Event horizon in different coordinate systems

[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 Schwarzschild 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?

Thanks in advance

 

[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.

 

[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 Schwarzschild coordinates, of course relative to an outside observer?

 

[Matterwave]

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 Schwarzschild 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?

 

[m4r35n357]

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 Schwarzschild coordinates, of course relative to an outside observer?

You might find http://physicspages.com/2013/11/24/painleve-gullstrand-global-rain-coordinates/ (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]

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]

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.

 

[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.

 

[George Jones]

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]

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."

 

[smoothoperator]

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]

"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]

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]

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]

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]

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]

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]

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).

 

[smoothoperator]

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.

 

[m4r35n357]

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.

 

[smoothoperator]

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.

 

[m4r35n357]

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]

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]

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.

 

[smoothoperator]

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.

The nub of my confusion is how something like the stoppage of time at the horizon can be a coordinate-dependent property. I mean if we change the coordinate system and conclude that a finite distance from us time isn't frozen, but a finite amount of time has passed (relative to our frame), does that still imply that at the point that represents the horizon the light will never reach the observer, despite the change of the coordinate system. The fact that in one coordinate chart the time is frozen and in another that time passes (at some distance from us) seems weird and I don't understand how can we conclude that the light will never reach the observer from the horizon point despite the changes of the coordinate system.

 

[PeterDonis]

The nub of my confusion is how something like the stoppage of time at the horizon can be a coordinate-dependent property.

As several people now have told you, it isn't. The horizon is a lightlike surface, and any lightlike surface, even in flat spacetime (as PAllen pointed out), has the property you are calling "stoppage of time" (though, as I've said several times now, that's not a good way to describe it--a better way is to say that the concept of "elapsed time" is not meaningful).

if we change the coordinate system and conclude that a finite distance from us time isn't frozen, but a finite amount of time has passed (relative to our frame)

As I pointed out in a previous post, even if you change coordinates so they are not singular on the horizon, and therefore you can now assign different values of the coordinate called "t" to different events on the horizon, that still does not mean that "t" on the horizon can be physically interpreted as "elapsed time". As I reiterated above, since the horizon is a lightlike surface, the concept of "elapsed time" is not meaningful for something that stays at the horizon (since that something must be a light ray), regardless of what coordinates you adopt. Please take some time and think carefully about what this means, since it seems to be at the root of your confusion. Pay particular attention to PAllen's post showing that the same thing happens on a lightlike surface in flat spacetime, so there is nothing special about a black hole's horizon in this respect.

does that still imply that at the point that represents the horizon the light will never reach the observer

The light from the horizon not being able to reach any observer outside the horizon is independent of coordinates, since it's implicit in the fact that the horizon is an outgoing lightlike surface, so yes.

 

[bcrowell]

So what's the alternative? If it's not a singularity, then what can it be in some different coordinate system?

"Singularity" is a technical term with a technical definition, and the definition is coordinate-independent. The definition is that there are incomplete geodesics. It sounds to me like you aren't referring to any specific definition of the term. We can't have a discussion of whether something is a singularity unless you're willing to work with the standard definition of the term.

 

[smoothoperator]

@PeterDonis, I apologize if I'm being annoying or repetitive, I'll examine the posts so far and try to grasp these new concepts and if I have further questions I'll ask them.

Thank you.

 

[Matterwave]

The nub of my confusion is how something like the stoppage of time at the horizon can be a coordinate-dependent property. I mean if we change the coordinate system and conclude that a finite distance from us time isn't frozen, but a finite amount of time has passed (relative to our frame), does that still imply that at the point that represents the horizon the light will never reach the observer, despite the change of the coordinate system. The fact that in one coordinate chart the time is frozen and in another that time passes (at some distance from us) seems weird and I don't understand how can we conclude that the light will never reach the observer from the horizon point despite the changes of the coordinate system.

I think if you go back and try to answer the question I posed to you in post #4, you will realize that in some coordinate systems (e.g. null coordinates), it doesn't make sense to talk about any of the coordinates as a "time".

And like I have been emphasizing since post #2, you are putting way too much emphasis on coordinate systems. Coordinate systems generally don't tell you anything about the rate at which time passes for different observers.

Again, as others have mentioned, the Schwarzschild coordinates happens to be adapted to a family of static observers and their times. But that's just the property of this particular coordinate system. And in this particular coordinate system, the "time coordinate" doesn't make sense on or inside the Event Horizon because there can be no static observers on or inside the Event Horizon.

Perhaps a better statement of "gravitational time dilation is infinite at the event horizon" is "gravitational time dilation is undefined at the event horizon".

