Transcript Document

Reissner–Nordström Expansion
Emil M. Prodanov
Dublin Institute of Technology
On a large scale, the Universe is isotropic and
homogeneous (Robertson–Walker) with energymomentum sources modeled as a perfect fluid,
specified by an energy density and isotropic
pressure in its rest frame.
This applies for matter known observationally to
be very smoothly distributed. On smaller scales,
such as stars or even galaxies, this is a poor
description.
We propose a microscopic description, modeling
the Universe in the radiation-dominated epoch
as a two-component gas with preserved global
Robertson–Walker geometry, but with local
Reissner–Nordström geometry.
The first component is a gas of ultra-relativistic
particles described by an equation of state of
an ideal quantum gas of massless particles (the
“normal” fraction).
The second component consist of very massive
charged particles.
The expansion mechanism is based on the
interaction between the “normal” fraction of the
Universe and the very massive charged particles:
the interaction is described purely classically in
terms of the Reissner–Nordström geometry of the
charged massive particles.
The candidates:
…
The Model
Hawking’s arguments:
The ultramassive charged particles are viewed
general-relativistically as naked singularities and
the interaction mechanism is the gravitational
repulsion of the naked singularities.
Naked singularities are particles of charge Q
greater than their mass M (in geometrized units G
= 1 = c) described by Reissner–Nordström
geometry.
For the electron, the charge-to-mass ratio Q/M is of
the order of 1021. In view of this, in the 1950s, the
Reissner–Weyl repulsive solution served as an
effective model for the electron.
We confine our attention to the local spherical
neighbourhood of a single naked singularity and
consider the Universe as multiple copies – fluid – of
such neighbourhoods (local Reissner–Nordström
geometry).
For a naked singularity of unit charge, the mass
cannot exceed 10-8 kg.
We show that the radii of the repulsive spheres of
the naked singularities grow in inverse proportion
with the temperature: the ultraheavy charged
particles “grow” as the temperature drops and drive
away the “normal” fraction of the Universe.
This repulsion results in power law expansion with
scale factor a() ∼ 1/2, corresponding to the
expansion during the radiation-dominated era.
If charge non-conservation of the naked
singularities is involved, then the gravitational
repulsion could be powerful enough to achieve
accelerated expansion that solves the horizon
problem and thus accounts for the inflation the
Universe: a() ∼ eH or a() ∼ n, with n > 1.
For temperatures below 1029K, quantum effects do
not play a role in the interaction between the two
fractions of the Universe.
We determine the “radius” of an ultraheavy charged
particle by calculating the turning radius of a
radially moving incoming (charged) test particle of
the “normal” fraction of the Universe, having ultrahigh energy kT >> mc2 (where m is the test particle’s
rest mass).
The proposed model is simplified significantly by
considering the incoming particles as collisionless
probes rather than involving their own gravitational
fields and by not considering the more general and
physically more relevant Kerr–Newman geometry.
We thus ignore the magnetic effects caused by
rotation of the centre, which drags the inertial
reference frames, and we also ignore the particles’
spins.
The Kerr–Newman metric in Boyer–Lindquist
coordinates and geometrized units is given by:
where:
The motion of a particle of mass m and charge q in
gravitational and electromagnetic fields is
governed by the Lagrangian:
Where λ – proper time  per unit mass m: λ =  /m,
and A – the vector electromagnetic potential,
determined by the charge Q and specific angular
momentum a of the centre:
The equations of motion for the particle are:
Where:
– conserved energy of the particle
– conserved projection of the
particle’s angular momentum on
the axis of the centre’s rotation
K is another conserved quantity given by:
With
 the -component
of the particle’s
four-momentum.
We consider the radial motion of a particle in
Reissner–Nordström geometry:
The radial motion is described by:
Where  = E/m is the specific energy of the particle.
Motion is possible only if
– non-negative.
Thus the radial coordinate of the test particle
must necessarily be outside the region (r- , r+)
where the turning radii are given by:
The loci of the event horizon and the Cauchy
horizon for the Reissner–Nordström geometry are:
The centre r = 0 however, can be reached by a
suitably charged incoming particle satisfying:
For example, a positively charged center (Q > 0),
can be reached by an incoming probe with specific
charge q/m  1.
The naked singularity can be destroyed if sufficient
amount of mass and opposite charge are fed into it.
We assume that the naked singularities have
survived such annihilation.
A positively charged naked singularity will never
be reached by incoming particles of small
negative charge (i.e. −1 < q/m < 0), neutral particles
(q = 0) and all positively charged particles (q > 0).
For these particles the centre is surrounded by an
impenetrable sphere of radius r 0(T ) = r + .
The radius r0(T) of that impenetrable sphere depends
on the energy  of the “normal” particles or the
temperature T of the Universe. For very high
energies, the centre’s “radius” can be written as:
With the drop of the temperature, the superheavy
charged particles increase their “size” in inverse
proportion and drive apart the “normal” fraction of
the Universe.
The Universe increases its size in inverse proportion
with the temperature:
a() ∼ r0() ∼ 1/T () ,
where a( ) is the scale factor of the Universe.
We therefore get the usual relation: aT = const or:
Consider the expansion rate equation without
cosmological constant:
The main contribution in the energy density 
comes from the electrostatic field of the ultraheavy
charged particles.
The energy density of the electrostatic field is
proportional to the square of the intensity of the
electrstatic field, that is, the main contribution to ρ
comes from a term proportional to Q4 /r04(T).
Therefore:
The solutions are:
and
and
At recombination ( ∼ 300 000 years), the free ions
and electrons combine to form neutral atoms (q = 0)
and this naturally ends the Reissner–Nordström
expansion: a neutral “normal” particle will now be
too far from a superheavy charged particle to feel the
gravitational repulsion (the density of the Universe
will be sufficiently low).
At recombination the volume of the Universe is:
Assuming that the repulsive spheres of the
ultraheavy charged particles are densely packed
during radiation domination, at recombination the
“radius” of such sphere would be:
The pseudo-Newtonian potential that describes
the interaction between particles of the two
fractions is:
We now cut off the potential of the interaction
between a naked singularity and a “normal” particle
with the same sign of their charges when r0(T)
reaches Rc.
For “normal” particles with specific charges
satisfying 0 ≥ sign(Q) q/m ≥ −1, we cut off the
potential at the point where gravitational
attraction and repulsion interchange:
Namely:
We are now in a position to model the Universe as a
van der Waals gas.
This is possible in the light of the deep analogies
between the physical picture behind the Reissner–
Nordström expansion model and the classical van
der Waals molecular model:
Atoms are surrounded by imaginary hard spheres and
the molecular interaction is strongly repulsive in
close proximity, mildly attractive at intermediate
range, and negligible at longer distances.
The pre-van der Waals equation is:
The correction parameter  is due to the extra
pressure, while the correction parameter  accounts
for the non-zero volumes of the “atoms”.
Note that only in the limit N/V  0, this equation
reduces to the van der Waals equation:
For the parameters  and  we get:
The equation of state is:
Here  = const and  tends to zero at Recombination.