Yes, but the flat metric "distance" that the OP is interested in (what the OP calls "coordinate distance") is the distance along a geodesic of the flat metric, not along a curve lying in the embedded curved space.Dale said:“distance” along a path is the same in the embedding space as in the curved space.
The problem is that one cannot say when the object falls in, in terms of time that makes sense to the outside observer.phinds said:You are missing the point. If appearance and reality don't match, there's something wrong with the appearance. The object falls in. Period. The fact that it does not appear to do so to a remote observer is irrelevant.
I'm not sure I agree. For example, imagine an external observer watching an electric charge fall toward a neutral Schwarzschild black hole. When the charge is at any finite height above the hole, the observer sees the charge's electric field as a complicated pattern of superposed multipole moments arising from the distorted spacetime around the hole. But as the charge approaches the horizon, the field smoothly transitions to the electric monopole field of a charged Reissner-Nordstrom black hole. So even though the observer never sees the charge actually pass the horizon, by monitoring the distribution of the E-field outside the hole could she not reasonably declare the time of in-fall to be the moment the external field is sufficiently close to a coulomb field?Demystifier said:The problem is that one cannot say when the object falls in, in terms of time that makes sense to the outside observer.
I don't entirely agree with this. You can't say when it crosses the horizon in Schwarzschild coordinates, I agree, because it left the region covered by Schwarzschild coordinates. But that isn't the only definition of "now" available to an external observer - merely one of the simplest, as it corresponds to radar time.Demystifier said:The problem is that one cannot say when the object falls in, in terms of time that makes sense to the outside observer.
There's a calculation of this here:Ibix said:I don't entirely agree with this. You can't say when it crosses the horizon in Schwarzschild coordinates, I agree, because it left the region covered by Schwarzschild coordinates. But that isn't the only definition of "now" available to an external observer - merely one of the simplest, as it corresponds to radar time.
There is a time for any external observer when the event "infalling object crosses the horizon" is no longer in their future light cone. Up until this time they might do something that stops the infalling object from entering the hole. But after this time it is too late. Even if all they need do is send a light pulse to the object asking it to turn on its arbitrarily powerful rocket motor, they are too late - the object crossing the horizon is no longer in their causal future.
renormalize said:I'm not sure I agree…. could she not reasonably declare the time of in-fall to be the moment the external field is sufficiently close to a coulomb field?
That’s two different but equally reasonable definitions, which I think demonstrates @Demystifier’s point.Ibix said:I don't entirely agree with this…. There is a time for any external observer when the event "infalling object crosses the horizon" is no longer in their future light cone.
I'd say it illustrates the opposite - there are many definitions that make sense.Nugatory said:That’s two different but equally reasonable definitions, which I think demonstrates @Demystifier’s point.
That is not clear. He can define time by his watch only along his world-line. For these questions he needs time for much more than his world line. And he can define a time function, whose level surfaces will constitute his convention of nows and at the events of his world line will have values equal to his watch readings, and it can cover parts of the space-time inside the black hole. This way he can say when the object entered the black hole.Demystifier said:The outside observer can reason as follows. I define time as the thing measured by my own local clock. With this definition of time, it takes an infinite time that a falling object reaches the horizon, so the falling object can never (where "never" means not after a finite time) enter the black hole interior. Moreover, according to classical general relativity, there is no experiment that I (the outside observer) can perform that would prove me wrong.
The earlier question was about light pulses that are sent from an outside observer, reflect back from the object falling towards the black hole, and then return back at the observer again. They apparently would take longer and longer time to return back, the closer the object comes to the event horizon, until they take infinitely long.Ibix said:No, the horizon is a finite distance away (or, at least, ##\int\sqrt{ g_{rr}}dr## is finite). It's just that the horizon is never at a time you'd call "now" using those coordinates.
This reasoning does not follow. That definition of time only assigns time to events on your worldline. To extend it anywhere else you must also adopt a simultaneity convention.Demystifier said:The outside observer can reason as follows. I define time as the thing measured by my own local clock. With this definition of time, it takes an infinite time that a falling object reaches the horizon
Fair enough. Is there a reasonable simultaneity convention which I, as an outside observer, can apply to the black hole interior?Dale said:This reasoning does not follow. That definition of time only assigns time to events on your worldline. To extend it anywhere else you must also adopt a simultaneity convention.
Depends on what you mean by reasonable.Demystifier said:Fair enough. Is there a reasonable simultaneity convention which I, as an outside observer, can apply to the black hole interior?
Sure. You can use the simultaneity convention of any of the following:Demystifier said:Fair enough. Is there a reasonable simultaneity convention which I, as an outside observer, can apply to the black hole interior?
Not only is the coordinate speed of light less than c, it is also anisotropic in Schwarzschild coordinates.Rene Dekker said:Is the coordinate speed of light close to the black hole less than c?
