Can dark matter potential arise from baby universes linked to black holes?
- From: "stargene" <stargene@xxxxxxxxxxxxx>
- Date: 8 Dec 2006 16:50:38 -0800
BLACK HOLES-->BABY UNIVERSES?-->DARK MATTER?
A full synopsis of a heuristic model ...
A number of theorists, including S. Hawking, have postulated
that black holes may contain or somehow lead to daughter or
baby universes on the other sides of their event horizons, per-
haps through the Alice in Wonderland doughnut hole singular-
ities which might live in them.
Though some, like Prof. Hawking, have withdrawn from that
idea, I believe it may be only a temporary retreat. Far from
being the ultimate 'dead end' state of mass/energy crushed into
nothingness, the black hole will be discovered to be the ultimate
quantum-transitional object. A gateway between, and mother
to, universes.
Somewhat like R. B. Laughlin's scenario "Emergent Relativity"
(arXiv:gr-qc/0302028), mine has the local velocity of light de-
scending toward ~zero at the event horizon of a black hole and
then actually increasing interior to it. But with a difference, in-
volving baby universes 'inside' black holes leaking their gravi-
tational potential back into our universe (which we perceive as
evidence of 'dark matter'). Also, as we near the event horizon,
the Planck 'constant' would increase to an ultimate maximum
of about 10^-15 joule-sec, strongly elevating the uncertainty
principle. Its magnitude would decline interior to the event
horizon. Ie: h-bar would diminish as we move both inward and
outward from that surface. The model also predicts an 'age' of
13.83 billion years for our universe, which is well within the error
bars of the recent WMAP result.
To be separately posted soon: an application of the model's
key assumptions to the gravitational red shift using an unusual
Klein-Gordon approach which is algebraically identical to the
correct, fully relativistic equation from general relativity and
which gives identical results down to the event horizon. The
point here is that the equality of the standard GR approach
and the K-G approach opens a window on and lends support
to a unique scheme of variation of fundamental 'constants',
not only with cosmic time, but also with local gravitational pot-
ential. This answers some objections to the very notion of
variation of fundamental parameters, including hbar, over time
or space. This result shows that hbar must increase greatly to
a maximum as we near the event horizon, which will powerfully
increase the impact of the heisenberg uncertainty principle on
the physics of the event horizon itself.
NB:
I emphasize here that the K-G relation is NOT offered as an
<alternative> to Einstein's well confirmed GR or his gravitational
red shift result. Indeed, his standard GRS equation can be alge-
braically (though tediously!) transformed to obtain the K-G view
and vice versa, without loss of information. My equivalent rend-
ering simply allows an unusual and self-consistent reinterpreta-
tion of some of the basic physics without doing violence to them.
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
THE GENERAL IDEA:
Below I posit that whenever any black hole forms, it mediates
the creation of, and remains linked to, a single, distinct daughter
universe (DU) which expands in its own frame, in some sense
'outside' of our universe. An important aspect of this picture has
the gravitational potentials from the DUs propagating back into
our universe, centered on all of the individual black holes. The
propagation of gravitation and particle wave functions between
universe branes is indeed one of the hallmarks of M-theory. A
suite of these potentials centered on black holes inside of the
solar circle, say, would then collectively create a 'dark matter
potential' which now lies on top of the potential from all of the
normal, luminous matter inside the solar circle, which we expect
from observed luminous matter distributions (see (D) and (D1)
below). Letting the dark matter potential be - Cd^2, the model
shows that the mass of any black hole and the potential assoc-
iated with its DU must obey
Mbh varies as (Cd)^4 .
For the derivation of this, see (C4,5,6) below. This relation is
also true for large collections of black holes and their net resul-
tant 'dark matter' potential.
Note the similarity between this relation and that of Kormendy,
Gebhardt et al (2002) astro-ph/0203468, from their observations
of an unexpected interaction between the masses of super-
massive black holes (Msmbh) living at the centers of galaxies
and the associated velocity v (or velocity dispersion) data for
the stars and gas clouds far outside their effective 'cusp' radii.
They found that
Msmbh varies as v^b,
and their best-fit value for b is ~ 4.02 . The present model may
may provide the beginnings of a theoretical basis for their
observations (see (C6) below).
- - - - - - - - - -
The black hole and the DU follow an unusual Klein-Gordon-
like relation, as though they were both essentially quantum
particle states. The square of the total energy E of the total
system (black hole + DU), sans Psi notation, obeys
E^2 = (MdCo^2)^2 = (MbhCo^2)^2 + (MdVCo)^2 ,
where Mbh is the black hole mass, Md the DU mass, Co the
field-free-velocity of light in our universe, and
V^2 = Co^2 - Cd^2.
