Re: black holes and singularity

From: MP (pet.antispam_at_onlinehome.de)
Date: 11/05/04 

Date: Fri, 5 Nov 2004 17:17:26 +0100

"Randy M. Dumse" <rmd@newmicros.com> wrote in message
news:howid.231$QK5.8844@eagle.america.net...
> "MP" <pet.antispam@onlinehome.de> wrote in message
> news:cmdq99$j7q$1@online.de...
> > Its unknown territory, so one has to be very careful not
> > to fall prey to one's own prejudices, what it self-evident
> > and what is not.
>
> Wonderfully said!
>
> It is always the case some self imposed "blinders" prevent us from
> making advances. It is not that we don't know something and then
> discover something totally unexpected, it is that we think we know
> something very well, that blinds us from something obvious (in
> retrospect). I call this thing in the way of progress, "The One Wrong
> Idea". For Aristotle, it was things in motion come to rest. For Newton
> it was there is an absolute rest frame. Etc. Such perfectly reasonable
> assumptions then halt progress for decades. Their perfectly reasonable,
> but wrong.

This is what history tells us. Yet the majority of researchers always
seem to fall into the same trap. It is the trap of overconfidence
(assuming the majority *must* be right, so no worth using ones own
brain where so many other brains have spoken)

It took an independent thinker as Galileo to risk his life, in order
to supplant the old (Aristotelean) ideas. It took an independent
mind such as Einstein's to realize that the (then still perfectly
reasonable) Newtonian idea of absolute space was flawed. Both Einstein
and Galileo were heavily attacked by some of the most respected
scientists of their time. Einstein was lucky to have Planck publish
his first paper. (Such as Ramanujan was lucky to have Hardy read
his letter. Two other high caliber mathematicians thought it was
crap.) Planck was a great physicist with a broad understanding
of *all* aspects of physics (a "generalist" so to say). Sadly this
cannot be said from the typical referee of today's science journal,
who is only good in his highly specialized domain of expertise.

IMO the "one wrong idea" that has stalled progress in the last few
decades (at least in classical GR) is the wide spread belief among
experts, that space-time singularities are actually realized in the
real physical world. As these experts do nothing else than study
the properties of singularities (and other things that are intimately
connected to singularities, such as event horizons, apparent horizons
and trapped surfaces) it is extremely difficult for them to acknowledge
even the possibility, that they might have been fundamentally wrong.

These experts are blind to any evidence (observational and theoretical)
which goes against their prejudices. They are not capable (or not
willing) to interpret the singularity theorems in a truly constructive
way: i.e. to *use* the singularity theorems to sort out the *physically*
relevant (singularity free) solutions from the vastly larger *mathematical*
solution space of GR.

Although they know that the solutions they use (BHs, FRW-universe)
must be approximations, which break down when the singularity is
reached, they are not interested in analyzing seriously, *where* the
approximation breaks down.

IMO the really important message of the singularity theorems is
this: [and this message is really not very difficult to see]

There is *no hope at all* to get rid of a space-time singularity by
modifying GR in a "small" (Planck size, GUT-size) region in the
"vicinity" of the singularity. Any such small modification cannot
work and the singularity theorems are the key to realize why:

There are four assumptions needed to prove the singularity theorems.
Put in simple words:
i) no time travel (no closed timelike curves)
ii) "positive" energy (strong energy condition)
iii) space-time is not overly symmetric
iv) a trapped surface exists somewhere in the space-time

The first three assumptions are very reasonable from a physical
perspective (even the strong energy condition - at least if all
matter is fundamentally constructed out of strings: string matter
fulfills the strong, the weak and the dominant energy condition
- although only marginally)

Assumption iv), however, is not self-evident at all and it is not
a "weak" assumption, such as Wald claims in his book (a very good
book, otherwise)

Assumption iv) is the *key* to understand the "sphere of influence"
of a singularity. It tells us that *if* there is a trapped surface
in the space-time, then under reasonable physical conditions (i-iii)
a singularity will *inevitably* form.

