Can math.atan2 return INF?

[Steven D'Aprano]

On Sunday 26 June 2016 09:40, Gregory Ewing wrote:

> Marko Rauhamaa wrote:
>> pdorange at pas-de-pub-merci.mac.com (Pierre-Alain Dorange):
>> 
>>>Near a black hole 3.7 seconds can last an infinite time...
>> 
>> Which phenomenon prevents a black hole from ever forming. Yet
>> astronomers keep telling us they are all over the place.
> 
> Astronomers have observed objects whose behaviour is
> entirely consistent with the existence of black holes
> as predicted by general relativity.

Indeed.

There's a common myth going around that black holes take an infinite amount of 
time to form, or another way of putting it is that it takes an infinite amount 
of time for something to fall into a black hole, and therefore "black holes 
can't really exist". This myth comes about because people don't fully 
understand the (admittedly mind-boggling) implications of General Relativity.

First, you must accept that *your* experiences are not the only valid 
experiences. Just because *you* never see the black hole form, doesn't mean it 
doesn't form. You just don't get to experience it yourself.

So, let's consider a thought experiment. Imagine three astronauts, Tim, Bill 
and Graham ("we do anything, anytime"). Tim and Graham are in separate space 
ships, dropping straight towards a black hole under free fall. Bill is watching 
them from a safely orbiting space station.

Graham drops towards the black hole, as does Tim. Neither can see the black 
hole directly, or at least they can't see the *inside* of the black hole, since 
no light can escape it, but they can see it by its effect on the starlight: the 
black hole acts like a great big lens, bending light. If they line up their 
space ships with the black hole directly between them and a distant star, they 
will see one of the more amazing sights of the universe: gravitational lensing. 
The (somewhat reddened) light from the star will be bent around the black hole, 
so that the star will appear to be a donut of light with a black centre.

As Graham falls, he pays close attention to the distance from his ship to the 
black hole. That's easy to do: he can tell how fast he is going thanks to 
Bill's space station, which transmits a steady radio signal for him, a steady 
clock sending one pulse per second.[1]

But as Graham gets closer and closer to the event horizon, he notices Bill's 
radio signals have a higher and higher frequency, and appear to be sped up... 
at the end, just before he loses his nerve and fires the retro-rockets before 
crossing the event horizon, the signals are coming thousands of pulses per 
second.

When Graham returns to the space station, he finds a *much* older Bill waiting 
for him. Bill insists he was sending one pulse per second, as agreed, but that 
Graham has been gone for many years. Graham insists that he has only been gone 
a few days, and Bill has obviously been partying very hard indeed to look like 
this after such a short time. But after sitting down with Albert Einstein's 
head[2], they reconcile their two differing experiences:

As seen by Bill, in Bill's frame of reference far from the black hole, Graham's 
experiences have been slowed down enormously. But Graham sees things 
differently: he experiences his own frame of reference at the same speed he 
always has, and see's *Bill's* frame of reference as being immensely sped up. 
Neither is "right" and the other is "wrong", neither frame of reference is 
privileged over the other. BOTH are right, even though they contradict each 
other. That's the nature of the universe we live in.

What about Tim?

Tim is so engrossed by the view of the gravitational lensing that he forgets to 
fire the retro-rockets, and before he knows it, he's crossed the event horizon 
and there's no going back.

For a sufficiently large black hole, he might not even have noticed the 
transition. From his perspective, he's still distant from the singularity 
(being so small and distant, he can't quite make out what it looks like), and 
space-time is still quite flat for a sufficiently large black hole. Tim can 
still see out, although the incoming light is getting bluer, and he's still 
receiving Bill's clock signals, though like Graham he sees them as drastically 
sped up.

If Tim has sufficiently powerful rockets with enough fuel, he could *not quite* 
escape: he could change his space ship's trajectory enough to avoid the 
hypothetical singularity for days, weeks, years, as long as the fuel lasts. But 
nothing he can do will allow him to escape the event horizon. (Well, maybe 
faster-than-light travel, if his hyperdrive will still work this close to a 
black hole.)

And as invariably as tomorrow follows today, he's getting closer to the 
supposed singularity, and long before he reaches it, both Tim and his space 
ship will be torn apart into atoms by the ever-increasing tidal forces, and 
then even the atoms torn apart. And then, well we really have no idea what 
happens if you try to squeeze an electron into a volume of space a trillion 
times smaller than a sphere with radius equal to the Planck Length...

Should Tim realise his fate, and decide that there's no point delaying the 
inevitable, his free-fall drop into the supposed singularity will be over in a 
relatively short amount of time, at least according to his own frame of 
reference. (For a stellar size black hole, the time from crossing the event 
horizon to reaching the supposed singularity is a small fraction of a second.)

For a large black hole where the Schwartzchild Radius is 1 a.u., and ignoring 
any relativistic corrections, it will take Tim six months of free fall to 
collide with the singularity. During that time he will have plenty of time to 
reflex on the curious way that Bill's space station is running faster and 
faster, sending clock ticks millions of times a second. (At least until he is 
torn apart by tidal forces.)

Meanwhile, back on Bill's space station, they're still faithfully sending out 
radio pulses at the rate of one per second. By looking in their most powerful 
telescopes, they can see Tim's spaceship falling into the black hole, strangely 
slowing down as it approaches, the light getting redder and redder, Tim's own 
radio signals getting further and further apart. Tim's spaceship appears to be 
*asymptotically* approaching the event horizon, in some sort of horrible 
version of Zeno's Paradoxes: each minute that goes by, Tim gets closer to the 
event horizon by a *smaller* amount as time slows down for him (as seen by Bill 
and Graham on the space station).

Graham decides to mount a rescue mission. He flies back towards the black hole, 
and experiences the same speeding up of signals coming from Bill. But no matter 
how closely he approaches the event horizon, Tim always appears to ahead of 
him, like the turtle to Achilles (from Graham's frame of reference).

As seen by Bill, the closer Graham gets to Tim, the slower he appears to be 
experiencing time, with his return clock ticks coming back one a day instead of 
one per second, then one a week, one a month, ...

Unless Graham is willing to cross the event horizon too, he cannot catch up to 
Tim. That's because, from Tim's perspective, Graham left the space station too 
late. Tim has already experienced crossing the event horizon, so unless Graham 
has a time machine, there's nothing Graham can do to reach Tim before he 
crosses the event horizon. No matter how close Graham gets to the event 
horizon, Tim will already be just a bit closer. Unless Graham is suicidally 
willing to cross the event horizon too, he's going to have to pull out (after, 
from his perspective, perhaps as little as a few hours of high-acceleration), 
and return to the space station, where he will find that Bill has experienced 
possibly many thousands of years.

So whose viewpoint is right? According to Bill and Graham, Tim is frozen just 
before the event horizon, infinitely red-shifted, his radio signals coming in 
infinitely slowly. But according to Tim, the opportunity for rescue is long 
gone. He long ago watched Graham's idiotic rescue mission ("don't bother 
Graham, I'm well past the event horizon, you can't save me now..."), the daring 
flight almost to the event horizon, and Graham's eventual return to the space 
station.

Although their perspectives are very different, neither is "more right" than 
the other.

So does the black hole form? Do objects cross the event horizon? From the frame 
of reference of such objects, yes, certainly, and in a fairly short period of 
time too. From our frame of reference, we seem them asymptotically approaching 
the event horizon, but never cross it. Both are equally correct.





[1] Or very possibly playing "A Walk In The Black Forest".

[2] In a jar.




-- 
Steve