Why strings are required to halt collapse (was: black holes and singularities)

From: MP (pet.antispam_at_onlinehome.de)
Date: 12/07/04 

Date: Tue, 7 Dec 2004 18:46:42 +0100

"Randy M. Dumse" <rmd@newmicros.com> wrote in message
news:gKatd.258$Pk5.3907@eagle.america.net...
> "MP" <pet.antispam@onlinehome.de> wrote in message
> news:conome$k9q$1@online.de...
> > Yet I don't really think that a matter-distribution confined to a thin
> > spherical shell can be stable
>
> I agree. Which is why I resist the idea of matter confined to a thin
> spherical shell, and think only of a radiation shell. (We don't need to
> reexamine if radiation should be called matter. You will always find I
> side with the idea it radiation is excluded from discussions of matter.
> I make that distinction so as not to forget the unique properties of
> radiation, and to remain open to more yet to be found.)
>
> > My answer is, there is no horizon. Rather there is matter in the
> > complete interior, growing in density when we approach the center,
> > Planck-density and Planck-temperature at the center. Only such a
> > matter-distribution (if supported by a string equation of state) has
> > the chance to be static for large compact objects, i.e. not to
> > collapse under its own gravity to a singularity.
>
> And this view for me evokes the vision of classical collapse, with the
> singularity theorems as the inevitable endstop.
>
> So, for you to convince me, you'd have to explain how the stringy state
> of matter can stop a classical collapse. Of course, your job is probably
> tougher than mine, because I've always been skeptical of sting theory,
> so I really don't know enough about it to be much help, and you might
> have an educational job on your hands, before being able to frame your
> argument in final terms.

I have been very skeptical of string theory, too. Have been much
more in favor of the loop guys and girls. Until I realized that the
singularity free solution for a compact object, that I found when
I was analyzing a particularly interesting class of spherically
symmetric metrics, actually is nothing else than a solution with
string type interior matter.

So my own solution proved my personal prejudices wrong.
But this is always a good sign, that you are not fooling yourself
too seriously. :-)

So now, because you asked, I will make some moderate
advertisement for the holographic solution.

As I already pointed out, the holographic solution is the simplest
solution of the field equations, and its properties are very much
concordant with the properties of the observable universe. But
here we are not interested in describing the universe, but rather
in the much more moderate question of what might be the
physically realistic end-point of gravitational collapse of a large
self gravitating object. [Of course there is nothing that forbids
the collapsing object to be very large, even surpassing the size
of the observable universe...]

The holographic solution provides such a physically realistic
alternative, because the (classical) holographic solution can
be interpreted as the *densest spherically symmetrical
configuration of strings possible*. It is not really too difficult to
see this. Let me try to explain:

The holographic solution has an interior matter-state which is
given by:

\rho = -P_r = 1/(8 pi r^2)
P_\perp = 0

\rho is the matter density,
P_r is the radial pressure
P_\perp is the tangential pressure

If you look at these equations closely, you will see that these
are the equations for a string (or rather a radial collection
of strings), aligned in the radial direction, where the string
tension falls off with distance r from the center as an
inverse square law: String tension \mu is nothing else than
negative radial pressure, e.g. \mu = 1/(8 pi r^2) = -P_r,
and strings have the property, that there is only tension along
the direction of the string itself, e.g. the transverse
tensions/pressures of a string are zero.

Now the classical holographic solution tells you, how the
string tension falls off with radius. You have to take this
dependence for granted (it comes from an exact solution of
the field equations, so nothing has been made up by MP,
except, of course, that MP believes in GR)

[Of course MP has some very strange, heretical views
about how classical GR must be understood: MP is of the
opinion, that nobody has properly understood *classical*
GR, who has not (yet) grasped the beauty of classical GR
expressed in a string context. Vice versa MP holds the
widely unsupported belief, that nobody has properly
understood string theory, if she has not (yet) understood
that *classical* GR requires strings in order to produce
sensible results, not only in high, but even more so in
*low* energy situations. End of digression :-)]

But when you know the string tension at any radial
position r (we know it from an exact solution of the
field equations) you can calculate the number of strings
puncturing any concentric sphere at radial position r.
It is quite easy, almost trivial:

We know we have a spherically symmetric solution with
an interior string equation of state. Symmetry allows only
one possible arrangement of the strings: They must all be
aligned radially with respect to center. Any other arrangement
would destroy spherical symmetry. Furthermore, radially
aligned strings with respect to the center is exactly what
the interior string EOS tells us: P_r = -\rho, P_\perp = 0.

Now imagine a spherical shell of matter [with matter, as
always ;-), I mean all types of mass-energy, including
radiation] at radial position r and proper radial extension dl.
dl is supposed to be very small. Lets calculate the total
energy within such a shell. There are two ways to do this.
First, lets calculate the energy via the classical holostar
equations:

E_tot = \rho V

V = 4 pi r^2 dl
\rho = 1/(8 pi r^2)

so that

E = dl/2

Now the second way: Here we make use of the fact that
the holostar has a string EOS. Lets take up this idea seriously.

