But later researchers started to claim that the event horizon is just a "coordinate singularity", which is a result from badly chosen coordinates.
A standard procedure to get rid of the "coordinate singularity" is to switch to Eddington-Finkelstein coordinates.
For a test mass falling into the black hole, the Schwarzschild coordinate time goes to infinity before the test mass reaches the horizon.
Eddington-Finkelstein coordinates are based on the tortoise radial coordinate r* which goes to minus infinity at the same rate as the Schwarzschild time coordinate grows to infinity.
One can extend Eddington-Finkelstein coordinates to cover the journey of the test mass through the event horizon and deeper. There is no singularity at the event horizon.
But does the extension make sense?
Extending time past the infinity
Let us imagine that we have an infinite sequence of observers ever closer to the event horizon. They monitor the time at which the test mass passes them on its way deeper.
The observers use the global Schwarzschild time in their clocks.
The observers register that the test mass never reaches the event horizon. The word "never" is appropriate here, because the observers can synchronize their clocks in a perfectly sensible way and can make their clocks to run at the same rate. Any interval of time has the same duration measured by any of the clocks.
It is as if the observers were in the flat Minkowski space and they would use standard clocks.
• ------> ● ● ●
photon observers
------------------> x
Imagine that the universe is the Minkowski space, and we have an infinite sequence of observers at ever larger distances from us. We send a photon along the x axis, and the observers monitor in the global Minkowski time when it passes them. The observers register that the photon never leaves the universe.
Does it make sense to extend the Minkowski global time t and and the x axis to cover the time "past the infinity"?
The proper time of the photon does not advance at all. We could say that the photon reaches the "edge" of the Minkowski space in a finite proper time. This is just like in the Schwarzschild metric, where the test mass reaches the event horizon in a finite proper time.
Extension past the infinity is a dubious thing. It is like changing the standard model of arithmetic to a non-standard model. We add on top of the natural numbers ℕ a copy of integer numbers ℤ.
Suppose that we in electromagnetism study a phenomenon where an electron comes ever closer to a certain position. Does it make sense to extend time past the infinity and calculate what the electron does after the infinite time has passed and it has reached that position? People would find the extension very strange.
Achilles and the tortoise
Photo George Hodan
The name of the tortoise coordinate refers to a claimed similarity of a test mass falling to a horizon and Achilles chasing the tortoise.
Is there a similarity?
Let Achilles and the tortoise run along the x axis. Achilles reaches the tortoise at x = 1.
Let us use synchronized clocks along the race path, such that the clocks agree about the duration of time intervals. The clocks will show that Achilles reached the tortoise in a finite time. There is no similarity to a test mass falling to a horizon.
To get an infinite duration for the race, we should use clocks which run the faster, the closer they are located to x = 1.
Conclusions
Changing coordinates and extending history past an infinite time of Schwarzschild coordinates does not seem sensible.
Extended coordinates are used to get the interpretation that the event horizon is a one-way membrane. We have argued in previous blog posts that a one-way membrane breaks thermodynamics, and leads to the black hole information paradox.
A dubious procedure leads to a model which breaks basic laws of physics. We conclude that extending coordinates past the infinity does not make sense.
The singularity at the event horizon is not just a coordinate singularity. The existence of the singularity shows that the model which is used to derive the behavior close to a horizon is bad. One cannot deduce the behavior of a test mass just from the Schwarzschild metric when the test mass is extremely close to the horizon.
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