Event horizon in different coordinate systems

[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.]

 

[Matterwave]

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.]

Ah ok, that makes sense. I could only recall that an observer reaches the singularity in finite proper time (as must be the case for geodesic incompleteness to occur), I could not recall the result of which signals he can still receive from the outside as he's falling in.

 

[PeterDonis]

when we define a metric in curved space-time, what have we defined in general?

pervect's response is a good one, but might be a bit "heavy", though it's already a condensed version of the paper he linked to, so I'll try to condense it a bit more. ;) A metric defines a geometry--a shape, composed of points and curves with particular relationships between them--distances and angles. So, for example, if you know the metric of the Earth's surface, you know the distance between any two points on it (say, the intersection of 5th Avenue and 59th Street in Manhattan, and Nelson's Column in Trafalgar Square in London), and you know the angle between any two curves where they intersect (say, the angle between the great circle--note that all distances are measured along great circles, i.e., geodesics--connecting the two points I just named, and the great circle connecting the North Pole with Rio de Janeiro, Brazil, at the point where they intersect). But you can use many different coordinate charts to describe the same geometry, so just knowing a geometry does not give you any coordinates. (Notice that I used no coordinates in describing the distances and angles above.)

The geometry of spacetime works the same way except that you have to add time to the specification of a "point", i.e., an event. So, for example, in spacetime there is a particular "distance" (actually a proper time) between, say, the event of a rocket in free fall flying past the Moon on its way to Mars, and the same rocket flying past Phobos just before it reaches Mars. Assuming the rocket has been in free fall the whole time, its path through spacetime will be a geodesic, and the proper time elapsed on its clock between the two events will give the spacetime "distance" between them. Also, the "angle" between a pair of timelike worldlines is the relative velocity between them when they intersect, so, for example, knowing the geometry of spacetime means knowing the angle--the relative velocity--between the above rocket and another rocket in free fall on its way from Venus to Jupiter that happens to pass the first rocket, when they pass. But again, just knowing the geometry--all the distances and angles--doesn't specify any coordinates (once again, I used no coordinates in my description of events above).