In string theory it is well known, that the energy of a single string
is given by string tension times length (this is just the definition
of a classical string). Therefore a single radial string segment of
proper length dl, lying radially within the thin shell, and puncturing
the concentric sphere at radial coordinate position r, has an
energy given by:

E_s = \mu dl = dl / (8 pi r^2)

The question is, how many strings are there within the thin shell?
The answer can be found easily by dividing the total energy of the
shell E = dl/2 by the energy of one single string. We find:

N = (dl/2) / E_s = 4 pi r^2 = A

Now N is the number of strings puncturing a concentric sphere,
and this number is proportional (in fact equal) to the proper
area of the sphere, A. As we are working in Planck units this
means that each radial string segment (e.g. each puncture)
occupies an area on the sphere equal to exactly one Planck
area.

So the radially outlayed strings are densely packed! Each
string occupies a transverse extension equal to one Planck
area. This holds *everywhere* in the interior, because r was
arbitrary.

[There are other ways to show N=A. For instance, take a
*flat* (massless) *** of "paper" with area A. Puncture this
flat *** with N strings, and distribute the punctures at
constant separation along the ***. Assign an individual
string with tension \mu = 1/(8 pi r^2) to each string.
Calculate the sum of the deficit angles induced by all of the
strings. Hint: \Delta phi = 8 pi \mu for any one string. The
individual deficit angles of the strings will curve the former
flat ***. Calculate how many strings you need in order
that the former flat *** closes up to a sphere. Hint: How
many punctures do you need, so that the sum of the deficit
angles of all punctures is equal to the solid angle of the
sphere: 4 pi. Alternatively you could choose the topology
of a torus for the closed ***. In this case you would have
to calculate N, so that you get the sum of all deficit angles
corresponding to the Euler characteristic of the torus.]

Having established that N=A [at least for the holostar
solution], the question is, why would such an arrangement
of strings prevent gravitational collapse? Why couldn't
the strings be closer, so that for instance N >> A at the
center? If there is no physical mechanism that ensures that
N <= A always, (and at most N=A, where we know that
the object is almost as compact as a black hole), we
couldn't prevent collapse to a singularity, could we?

So is there a mechanism that ensures N <= A? The answer
can only be given by string theory. I am not an expert in string
theory, but I think I can guess the answer. The answer comes
from M-theory. In "normal" string theory (without M-theory)
strings don't necessarily have any transverse extension. Rather
they are thought to be infinitely thin. Not so in M-theory. One
way to view M-theory [as I understand it - maybe wrongly]
is to envision that the extra dimension of M-theory (i.e. going
from 10 space-time dimensions of string theory to 11 space-time dimensions
of M-theory) can be attributed to the string itself:
The extra dimension rolls up as a circle, and thus "transforms"
the infinitely thin string to a tube of finite transverse extension,
the extension given by the area of the circle in the 11th
dimension. The proper circumference of the circle is thought
to be of order Planck length [by most string theorists], so the
cross-sectional area of the tube would be of order Planck
area. If this picture is essentially correct, no string can be
compressed beyond this minimum transverse tube size.
[note that we require M-theory to realize this important
fact!]

Now in the classical holostar solution you can show, that *if*
the string tension is given by the radial pressure \mu = -P_r =
-1/(8 pi ^2), then the cross sectional area of the tube is exactly
equal to the Planck area. [We did this above] This "prediction"
of the classical holostar solution, i.e. a prediction of *classical*
GR, matches up nicely with the expectations from string theory.
So I guess it is fair to say, that in the holographic solution the
strings *are* densely packed [in a sense that requires M-theory
to work properly], and that this is the reason why the solution
cannot collapse to a singularity, although the object is almost
as compact as a black hole. The very string nature of matter
forbids complete collapse. And it ensures that the most compact
object possible in the real physical world [as described by
GR *and* by string theory!] is *not* a singularity, sitting within
a region filled with trapped surfaces, and of roughly Planckian
extension [but with arbitrary mass, which could exceed the
mass of the observable universe by far], but rather the most
compact object possible in real sensible physics is an object,
where the matter fills the whole interior and extends slightly
beyond the "horizon".

Well, if MP were the only one who thinks so, he most likely
would register as crack pot. A devious crack pot, but none-
theless. Fortunately [or unfortunately?!] MP is not the only
one, who thinks so. See the works of Samir Mathur. I
guess at least he can be considered to be a respectable
scientist.

Of course things are not as simple as MP has just tried to
explain. The real world is [slightly] more complicated. But
I will say no more on how things really area, because I have
not yet gotten out that paper which addresses the more
subtle issues...

Suffice it to say: If anybody of the onlisteners wants to beat
MP [who abhors writing papers, and wouldn't mind to see
another good paper on the subject], here is a hint: Look closely
at the ideas of Samir Mathur. His point of view is necessary
to make progress.

Best MP

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