 

[PeterDonis]

in this particular coordinate system, the "time coordinate" doesn't make sense on or inside the Event Horizon because there can be no static observers on or inside the Event Horizon.

More precisely, it doesn't make sense as a "time" coordinate. Inside the horizon it's a perfectly good coordinate, just not a timelike one. (It's singular on the horizon, so it isn't a good coordinate there of any kind.)

 

[Matterwave]

More precisely, it doesn't make sense as a "time" coordinate. Inside the horizon it's a perfectly good coordinate, just not a timelike one. (It's singular on the horizon, so it isn't a good coordinate there of any kind.)

I think mentioning this will only confuse the OP. But in any case, one can not smoothly connect the inner patch of Schwarzschild coordinates with the outer patch because one will hit the coordinate singularity at the event horizon. There is no physical connection between the two coordinate patches. In addition, one rarely uses the interior Schwarzschild coordinates to analyze physical problems (correct me if I'm wrong on this). It is usually much more convenient to use any of the other coordinate systems (e.g. Eddington-Finkelstein or Kruskal-Szekeres Coordinates). As such, I think I am roughly justified in only considering the exterior patch of the Schwarzschild coordinates, and considering the disconnected inner patch to be a different beast altogether.

 

[PeterDonis]

one can not smoothly connect the inner patch of Schwarzschild coordinates with the outer patch because one will hit the coordinate singularity at the event horizon

Yes, agreed. They are two disconnected coordinate patches.

one rarely uses the interior Schwarzschild coordinates to analyze physical problems (correct me if I'm wrong on this)

I haven't seen it used much in actual solutions, but it's implicitly used a lot in pop science discussions that talk about "time and space being swapped" (or words to that effect) inside the horizon. That's not to say I am thrilled by those pop science discussions, but they spawn threads here on PF fairly often.

 

[pervect]

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?

Do you mean the world-tube of the event horizon, perhaps? I'm surprised nobody has mentioned it in this thread yet, but the event horizon is a null surface, i.e. lightlike.

There is no coordinate system in which light has a proper time, hence there is no surface in which the event horizon has proper time. The world tube of a point on the event horizon and is well defined - it is what we call a null geodesic, what might informally be called the path of a light beam. The event horizon as a hole is a collection of a set of such worldlines.

So this whole idea that "time stops at the event horizon" is misleading, it's similar to the idea that "time stops for light" which is also misleading (and we have a FAQ on that), though it is true in some sense that time stops in the limit as one approaches the speed of light even though there is no sensible notion of a material body going at the speed of light. )

I rather suspect that the fundamental issue is misleading people here and is likely to be misleading smoothoperator is the habit of thinking that time is an absolute thing, when we know from special relativity that it's not. For instance, you are right now moving at .9999999c in some frame relative to some ultra-relativistic particle , but in spite of this fact, time is not stopped for you. In your own frame (rather than the hypothetical ultrarelativistic particles frame), your time is not even slowed.

The idea that time is not absolute, but relative, also known as "the relativity of simultaneity", is an important background concept needed to understand special relativity. It's got a detailed writeup in the "Einstein's train" thought experiment.

Someone falling into a black hole will, using local clocks and rulers, see the event horizon pass them at light speed - because it is a lightlike surface. It is never possible to hover at the event horizon, just as it's not possible for someone to reach the speed of light. It is possible to hover arbitrarily close to the event horizion, which can give the naieve reader an impression that time slows.

But someone free-falling into the event horizon rather than attempting to hover above it will not see time as stopping, they won't (for instance) receive any infinitely blueshifted signals before they reach the event horizon. They won't notice anything more odd than you do, traveling at .9999999c relative to a frame in which some fast-moving particle is at rest.

 

[Matterwave]

I haven't seen it used much in actual solutions, but it's implicitly used a lot in pop science discussions that talk about "time and space being swapped" (or words to that effect) inside the horizon. That's not to say I am thrilled by those pop science discussions, but they spawn threads here on PF fairly often.

This is just a totally horrible description of what's happening inside the event horizon..."time and space being swapped" makes literally no sense. I would hope that we can try to stay away from this.

@pervect why is your whole post a hyperlink to Peter's profile? o.o

 

[pervect]

This is just a totally horrible description of what's happening inside the event horizon..."time and space being swapped" makes literally no sense. I would hope that we can try to stay away from this.

@pervect why is your whole post a hyperlink to Peter's profile? o.o

I'm not sure what you mean by this? If it's a literal hyperlink, I messed up somewhere. I suspect you mean a certain similarity of ideas and presentation, I'd say that this similarity evolved after many, many threads (hundreds, or more, I'd guess) on the issues.. Of course, this doesn't explain why we don't agree on everything :).

