Merging black holes and event horizons
Robert Low
black hole formation.
A sherically symmetric collection of matter collapses, and
at some stage of the collapse, an event horizon forms.
The event horizon starts off as a point, and thereafter is a
smooth sphere which grows until it surrounds all the matter,
and thereafter remains unchanging. This is OK for a
spherically symmetric, asymptotically flat universe.
More interesting: what happens if there are two collapses
into black holes which subsequently merge?
I'd kind of assumed that each black hole formed in much the
same way as the isolated black hole, maybe with a little
more complication about the initial formation of the event
horizon. Then each event horizon would settle down to a
smooth sphere, and that after a while the event horizons would
get close together, and there'd be some funny business
as they merged (maybe they'd both grow like teardrops,
and the pointed ends would meet to form a single sphere).
But I hadn't really thought about it too much.
It turns out that things aren't like that at all.
At least, not subject to some reasonably plausible
assumptions on the nature of the space-time. What I'm
about to describe follows from the paper:
Masaru, S (1998) Topological appearance of event horizon,
Progress in theoretical Physics Vol 99 1--32.
First, assumptions:
1/ Eventually, the event horizon will settle down to a smooth
sphere.
2/ The set of points at which the event horizon is not smooth
is compact. (This comes down to saying that for any way of
slicing the space-time up into space and time, the whole
non-smooth region lies between two slices.)
From these assumptions, one can show that the set of points
at which the event horizon fails to be smooth has two rather
remarkable properties. First, it is acausal (i.e. no point
in this set lies either to the future or to that past
of any other). Second, it is path connected (i.e. any two
points in the set can be connected by a curve which lies
in the set).
First consequence: you can't find a slicing of space-time
in which two black holes with smooth event horizons form,
and subsequently the event horizons merge. Since the
non-smooth portion is connected, the event horizons of
the two black holes have singularities all the way from
formation to merging.
Second consequence: since the non-smooth region is acausal,
you can't unambiguously say that two black holes form and
then merge. You can also slice space-time up in such a way
that what seems to happen is that a single black hole forms
all at once, and the event horizon is a sphere right from
formation.
This really drives home the fact that the event horizon
is a global, not a locally defined object. No matter how
far apart the collapsing balls of matter are, the event horizon
that forms round each is significantly affected by the
other. It also reminded me, at least, that what 'obviously'
happens can be far from right...
C. Hillman
Re Robert Low's very interesting reply to Andrew's query--- this is
-exactly- the local versus global distinction again.
Chris Hillman
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Thanks!
Nathan Urban
> More interesting: what happens if there are two collapses
> into black holes which subsequently merge?
> I'd kind of assumed that [...]
> It turns out that things aren't like that at all.
Wow. That's what I'd always assumed too, though I was always uneasy about
it because I knew that I hadn't seen any hard results on this scenario,
and these things are notoriously tricky. I'm going to have to get a
copy of that paper. Some of it sounds very nonintuitive, like:
> Second consequence: since the non-smooth region is acausal,
> you can't unambiguously say that two black holes form and
> then merge. You can also slice space-time up in such a way
> that what seems to happen is that a single black hole forms
> all at once, and the event horizon is a sphere right from
> formation.
Just goes to show you how subtle GR can be.
George Dishman
[snipped]
>First consequence: you can't find a slicing of space-time
>in which two black holes with smooth event horizons form,
>and subsequently the event horizons merge. Since the
>non-smooth portion is connected, the event horizons of
>the two black holes have singularities all the way from
>formation to merging.
>
>you can't unambiguously say that two black holes form and
>then merge. You can also slice space-time up in such a way
>that what seems to happen is that a single black hole forms
>all at once, and the event horizon is a sphere right from
>formation.
Fascinating! For a poor layman like myself, does that
mean that we can never see two black holes merge? If
so, that would seem to rule out such events as a
possible cause of GRBs, or require at least three
holes.
--
George Dishman
Give me a small laser and I'll move the sun.
The arrow of time points in many directions.
The FAQ: <http://math.ucr.edu/home/baez/physics/relativity.html>
Robert Low
George Dishman <geo...@briar.demon.co.uk> wrote:
>>Second consequence: since the non-smooth region is acausal,
>>you can't unambiguously say that two black holes form and
>>then merge. You can also slice space-time up in such a way
>>that what seems to happen is that a single black hole forms
>>all at once, and the event horizon is a sphere right from
>>formation.