Cd is the field-free-velocity of light in the DU itself. It is the
latter,
as -Cd^2 , which will propagate back out into our universe, for-
ming the 'dark matter' potential.
Md x V implies a momentum p for the entire DU in some sense,
as it travels at velocity V 'away' from our universe (whether as an
object in an evolving space-time contained entirely within the
black hole's event horizon or as an essentially separate new
'brane' parallel to our own 'brane' in an M-theory sense, the
model cannot resolve). Regardless of the true topology how-
ever, the momentum p is relativistic, and follows
Md = Mbh / (1- (V^2/Co^2))^.5 ,
as though indeed a rest-mass, equivalent to Mbh, were hurtling
away from our universe at velocity V with a resulting relativistic
mass of Md.
Additionally the model requires that
MbhCo = MdCd ,
which implies a conservation of linear momentum across the
event horizon as "interaction-cross-section".
- - - - - - - - - -
The derivations of the above are straightforward, but lack of space
dictates against this here. It's pretty clear that this implies the
possibility of an infinite sequence of proliferating universes, ours
then being a DU itself arising from a parent universe, which simi-
larly was one of many DUs stemming from a preexisting super-
parent universe and so on, all the way up. But not infinitely so
all the way down, since the total number N of fermions (~proton
and electron pairs) falling into a given black hole's event horizon
always turns out to be the number N of such fermions emerging
into the DU 'on the other side'. The number N cannot be subdivi-
ded arbitrarily. Whether this also means that infalling fermions
actually pass <through> the horizon and then emerge as the new
DU"s fermions is arguable. I suspect not though.
At any rate, several things following from the model suggest that
there is some chance that the model has some tenuous contact
with reality:
(A) Every universe has a total mass-equivalent (dark + luminous)
(1) Mu= (hbarCo)^2 / (G^2 Mn^2 Me) =Mpl^4 / (Mn^2 Me),
where hbar is the reduced Planck constant and Mn is the sum
of the proton mass (Mp) and electron mass (Me). G is Newton's
constant and Mpl is the Planck mass; for self consistency the
model indeed requires both of these and also hbar x Cd to be
true trans-universal constants. As an example of (1) the total
mass equivalent for our universe would then be 8.8*10^52 kg.
(B) Each such universe has a radius-of-curvature (some would
prefer to say a 'distance to the cosmic event horizon') Ru (or Rd
for daughter universes in general) given by
(1) Ru = 2 hbar^2 / (GMn^2 Me) , see (F) below for some impli-
cations of this.
And
(2) Co^2 = 2GMu / Ru .
Ru / Co gives an effective 'age' t for each universe as measured
in any self consistent local units. But more precisely, t gives a
typical crossing-time for a photon of light. In the case of our uni-
verse, the model gives
t = 4.364 *10^17 seconds or 13.83 billion years.
This is within the error bars of the recent WMAP result where the
age for our universe is found to be 13.7 (+/- .2) Gyrs.
(C)
A little algebra shows that (A1) and (B1,2) reduce to
(1) Md = Mpl N^.75 (Mn'/Me')^.25
(2) Rd = (R*) N^.25 (Mn'/Me')^.75
(3) Cd^2 = 2G Md/Rd = (2G Mpl / R*) N^.5 (Me'/Mn')^.5
where N is identically the total number of (proton + electron)
pairs which were incorporated into the black hole and also the
ratio Md / (Mn') for the DU. Mn' is the (proton + electron) mass
sum and Me' is the electron mass as measured in the DU. For
the definition of R*, see (E) below. A smaller Mbh will yield a
smaller DU value for Md; but smaller mass DUs will also have
fewer such fermions but the masses Mp' and Me' of their protons
and electrons will be <larger>. More specifically
Mbh varies as Md^4/3
and
Mbh varies as 1 / Mn'^4 .