 

[Matterwave]

I'm not sure what you mean by this? If it's a literal hyperlink, I messed up somewhere. I suspect you mean a certain similarity of ideas and presentation, I'd say that this similarity evolved after many, many threads (hundreds, or more, I'd guess) on the issues.. Of course, this doesn't explain why we don't agree on everything :).

No I mean, literally I see your whole post as blue, and it's a link to Peter's profile...

 

[PeterDonis]

This is just a totally horrible description of what's happening inside the event horizon..."time and space being swapped" makes literally no sense. I would hope that we can try to stay away from this.

You're preaching to the choir. :) Unfortunately, plenty of people come to PF having seen or read misconceptions like these, and we try to straighten them out as best we can.

 

[PeterDonis]

literally I see your whole post as blue, and it's a link to Peter's profile...

I did too, so I used magic Mentor powers and edited it to delete the spurious user tags that had got in there somehow.

 

[pervect]

No I mean, literally I see your whole post as blue, and it's a link to Peter's profile...

I attempted to create a converastion with you, me, and Greg Bernhardt on this issue - if there's a better way of resolving it let me know.

[add]I see it was already resolved. Nevermind!

 

[smoothoperator]

So what's the perspective of the observer free falling in the black hole, I mean from a simultaneity perspective? I know that he travels pass the horizon into the hole in a finite amount of time,which clearly means that the time isn't 'null' at the horizon for him. Or is it? As he's relatively close to the horizon the events between the horizon and him pass slower than his proper time, but what about the horizon itself and the inside of the black hole. Are those segments somehow not in his reference frame until he reaches the hole?

 

[PeterDonis]

So what's the perspective of the observer free falling in the black hole, I mean from a simultaneity perspective?

Simultaneity is a convention, so there is no unique answer to your question as you ask it. But see below.

I know that he travels pass the horizon into the hole in a finite amount of time

Correct.

which clearly means that the time isn't 'null' at the horizon for him. Or is it?

He's traveling on a timelike worldline, not a null worldline, so no, time is not "null" at the horizon (or anywhere else) for him. To him, the horizon is an outgoing light ray that passes by him at the speed of light.

As he's relatively close to the horizon the events between the horizon and him pass slower than his proper time

Which events? Along what worldline? With what simultaneity convention?

You appear to be under the misconception that there is some absolute meaning to "the passage of time" or "the rate of time passing". There isn't.

what about the horizon itself and the inside of the black hole. Are those segments somehow not in his reference frame until he reaches the hole?

It depends on how he chooses his "reference frame", i.e., what coordinates he chooses to adopt. If he adopts Painleve coordinates, for example, the horizon and the region inside it are covered by those coordinates, so they are in his "reference frame", yes. But if he adopts Schwarzschild coordinates, they're not, at least not until he reaches the horizon, finds out that the coordinates he's been using are singular there, and has to switch coordinates.

You appear to be under the misconception that there is some unique choice of "reference frame", i.e., coordinates, for any given observer. There isn't. As Matterwave said in earlier posts, it's very important not to put too much emphasis on coordinates; it's better to focus on the physics, i.e., on actual observables. Instead of thinking about the free-falling observer's "reference frame", think about what he would actually observe. For example, what light signals would he see coming from other observers (for example, from an observer "hovering" at a constant altitude close to the horizon)? What light signals would they see coming from him?

 

[smoothoperator]

So can we compare the different conventions that observer in GR uses to the choice of different conventions in SR in a non-inertial frame? I mean, as far as my thought process goes, when we define a metric in GR we determine the gravitational time dilation and simultaneity, which is closely related to the concept of coordinate time and different simultaneity conventions in a non-inertial frame in flat space?

 

[WannabeNewton]

So can we compare the different conventions that observer in GR uses to the choice of different conventions in SR in a non-inertial frame?

Compare in what sense? Simultaneity conventions are certainly complicated in flat space-time for non-inertial frames but they become even more complicated in curved space-times because simultaneity conventions are strongly bound by causal structure and topology so unless you specify what you mean by compare, the question is quite ambiguous.

I mean, as far as my thought process goes, when we define a metric in GR we determine the gravitational time dilation and simultaneity, which is closely related to the concept of coordinate time and different simultaneity conventions in a non-inertial frame in flat space?

A metric simply determines the local Einstein simultaneity convention for any timelike worldline, which is not surprising as local Lorentz frames are approximations of flat space-time. This allows one to determine a local 3+1 split of the tangent space at any event on the observer's worldline into a "space" + "time" part. But the metric does not uniquely specify, in any sense, a simultaneity convention for coordinate systems comoving with observers or global simultaneity conventions for congruences of observers. The same goes for flat space-time with the exception of inertial frames.