For a radial null path the Schwarzschild metric tells us that ##0=\left(1-\frac{R_S}r\right)dt^2-\left(1-\frac{R_S}r\right)^{-1}dr^2##, which means that in these coordinates the coordinate speed of light is ##\frac{dr}{dt}=\pm\left(1-\frac{R_S}r\right)## (minus sign for inward-going rays). This tends to zero as ##r## approaches ##R_S##.Rene Dekker said:Is the coordinate speed of light close to the black hole less than c?
Yes, I see it myself now. It is ## (1 - \frac{r_s}{r}) c## in the radial direction, correct?Dale said:Not only is the coordinate speed of light less than c, it is also anisotropic in Schwarzschild coordinates.
In Kruskal–Szekeres coordinates the radial speed of light is ##c## everywhere, i.e. inside, outside and on the event horizon (though not the tangential speed).Rene Dekker said:I would have thought that it would be more practical to always use a coordinate system where the coordinate speed of light is c everywhere. But I guess in such a system it is not possible to describe things like a black hole.
You are still fundamentally confused by the relationship between a geometry, which is the same geometry in all coordinate systems; and coordinates.Rene Dekker said:Yes, I see it myself now. It is ## (1 - \frac{r_s}{r}) c## in the radial direction, correct?
I would have thought that it would be more practical to always use a coordinate system where the coordinate speed of light is c everywhere. But I guess in such a system it is not possible to describe things like a black hole.
Yes, that is correct.Rene Dekker said:Yes, I see it myself now. It is ## (1 - \frac{r_s}{r}) c## in the radial direction, correct?
There are fundamental limitations to the types of spacetimes for which such a coordinate system is possible. And even in well-behaved spacetimes often such coordinates cannot cover the entire spacetime.Rene Dekker said:I would have thought that it would be more practical to always use a coordinate system where the coordinate speed of light is c everywhere. But I guess in such a system it is not possible to describe things like a black hole.
What I am definitely confused by, is the habit of changing coordinate system if they don't suit your needs. In my mind, coordinates are related to measurements, and you can only (and must) change a coordinate system, if it does not correspond to the measurements you are making.PeroK said:You are still fundamentally confused by the relationship between a geometry, which is the same geometry in all coordinate systems; and coordinates.
Yes, but this is modern physics (relativity) and we have not separate time and space, but a single spacetime continuum! Spacetime distances are invariant. But, unlike classical physics, spatial distances and time intervals are not invariant. They are coordinate dependent.Rene Dekker said:I can fully understand that different coordinate systems can describe the same reality in different ways, like cartesian and polar coordinates. But the measurements (distances, proper time, etc) that you calculate with those coordinates should be the same. The should reflect reality.
The point is that "now, over there" doesn't have any possible assumption-free definition. You need to decide what you mean by "now, over there", which you can do in several different ways. And once you've decided what you mean you can set up a measurement process that reflects that definition - or simply define a mathematical transform between convenient physical measurements and convenient coordinates.Rene Dekker said:Like in this discussion, when one says: "there is no event on the event horizon that corresponds with 'now' for the remote observer". And then somebody else says: "But we can just choose another coordinate system, and then there will be a 'now' on the event horizon". That is something I don't yet grasp.
And that is something you will have to unlearn in GR, because it is simply not possible to always find coordinate charts in a curved spacetime that work the way your intuitions say they should work.Rene Dekker said:In my mind, coordinates are related to measurements
It is.Rene Dekker said:The event horizon is either part of reality or it isn't.
This is the part you will need to unlearn. I understand that your intuitions are telling you this should be true. That means you need to retrain your intuitions.Rene Dekker said:If it is part of reality, there should be events that happen at the same time as events on Earth.
That's not what you're doing. The horizon is there regardless of what coordinates you choose; it's part of reality. But "simultaneity" is not part of reality. It's a human convention.Rene Dekker said:I cannot understand that you can simply whip that into or out of existence by changing coordinate systems.
Yes.Rene Dekker said:And yes, I understand that simultaneity is a much more fluid concept in GR than it is in Newtonian physics
No. SR has the same "fluidity" about simultaneity that GR does. Unfortunately many treatments of SR do not properly emphasize this fact.Rene Dekker said:and Special Relativity.
Because simultaneity is not a "real thing"; it's a human convention. Changing coordinates just means changing the convention.Rene Dekker said:my mind cannot follow why you can just change that by choosing another coordinate system, without making any physical changes in reality, or taking another observer's viewpoint.
The counterintuitive thing here is buried in that phrase "at the same time". "At the same time" is just an informal way of saying "has the same time coordinate", so its meaning depends on the way that we assign time coordinates to events.Rene Dekker said:If it is part of reality, there should be events that happen at the same time as events on Earth. I cannot understand that you can simply whip that into or out of existence by changing coordinate systems.
You have already been taught that this is incorrect. Coordinates are basically just arbitrary labels, and they do not have any intrinsic relationship to measurements.Rene Dekker said:What I am definitely confused by, is the habit of changing coordinate system if they don't suit your needs. In my mind, coordinates are related to measurements
You can adopt any smooth and invertible mapping as your coordinate system at any time for any reason or no reason. Usually we pick a coordinate system for convenience. Usually convenience is determined by how well it simplifies the math.Rene Dekker said:and you can only (and must) change a coordinate system, if it does not correspond to the measurements you are making.
