But a distant observer is any observer outside of the event horizon.
The crucial point here is to understand that a coordinate system is basically the “point of view” of a specific observer – in the case of Schwarzschild coordinates, that would be a stationary observer very far away from the black hole. All observers agree that the in-fall happens in a finite amount of proper time; what differs are only the coordinate times. This isn’t simply a case of what the distant observer sees, from their perspective it’s always possible to for an object to accelerate away. The proper time however, i.e. what a clock physically records as it falls, is buy a essay paper
quite finite, even in Schwarzschild coordinates, as shown below.The physical event takes place regardless of which coordinate system you choose, it’s not a frame-dependent object. The proper distance from some shell r1 down to the event horizon, in Schwarzschild coordinates, isThe proper time this in-fall takes, still in Schwarzschild coordinates (geometrised units), isBoth of these – proper distance as well as proper time – are finite. In other words an object can never reach the event horizon until you do, and you can’t until another object does, and so on.The Schwarzchild coordinates do tell us something about the geometric length of a world-line extending towards an event horizon. I’m not talking about a hypothetical observer at infinite an distance, I’m talking about a real observer at any distance from the horizon. Yes I know the difference between coordinate dependent time and proper time, I’m also well aware of what a change in coordinate systems can do in regards to how time is measured and what it most definitely can’t do.The time interval between events and the order in which they occur is an arbitrary choice of how you choose to measure the system but if an event does occur in one coordinate system then it’s a physically real event that happens in every coordinate system (although the event may be outside of the coordinates used it will be in that coordinate system).
This is what the Schwarzschild coordinates clearly show any coordinates that allow an object to reach an event horizon in a finite amount of proper time are in direct contradiction to this coordinate system. What is not finite is the coordinate in-fall time for a Schwarzschild observer, which corresponds to what he measures on his own clock that remains stationary infinitely far away from the event horizon. Simply put, the Schwarzschild observer never sees anything reach the horizon (he just sees it slowing down, dimming, and being red-shifted to invisibility), but the test particle itself still reaches it in finite time, as demonstrated above.
If you want to be absolutely sure, you can also explicitly calculate the in-fall geodesics from the geodesic equation, and then integrate up to obtain their geometric length. There is no contradiction. therefore no amount of proper time on any watch allows an object reach the horizon. This measurement is equivalent to the geometric length of a radial geodesic, so it being finite means that things do indeed reach and cross the horizon, even though a Schwarzschild observer never sees that happening. The proper distance from some shell r1 down to the event horizon, in Schwarzschild coordinates, isThe proper time this in-fall takes, still in Schwarzschild coordinates (geometrised units), is I don’t doubt the result of the equations you’re using, I doubt their applicability to this situation.
The proper time it takes for something to fall to the event horizon (and onwards into the black hole) is finite and well defined. The Schwarzchild coordinates do indeed tell us something about the geometric length of a world-line extending towards an event horizon. So yes, an event that does not happen for one observer may indeed happen for another observer – because we are in a curved spacetime.There is no contradiction here either, because the in-fall time as measured by a co-moving clock is a proper quantity, i.e. something that all observers agree on, whereas the infinity of the Schwarzschild observer is merely an artefact of the coordinate system, and hence observer-specific.To put it succinctly, in order to check whether the horizon can be reached in a finite amount of time, you need to look at a watch in free fall, not at a watch stationary at infinity.
That is incorrect, see the maths in the last post. This is done in (e.g.) T. The coordinate in-fall time diverging to infinity for Schwarzschild coordinates physically means that such observers never see/measure anything reach the horizon.However, Schwarzschild coordinate time does not tell us anything about the geometric length of a world line extending down to the horizon. But it does happen from the perspective of the in-falling observer himself, and it does so in finite time.
The geometric length of the world lines is finite. Do note though that Schwarzschild coordinate measurements do not correspond to anyone’s physical clock or ruler in the real world.P.S. Time dilation and length contraction on the other hand are observer-dependent quantities, and both diverge at the horizon in the case of Schwarzschild observers. In curved space-times, “time” is a purely local concept, so a far-away clock cannot tell us anything about what happens locally at the horizon, or at any point in between.Understanding the difference between coordinate quantities and proper quantities is absolutely crucial in General Relativity.
Post by Markus Hanke:This is a very common misconception, but it is wrong. It tells us that due to time dilation and length contraction, the world-line is infinite in length before ever reaching the horizon No. So an object can never reach an event horizon from any distance under any amount of acceleration, gravitational or otherwise. Also see Baez (2nd paragraph) for a plain-text explanation : What happens to you if you fall into a black holesP.S. But that does not necessarily mean that everyone can observe, measure, or calculate that event. it never ever happens from the perspective of a distant observer Correct.