It follows that as matter falls into a black hole, increasing its mass,
the mass of its DU must also increase commensurately as new
fermion pairs appear within it. An open question is: does DU
time itself move forward only as actual matter falls into its linking
BH, and effectively freeze otherwise? At any rate, as an exam-
ple, our universe will have been increasing in mass, as well as
spatial extent, from its very beginnings, with ever new mass par-
ticles emerging into our spacetime, initially as near-Planck mass
particles (but with energy-contents only about 1 / 10^19 that of a
typical neutron in the lab.) A rough estimate of our universe's
rate of mass increase is
dMu / dt = Co^3 / G
or roughly about one Planck mass per unit of Planck time, or
about 4*10^35 kg/sec ! (Question: Where might new fermions
show up in the universe...randomly anywhere? In the great
voids between the clusters and superclusters of galaxies and
then slowly falling gravitationally into the void walls to add to
galaxy cluster masses?)
Thus, when Mbh is roughly the Planck mass (say at time zero
of the universe, or at the center of an accreting neutron star just
surpassing its maximum mass), its generated Md is roughly Mpl
and increases from there; similarly Mp' and Me' are also near
Mpl and each <decreases> from that point onward, as new mass
enters the BH.
Since Mbh = NMn and a fair assumption is that the ratio Mn/Me
is a true constant (=1837.15) over all universes, this gives for
different masses of black holes in our universe the relationship
(4) Mbh1 / Mbh2 = N_1 (Mn) / (N_2 (Mn)) = N_1 / N_2
Then, using (C3), we have that
(5) (Cd_1)^2 / (Cd_2)^2 = (N_1)^.5 / (N_2)^.5 .
Squaring both sides gives us
(6) (Cd_1)^4 / (Cd_2)^4 = Mbh_1 / Mbh_2 ,
which is nearly identical to the empirically observed result by
Kormendy, Gebhardt and others, which should not be surprising
since a dark matter potential term obeying (C6), and being cum-
ulative through the likely distributions of BHs in our galaxies,
should easily dominate a galaxy's velocity curve. See relation
(D1) below.
(D) An important point here is that, given some rough, general
assumptions, one can actually calculate the net effect of a large
number of 'dark matter' gravitational potentials (DMPs) as they
propagate back out into our universe from their respective DUs,
affecting the orbits of stars in our galaxy. We make a fair guess
for the total number of black holes interior to the solar circle (our
sun's distance from the center of our galaxy), to estimate the net
DMP: say ~ 300,000,000 BHs (about 1/3 of the Bethe and Brown
(3/1994 ApJ) estimate for the whole galaxy), with an average
mass of ~ 8 Msol (the ~3 million-Msol SMBH at SgrA* will not
affect this greatly.) This gives a rough total BH mass interior to
the solar circle of ~ 4.8*10^39 kg.
I then assume that this total population of BHs inside the solar
circle acts collectively as if it were a single gravitating BH with
the same mass and then ask what would be the resultant DMP
from its effective DU. Using (C3), this gives an N = 2.86*10^66,
and assuming Mn/Me = 1837, this gives
- Cd^2 = - 2.09*10^10 (m/sec)^2 .
All by itself this would set up a ~144 km/sec velocity for galacto-
centric orbits at the distance of the solar neighborhood, and act-
ing together with the normal luminous potential at r_sol would
yield close to the observed extra velocity which we currently
attribute to an as yet unidentified, and supposedly WIMPy, 'dark
matter' existing in this universe.
More generally, the total gravitational potential Phi_r, (ie: normal
potential + 'dark' matter potential), generated at a distance r from
the center of a fairly smooth, radially symmetric distribution of both
normal matter and the dark matter DUs mediated by black holes
within that distribution, may be roughly approximated by
(1) Phi_r = (- GM / r ) - 2GMd / (r + Rd) ,
where M is the total normal, luminous mass (eg: stars + gas) with-
in r. Md is the mass of the single net effective DU to be associated
with the total mass of all of the contained black holes inside of r,
again as though they were one supermass at the center. Md can
be derived from (C1). Rd is derived from (C2) and is the effective
'radius of curvature' for that DU. On its own, the first term on the
right in this relation would yield an increasingly Keplerian velocity
curve as r increases. Notice however that the second term auto-
matically yields an overall flattened velocity curve which declines
only gradually at great distances from a typical galactic center,
when r approaches and then surpasses Rd.
