Science > Physics > Fractal String Theory and Black Hole Singularities
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30 Sep 2005 05:13:27 PM |
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Fractal String Theory and Black Hole Singularities |
1. Introduction
---------------
In this post I describe a simple recursive model for how
strings and branes interact and maintain their stability,
based on a surprisingly commonplace principle. This leads
to a compelling, and equally simple, way of thinking about
gravity and mass.
As a first striking example of the model in operation,
we describe, in section 4, the detailed structure of a
black hole singularity and how it maintains this and
grows as the hole accretes mass/energy.
(If the prospect of reading this entire post is too much,
as I can well understand it may be, or if you dear reader
spot what seems like an amateurish turn of phrase or even
kook warning sign, please at least read section 4, and ever
afterward laugh at the crudity of black hole singularies
being described as "points" and of "infinite density" !)
The model can be applied equally to (fundamental) fermions
and in conjunction with SR suggests how rest mass could be
relativistic mass induced by intrinsic motion within the
fermion, and so on at lower scales, so that "true" rest
mass becomes evanescent, even if a useful fiction.
At the opposite end of the scale, the model suggests that
the expansion of our entire Universe (understood to mean
the "visible" Universe, outside black holes, and all the
observer-connected regions beyond our causal horizon)
is an inevitable consequence of gravity acting over
long distances and will accelerate.
Furthermore, it indicates that our Universe started life
as a singularity in a black hole which evaporated as the
Universe (with the same fundamental physical laws as ours)
containing it expanded past a certain rate, and that the
evolution of our Universe is, in broad brushstrokes and
in terms of 2D surfaces as a vague indication of higher-
dimensional equivalents, from a torus to a sphere, just
as in M Theory a single toroidal brane can decay to a
sphere [see "The Elegant Universe" by Brian Greene,
Fig 13.3 on page 327].
Reading Greene's book all through for the first time last
week, I learned that the idea of the Big Bang originating
in a black hole singularity was suggested by Lee Smolin,
who also ingeniously and plausibly elaborated it into a
form of inanimate natural selection of universes whose
physical laws favour the formation of black holes. See:
http://www.edge.org/3rd_culture/smolin_susskind04/smolin_susskind.html
Probably most ideas in this post, such as rest mass being
relativistic mass due to internal motion, have been put
forward by others before (and I'd be very interested if
these are pointed out); but I'm not aware of the idea of
parallelism having been applied to them all as an uniting,
underlying theme.
2. Note to the Reader
---------------------
Let me assure readers worried at the outset by that
phrase "commonplace principle" that my model does
not in any way seek to contradict established
and generally accepted theories such as QM and GR,
quite the reverse.
Even if it is right on the button and allows one
to think about and describe (and hopefully in due
course formalize) sub-microscopic phenomena in
deterministic terms, we as observers are made of
and surrounded by strings and branes and thus the
inescapable probabilistic nature of QM still applies
to us, just the same as the literally inescapable
nature to us of a black hole!
As an aid to understanding and intuition, the descriptions
of strings and branes below are expressed in terms of 3D
space. But as formal string theories operate in 11 or more
dimensions, this is likely to be an oversimplification.
However I trust the reader will not reject it out of hand
on that basis alone.
Also, the reader will have noticed the absence of any
equations, which to many is another worrying sign. I
think the ideas are almost specific enough now for a
crack at the Maxwell equations and SR; but I'll admit
I'm not quite there yet.
I'm also all too aware of "gray areas", the most prominent
being how exactly the "group waves", cavalierly referred
to throughout, tie in with familiar concepts.
3. What is Gravity and How do Strings and Branes Relate?
--------------------------------------------------------
My proposed answer to the first suggests a consistent
answer to the second, and to other fundamental questions,
provided we assume that branes form an infinite heirarchy,
in which a brane is itself composed of elongated branes
or "strings" at lower levels in the heirarchy.
Gravity is then simply a manifestation of the tendency
of these strings to minimize their "energy" by aligning
in parallel layers.
A change in the relative orientation of two strings
changes that of their constituent strings relative to
neighboring strings at the same smaller scale, and a
cascade of orientation changes fans out through the
latter (and _their_ constituent strings ad infinitum)
to minimize _their_ energy.
This is somewhat analogous to the group velocity of two
out-of-phase waves increasing as the phases approach
equality.
For convenience I use the word "energy" in this context
throughout; but whatever is "shed" by strings becoming
more parallel may not correspond exactly to energy and
perhaps this word should be considered simply a label
to indicate what is going on.