Therefore the occurrence of a *trapped surface* in the space-time is
absolutely *equivalent* to the occurrence of a *singularity* in the
space-time (from a reasonable physical perspective that acknowledges
conditions i-iii to hold in any physically relevant space-time). Both
things are geometrically equivalent.

Therefore in order to get rid of a singularity, you have to get rid
of *all* trapped surfaces in the space-time. It doesn't suffice
to just cut out a "small" space-time region in the vicinity of
the singularity and replace it by whatever strange beast a future
theory of quantum gravity might come up with. This simply cannot
work, because already the *classical* theory tells us without any
doubt, that no (geometric!) theory of quantum gravity will be able to
get rid of the singularity, if it doesn't at the same time manage
to get rid of the true (purely geometrical!) cause of the
singularity: the trapped surfaces.

[An alternative would be to forsake geometry completely in a future
theory of quantum gravity, but then it would be quite difficult to
define what one would be talking about]

Or to say it somewhat differently: The roots of the singularity
don't have *anything* to do with the (apparent) failure of GR in
the "near" vicinity of a singularity. The roots lie much deeper
and they are geometrical. They lie in the classical theory itself,
i.e. in the *geometrical properties* of a space-time region far far
away from a singularity, a region where the classical theory is
*perfectly well defined*! If there is a trapped surface in such a
space-time region a singularity will inevitably form, for purely
geometrical reasons, and no theory of quantum gravity (which seeks
to modify GR only at high energies) can change this. [the argument
may break down, if there are only trapped surfaces in space-time
regions where we cannot be certain, that GR is still valid. But
for a large black hole we know for certain, (at least this is the
common lore), that the *local* physics in an inertial frame passing
the event horizon is perfectly well defined.]

I sometimes wonder about colleagues, that phantasize about some -
yet unknown - properties of of a - yet to be found - theory of
quantum gravity coming to the rescue, "explaining" the singularities
(and leaving the event horizon and the interior vacuum space-time
of the black hole solutions essentially intact). What different
are such phantasies from the phantasies of kooks (except possibly,
that a kook *can* not understand, whereas the other *will* not)?

At least Steven Hawking has realized, that it is the event horizon
itself, that has to be banished. Look carefully at what he said at
GR 17 in Dublin: The *trivial* metrics *dominate* the path-integral.
whereas the contribution of the *non-trivial* metrics is *negligible*.
(in the saddle-points approximation that Hawking was using).
A trivial metric is a metric *without* event horizon. If Hawking
is right, the metrics *with* event horizon make practically no
contribution to the path-integral. It is not difficult to figure
out, what this means: Consider the flight of a canon ball from
A to B. The path-integral approach tells us, that quantum mechanically
we have to consider all possible paths (but most of these paths
make no significant contribution to the path-integral). When we
calculate the classical trajectory, what do we do with paths that
make no contribution to the path integral? Do you consider the
(quantum mechanically) possible trajectory of the cannon ball flying
to the moon and back again? So lets apply this to possible space-
time metrics. What would the classical space-time metric be, as
determined from the path-integral approach? Would it be a metric
with *no* contribution to the path-integral, or one with *significant*
contribution to the path-integral? I guess, it is not too difficult
to figure out.

>
> > However, I must
> > admit that *if* there is a horizon in the space-time, "directed"
> > radiation on the horizon is a viable possibility. I wonder about
> > the stability of such a matter-state, though...
>
> Well, current thought is (all) geons are not stable. As you might know,
> Wheeler loved his geon idea (not that he put the "on the horizon"), but
> Einstein told him it would not be stable. (Like a pencil on its tip,
> falling either to collapse or dissolution. I'm getting very tired of
> being told its like a pencil on its tip! yet it is a good description of
> the instability. just repeated too often.)
>
> > any *surface* by arguing, that the metric would have to become
> > discontinous, a thing I am not (yet) willing to accept.
>
> Well, intuitively you might have the same problem with empty Schw.
> vacuum outside a massive body, then the sudden change to the interior
> solution of the mass of the star. The same would be true with a compact
> star surface. There is an abrupt change between the two metrics. But
> that doesn't mean it is discontinuous, that they cannot be matched at
> that surface.
>

I don't have a problem with an abrupt change in properties over a
boundary, such as the boundary of a star, a table or a water-drop.
But there is a difference between

i) a surface layer with non-zero *mass-energy* (or pressure
*perpendicular* to the surface) and

ii) a surface layer with pressure *tangential* to the surface.