 

[PeterDonis]

when we define a metric in GR we determine the gravitational time dilation

Sort of. The concept of gravitational time dilation is only well-defined for certain kinds of metrics, the ones that have a time translation symmetry--basically, there has to be a family of observers who each see the metric as unchanging along their worldlines. For those metrics, yes, once you've defined the metric you've determined gravitational time dilation.

and simultaneity

No. Simultaneity is a convention, and there are many different simultaneity conventions that are compatible with a given metric.

which is closely related to the concept of coordinate time and different simultaneity conventions in a non-inertial frame in flat space?

Coordinate time in flat spacetime (not flat "space") has nothing to do with gravitational time dilation; there is no gravitational time dilation in flat spacetime.

Simultaneity is always a convention, even in an inertial frame; the correct way to say it is that the term "inertial frame" implies adopting a particular kind of simultaneity convention (the term "Einstein simultaneity" or "Einstein clock synchronization" is often used to describe this kind of convention), but an inertial (free-falling) observer is not required to use an inertial frame to describe physics, so simultaneity is still a convention even for inertial observers in flat spacetime.

 

[LitleBang]

To me the only time that makes any sense is local time. If you are at the event horizon your time is normal why worry about what it looks like to an outside observer? To an entity at the horizon events in the outside Universe appear to be occurring at an infinitely fast rate.

 

[Matterwave]

To an entity at the horizon events in the outside Universe appear to be occurring at an infinitely fast rate.

This is only true if that "entity" can hover at the event horizon, which he can't. A free-fall observer into the black hole will not see the outside universe evolving at an "infinitely fast rate". He will get signals from the outside as they fall in behind him. He won't get all the signals at once because signals which fell in later will reach him later. As the observer nears the (central) singularity, I can't recall if all future signals pile up behind him as he arrives "at the singularity". Perhaps there the outside universe will all "happen at once" I'm not sure. One can analyze this problem using the Kruskal coordinates, but I haven't done so in a quite long time. Hopefully someone can come give a more complete answer than me.

 

[smoothoperator]

Sort of. The concept of gravitational time dilation is only well-defined for certain kinds of metrics, the ones that have a time translation symmetry--basically, there has to be a family of observers who each see the metric as unchanging along their worldlines. For those metrics, yes, once you've defined the metric you've determined gravitational time dilation.

No. Simultaneity is a convention, and there are many different simultaneity conventions that are compatible with a given metric.

Coordinate time in flat spacetime (not flat "space") has nothing to do with gravitational time dilation; there is no gravitational time dilation in flat spacetime.

Simultaneity is always a convention, even in an inertial frame; the correct way to say it is that the term "inertial frame" implies adopting a particular kind of simultaneity convention (the term "Einstein simultaneity" or "Einstein clock synchronization" is often used to describe this kind of convention), but an inertial (free-falling) observer is not required to use an inertial frame to describe physics, so simultaneity is still a convention even for inertial observers in flat spacetime.

So, when we define a metric in curved space-time, what have we defined in general? These coordinate systems that are so oftenly mentioned confuse me very much.

 

[pervect]

So, when we define a metric in curved space-time, what have we defined in general? These coordinate systems that are so oftenly mentioned confuse me very much.

The paper I like best on this issue is "Precis of General Relativity", http://arxiv.org/abs/gr-qc/9508043. It's a rather abstract read. I'll attempt to simplify it some, the simplifications will hopefully be more helpful in understanding than they will be hurtful in the lost of preciseness.

We have three things we want to define - not necessarily a rigorous definition, but one of enough information that we can communicate sensibly about them. These three things are: metrics, coordinates, and physical measurements.

A metric is fundamentally a map of space-time. Coordinates are arbitrary labels on the map (like grid markers, B7, on a hopefully familiar 2d paper map of space - except that the coordinates in GR are all numeric rather than a letter-number combination like I used above. Those are 2 of the 3 things we want to define, the last thing we need to define are physical measurements. On the paper map, we can take (for our purposes) physical measurements as being the bearings of landmarks. Then by triangulation, with enough physical measurements (bearings), you can find your location (and the coordinates you labelled that location with on the map).