Whatever is shed must be subject to "compressibility"
constraints of some kind, on account of the ubiquity
of strings at all scales and positions (so that no
pair of strings can ever be considered in isolation),
and maybe simply formalizing this constraint would
be enough to pin down the exact laws governing their
interactions.
Flat and empty spacetime can be pictured in the familiar
way as a sea of branes, at all scales below a given one,
with a symmetric distribution of orientations.
A gravitational field or, equivalently, curvature of
this spacetime is then a skew in this distribution
towards (locally) one direction.
Mass can then be identified as originating in optimal
or limiting everywhere-parallel string alignment, for
example a torus comprising strings wrapped in parallel
bands with at least one "twist".
(Section 4 explains why strings can only align in branes
rather than solid "bundles".)
Between this limiting brane and a distant symmetric
"random" distribution of orientations is a continuous
gradation of "parallelness", and if two limiting branes
approach then they can obviously increase the overall
"parallelness" of their combined consituent strings and
those in the vicinity by approaching nearer still and of
course changing their orientation to become more parallel.
In M theory, strings and branes have a "frequency". But
in my model, detailed in Section 4, a brane comprises
parallel bands of strings (which are themselves elongated
reconnected branes) and its natural "banded" structure can
perhaps be interpreted as a frequency, the more so as the
banded structure of each string (on account of _its_
constituent substrings) must cause the strings to
"rotate in place". Don't worry - all will become
clear in a few lines ...
The next section gives an example of a brane, describing
in detail how it is constructed from smaller branes, and
exactly how branes interact and combine.
4. Black Hole Singularities
---------------------------
Since the surface of a sphere cannot be "combed", i.e.
contain an everywhere parallel vector field, which our
model assumes mass is, in the limit, and a black hole
is an example par excellence of mass, it is natural
to assume that the singularity at its centre is a
torus and see where this leads.
In practice, we can rule out the ridiculous idea of a
"point of infinite density". I'm sure every physicist
must realize this is an idealization, whatever the
popular science books may claim.
(Curiously, a rapidly spinning black hole develops
outside the main event horizon a disjoint wedding-ring
shaped region called an ergosphere, which is vaguely
suggesive evidence that the assumption of a toroidal
singularity is not so far-fetched, and there's more
evidence to come..)
Before examining the structure of the singularity itself
in detail, consider what happens when a single fundamental
fermion ("particle with mass") falls into the black hole.
As this particle contains mass, we assume in keeping with
our model that it is also at heart torus shaped. In flat
spacetime, barring sub-atomic interactions and so forth,
this torus is "round" in the way one normally pictures
a torus.
But consider what happens as it approaches the singularity,
where the smaller-scale strings comprising spacetime are
increasingly parallel - To maximize its alignment, and
the symmetry of this alignment over its entire surface,
the particle must start elongating and aligning lengthwise
with the predominant sub-string direction until it starts
to resemble a primitive worm, and then something even
stranger starts happening ..
At both ends of the particle, as a result of elongation
of the torus a circular ridge has started to form.
Assuming the constituent strings of the torus form
closed loops with at least one twist, they cannot
maintain exact parallelism at the top of this ridge,
and they thus become locally "skewed" across it.
So to minimize overall energy, sub-strings of comparable
frequency from the surrounding space must start being
deposited on the ridges, and the particle grows at the
ends like a crystal!
(This is analogous to the phenomenon of "mass inflation",
whereby vast amounts of mass/energy can be created inside
black holes by crossing shells of light causing a spacetime
fold. See http://en.wikipedia.org/wiki/Werner_Israel )
That's not all though - Recall that the particle, or string
as is now fast becoming, is increasingly "held in place"
by the increasingly everywhere parallel sub-strings comprising
the spacetime background as it approaches the singularity.
So this parallel field actually guides the "crystal growth"
of the string as if on rails until the two ends meet and
seemlessly join!
In short, the particle has been transformed from a stubby
torus to an elongated "orthogonal" torus that will now
shrink down to merge with the singularity by wrapping
round it with the same number of twists as all of its
existing constituent strings. But there's a catch..
Each string comprising the singularity is itself made
up of twisted bands of circular strings (elongated tori),
and to preserve local parallelness between the bands in
the strings that form the singularity, the bands of
these strings strings must _alternate_ like a tweed
pattern.