[As examples from the model, and assuming Mn/Me has the same
value in all DUs as in our universe, we can look at the dark matter
potential projected in their immediate vicinity by two different DUs
linked to two different BH masses, using only the second term in
(D1):
If BH = 8 Msol (fairly typical BH)
then Md = 4.35 *10^36 kg
Rd = 4.8 *10^20 m = 50.7 thousand LY
- Cd^2 = - 1.21 *10^6 (m/s)^2
If BH = 3 *10^6 Msol (SgrA*)
then Md = 6.6 *10^40 kg
Rd = 1.19 *10^22 m = 1.26 million LY
- Cd^2 = -7.42 *10^8 (m/s)^2
Cd = 27 *10^3 m/s
That is, if all normal sources of gravitational potential (ordinary
matter and all black holes, including even the ordinary potential
from SgrA* itself) were to be magically transformed away from
the region of the galactic core, a small test mass in circular orbit
around Sgr.A* would then STILL be seen to move at an almost
constant velocity of ~27 kilometers per second at ALL values of
r from the galaxy center out to the edge of the core region - - a
nearly flat velocity curve in apparent violation of Keplerian orbits.
Such a flat curve would however not be seen in reality due to
the presence of normal masses and the DMPs from even more
BHs in the core region.]
Lastly, using a little algebra and seeing that in (r + Rd) in (D1)
Rd dominates overwhelmingly for r out to borders of most gal-
axies, lets us estimate the total mass of BHs inside a given r
and, assuming an average BH mass of say ~ 8 - 10 Msol, even
the total number of BHs interior to r in a galaxy. This would only
require knowing the gravitational potential W at r, as expected
from normal luminous matter inside r, and also the actually ob-
served potential U at r, due to the additional presence of some
mysterious 'dark matter' interior to r.
The resulting relation for total-BH-mass inside a given r is
(2) Mbh_r = (Mn^2 / Me) [R*(U-W) / (2GMpl) ]^2 ,
where | U-W |^.5 is approx. = Cd, the velocity of light in the net
single fictitious 'DU' resultant from all the actual DUs linked to
BHs interior to r. This approximation works here because as
noted, r can be neglected in (D1) for intragalactic calculations
of the DMP.
A good schematic way of visualizing the relationship between
a DU, its linking BH and an observer (or mass particle) in our
universe is the following:
(DU)<----------Rd----------->(BH)<-----r------>(observer)
keeping in mind that the "(DU) <---Rd---> " part is literally 'out-
side' of our universe, while the "(BH), <---r--->(observer)" are in
our universe. Thus the extra gravitation from the DU is always
acting through the locus of the BH, and acts as if it were coming
from an object (the DU) which is always a distance Rd directly
*behind* the BH. So as far as any test particle in our universe
is concerned, the daughter universe is always a distance Rd
on the opposite side of the black hole. This would be true of
any and all observers located at any points outside of that BH.
In this sense the black hole acts as a window, lens or hole be-
tween the two different spacetimes and, if true, this schematic
relationship would seem to qualify as an invariant.
(E) R* is a conjectured mini-max quantity which may give the
Schwarzschild radius of a minimum-allowable-mass for a black
hole derived through stellar collapse. It also represents a con-
jectured maximum value for twice the Planck length
Rpl = (hbar G / C^3)^.5
at the event horizon of the black hole. This follows from the in-
crease in value of the Planck 'constant'. Thus, in mirror fashion,
the fundamental Planck length, defining the atomicity of space-
time itself, decreases and becomes ever finer as we move away
from the event horizon, both outward and inward. At the event
horizon, the Planck length will then be about 2,734 meters and
R* = 2 hbar^2 / (G Mn^2 Mpl) = 5,468 meter.
[This corresponds to a mass Mbh of 1.84 Msol for the minimum
mass of a stellar-formed black hole, in approximate agreement
with some estimates, including that of Bethe and Brown, but be-
low the usually cited maximum possible mass for a neutron star
~ 3 Msol , (the Tolman-Oppenheimer-Volkoff limit, based on the
most general considerations, regardless of its interior equation
of state.)]
The increase of the fundamental Planck length to a macroscopic
size, on the order of the above Schwarzschild radius, is also re-
quired for self consistency to satisfy energy conservation etc. in
the K-G approach to the gravitational red shift referred to above,
and may be compatible with the idea of a particle's characteristic
spatial extent spreading throughout the event horizon-as-mem-
brane (a la Susskind et al), as the particle approaches the event
horizon. It would signal an ultimate and possibly overwhelming
dominion of quantum phenomena over classical ones even over
the entire macroscopic event horizon surface-as-cross-section.
(F)
Note: several fundamental parameters must change over cosmic
time, as follows from equation (B1) for example, since if the uni-
verse is expanding and its radius-of-curvature is commensurate
with that expansion, then indeed one or more factors on the right
hand side of (B1) must also change. So far, self consistency at
all stages of a DU's evolution require that some values increase
and some decrease from initial set-values by the square root
of the Gravitational-Fine-Structure (GFS) between two protons
where
sqrt(GFS) = {(hbar Co)/ (GMp^2)}^.5,
which is currently ~1.3*10^19 . This would also mean that our
current GFS will continue to increase very slowly and mono-
tonically.