Indeed, the measurements are invariant and any coordinate chart that covers a region will agree on the outcome of any measurements in that region. The problem is that the Schwarzschild coordinate chart does not cover the event horizon. So it cannot be used to make any statements about the horizon.Rene Dekker said:But the measurements (distances, proper time, etc) that you calculate with those coordinates should be the same. The should reflect reality.
It is part of reality, but coordinates are not. So it is not a reflection on reality if a given coordinate chart does not cover the event horizon.Rene Dekker said:The event horizon is either part of reality or it isn't.
There have been studies that show that the single most difficult concept in learning relativity is understanding that the concept of “at the same time” is a human concept. It is not part of reality. The idea of “now” is part of the coordinates (simultaneity) so it is not a part of nature. This is a hard thing to accept, but it is well founded both experimentally and theoretically.Rene Dekker said:If it is part of reality, there should be events that happen at the same time as events on Earth
When you visit a new city do you use their addresses or what they would be in your home town? (Let's see...12 North Avenue would be 1,005,872 South Street and....)Rene Dekker said:What I am definitely confused by, is the habit of changing coordinate system if they don't suit your need
I see your point. However, I would like to point out that these simultaneity conventions cannot be interpreted as synchronization conventions. A synchronization convention is an experimental procedure, while these simultaneity conventions cannot be used to experimentally synchronize two clocks, one outside and the other inside the black hole. It does not imply that these simultaneity conventions are wrong, but to avoid confusion (including my own) it is important to keep in mind the difference between simultaneity conventions and synchronization conventions.Dale said:Sure. You can use the simultaneity convention of any of the following:
https://en.m.wikipedia.org/wiki/Kruskal–Szekeres_coordinates
https://en.m.wikipedia.org/wiki/Lemaître_coordinates
https://en.m.wikipedia.org/wiki/Eddington–Finkelstein_coordinates
https://en.m.wikipedia.org/wiki/Gullstrand–Painlevé_coordinates
Or any other simultaneity convention you like. Your simultaneity convention is arbitrary.
They're distinct from synchronization conventions available to permanently external observers, yes. Free-falling observers (see Gullstrand-Painleve coordinates) may well synchronise clocks using these conventions, since they cross the horizon themselves. They can't complete the synchronisation process while the clocks are on opposite sides of the horizon, but they can cross the horizon before, during, or after it.Demystifier said:I see your point. However, I would like to point out that these simultaneity conventions cannot be interpreted as synchronization conventions. A synchronization convention is an experimental procedure, while these simultaneity conventions cannot be used to experimentally synchronize two clocks, one outside and the other inside the black hole. It does not imply that these simultaneity conventions are wrong, but to avoid confusion (including my own) it is important to keep in mind the difference between simultaneity conventions and synchronization conventions.
Coordinates are not related to measurements. They are just arbitrary parameters mapping the spacetime manifold locally to open sets of ##\mathbb{R}^4##. From a mathematical point of view the "general covariance" of relativity, i.e., the local diffeomorphism invariance, is a gauge symmetry and in this sense coordinates are gauge-dependent mathematical entities and thus cannot be directly observables.Rene Dekker said:What I am definitely confused by, is the habit of changing coordinate system if they don't suit your needs. In my mind, coordinates are related to measurements, and you can only (and must) change a coordinate system, if it does not correspond to the measurements you are making.
Exactly! I.e., only (local) gauge-invariant quantities (tensors) directly describe observables. It's as in electrodynamics: The four-potential is not directly an observable, because it's gauge dependent. You can calculate observables from it, i.e., the electromagnetic field, which is a tensor, which is independent of the choice of gauge.Rene Dekker said:I can fully understand that different coordinate systems can describe the same reality in different ways, like cartesian and polar coordinates. But the measurements (distances, proper time, etc) that you calculate with those coordinates should be the same. The should reflect reality.
Oh, interesting. I have never seen this distinction before. It is a reasonable distinction, but it is novel to me.Demystifier said:I see your point. However, I would like to point out that these simultaneity conventions cannot be interpreted as synchronization conventions. A synchronization convention is an experimental procedure
I would need to see a proof that this is true. Of course, you cannot use Einstein's synchronization procedure to do so, but to say that there exists no possible experimental synchronization procedure seems like quite a stretch.Demystifier said:these simultaneity conventions cannot be used to experimentally synchronize two clocks, one outside and the other inside the black hole
It seems that you are right. Originally I thought that synchronization should involve an exchange of information between two clocks, but there is actually no need for that, one way information should be enough. If the master clock is outside the black hole, it should work. More precisely, it would work for physical black hole created in a gravitational collapse, but it would not work for a maximal Kruskal extension containing both the black and the white hole.Dale said:I would think that it suffices to have one master clock broadcast a time signal and other clocks adjust their clocks accordingly by solving the null geodesic equation in the given coordinates.