At least in the spherically symmetric case (and I guess also in
the more general case) any surface layer of the first type (non
zero energy-density, non zero pressure *perpendicular* to the
surface) will induce a discontinuity in the metric, whereas any
surface layer of the second type (non-zero pressure *within*
the surface) will not produce such a discontinuity.

The second type of surface layer is well known: The tangential pressure
in the surface doesn't arise because there is mass-energy within the
surface, it arises because there is a boundary between two different
phases of matter (with an abrupt - yet finite! - change in pressure
and/or energy-density across the two phases). A good example is the
surface tension of a drop of water. The tension is not produced
because there are actually molecules in the boundary. The surface
tension owes its existence due to the pressure-difference in the two
phases: Surface_tension = (R / 2) pressure_difference according
to the well known Kelvin-Helmholtz relation.

However, the first type of surface layer (with "real" mass-energy in
an infinitesimally thin surface) doesn't seem to exist in the real
physical world (at least as far as I know). Therefore I am somewhat
skeptical about surface layers of the first type, and the fact that
such surface layers introduce a discontinuity in the metric (whereas
the second type of surface layer does not) enhances my skepticism.
I might be more inclined to relax my skeptical attitude, if there
were a good method to calculate the (finite) step in the metric. But
it appears to me, that this is only possible by some sort of embedding
in a higher-dimensional space.

> > I haven't thought about radiation with an equation of state
> > of the type T_\mu_\nu = diag(e, e, 0, 0). Seems somewhat
> > similar to the gravastar EOS in the shell used to match exterior
> > Schwarzschild to interior deSitter (however, the gravastar
> > has e in all *four* diagonal slots)
>
> Well, I'm not sure of your analysis. I would have thought the correct
> view would be the one with outwardly directed radiation. I don't think
> you need the inwardly directed radiation. The inbound radiation at the
> shell would be on its way to the singularity, and after all, that's what
> we're trying to avoid. (I think all inbound radiation reflects on the
> surface and thereby transfers its momentum to the body, and redirected
> outward, takes its place on the shell.)

I thought so too, first. But consider this:

There are also the white-hole solutions. These are characterized by
the property, that the *ingoing* radiation remains stationary at
the horizon. A white hole is a perfectly reasonable solution of the
field equations. [The reason that we rather believe in black holes
(and not so much their time-reversed counter-parts) might have to
do with the wide spread belief, that singularities must form in
"physically realistical" gravitational *collapse*]

You want a stationary solution, don't you? This means that you have
to restore time-symmetry.

One could try to achieve this, by "superimposing" a black hole and
a white hole solution. Assuming your "photons on the horizon" idea
would work (I am still skeptical!), and the interior space-time
becomes singularity free (I guess you wan't it to be plain Mink),
it might just be the right thing to do, to superimpose a white
and a black hole solution to achieve a *stationary* state.

Think also from this perspective: You (probably) wan't to place the
photons on the horizon in order to abolish the interior "center
of attraction", i.e. the singularity of a spherically symmetric
black hole. But if your "photons on the horizon" really would manage
to succeed in abolishing this "center of attraction" (which I still
doubt), there is no preference for a black hole horizon over to a
white hole horizon. The only "center of attraction" remaining
would be the horizon itself, so that photons moving in both
directions "away" from the horizon don't seem that weird an idea
any more...

Of course, you would have to find a solution of the field equation
that actually describes such a state. I still doubt, that such a
solution exists.