I suggest that this explains why strings form branes
rather than "solid" bundles, as one might naturally
expect assemblies of parallel strings to do - Being
banded, in either of two possible handednesses, in
orthogonal cross section two adjacent strings in a
brane are analogous to linked cog wheels, with the
handedness corresponding to a clockwise or anti-
clockwise rotation, and if a third cog wheel is
added linked to both then either direction of
"rotation" (in our analogy) is incompatible.
Although the word "rotation" is quoted, perhaps it is
not so-far fetched to picture the strings forming the
singularity brane as rotating "in place", not unlike
those cogwheels after all (God, this must sound so
kookish; but bear with me - we're almost there!)
It only remains to explain how our single shrinking
former fermion merges into this singularity torus
while preserving the latter's alternating pattern.
Well that's easy: In becoming the elongated torus,
when the ends join, the string is actually a _pair_
of nested toruses with opposite handedness!
So in a final flurry, the singularity "draws out" the
inner torus through the outer one and the pair take
their place in the alternating pattern, for all the
world like DNA splitting into two complementary parts!
Also, somehow, our former fermion also needs to ensure
that the number of twists of its constituent strings
matches that of the number of substring twists in the
strings comprising the singularity torus (the same for
all strings). I imagine this, if necessary, must occur
as part of the "flurry" referred to above.
Note that the nested toruses of the string only have
opposite handedness in the context of being wrapped
round the singularity torus or, more generally,
constrained in their orientation by parallelism
of surrounding sub-strings - If they were free to
float in flat spacetime then one torus would only
have to flip through half a revolution to become
indistinguishable from the other. But as we'll see
in the next section, for a fermion, nearby spacetime
isn't anywhere close to being flat..
5 So How does Relativistic Mass Fit into This Picture?
------------------------------------------------------
Like a black hole singularity, a fundamental fermion
in my model is a toroidal parallel-string brane,
surrounded by the sea of substrings whose orientation
can be pictured as a vector field that graduates away
from being parallel until it merges with the random
(for flat spacetime) "background" sea of substrings
and sub-substrings etc.
If the Fermion is in motion relative to an observer
(the latter being themselves ultimately just an
assembly of strings) then SR applied to the strings
says they will appear more foreshortened, and hence
"more parallel", to the observer.
This effect is most pronounced, or at least can be
expected to have the most effect, nearest to the
Fermion, where the surrounding substrings are the
most parallel, and increases the probability of
the substrings spontaneously crystalizing into
a string that merges into the torus comprising
the fermion, in the same way as described for
the singularity in Section 4, thus increasing
its apparent mass to the observer.
But now we have the baffling scenario in which
the number of strings comprising a brane depends
on the relative motion of the observer!
In the limiting case of the "observer" _being_
one of the strings comnprising the fermion
torus brane, they would presumably only
perceive their two neighboring strings. So
in zero relative motion, and including no
"internal" string motion, the fermion would
appear almost massless, which has implications
for the interpretation of rest mass..
Since in my model all particles and fields are
ultimately assemblies of strings and changing
parallelism relations between these (mediated
by smaller-scale strings), and since we know
that some of these particles and fields are
massless, I suggest that ultimately fermions
are in some form of internal relative motion
of their constituent strings and their entire
rest mass is the the many-orders-of-magnitude
smaller rest mass of their constituent strings
inflated via the rules of SR by this motion and
the result averaged.
This internal motion could be the constituent
strings of the limiting torus "twisting in place"
and/or "whirling in place" in a perpetual attempt
to cancel a second-order deviation from parallelism
among the constituent substrings of adjacent strings,
exactly as described in Section 4.
6 The Big Bang and Evolution of the Universe
--------------------------------------------
So far we have focused on the local effects of
string parallelism. But there is also a global
aspect, because a torus is not the only low-
dimensional shape that supports a limiting
everywhere-parallel vector field:
At the cost of decrementing its genus by one,
but (in my model) gaining an extra intrinsic
dimension, a torus brane can flip into a larger
sphere in which the constituent strings all point
radially outward, vaguely reminiscent of a popcorn
grain going pop and evolving from a compact object
to a larger fluffy form.
In the latter configuration the strings are only
parallel in the limit as the sphere expands to
infinite size, although since we have established
(or at least reasoned) that strings cannot collect
in true compact 3D bundles it appears that they
must arrange themselves as a foliation of radial
sheets somewhat resembling the vertical sheets
of an aurora borealis.
