## On the Approach to a Black Hole

In this note we consider “approach to a black hole” in two senses. First the experience of approaching the event horizon of an existing Schwarzschild black hole and then the evolution of a sphere of gas that is almost dense enough to form a black hole as it evolves towards black hole status.

1. An extended object approaching the event horizon of an existing black hole experiences extreme tidal distortions if the black hole is of modest size, but as the size of the BH increases the tidal forces near the event horizon decrease so that for a supermassive black hole at a galactic centre they may in fact be quite negligible. This has led to the claim that a human traveller might enter the black hole while maintaining consciousness and not noticing anything untoward. That this is wrong might seem obvious if we consider what an observer passing through the event horizon feet-first would experience. The argument in support of this wrongness is that if at any point (on the timeline of the head of the observer) the feet of that observer are inside the event horizon while the head is still outside, then the feet would have to be invisible to the head. So the head must either see the feet disappear or see itself reaching catching up with the feet – either of which would surely be noticeable. The problem with that analysis is that what the head actually sees is the feet at some time in the past and those past feet can still have been outside the event horizon, while the feet just inside the horizon will indeed be seen by the freely falling head sometime after it too passes through the horizon.

[Some (including sometimes me) have suggested that the location of the event horizon is not fixed and is actually relative to the observer, but the set of events with r=2M in Schwarzschild coordinates is a well defined tube in spacetime and has the property that any event inside that tube has a forward light cone contained entirely within it – and this topological property is quite independent of what coordinates are used to describe the situation.

Others have cited a theorem to the effect that there is no local property of the metric that distinguishes the event horizon and claim that this means that passage through the event horizon is not noticeable. But the human observer is an extended object, and the mathematical concept of a local property involves only the fields and their derivatives at a single point (so the negation of “local” is not “global” and properties relating the distinct parts of even a very small object are not actually “local”). Indeed if the non-local aspect were sufficient to make the event horizon unnoticeable then that would also apply in the situation of a small black hole where the tidal forces really were destructive – and most human observers would surely notice that! But, in any case we believe that the alleged theorem has been mischaracterized in those discussions and that local evidence of the event horizon is available.]

In part 1 below we will try to establish what a human-scale observer would actually experience on approaching the event horizon of a black hole in the case where the schwarzschild radius is large enough that tidal forces would not be an issue – but we shall do so only for the case of a pure schwarzschild black hole with no surrounding matter (aside from the freely falling observer with purely radial velocity)

2. A related problem is whether or not a human could survive in the midst of a low density gas or stellar distribution of such extent that the total mass was almost sufficient to create an event horizon (and of what the “experience” would be like as the overall mass of material contracts so that an event horizon does get created). We will address this situation in part 2 below – but only in the unrealistic case where the initial configuration is a ball of uniform density.

In both cases our “observer” will be simplified to consist of just two point masses joined by a massless rod of fixed length oriented radially relative to the centre of mass of the system. And in both cases the key quantity that will be used is the red shift per metre for radiation propagating in the radial direction.

Part 1 (yet to be written)

Part 2 (yet to be written)