>
> Of course, the gravastar has a BEC shell of matter (Bose Einstein
> Condensate for those not familiar), rather than an ordered shell of EM.

Is this really true? Yes, I've read the papers. M&M *say* it is a
Bose Einstein Condensate. But is it really? Maybe I don't understand
enough about BECs, but I haven't found any good evidence (except
the assumption that M&M *know* what they say) which could truly
convince me that the properties of the shell are those of a BEC.

> Otherwise there are many surprising similarities. Both have interiors
> swept clear of matter/radiation, and no internal singularity. I have
> difficulty imagining how they come up with a physical mechanism to allow
> this sweeping. So I see the radiation shell construct as more
> intuitively physical, like a "bubble of mass"; or "mass without mass" to
> quote Wheeler.

With respect to the radiation shell construct it will remain an intuitive
idea, until somebody comes up with a genuine (stationary) solution.

With respect to the physical mechanism that "sweeps" free the interior
of a gravastar, I could imagine such a mechanism along the following
lines: The event horizon (in contrast to the apparent horizon) always
grows smoothly outward, and starts to form in the center of a collapsing
object. M&M assume that the BEC forms shortly before the event horizon
"appears", which would be in the center of a collapsing star (at
almost Planck-densities). Now assuming that space-time is "regular"
outside and *inside* of the "almost" event horizon (this is what the
gravastar solution was constructed for - to provide a regular interior),
it is not difficult to see that the gravastar's BEC-condensate membrane
(which will "follow" the track that the event horizon would have taken,
if it were there) is located at the *minimum* of the potential energy.

[This has to do with the fact, that in any spherically symmetric solution
the potential energy is proportional to the time-coefficient of the metric,
which is zero at the event-horizon, and due to the (assumed) regularity of
the metric must be *positive* everywhere else. This guarantees
a local minimum of the potential energy at the event horizon. Physically
it seems obvious, that the event horizon as the region of "infinite
red-shift"
must be a local minimum, if the space-time is regular everywhere else]

So as the event horizon (or rather the BEC-membrane) sweeps through the
space-time (outwardbound from the center) it will "trap" all matter in its
(hopefully) deep potential well, until there is no matter any more to
"sweep" up. This is, when the final stationary configuration is reached.

Mind me, I don't believe in gravastars. But if I am wrong and gravastars
exist, this probably would be the mechanism by which the interior is
"swept free" and the matter assembles in the NEC-membrane (however,
don't ask me where the fermions have gone. This is one of the major
weaknesses of the gravastar...)

>
> Yes, Valerio (my occasional research partner) and I have long considered
> such constructions with dS and AdS interiors, and wondered how to match
> them. We did match a Schw. exterior with a shell to a Mink. interior (as
> you would expect from Birkhoff) everywhere *except* at the horizon,
> which was frustrating.
>
> > I would not like to allow a discontinuous metric, unless absolutely
> > necessary. Maybe as a "last resort", but I can't see this yet.
>
> Well, I can explain why I think it might be necessary, if it is of
> interest. But let me save that for another post. Here's a thought for
> you though to get you moving in that direction, can you show me a
> propagating sudden change in two gravitational fields that doesn't have
> a photon (classically em-wave) between them?

I fear, I cannot start moving. I don't know too much about gravitational
radiation, so you will have to be more explicit.

Best wishes, MP.

Relevant Pages

  • Re: A partial black hole?
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    (rec.arts.sf.science)
  • Re: Event horizon / black holes and Schwarzschild metrics
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    (sci.physics.research)
  • Re: Event horizon / black holes and Schwarzschild metrics
    ... Look into the works of Hawking & Ellis and their singularity theorems. ... be behind an event horizon without even knowing it. ... People talk glibly about how quantum corrections will be needed ... what sort of corrections might be applied there, ...
    (sci.physics.research)
  • Re: Misinterpretation of the radial parameter in the Schwarzschild solution - a response from St
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  • Re: A good ref for g-wave theory.
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    (sci.physics.relativity)