As expected, the result – i.e. proper time – matches the expressions given earlier, and is quite finite. But a distant observer is any observer outside of the event horizon. Strictly speaking, a Schwarzschild observer does not even exist in the real world, the coordinate time is simply a bookkeeping device, and does not correspond to anyone’s physical clock reading.See Taylor/Wheeler “Exploring Black Holes” chapter 3, as well as Misner/Thorne/Wheeler “Gravitation” chapter 31, for the precise derivation and physical justification of the above expressions. It’s just that the Schwarzschild observer at infinity cannot see, measure, or calculate it using his own clocks and rulers, since his notions of time and space or not the same as the ones locally at the event horizon.Let’s look at this in more detail.
Posts by Markus Hanke: It takes takes an infinite amount of proper time in Schwarzchild coordinates for an object to reach an event horizon No, it takes an infinite amount of coordinate time for the Schwarzschild observer, meaning that he never sees or measures anything reaching the horizon. Using equations derrived from the perspective of a distant observer, the proper time required on their clock for any object to reach the horizon is infinite (so is the distance in space because of length contraction but sticking with time is fine to keep it simple). This is not a contradiction – they are both right, since measurements of time and space are purely local quantities. If we examine the point of view of an observer in free fall towards the black hole, then we will find that the horizon is reached – and crossed – in a finite amount of time, as measured by a co-moving clock.
The proper measurements on the other hand correspond to what a clock and ruler physically measure when they actually travel along the free-fall geodesic, which is of finite length – all observers agree on this, including the Schwarzschild observer.Since both the above expressions are covariant quantities, if they are finite in one coordinate system, they are finite in all coordinate systems.So, while the Schwarzschild observer argues that – according to his own clock – the in-falling test particle never reaches the horizon, a clock co-moving with the particle disagrees, and reaches the horizon in a finite time as given above. Changing between two valid coordinate systems doesn’t allow an event that can never happen in one to happen in another. Since Schwarzschild observers are by definition infinitely far away from the horizon, it is not really a surprise that their coordinate in-fall time diverges.
It can’t be anything to do with the distant observer accelerating away from the horizon in order to maintain a constant distance as opposed to the falling object only being under gravitational influence because if a distant observer were to fall towards the black hole they still could never observe the closer object reaching the horizon.It can’t be possible for a falling observer to reach the event horizon from the perspective of a more distant falling observer either because the more distant observer could accelerate away and the closer object would have to reemerge from inside the horizon. Apologies for the bad formatting of the maths above, the LaTeX rendering isn’t brilliant on this old forum software. if an event does occur in one coordinate system then it’s a physically real event that happens in every coordinate system An event is physically real regardless of any coordinate system. It tells us that due to time dilation and length contraction, the world-line is infinite in length before ever reaching the horizon. The coordinate labels are an arbitrary choice. The Schwarzchild coordinates are valid though and show that time dilation and length contraction approach infinity at the horizon.A coordinate system that allows an object to cross the event horizon is in direct contradiction to the Schwarzchild coordinates.
If an event never occurs in one coordinate system than it never happens in any coordinate system.It takes takes an infinite amount of proper time in Schwarzchild coordinates for an object to reach an event horizon (proper time as in the time measured on the distant observer’s watch) so it never ever happens from the perspective of a distant observer. That means however long the black hole hole exists for, it can never be too late for an object to escape and therefore no amount of proper time on any watch allows an object reach the horizon.If it were possible to reach an event horizon from the perspective of a falling observer then at what point does the Schwarzchild coordinate system become invalid, how close to the event horizon? Schwarzchild coordinates must always be invalid if this is the case and it’s just a matter of degree, they can’t suddenly stop working at a certain distance. See the expressions in the last post, and refer to the sources I quoted for their detailed derivation and explanation. It takes takes an infinite amount of proper time in Schwarzchild coordinates for an object to reach an event horizon No, it takes an infinite amount of coordinate time for the Schwarzschild observer, meaning that he never sees or measures anything reaching the horizon.
The proper time however, i.e. what a clock physically records as it falls, is quite finite, even in Schwarzschild coordinates, as shown below.The physical event takes place regardless of which coordinate system you choose, it’s not a frame-dependent object. How close to the black hole does an observer have be in order to ‘see’ an object crossing the horizon? It doesn’t make any difference how close it is.No matter how long a black hole lasts for, it will never be enough time for an object to reach the horizon from a distant observer’s perspective.
It’s just that the Schwarzschild observer at infinity cannot see, measure, or calculate it using his own clocks and rulers, since his notions of time and space or not the same as the ones locally at the event horizon.Let’s look at this in more detail. Fliessbach, “General Relativity”.