# Event horizon in different coordinate systems

## Main Question or Discussion Point

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?

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Matterwave
Gold Member
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.

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?

Matterwave
Gold Member
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?
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?

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?
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.

PeterDonis
Mentor
2019 Award
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?
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.

how is this related to the gravitational time dilation?
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.

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?
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.

bcrowell
Staff Emeritus
Gold Member
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.

@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.

George Jones
Staff Emeritus
Gold Member
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.

George Jones
Staff Emeritus
Gold Member
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.
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."

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."
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.

PeterDonis
Mentor
2019 Award
"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."
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.

PeterDonis
Mentor
2019 Award
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),
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.)

Nugatory
Mentor
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.
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.

George Jones
Staff Emeritus
Gold Member
since gravitational time dilation is defined relative to the timelike Killing vector field
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.

PeterDonis
Mentor
2019 Award
I use observer 4-velocities to calculate gravitational time dilation.
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.

Far from a Schwarzschild black hole, the timelike Killing vector
is (almost) the same as the 4-velocity of a hovering observer.
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.

bcrowell
Staff Emeritus
Gold Member
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.
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.

bcrowell
Staff Emeritus
Gold Member
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.
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).

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.)
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.

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.
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.

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.
I did look it up, but honestly I didn't understand much because mainly there are so many formulas and nowhere in the text a straightforward answer has been mentioned.

OK, I realize it's not easy stuff, but I still think it is the simplest answer your questions. Even ignoring the equations you might be able to get something out of the words.

OK, here's my final offer on these coordinates ;)

PeterDonis
Mentor
2019 Award
If it's not a singularity, then what can it be in some different coordinate system?
A coordinate singularity is not a physical thing; it's just a problem with that particular system of coordinates. The North and South Poles are coordinate singularities in our usual latitude/longitude system of coordinates on the Earth's surface; but that doesn't mean there's anything wrong with them physically. There are other coordinates we can use for the Earth's surface that do not have coordinate singularities at the poles. But no choice of coordinates can change the shape of the Earth's surface at the poles.

Similarly, we can choose coordinates that are not singular at the event horizon (two examples have already been given in this thread). Such coordinates will allow you to label different events on the horizon with different coordinate values. But no choice of coordinates can change the physics of the horizon. See below.

time frozes at the event horizon from some observer's perspective
Read my post #13 again. The property of the event horizon that you are referring to when you say "time is frozen" there (which is not really the right way to say it, as I explain in that post), that it is an outgoing lightlike surface, is independent of coordinates.

it seems to me that an alternative would be that the time doesn't froze, or that some time passes at the event horizon?
No; again, as I said in post #13, the fact that the concept of "elapsed time" is not meaningful for a light ray is independent of coordinates. Since any object which stays at the horizon must be an outgoing light ray, the concept of "elapsed time" is not meaningful for any object that stays at the horizon. As above, this property is independent of coordinates.

Note, by the way, that this property means that, even if you adopt coordinates that are not singular at the horizon, calling the coordinate that changes along the horizon the "time" coordinate is somewhat problematic. For example, in Painleve coordinates, each event on the horizon has a different Painleve coordinate time. However, this does not mean the concept of "time passing" is meaningful for an object that stays at the horizon, because, as above, such an object must be an outgoing light ray.

In other words, the reason we usually call a coordinate a "time" coordinate is that a curve along which only that coordinate changes is timelike. For Painleve coordinate time, that's true outside the horizon. But on the horizon, a curve along which only Painleve coordinate time changes is null (lightlike), not timelike; and inside the horizon, such a curve is spacelike. So even though Painleve coordinates are not singular on the horizon, you still have to be careful how you interpret them physically on or inside the horizon.

WannabeNewton
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?
The event horizon is a surface in space-time that constitutes the causal boundary between the exterior and interior of the black hole. It is a geometric object existing independently of any coordinate system you choose to describe it. Now, your problem comes from putting too much emphasis on the existence of a global time coordinate in GR. The global time coordinate in Schwarzschild coordinates can be understood as follows.

There exists a family of observers situated at rest (relative to infinity) at each point exterior to the event horizon of the black hole and each of these observers carries a clock. By synchronizing their clocks with one another using radar, which involves taking into account gravitational time dilation, they can build a global time coordinate $t$ constituting the time at each point in space of the global reference frame formed by these observers. Thus if an observer at rest within some point outside the event horizon wants to know the time near the event horizon as read by the observer at rest there then the first observer simply notes the time $t$ read by their clock and uses the fact that this clock is synchronized with that of the second observer.

But no such observer exists on the event horizon itself because it is a null surface. Only light (null curves) can remain tangent to it. As such it doesn't make any sense to ask what the time $t$ is on the event horizon through the aforementioned clock synchronization. Indeed time does not "freeze" at the event horizon for these static observers. All that happens is, if one has a particle falling freely into the black hole that emits light at regular pulses towards the distant static observer then in the limit as the particle approaches the event horizon, the light gets more and more redshifted when it arrives at the observer's location. Therefore the observer sees the particle getting dimmer and dimmer to the point where it appears frozen at the event horizon because no information about the particle after it has passed the event horizon arrives at the observer due to the infinite redshift.

The coordinate singularity at the event horizon in Schwarzschild coordinates is again due to it being adapted to the family of static observers whose tangent field does not extend to the event horizon while still remaining time-like. One can switch to any of the other coordinates already mentioned in the thread which are well behaved at the event horizon but these coordinates need not be adapted to any family of observers such as retarded null coordinates or Kruskal coordinates. In such a coordinate system there is no synchronized global time coordinate because one needs an observer congruence comoving with the coordinates in order to even define such a $t$. Thus it doesn't make sense in these coordinates to ask if time still "freezes" or doesn't "freeze" at the event horizon. All one can say is the coordinates are no longer singular there.

PAllen
2019 Award
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.
Time freezing, if you will, is property of light independent of any coordinates, with or without gravity. More accurately, as Peter said, proper time (e.g. clocks) are undefinable for light (more generally, for null world lines). This has nothing to do with coordinate singularities or black holes. Note further, that a coordinate singularity can be induced to occur anywhere in any space-time. In particular, the arguably most natural coordinates for uniformly accelerating rocket in empty space produce a coordinate singularity at the Rindler horizon (c2/a behind the rocket, where 'a' is the acceleration experienced by a rocket occupant). This horizon represents what you would like to call 'time stoppage' for the uniformly accelerating rocket. A clock dropped from the rocket undergoes infinite redshift and 'freezing' on approach to this distance from the rocket. Light emitted at this distance from the rocket can never catch the rocket. This is all special relativity, that carries over into general relativity.

Meanwhile, for the clock dropped from the rocket, the rocket's continued acceleration is irrelevant, and its crossing of the rocket's Rindler horizon is meaningless except that it can no longer send a message to the rocket once it has crossed (but the rocket and still send messages to it).

All these facts are the same about distant observers versus free fall observers for a BH horizon.

I think(?) the nub of your confusion is that you think time freezing for some observer observing something at a distnace is a deep fact of nature. It isn't, period. It is just a fact about the relationship between the observer and the observed that can happen in empty space with no black hoels.