is stated that an event horizon doesn't need to be a singularity (that time doesn't have to be frozen on the event horizon), but if that's true, that means that time can pass on a black hole relative to an outside observer which is in contradiction with some posts here, and I tend to believe that it's wrong and that PeterDonis is right.

12. Jan 15, 2015

### Staff: Mentor

The observer inside can see light coming from the watch of the observer outside, yes; but since gravitational time dilation is defined relative to the timelike Killing vector field, and that KVF at and inside the horizon is not timelike, I don't think that whatever the observer inside observes can be used to define a meaningful gravitational time dilation, at least not in any invariant sense.

13. Jan 15, 2015

### Staff: Mentor

You're mixing up different things here. The event horizon does not have to be a coordinate singularity; whether or not it is depends on the coordinates you choose. But the event horizon is an outgoing lightlike surface, so anything that stays at the event horizon must be an outgoing light ray, and the concept of "proper time" is not meaningful for light rays (this is the correct way to state what you refer to as "time being frozen"), so anything that stays at the event horizon cannot have a meaningful concept of "proper time", and that's true regardless of what coordinates you choose. (Note, however, that things that fall through the horizon and reach the region inside can have a meaningful proper time.)

14. Jan 15, 2015

### Staff: Mentor

All properties of the black hole are the same in all coordinate systems, except that some coordinate systems cannot be used to make some calculations at some points. When this happens, it's a problem with the coordinates and has nothing to do with the actual physics of what's going on at that point. For example, the earth's north and south poles are just points on the surface like any other, but the concept of longitude stops working there - that's a problem with the way that we've defined longitude, not anything special about those points.

15. Jan 15, 2015

### George Jones

Staff Emeritus
Maybe we just have different definitions, but I don't agree with this. I don't have time to write much, as I have two deadlines I have to meet at work today.

I use observer 4-velocities to calculate gravitational time dilation. Far from a Schwarzschild black hole, the timelike Killing vector $\partial / \partial t$ is (almost) the same as the 4-velocity of a hovering observer.

16. Jan 15, 2015

### Staff: Mentor

But they have to be 4-velocities of observers who are following integral curves of the timelike KVF, right? Otherwise I would argue that you are using the term "gravitational time dilation" in a nonstandard way.

To clarify somewhat: you can define gravitational time dilation for an observer who is not following an orbit of the KVF--for example, an observer in a circular free-fall orbit. But then you have to separate out the gravitational time dilation from the kinematic time dilation; the gravitational time dilation is the difference between an observer at rest at infinity and an observer at rest at the altitude in question (e.g., the altitude of the circular orbit), and the kinematic time dilation is the difference between an observer at rest at that altitude and the observer in question (e.g., the observer in a circular orbit). At least, that's my understanding of the standard usage. And at or inside the horizon, there aren't any observers at rest at any altitude, so there's no way to carry through what I just described.

Yes, but closer to the horizon, it's not; and at or inside the horizon, that KVF is no longer timelike so it can't be parallel to the 4-velocity of any observer.

17. Jan 15, 2015

### bcrowell

Staff Emeritus
Yes, but there are a couple of caveats I would add. (1) What they're learning from this depends partly on their own motion. That's different from the situation outside the horizon, where we can make our observer static and think of the measurement as telling us something about the properties of a fixed location in space. (2) I don't think there's any reciprocity here in the sense that you would have if comparing two static observers. When comparing static observers A and B, if A says B is slow by a factor of 7, then B says A is fast by a factor of 7.

Because of these issues, I think this is really not comparable to measuring gravitational time dilation.

18. Jan 15, 2015

### bcrowell

Staff Emeritus
The no-hair theorems tell us that a Schwarzschild black hole has only one adjustable parameter that is coordinate independent, and that's its mass.

It also has some coordinate-independent properties that are generic to all Schwarzschild black holes. These include spherical symmetry, asymptotic flatness, the presence of an event horizon, and geodesic incompleteness (i.e., the existence of the central singularity).

Some other coordinate-independent properties can be derived from the mass. An example would be the value of certain scalar measures of spacetime curvature at the horizon (i.e., basically how strong tidal forces are there).

19. Jan 15, 2015

### smoothoperator

So what's the alternative? If it's not a singularity, then what can it be in some different coordinate system? I mean, time frozes at the event horizon from some observer's perspective, and it seems to me that an alternative would be that the time doesn't froze, or that some time passes at the event horizon? Can you please explain this since it's the biggest source of my confusion.

20. Jan 15, 2015

### m4r35n357

Investigate the Gullstrand-Painleve coordinates, and the River/Waterfall Model, that I mentioned earlier. Seriously, these are as simple as it gets. I don't have the time to go into more detail, but perhaps someone else can.