In GR, we can idealize the physical measurements as consisting of the readings of physical clocks, and the transmission and reception of radar signals (which can carry timestamp information from the clocks). We will assume for simplicity the signals propagate in a vacuum. If the signals don't propagate in a vacuum, nothing fundamental changes except that everything gets much more complicated and hard to explain. The theoretical model needs not only a map of space-time, but a map of the characteristics (velocity, density, composition) of the matter that the space-time contains, and details of how the presence of this matter affects the propagation of the specific signals you used. We really regard the underlying properties of the space-time itself as being fundamental, the presence of the matter is an experimental distraction that we need to compensate for.

How might we use these very limited tools to measure distance? Well if we send out a radar signal and the clock reads 0, and we receive a reflection of the radar signal and the clock reads 1 second, we know that when the clock read .5seconds, the object in question was half a light second away.

How do we use these very limited tools to measure coordinates? GPS, which the paper is about, serves as a good example. We send out signals from at least four reference satellites. If the reference satellites were fixed, we'd have an easier job, but it doesn't matter if they move as long as we know how they move. Then if the signals encode their transmission time, all we need to know is the reception time of the 4 signals to triangulate our position in space-time.

Now, one interesting point the paper makes is that specifying a metric operationally specifies the coordinates. While I suspect that the reaction of the average lay-person to being given a metric is one of confusion, rather than one of saying "Oh, this defines our coordinates", a metric does operationally specify coordinates.

The way the metric specifies the coordinates is rather similar to the way having a good 2d paper map of a tract of land specifies coordinates. You make measurements (reception and transmission times of radar signals in the space-time case, triangulation of landmarks in the paper map case) and you can operationally find where you are on the map, then you use the map to read out your coordinates. Recall that the coordinates are just the labels you've put on the map. So, once you've put the labels on the map, then when you make physical measurements, you can determine where you are on the map, and then you can communicate this information concisely by the labels you've put on your map, the coordinates.

There is one thing I've skipped over here, which it is assumed by the author of the above paper that you already knew, and which might not be obvious. This is that the space-time map specifies not distances, not times, but the observer independent "Lorentz Interval" that is the square root of distance^2 - time^2, the only observer-independent interval that special relativity has.

Now suppose we ask - how do we survey space-time? We often survey land , to draw an accurate map, what are some of the ideas we use to survey space-time? All we need to do is to have a large number of observers, who constantly exchange radar signals, and who compute the Lorentz interval between any pair of events. The metric can be viewed as a mathematical model that gives the Lorentz interval in terms of coordinate displacements, as well as being viewed as a map. This is how the map works, saying that the space-time map gives us the Lorentz intervals between all nearby points is similar to saying that a paper map gives the distances between all nearby points. The point is that the distances on the paper map are (in this non-relativistic context) observer -independent, and the Lorentz intervals are the equivalent observer-independent concept in the space-time case..

The abstraction here is all this focus on the Lorentz interval, which will become a familiar and friendly quantity if you seriously study relativity, but may not start out that way. If you had just the Lorentz interval between all pairs of possible points, how do you turn this into (for example), distances and times?

What you need to interpret the information about the Lorentz interval into space and time displacements are coordinates. These used to be just the labels on the map (in our previous discussion), but now we are applying them in a semi-physical way. Given coordinates, you can break the observer independent Lorentz interval into two parts, one part that is spatial only, the length of a path that has a constant time coordinate, and another path that has constant spatial coordinates, and progresses only through time. So by using your coordinates, you can split space-time into space, and time.

I think it might be less circular to say that you need information about the simultaneity, rather than a coordinate choice, to break space-time into space and time in a familiar way. Unfortunately, this got difficult to explain, so I went with the former view as being easier, and just about as good.

 

[PAllen]

This is only true if that "entity" can hover at the event horizon, which he can't. A free-fall observer into the black hole will not see the outside universe evolving at an "infinitely fast rate". He will get signals from the outside as they fall in behind him. He won't get all the signals at once because signals which fell in later will reach him later. As the observer nears the (central) singularity, I can't recall if all future signals pile up behind him as he arrives "at the singularity". Perhaps there the outside universe will all "happen at once" I'm not sure. One can analyze this problem using the Kruskal coordinates, but I haven't done so in a quite long time. Hopefully someone can come give a more complete answer than me.

On approach to the singularity, there is no pileup or unusual feature of received signals. There is asymptotically a well defined last signal received from some outside source on approach to the singularity, and it is not very long after the signal received from said outside source on free faller's horizon crossing. The catastrophe for an infaller is tidal - infinite stretching in the direction of the extra spacelike killing vector (extra being in addition to the two angular killing vectors), and compression in the other spatial directions - a little ball approaches death as a line. That, and 'no future' - geodesic incompleteness. [This is, of course, for the ideal SC BH, which doesn't exist in nature; real BH interiors are much more complex and not known, in that the Kerr interior is unstable against perturbation.]