Thus, while Mpl, G and R* would remain as true constants, both
within and between universes, the GFS will steadily increase
from approximate unity, Co will increase by the factor GFS from
a DU formation-time set-value of about 2.3*10^-11 m/s. Or to put
in dimensionless units, Co started at unity and is now 1.3*10^19.
Similarly, at formation-time, h-bar will have been ~1.37*10^-15
joule-sec, or in dimensionless units, near unity and is now down
to ~ 1 / 1.3*10^19. The possibility of a greatly increased h-bar at
an event horizon would have tremendous implications for all
quantum phenomena since the uncertainty principle would be
kicked into high gear on a macroscopic scale over the entire
'cross-section' of the black hole as a 'gateway' into the creation
of a new universe. Virtually all of classical physics would take
a back seat here and would only take center stage on macro
scales, as emergent phenomena, farther away from the event
horizon, both inward and outward.
At time zero, both Mn' and Me' were close to the Planck mass,
and decreased steadily to their current values and will continue
to decrease indefinitely. Relations (C1,2,3) raise a question:
does Mn / Me also change? But as it now stands, the model
cannot answer this. I should emphasize that these parameter
changes will normally be masked by commensurate changes
in local standard clocks, measuring rods etc. and will only show
themselves in dimensionless numbers, such as the GFS and the
cosmic red shift of light, which lets us compare wavelengths of,
say, H-alpha light from distant sources to local H-alpha light in
the lab.
If the GFS changes from ~ unity in our distant past, this implies
that the ratios of the various physical interactions may also have
been changing from near unity. That is, the gravitational inter-
action would have been the same as now, since G is a constant.
But the other three interactions would have been far weaker then,
having the same magnitude as the gravitational interaction, and
would have steadily increased with time to their present strengths.
This would have deep implications for ancient nucleogenesis
and star formation nearer to the Big Bang. It should also have
implications for the physics of all new matter emerging into our
universe ever since then.
Note:
R* = 2 hbar^2 / (G * Mn^2 * Mpl) = 5,468 m, an apparent con-
stant iff Mn = Mpl at time zero, since as a little algebra shows
that at that point (at the event horizon) R* = 2(hbar*G / Co^3)^.5
= 2Rpl. This would also mean that hbar and Mn decrease from
their event horizon set-point values at the same rate as the DU
develops, or hbar varies as Mn.
Some Questions....
The model raises many questions: Eg: How do fermion pairs
emerge in the DU? In this regard, I suspect that one effect of
a greatly increased uncertainty principle would be a de-local-
ization of the black hole's event horizon itself. That is, some
aspects of the event horizon-as-membrane may map stochas-
tically as large quantum fluctuations from the 'classical' entity
located at the Schwarzschild radius to places well within the
DU and acting as quantum-tunnel white holes, for emergent
matter coming into the DU.
When two BHs coalesce, what happens to their DUs? Can
the model say anything specific about dark energy (eg: does
it have to do with the conservation of linear momentum implied
by MbhCo = MdCd shown in the K-G section above?) In the
anatomy of the metric, how does the DMP propagate and how
does matter in our universe even recognize it? Also, a <very>
rough calculation shows that the sum over all of the masses of
all Md's linked to a plausible estimate of the total number of
black holes in the universe (stellar, intermediate and super-
massive) is not too far (by an order of magnitude?) from the
total mass Mu for our universe given in (A) . A similar rough
relation exists between the volume of our universe and the
sum over all of the volumes of all the DUs. Is this the flip side
of the observation that most of the universe mass is in the
form of dark matter/energy? Has the bulk of the universe
normal matter already gone over into DU formation through
BHs?
The last raises a further question... If the vacuum is associated
with a bose-einstein-condensate (BEC), as many theorists posit,
can there logically exist, in principle, an anti-BEC, in the sense
of particles and anti-particles in QFT. And if so, can such an
anti-BEC be identified with an anti-space-time, which if joined
somehow with its BEC space-time counterpart, would produce
a net null-or-non-space-time? That is, would the reality of space-
time as a BEC logically imply the existence of its opposite, and
further imply that both come from the same 'nothing'... a back
door answer to the old query 'why is there something rather than
nothing at all'?
(Non pauca, sed non matura!)
.
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