>
>Fascinating! For a poor layman like myself, does that
>mean that we can never see two black holes merge? If
No. You could still see two distant stars collapse down
and red-shift out of sight, and then see the accretion
disks fall together and merge into a single object. (Can't
win, can you?)
However, that picture is to a large extent an artefact of
you position in the universe (i.e. distant from both
stars).
Remember that in the vicinity of the event horizon the
light cones are highly tilted (in a representation such
as you get from Eddington-Finkelstein coordinates) so
that your notion of space-time doesn't extrapolate well
to that region. Even although to an distant observer
the two black holes form separately and then ages later
merge into a single one, due the the space-time curvature
near the black holes, it turns out that the first appearance
of the event horizons is spacelike separated from their
coalescence. And this means that you can slice up space-time
in such a way that the events are simultaneous. But that
slicing is not at all obvious to a distant observer,
though it is fairly natural to one who lives near the
even horizon.
>so, that would seem to rule out such events as a
>possible cause of GRBs, or require at least three
>holes.
No; but it does mean that you have to be extremely careful
when you try to talk about what causes what, since events
that you think are timelike separated may in fact be
spacelike separated.
chor...@my-dejanews.com
> Masaru, S (1998) Topological appearance of event horizon,
> Progress in theoretical Physics Vol 99 1--32.
> in which two black holes with smooth event horizons form,
> and subsequently the event horizons merge. Since the
> non-smooth portion is connected, the event horizons of
> the two black holes have singularities all the way from
> formation to merging.
> you can't unambiguously say that two black holes form and
> then merge. You can also slice space-time up in such a way
> that what seems to happen is that a single black hole forms
> all at once, and the event horizon is a sphere right from
> formation.
These are both immediate consequences of the standard "trousers"
picture of merging black holes. Note also that if we assume
a closed universe, then, according to the strict global definition,
no black holes can even exist, because there is no future null
infinity. Also, if we consider approximate local notions of
"black holes" in a closed universe, there is really only one, in
the sense that they all share a common event horizon with each
other (not to mention the rest of the universe). This was all
explained in detail a year or two ago in sci.physics, and can
certainly be found in various standard GR texts. It seems
strange that this is presented as "progress" in theoretical
physics. Seems more like an exposition or popularization.
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Robert Low
>mtx...@coventry.ac.uk (Robert Low) wrote:
>> Masaru, S (1998) Topological appearance of event horizon,
>> Progress in theoretical Physics Vol 99 1--32.
>
>These are both immediate consequences of the standard "trousers"
>picture of merging black holes.
Oh? Maybe you could favour me with a quick outline of
how the results follow immediately from that standard picture?
[These results being acausality and connectedness of the
non-smooth subset of the event horizon.]
The only 'picture' I've seen is the one in Hawking&Ellis
which does not address the issue of the differentiability
of the crease set, or the causal relationship between the
formation of black holes and their 'subsequent' coalescence;
or at least, if it does, I missed that bit of the discussion.
Page references to Hawking&Ellis, or whatever other
standard text covers this in detail would be fine.
> Note also that if we assume
>a closed universe, then, according to the strict global definition,
>no black holes can even exist, because there is no future null
>infinity.
Oddly enough, there is a significant amount
of interest in asymptotically flat space-times. Maybe you
don't think it counts as physics, but kindly let the rest
of us poor misguided types play with the shiny pebbles which
we like best.
Are you familiar with the recent literature on topological
censorship? In particular with the discussion about the
change in topology of event horizons?
>explained in detail a year or two ago in sci.physics, and can
>certainly be found in various standard GR texts.
I'd certainly like to know which standard GR texts contain
a detailed discussion of the differentiability of the event
horizon and how different time-slicings give different
pictures of topology change for event horizons. I'm not saying
that these properties are hard to get to; they aren't. But they
certainly aren't covered in any standard textbooks that I've seen.
> It seems
>strange that this is presented as "progress" in theoretical
>physics. Seems more like an exposition or popularization.
Well, how about going and reading the paper rather than
making up your mind based on the one facet I picked out
because it was relevant to a question that had been
asked recently? You may remain convinced that the paper
is just a popularization, but at least you'll have some
evidence.
It isn't quite compulsory to be so snotty, not even in
this newsgroup, Mr Anonymous.