In summary, whatever the exact configuration
approached, the fractal string model strongly
indicates that "our" Big Bang originated as
no more than a black hole singularity in a
Universe containing ours and at the next
higher string scale.
That Universe expanded, at an ever accelerating
rate, in the process of reconfiguring _its_
strings into the spherical minimum energy
configuration, until the light-speed causal
horizon shrank to the size of the black hole's
event horizon, whereupon the hole evaporated
and the toroidal singularity within was somehow
disrupted or distorted sufficiently for it to
"pinch off" at one or more points and start its
evolution to the limiting sphere (or, properly
speaking, hypersphere).
In the same way as the higher-level Universe,
part of this evolution of our Universe involves
forming black holes to incubate possible future
Universes to be released in their turn when ours
has expanded sufficiently.
I suggest that the _reason_ the (or "a") Universe's
expansion rate accelerates is that as strings become
more parallel the group velocity of pairs of parallel-
converging strings increases.
The same idea may explain the supposed early inflation
stage, because while the initial toroidal singularity
was largely intact, and its constituent strings almost
parallel, group waves must have darted away tangentially
at very high speed (many times faster than light speed,
which is not a limit for expanding spacetime itself,
or for group waves for that matter).
It is reassuring to believe our Universe is in a very
early stage of this limiting spherical parallelism
state, which probably will not become noticeable or
even measurable, for dozens of billions of years,
if not a much larger power of ten.
7 Conclusion
------------
These days we are indoctrinated from the cradle up
with the dichotomy between "everyday size equals
classical equals continuous" versus "subatomic
equals quantum equals weird".
While there's no disputing the tenets of QM, and
experimental evidence and practical applications,
who is to say that quantum effects may not themselves
be emergent properties of an even smaller continuous
regime, and vice-versa at smaller scales still, just
as chaos diagrams often seem to have multiple "blurred"
bands with curious discrete threadbare layers between?
I know this idea, or the first quantum-from-continuous
stage at any rate, has been explored in the form of
"hidden variable" theories and suchlike; but my fractal
string model, suitably refined, may provide a more
consistent and dare I say successful means to tackle it.
Cheers
John R Ramsden (jhnrmsdn@yahoo.com.uk)
^
remove m from com to reply
(Google "From" address is defunct, as a spam trap)
.
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| User: "" |
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| Title: Re: Fractal String Theory and Black Hole Singularities |
30 Sep 2005 07:54:14 PM |
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wrote:
1. Introduction
---------------
In this post I describe a simple recursive model for how
strings and branes interact and maintain their stability,
based on a surprisingly commonplace principle. This leads
to a compelling, and equally simple, way of thinking about
gravity and mass.
xxein: There are millions of such invented physical explanations, but
matter is nothing but a cohesive lump form of energy. It only acts
like it is something separate like matter. In this way, it is a
compound and not a suspension. It only looks like that to us in the
macro way.
The idea of strings being self-connected to themselves is not a bad
scenario or analogy, but it restricts free thought of how they might be
connected in a non-topological manner; even a dynamic and ever-changing
connectedness. But, I don't wish to suggest anything GR-ish.
Recursiveness is fine. It is understood in many ways as logically
acceptable from pure math to the Mandelbrot and Julia sets that can be
produced as graphics (with the elements of time and cause and effect).
But unless we "guess" the C&A correctly there will exist the elementary
chaos that only filters out when it becomes recognizably macro. But by
that stage of recognition, we have what we call matter.
I think that we should first describe energy in terms only of itself
(with its dynamic behavior) and only then proceed to make dumplings.
This would eliminate almost all frivolous mass-energy connotations
arising from the notion of a separateness.
I know that some will say that this is what GR does with e = mc^2, but
it does not smooth out the lumps in a progressive or digressive manner.
We have a 3-D + 1-T. Any higher dimension is not necessary except to
fulfill a mathematician's wet dream. I don't say that the math is not
there; I prefer to think of it as description without cause or just a
general behavior like GR. We can add as many dimensions as we like to
any behavior, but only certain ones are C&A first primed.
If all else fails to be feasible, investigate Enron. It is a lot
easier than coming up with the answer that satisfies the reality of
universal physics.
It only takes val(x) to believe in a truth. It takes val(x^2) to
suspect an untruth out of a 'given' truth. It takes val(x^3) to
realise that a belief may not be a truth. Truth may seem stranger than
fiction, but belief whips truth's *****.
You'll have to make strings a lot better for me to consider them viable
as such. There is still hope for our understanding this universe.
Have a happy.
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